Transit-oriented development in an urban rail transportation corridor

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Transit-oriented development in an urban rail transportation corridor Ya-Ting Peng a, Zhi-Chun Li a,∗, Keechoo Choi b a b

School of Management, Huazhong University of Science and Technology, Wuhan 430074, China Department of Transportation Engineering, Ajou University, San 5 Woncheon-Dong, Yeongtong-Ku, Suwon, 442-749 South Korea

a r t i c l e

i n f o

Article history: Received 30 June 2016 Revised 21 March 2017 Accepted 21 March 2017 Available online xxx Keywords: Transit-oriented development Rail transportation corridor Households’ residential location choices Housing market Public and private regimes Social welfare

a b s t r a c t Transit-oriented development (TOD) has been recognized as an important avenue for creating a green transportation system. This paper addresses TOD investment issue in terms of the location, number and size of the TOD zones along a rail line. An urban system equilibrium problem with TOD investment is first formulated. Two social welfare maximization models, which take into account different investment regimes for TOD projects (i.e., public and private), are then proposed for optimizing TOD investment schemes along a rail line and train service frequency on that line. In the public regime model, the government is responsible for the investment cost of TOD projects, which is borne by the private property developers in the private regime model. The proposed models explicitly consider the interactions among the government, property developers and households in the urban system, together with the effects of the TOD investment on households’ residential location choices and housing market. The population thresholds for investing in a TOD project under the public and private regimes are also identified. The findings show that the TOD investment can cause population agglomeration at the TOD zones and a compact city; households and the society can benefit from the TOD investment; and the private TOD investment regime outperforms the public regime in terms of total social welfare of urban system. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction In response to rapid growth in the number of motorized vehicles and thus in traffic congestion, transit-oriented development (TOD) has been suggested as an effective tool for mitigating auto-related problems (e.g., traffic congestion and environmental issues) and controlling excessive urban sprawl (Cervero et al., 2004; Cervero and Day, 2008; Papa and Bertolini, 2015). TOD refers to medium- and high-density housing along with complementary public uses, jobs, retail, and services in mixed-use development around transit stations (Calthorpe, 1993). The TOD investment projects, as an important avenue for creating a green transportation system, have received considerable attention from the relevant authorities of many countries or regions in the world, such as United States, South Korea, and Hong Kong (Lund, 2006; Cervero and Day, 2008; Loo et al., 2010; Sung and Oh, 2011). Recently, the Chinese government has been developing the TOD projects in some densely populated cities, such as Guangzhou, Shenzhen, and Wuhan. The TOD schemes can attract households to reside nearby transit stations by providing them with amenities for convenience, enjoyment, or comfort, and thus improve the land-use efficiency and the value of properties nearby transit stations ∗

Corresponding author. E-mail address: [email protected] (Z.-C. Li).

http://dx.doi.org/10.1016/j.trb.2017.03.011 0191-2615/© 2017 Elsevier Ltd. All rights reserved.

Please cite this article as: Y.-T. Peng et al., Transit-oriented development in an urban rail transportation corridor, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.03.011

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Table 1 Contributions to the land use and transportation models.

Citation

Household’s residential location choice

Housing market

Transit service optimization

TOD design in terms of location, number and size

TOD investment regime (public or private)

Eliasson and Mattsson (20 0 0) Chang and Mackett (2006) Martínez and Henríquez (2007) Li et al. (2012b) Li et al. (2012c) Li et al. (2013) Ma and Lo (2013) Li et al. (2015) Ng and Lo (2015) Wang and Lo (2016) This paper

    ×     × 

× ×   ×     × 

× × ×   ×  × ×  

× × × × × × × × × × 

× × × × × × × × × × 

Note: “” means that the associated item is considered, whereas “×” means that the associated item is not considered.

(Kay et al., 2014). The pedestrian-friendly environment inside TOD zones encourages residents to commute by public transit modes, which can effectively alleviate the auto-related problems (Cervero, 2007; Loo et al., 2010). However, the development of TOD neighborhoods around transit stations may be costly due to investment in the public amenities inside the TOD zones (e.g., pedestrian lane design, school, hospital, shopping centers and sports facilities) (for more details, see Nelson and Niles, 1999; Lee et al., 2016). This leads to a tradeoff between cost and benefit generated by the TOD investment projects. In the TOD projects, the location, number and size of the TOD zones along a rail transit line play an important role in the cost and benefit analyses of the TOD investment. As a matter of fact, different locations, numbers or sizes of TOD zones on a rail transit line imply different residential distributions along that rail line and thus different passenger demands. They also imply different housing supplies and land development costs for the TOD facilities due to differential land values along the rail line and thus different costs and benefits of the TOD investment projects. Therefore, there is indeed a need to determine the optimal location, number and size of the TOD zones on a rail transit line such that the TOD investment projects are economic viable and cost-effective from an input-output perspective. The present study addresses the TOD investment issues for strategic planning purposes. In the literature, there are some studies involving TOD design or investment issues (see e.g., Cervero, 1994; Bernick and Cervero, 1997; Cervero and Kockelman, 1997). However, these previous related studies mainly focused on the strategies and principles/rules for TOD design. For example, Cervero and Kockelman (1997) summarized the TOD strategies and design principles as high-density, mixed land use and pedestrian-friendly development around transit stations. Lund (2006) conducted a survey to find out the motivations that people move into the TOD zones and their travel mode choices. Cervero (2007) addressed the effects of TOD investment on the transit ridership of residents inside the TOD zones by using a statistical analysis method. Loo et al. (2010) revealed the important factors of affecting the transit ridership inside a TOD zone through a comparison of the case studies of New York and Hong Kong. Kay et al. (2014) carried out a hedonic regression analysis to examine the effects of the TOD investment and the amenities provided by the mixed land development on residential properties nearby the transit stations. However, these existing relevant studies usually adopted empirical and/or statistical approaches to address the TOD investment related issues, and few studies adopted an analytical modeling method. An exception is Lin and Gau (2006), which developed a multi-objective model to determine the optimal investment intensity (or the ratio of floor space to site space) inside a given TOD zone. However, their study only focused on a specific TOD station on a rail line, and did not concern the optimization problems of the location, number and size of the TOD zones along a rail line. The present study aims to fill the gap by developing an analytical model for investigating the TOD investment issues along a rail corridor. In the literature, the interactions between urban land use and transportation system have been widely studied. For the convenience of readers, we have provided in Table 1 some principal contributions to the land use and transportation research, in terms of household’s residential location choice, housing market, transit service frequency/fare optimization, and the TOD investment. Table 1 shows that the previous related studies have mainly focused on urban land use (in light of residential location choice and housing market) and/or transit service optimization issues. In particular, the studies of Li et al. (2012b), Ma and Lo (2013) and Li et al. (2015) incorporated the effects of rail line investment on the value of properties, households’ residential location choices and housing market along the rail line. However, these relevant studies were not concerned with the TOD design problems in terms of the location, number, and size of TOD zones, together with the interactions among the TOD investment, households’ residential location choices, and housing market. Some empirical studies (see e.g., Lund, 2006; Hess and Almeida, 2007; Duncan, 2011; Dröes and Rietveld, 2015) have shown that the TOD schemes can affect households’ residential location choices and passenger demand distribution along the rail line, which in turn affect the decisions on the TOD investment and the train service frequency along the rail line. It is, therefore, important to consider such interactions among the TOD investment, households’ residential location choices and train service frequency in the TOD design problems. Please cite this article as: Y.-T. Peng et al., Transit-oriented development in an urban rail transportation corridor, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.03.011

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3

Fig. 1. The rail line configuration and TOD zones along the rail corridor.

The government Objective: maximizing social welfare of urban system (Decision variables: location, number and size of TOD zones, and train headway) TOD design scheme and train headway

Land value, passenger demand, utility level, and city boundary

Housing market equilibrium Households Objective: maximizing utility (Decision variables: residential location and housing/non-housing consumption)

Housing demand Housing supply

Property developers Objective: maximizing net profit (Decision variables: capital investment density)

Fig. 2. Interactions among three types of agents in the urban economy.

In addition, these existing related studies never concern the source of the capitals for TOD projects. The TOD facilities usually consist of housing and amenities (e.g., shopping centers or sports facilities that are provided for people’s convenience, enjoyment, or comfort). In the cities of mainland China, the housing inside the TOD zones is usually invested and built by the private property developers. The amenities (e.g., shopping centers or sports facilities) inside the TOD zones, as a public infrastructure, are usually invested by the government. However, there are also some cases that the amenities inside the TOD zones are invested by the private property developers. In this paper, the investment regimes that the amenities are invested by the government and by the private property developers are referred to as the public and private investment regimes, respectively. It is, thus, meaningful to examine the effects of the amenities’ investment regimes on the TOD investment decisions. For ease of presentation, “TOD’s amenities” and “TOD”, and thus “amenities’ investment” and “TOD investment” are used without difference in this paper. In view of the above discussions, this paper proposes analytical models for investigating the TOD investment issues in a rail transportation corridor. The problems to be addressed in this paper are stated as follows: given a rail line configuration (as shown in Fig. 1), how does the government determine such decision variables as the optimal location, number and size of the TOD zones along a rail line and the train service headway on that rail line so as to maximize the social welfare of the urban system? What are the effects of the TOD investment regimes on the TOD design? The main contributions of this paper are twofold. First, we present an urban system equilibrium problem with the TOD investment consideration, in which the residents inside and outside of the TOD zones are differentiated by their utility functions. Compared to the residents living outside the TOD zones, the residents living inside the TOD zones enjoy an extra amenity utility due to a better accessibility to the train stations and the constructed amenities inside the TOD zones. The households’ residential location choices, land value across the city, and the housing market in terms of housing rental price and space can be endogenously determined by the urban system equilibrium problem. Second, TOD design models under the public and private investment regimes are proposed, respectively. In the public and private regimes, the TOD investment cost is, respectively, borne by the government and the private property developers. The interactions among the government, property developers, and households are explicitly considered, as indicated in Fig. 2. Specifically, the government seeks to maximize the total social welfare of the urban system by determining the optimal location, number and size of the TOD Please cite this article as: Y.-T. Peng et al., Transit-oriented development in an urban rail transportation corridor, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.03.011

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Fig. 3. The shape of amenity function (x).

zones along a rail line corridor and the optimal train headway on that rail line. The private property developers aim to maximize their own net profit by providing the residents (or households) with housing. Households choose their residential locations to maximize their own utility within their budget constraints. The effects of the TOD investment on the urban system are examined, together with a comparison of public and private TOD investment regimes and no TOD investment. Sensitivity analyses of some factors that affect the TOD investment decisions are also conducted, including the TOD investment cost and the total number of households. The remainder of this paper is organized as follows. The next section describes some basic assumptions. Section 3 formulates basic components of the models, including passenger travel cost, passenger demand for each railway station, households’ residential location choices, property developers’ housing production behavior and housing demand-supply equilibrium. Section 4 presents TOD design models under different investment regimes for determining the optimal location, number, and size of the TOD zones and the train headway on the rail line. A heuristic solution algorithm for solving the proposed models is also proposed. In Section 5, a numerical example is provided to illustrate the applications of the proposed models and the model properties. Finally, Section 6 provides conclusions and recommendations for further studies.

2. Assumptions To facilitate the presentation of essential ideas without loss of generality, this paper makes the following basic assumptions. A1. The city is assumed to be linear, closed, and monocentric, which implies that the population of the city is exogenously given and fixed and all job opportunities are located in a highly compact central business district (i.e., CBD). It is also assumed that the land is owned by absentee landlords. The value of land at/beyond the city boundary equals the agricultural rent or opportunity cost of the land. These assumptions have been widely adopted in the previous studies (see e.g., Alonso, 1964; Muth, 1969; Mills, 1972; Fujita, 1989; O’ Sullivan, 20 0 0; McDonald, 20 09; Li et al., 2013, 2015; Li and Peng, 2016; Li and Guo, 2017). A2. There are three types of agents in the urban economy: the government, households and private property developers. The government aims to determine the location, number and size of TOD zones along a rail line corridor and the train headway to maximize the total social welfare of the urban system. The private property developers seek to maximize their own net profit generated by housing supply. Each property developer follows a Cobb-Douglas housing production function (see e.g., Beckmann, 1974; Quigley, 1984; Li et al., 2013, 2015; Li and Guo, 2017). A3. All households are assumed to be homogeneous, implying that the income level is identical for all households. Each household has a quasi-linear form of utility function (see e.g., Song and Zenou, 2006; Kono et al., 2012). A household’s income is spent on transportation, housing and non-housing goods. The objective of each household is to maximize its own utility by choosing a residential location, size of housing space, and amount of non-housing goods within its budget constraint (see e.g., Solow, 1972, 1973; Beckmann, 1969, 1974; Anas, 1982; Fujita, 1989). A4. The TOD facilities provide the residents living in the TOD zones with amenities for convenience, enjoyment, or comfort. The amenity level decreases with the distance from the train station (i.e., the center of the TOD zone). In this paper, the following form of amenity function is adopted: (x ) = 1 + a1 e−a2 |D j −x| , where Dj is the distance from TOD station j to the CBD and a1 and a2 are positive parameters, which can be calibrated by survey data (see e.g., Wu and Plantinga, 20 03; Wu, 20 06; Kovacs and Larson, 2007). The amenity function is discontinuous with the TOD’s boundary as discontinuous/interruption point, as shown in Fig. 3. Please cite this article as: Y.-T. Peng et al., Transit-oriented development in an urban rail transportation corridor, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.03.011

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A5. This study mainly focuses on the commuters’ home-based work trips, which is a compulsory (or obligatory) activity, and thus the household’s number of trips is not affected by various factors, such as the household’s income level and/or housing rents. The average number of commuters (or workers) per household is assumed to be exogenously given and fixed, and is represented as η. Every day, each worker makes a round commuting journey between his/her place of residence and workplace located in the CBD. Thus, the average daily number of trips to the CBD per household is η. For example, η = 1 means that each household makes an average of one trip to the CBD per day. This assumption of one worker per household has also been adopted by the previous related studies (see e.g., Anas and Xu, 1999; Song and Zenou, 2006; Li et al., 2013, 2015; Li and Guo, 2017). A6. An exponential elastic demand function is used to capture passengers’ responses to the quality of the rail transit service (Ortuzar and Willumsen, 2001), which is measured by a generalized travel cost that is a weighted combination of the access time to station, waiting time at station, in-vehicle time, and fare. The responses include the decision to switch to an alternative transportation mode (e.g., auto or bus) and the decision not to make the journey at all (Lam and Zhou, 20 0 0; Li et al., 2012c). In order to capture the effects of the TOD’s amenities, without loss of generality it is assumed that the passengers residing inside the TOD zones are less sensitive to the rail travel cost than those who reside outside the TOD zones (Lund et al., 2004; Lund, 2006; Cervero, 2007). 3. Components of the models Consider a linear rail line corridor in which TOD projects may be introduced. Referring to Fig. 1, the rail line is described by an ordered sequence of stations, represented as {1, 2, …, M + 1}. Ds represents the distance between station s and the CBD, and D1 represents the length of the rail line. The TOD investment divides the rail line corridor into two kinds of regions: internal regions inside the TOD zones and external regions outside the TOD zones. We denote J as the catchment area of all the internal regions and J¯ as the catchment area of all the external regions. Let B be the distance of the city boundary from the CBD (i.e., the corridor’s length). Thus, J ∪ J¯ = B holds. Let Xs be a 0–1 indicator variable which represents whether station s is selected as a TOD station, s = 1, 2,…, M. Xs = 1 if station s is selected as a TOD station, and Xs = 0 otherwise. Let s be the size (or radius) of TOD station s. If station s is a TOD station, then the catchment area of TOD station s can be represented by [Ds − s ,Ds + s ]. Consequently, J can further be expressed as

J = ∪M s=1 [Xs (Ds − s ), Xs (Ds + s )].

(1)

In the following, we in turn formulate the passenger travel cost, passenger demand for each station, households’ residential location choices, property developers’ housing production behavior and housing market equilibrium. 3.1. Passenger travel cost Let x denote the distance of a location from the CBD and c(x) denote the travel cost of a passenger from location x to the CBD. It comprises the walking/access time to the railway station, the wait time at the station, the in-vehicle travel time and the fare, expressed as

c (x ) = τa As (x ) + τwWs + τt Ts + Fs , ∀x ∈ [0, B],

s = 1, 2, . . . , M,

(2)

where As (x) is the average passenger access time to station s from location x, which is dependent on the distance of location x from station s. Ws is the average passenger wait time at station s. Ts is the average passenger in-vehicle time from station s to the CBD. Fs is the fare for traveling from station s to the CBD. τ a , τ w , and τ t are the values of the access time, wait time, and in-vehicle time, respectively. The average passenger access time As (x) depends on the walking distance between location x and station s and the average passenger walking speed Va . It is expressed as

As ( x ) =

|Ds − x| Va

,

∀x ∈ [0, B], s = 1, 2, . . . , M.

(3)

With an assumption of constant headway between trains and a uniform random passenger arrival distribution (Lam and Morrall, 1982), the average passenger wait time at station s, Ws , can be estimated by

Ws = 0.5H,

∀s = 1, 2, . . . , M,

(4)

where H is the train headway. The passenger in-vehicle time, Ts , from station s to the CBD can be calculated by

Ts =

Ds + κ0 (M + 1 − s ), Vt

∀s = 1, 2, . . . , M,

(5)

where Vt is the average train travel speed, and κ 0 is the average train dwell time at a station, which can be calibrated by survey data. The first term on the right-hand side of Eq. (5) is the non-stop line-haul travel time along the corridor and the second term is total train dwell time at all stations on the corridor. Please cite this article as: Y.-T. Peng et al., Transit-oriented development in an urban rail transportation corridor, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.03.011

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The train ticket fare Fs from station s to the CBD is defined as a linear function of the distance of station s from the CBD, similar to Li et al. (2012b, c). It is represented as

Fs = f0 + f Ds ,

∀s = 1, 2, . . . , M,

(6)

where f0 and f are the fixed and variable components of the train ticket fare, respectively. In light of the above, the annual passenger cost, ϕ (x), at location x can be expressed as

ϕ (x ) = 2ρ c(x ) = 2ρ (τa As (x ) + τwWs + τt Ts + Fs ), ∀x ∈ [0, B], s = 1, 2, . . . , M,

(7)

where “2” denotes a round trip between the CBD and location x, and ρ is the average number of trips to the CBD per household per year, which can be calibrated by survey data. 3.2. Passenger demand for each station Note that any two consecutive stations on any segment of the rail line are competing for passengers between those two stations. Consequently, there is a passenger watershed line that divides the line segment between two adjacent stations into two sub-segments (see Fig. 1). The passengers in the two sub-segments use the upstream and downstream stations of the line segment, respectively. Let ls be the passenger watershed line between stations s and s + 1, and Ls be the distance of the passenger watershed line ls from the CBD. According to Vuchic and Newell (1968), Vuchic (1969) and Li et al. (2012b), the passenger shed line ls is located so that the walking time from the shed line to the downstream station s + 1 equals the sum of the walking time from the shed line to the upstream station s and the riding time from station s to s + 1, that is,

Ls − Ds+1 Ds − Ls Ds − Ds+1 = + , Va Va Vt

∀s = 1, 2, . . . , M.

(8)

From Eq. (8), we obtain

Ls =

Vt + Va Vt − Va Ds + Ds+1 , 2Vt 2Vt

∀s = 1, 2, . . . , M,

(9)

where DM + 1 = 0. The catchment area of station s is thus [Ls ,Ls − 1 ] for any s = 1, 2,…, M. Let q0 (x) be the daily density of potential (latent) passenger demand (i.e., the potential number of passengers per unit of distance) at location x. According to A5, the average number of daily trips per household is η. Hence, we have q0 (x) = ηn(x), where n(x) is the household residential density at location x defined later. The passenger demand for the rail line service is elastic because it is sensitive to the travel cost by rail. Following A6, an exponential elastic demand density function is adopted to model the effects of the passenger demand elasticity. However, the passengers residing outside the TOD zones are more sensitive to the rail travel cost than those residing inside the TOD zones. The elastic demand density functions inside and outside the TOD zones are specified as



q (x ) =

q0 (x ) exp (−ω1 c (x ) ), q0 (x ) exp (−ω2 c (x ) ),

∀x ∈ J, ∀x ∈ J¯,

(10)

where q(x) is the daily density of (actual) passenger demand for rail transit travel at location x. ω1 and ω2 are parameters that reflect the demand sensitivity to the rail travel cost inside and outside the TOD zones, respectively. Herein, ω 1 < ω2 holds, implying that the demand elasticity of the passengers for rail service inside the TOD zones is lower than that outside the TOD zones, as assumed in A6. Hence, the daily number of passengers boarding trains at station s, denoted as Qs , can be calculated by



Qs =

Ls−1 Ls

q(x )dx,

∀s = 1, 2, . . . , M,

(11)

where Ls can be given by Eq. (9) and L0 = B. 3.3. Households’ residential location choices We first define the households’ utility function. As stated before, the TOD facilities provide residents living inside the TOD zones with public amenities (e.g., shopping centers or sports facilities), which incur an extra utility (called amenity utility in this paper) to the residents inside the TOD zones compared to those living outside the TOD zones. Following Song and Zenou (2006) and Kono et al. (2012), we assume in this paper that the household utility function takes a quasi-linear form, but with an additional consideration of the amenity utility for the households inside the TOD zones. The households’ utilities are specified as

U (x ) = z(x ) + α log g(x ) + β log (x ),

α , β > 0,

(12)

where U(x) is the utility of the households at location x. z(x) is the consumption of non-housing goods at location x, of which the price is normalized to 1. g(x) is the consumption of housing goods at location x, measured in square meters of floor space. α and β are positive constants. As specified in A4, the amenity level, (x), at location x is determined by Please cite this article as: Y.-T. Peng et al., Transit-oriented development in an urban rail transportation corridor, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.03.011

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 (x ) =

1 + a1 e−a2 |D j −x| , 1,

∀x ∈ J, ∀x ∈ J¯,

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(13)

where a1 and a2 are positive parameters, which can be calibrated by survey data. Note that when the amenity level, (x), at location x is 1 (i.e., the locations outside the TOD), the amenity utility, β log (x), at location x equals zero. According to A3, each household within the city chooses a residential location and housing / non-housing consumptions that maximize its utility subject to the income budget constraint. The household utility maximization problem can be expressed as

max U (z, g) = z(x ) + α log g(x ) + β log (x ),

(14)

subject to

z ( x ) + p( x ) g ( x ) = Y − ϕ ( x ) ,

∀x ∈ [0, B],

(15)

where p(x) is the average annual housing rental price per unit of housing area at location x, Y is average annual income of households in the city, and ϕ (x) is the average annual travel cost of a household living at location x. Substituting z(x) in Eq. (15) into Eq. (14) and setting the derivative of U(•) with regard to g equal to zero (i.e., dU /dg = 0), one obtains

p( x ) g ( x ) = α ,

(16)

z (x ) = Y − ϕ (x ) − α .

(17)

and

When the households’ residential location choice equilibrium state is reached, all households in the city have the same utility level regardless of their residential locations. Let u be the common utility level of all households at the equilibrium state. Substituting Eq. (17) into z(x) + α log g(x) + β log (x) = u, we have

g(x ) = exp

1

α

 (u − Y + ϕ (x ) + α − β log (x )) .

(18)

Substituting Eq. (18) into Eq. (16), we obtain

 1  (u − Y + ϕ (x ) + α − β log (x )) . α

p(x ) = α exp −

(19)

Eqs. (17) and (18) describe the equilibrium consumption of non-housing goods and the equilibrium amount of housing floor space per household at location x, respectively. Eq. (19) defines the equilibrium housing rental price per unit of housing floor space at location x. Given the common utility level u, TOD investment scheme and the train headway, one can then obtain the rail travel cost ϕ (x) by Eq. (7) and the amenity level (x) by Eq. (13). Subsequently, the equilibrium consumption of non-housing goods z(x), the equilibrium amount of housing floor space per household g(x), and the equilibrium housing rental price p(x) per unit of housing floor space at location x can be determined by Eqs. (17)-(19), respectively. 3.4. Property developers’ housing production behavior According to A2, the following Cobb-Douglas form of the housing production function is adopted by the property developers. θ

h ( S ( x ) ) = θ1 ( S ( x ) ) 2 ,

θ1 , θ2 ∈ (0, 1 ), x ∈ [0, B],

(20)

where h(S(x)) represents the housing supply per unit of land area at location x, S(x) denotes the capital investment per unit of land area at location x, and θ 1 and θ 2 are positive parameters. 3.4.1. Property developers’ profit under public TOD investment regime As previously pointed out, the public TOD investment regime in this paper implies that the TOD project is invested by the government. That is, the government is responsible for the TOD investment cost (i.e., the construction cost of amenities inside the TOD zones). Therefore, under the public regime, the TOD investment cost is included in the government’s objective of social welfare maximization, which is defined later. The property developers under the public investment regime receive the revenue from housing supply along the rail line and pay the land rent and the opportunity cost of the capital used for housing construction. The net profit of the property developers under the public TOD investment regime, represented by ψ PUB (x), is thus expressed as

ψPUB (x ) = p(x )h(S(x ) ) − (r (x ) + kS(x )), ∀x ∈ [0, B],

(21)

where the rental price per unit of housing space, p(x), is given by Eq. (19). k is the price of capital (i.e., the interest rate). The term p(x)h(S(x)) on the right-hand side of Eq. (21) denotes the total revenue received by housing supply. These two terms, r(x) and kS(x), in the bracket are the land rent cost and the capital cost, respectively. Please cite this article as: Y.-T. Peng et al., Transit-oriented development in an urban rail transportation corridor, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.03.011

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3.4.2. Property developers’ profit under private TOD investment regime For the private investment regime, the TOD investment cost is undertaken by the private property developers. The TOD investment cost is closely related to the value of the land used for constructing the amenities and the amenity level. In general, the higher the value of the land or the amenity level is, the higher the TOD investment cost is, and vice versa. Without loss of generality, we assume that the TOD investment cost at location x inside a TOD zone is a linear function of the land value and the amenity level at that location. Specifically, the TOD construction cost π (x) at location x is given by

π (x ) = c0 + c1 r (x ) + c2 (x ), ∀x ∈ J,

(22)

where r(x) is the land value at location x, which will be defined later. c0 is the fixed component of the TOD investment cost. c1 and c2 are the variable components of the TOD investment cost, which are relevant to the land value and the amenity level, respectively. Let ψ PRI (x) represent the net profit of the property developers due to the provisions of the housing and/or TOD amenities at location x under the private investment regime. For a location x within a TOD zone, the private property developers have to pay the TOD investment cost. ψ PRI (x) can be defined as



ψPRI (x ) =

p(x )h(S(x ) ) − (r (x ) + kS(x )) − π (x ), p(x )h(S(x ) ) − (r (x ) + kS(x )),

∀x ∈ J, ∀x ∈ J¯,

(23)

where the TOD investment cost, π (x), at location x is defined by Eq. (22). 3.4.3. Profit maximization problem for property developers From A2, the private property developers in the housing market aim to determine the capital investment intensity to maximize their own net profit. The profit maximization problem under the public or private regime can be expressed as

max S

ψPUB (S ) or ψPRI (S ).

(24)

The first-order optimality condition of maximization problem (24) yields

p(x )θ1 θ2 Sθ2 −1 − k = 0.

(25)

From Eq. (25), we obtain the equilibrium capital investment intensity as follows.



S(x ) = p(x )θ1 θ2 k−1

1 / ( 1 − θ 2 )

.

(26)

It should be pointed out that the expression for the equilibrium capital investment intensity S(x) is identical for the public and private TOD investment regimes. However, their values for a given location x are generally different because the value of the housing rental price p(x) at location x is generally different for these two regimes. This will be illustrated in the numerical study section. Note that under the perfect competition, the property developers earn zero profit (i.e., ψ PUB = ψ PRI = 0). The equilibrium land values under the public and private investment regimes can thus be, respectively, expressed as

rPUB (x ) = k and

1 θ2

−1



p(x )θ1 θ2 k−1

1/(1−θ2 )

,

∀x ∈ [0, B],

⎧    1/(1−θ2 ) ⎨ k 1 − 1 p(x )θ1 θ2 k−1 − c0 − c2 (x ) /(1 + c1 ), θ2 rPRI (x ) = ⎩k 1 − 1 p(x )θ θ k−1 1/(1−θ2 ) , 1 2 θ2

(27)

∀x ∈ J,

(28)

∀x ∈ J¯.

From Eqs. (27) and (28), one can immediately obtain the following property, which reveals the relationships among the land value, the housing rental price, and the TOD investment cost. Proposition 1. (i) For the public and private TOD investment regimes, as the housing rental price p(x) increases, the land value r(x) increases, and vice versa. (ii) For the private TOD investment regime, given the housing rental price p(x), as c0 or c2 in the TOD investment cost function (22) increases, the land value r(x) inside the TOD zones decreases, and vice versa. The proof of Proposition 1 is provided in Appendix A. It should be pointed out that the effects of c1 in the TOD investment cost function (22) on the land value r(x) is ambiguous because the relationship between k( θ1 − 1 )( p(x )θ1 θ2 k−1 ) 2 and c0 +c2 (x) is indeterminate.

1/ (1−θ2 )

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3.5. Housing demand-supply equilibrium The equilibrium household residential density, n(x), at location x according to Eqs. (18), (19), (20) and (26) can be given by



n (x ) =



u − Y + α + ϕ (x ) − β log (x ) h (S (x ) ) = σ exp − , g( x ) α ( 1 − θ2 )

(29)

θ

where σ = (θ1 (αθ2 k−1 ) 2 )1/(1−θ2 ) is a constant. At the equilibrium, all households in the city are inside the urban area, which can be given by



J

n(x )dx +



J¯

n(x )dx = N.

(30)

The first and second terms on the left-hand side of Eq. (30) represent the numbers of households inside and outside all the TOD zones, respectively. From J ∪ J¯ = B and substituting Eq. (29) into Eq. (30), we have



B 0



σ



u − Y + α + ϕ (x ) − β log (x ) exp − dx = N. α ( 1 − θ2 )

(31)

According to A1, the equilibrium rent per unit of land area devoted to housing at the fringe of the city equals the exogenous agriculture rent or the opportunity cost of the land, i.e., r(B) = rA . Combining Eqs. (19) and (27) and r(B) = rA , one obtains



u − Y + α + ϕ (B ) α (1 − θ2 )σ exp − α ( 1 − θ2 )



= rA .

(32)

Given the households’ annual income Y, the annual travel cost ϕ (x) and the amenity level (x), one can determine the equilibrium household utility level u and the city boundary B by solving the system of Eqs. (31) and (32) and thus the functions z(x), g(x) and p(x) in terms of Eqs. (17)–(19), and S(x), r(x) and n(x) in terms of Eqs. (26)–(29), respectively. The closed-form solutions of the equilibrium household utility level u and the city boundary B are given as follows (its proof is provided in Appendix B). Proposition 2. Given the annual travel cost ϕ (x), household income Y and the opportunity cost of land rA , the common utility u and city boundary B can be, respectively, given by



u = Y − α + α (1 − θ2 ) log and

B = D1 + where

Va



τa

σ (α (1 − θ2 )Va 1 + 2ρτa 2 ) 2ρτa N + Va rA



 α ( 1 − θ2 ) α (1 − θ2 )(2ρτa N + Va rA ) log − c ( D1 ) 2ρ rA (α (1 − θ2 )Va 1 + 2ρτa 2 )

(33)

(34)

 M    M 

ϕ ( LM ) ϕ ( Ds ) ϕ ( Ls ) ϕ (Ls−1 ) 1 = 1 − exp − + 2 exp − − exp − − exp − (35) α ( 1 − θ2 ) α ( 1 − θ2 ) α ( 1 − θ2 ) α ( 1 − θ2 ) s=1 s=2

and

2 =

M

s=1

 Xs

D s + s

D s −s



ϕ (x ) exp − α ( 1 − θ2 )





β log (x ) exp − 1 dx. α ( 1 − θ2 )

(36)

The following proposition further reveals the effects of total number of households, and location and size of TOD station on the household utility level and city boundary. Its proof is given in Appendix C. Proposition 3. The household utility level u increases with the size (or radius) of this TOD station j , but decreases with the total number of households in the city N and the distance of the TOD station from the CBD Dj . The city boundary B increases with N and Dj , but decreases with j . These effects are summarized in Table 2. 4. TOD design models under public and private investment regimes In this section, two models, a public and a private regime, are presented for the TOD design along a rail transit line. As previously stated, the layout and design of the TOD system are made by the government. However, the TOD investment cost may be undertaken by the government or the private property developers in practice. From the perspective of the Please cite this article as: Y.-T. Peng et al., Transit-oriented development in an urban rail transportation corridor, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.03.011

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Y.-T. Peng et al. / Transportation Research Part B 000 (2017) 1–22 Table 2 Effects of total number of households, location and size of TOD station on the household utility level u and city boundary B.

u B

N

Dj

j

− +

− +

+ −

Note: “+” means a positive correlation and “−” means a negative correlation.

planner (i.e., the government), the objective of the TOD design is to maximize the social welfare of the urban system by determining the optimal location, number and size of the TOD zones along a rail line corridor and the train headway on that rail line. The social welfare is the total benefits of all parties in the urban system, and is defined as the sum of the total utility of all households in the urban system, the aggregate land rent received by the absentee landlords, and the total fare revenue, minus the train operating cost and the TOD investment cost (only for the public investment regime in which the government pays the TOD investment cost). Let U represent the total utility of all households in the city. It can be given by

U = uN,

(37)

where the equilibrium household utility u can be calculated by Eq. (33). Let RENT represent the total land rents for constructing the rail line or housing and FARE represent the total fare revenue generated by the rail operations, respectively. They can be, respectively, expressed as



RENT =

B

0

(r (x ) − rA )dx,

(38)

and

F ARE = 2ρ

M

Fs Qs ,

(39)

s=1

where the rent or value r(x) per unit of land area at location x is determined by Eqs. (27) and (28) for the public and private regimes, respectively. The rail fare, Fs , from station s to the CBD is given by Eq. (6). The passenger demand, Qs , for station s is given by Eq. (11). Let COPE represent the annual train operating cost, which comprises the fixed operating cost and variable operating cost (Li et al., 2012b, c), expressed as

COPE = λ0 + λ1

 H

,

(40)

where λ0 is the annual fixed operating cost and λ1 is the annual operating cost per train.  represents the vehicle round journey time, and thus  H represents the fleet size or number of vehicles on the rail line. The round journey time  consists of the terminal time, the line-haul travel time and the train dwell time at stations. It is given as

 = ζ t0 + 2(t1 + t2 ),

(41)

where t0 is the constant terminal time on the circular line and ζ is the number of terminal times on the line. t1 and t2 are, respectively, the total line-haul travel time and total dwell time at stations (i.e., t1 = D1 /Vt and t2 = κ 0 M). Let CTOD represent the total construction cost of all the TOD zones along the rail line. According to Eqs. (1) and (22), CTOD can be expressed as

 CT OD =

J

π (x )dx =

M

s=1

 Xs

D s + s D s −s

(c0 + c1 r (x )+c2 (x ))dx,

(42)

where the TOD construction cost, π (x), at location x is determined by Eq. (22). The total catchment area J of all the TOD zones along the rail line is given by Eq. (1). In light of the above, the social welfare maximization problem under the public and private TOD investment regimes can be formulated as

 U  + RENT + F ARE − COPE − CTOD , for public regime, U + RENT + F ARE − COPE , for private regime, ⎧   B M uN + 0 (r (x ) − rA )dx + 2ρ s=1 Fs Qs − λ0 + λ1  ⎪ H ⎪ ⎨  D s + s  = − M for public regime, s=1 Xs Ds −s (c0 + c1 r (x )+c2 (x ) )dx, ⎪ ⎪   B M ⎩  uN + 0 (r (x ) − rA )dx + 2ρ s=1 Fs Qs − λ0 + λ1 H , for private regime,

max SW (X, , H ) =

(43)

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s.t. Constraints (30)–(32), M

μQ s ≤

s=1

KV EH , H

¯, Xs  ≤ s ≤ Xs 



Xs =

1, 0,

(44)

∀s = 1, 2, . . . , M,

if s is selected as a TOD station, s = 1, 2, . . . , M, Otherwise,

(45) (46)

where SW(·) denotes the total social welfare of the urban system. X = (Xs ,s = 1, 2,…, M) is the vector of the 0–1 indicator  variables. M s=1 Xs is thus the total number of the TOD stations along the rail line.  = (s ,s = 1, 2,…, M) is the vector of the sizes (or radii) of the TOD stations along the rail line. H is the train headway on the rail line. Constraints (30)–(32) represent the housing market equilibrium. μ is the peak-hour factor, i.e., the ratio of peak-hour demand to the daily passenger demand, which is used to convert the passenger volume from a daily basis to an hourly basis. μQs is the peak-hour passenger demand at station s. KVEH is the capacity of vehicles (i.e., the maximum number of passengers allowed in a vehicle, both seated and standing). Eq. (44) is the line capacity constraint, which guarantees that the rail service supply satisfies the asso¯ in Eq. (45) are the lower and upper bounds for the TOD size, respectively. ciated (peak-hour) passenger demand.  and  Eq. (45) reveals that s > 0 if and only if Xs = 1 (i.e., station s is a TOD station), and s = 0 otherwise. The social welfare maximization model (43)–(46) is a mixed 0–1 integer programming problem, which is usually nonlinear and non-convex, making it difficult to find its global optimal solution. In the following, a heuristic greedy approach M ∗ ∗ ∗ is used to find the optimal number s=1 Xs , optimal location X and optimal size  of the TOD stations along the rail line, where the asterisk represents the optimal solution of associated variable. The basic idea behind the greedy approach is to decompose the optimization problem with M candidate TOD stations into M sub-problems, and then sequentially solve the M sub-problems with one sub-problem at one time. Note that the number of the TOD stations is an integer variable, which makes it difficult to solve. Fortunately, the number of the TOD stations on a rail line is a finite number. Therefore, a simple approach to find the optimal number of the TOD stations is to compare the resultant objective function values with different numbers of the TOD stations. The radius of the TOD station is a continuous variable usually within a range of 100 m to 10 0 0 m (see the definition of TOD in Calthorpe (1993)). For simplification, in this paper we treat it as a discrete variable with a range from 100 m to 10 0 0 m, and the step size for searching its solution is set as 10 m. The step-by-step procedure of the heuristic greedy approach is described as follows. Step 1. First loop operation. Let j be the counter of the TOD stations, and j begins with 1, i.e., j = 1. Step 2. Second loop operation. Determine the optimal location and size of the jth TOD station. Step 2.0. Choose an initial location Xj(1) and an initial size j(1) for the jth TOD station. Calculate the amenity level j(1) and set the outer loop iteration counter ξ = 1. Step 2.1. Third loop operation. Choose an initial value for the train headway H(1) . Set the inner loop iteration counter to i = 1. Step 2.2. Calculate the annual travel cost ϕ (i) , and then determine the common utility u(i) and the city boundary B(i) by Eqs. (33)–(36). Calculate the values of the vectors p(i) , g(i) , S(i) , r(i) and n(i) by Eqs. (18)–(20) and (26)–(29), and then the aggregate land value RENT(i) , the total fare revenue FARE(i) , the annual train operating cost COPE (i) and the total TOD investment cost CTOD (i) by Eqs. (38)–(42), respectively. Calculate the resultant passenger demand, Qs (i) , for railway station s by Eq. (11). Step 2.3. Check whether the resultant passenger demand exceeds the rail service supply, i.e., whether the capacity constraint Eq. (44) is satisfied. When Eq. (44) is active (or binding), then solve the auxiliary train headway Hˆ (i ) =  (i ) KV EH / M s=1 μQs . Step 2.4. Update the train headway according to H (i+1 ) = H (i ) + (Hˆ (i ) − H (i ) )/i. Step 2.5. Convergence check for the third loop operation. If the relative gap H (i+1 ) − H (i ) /H (i )  is smaller than a prespecified tolerance, then stop. Otherwise, set i = i + 1 and go to Step 2.2. Step 2.6. Convergence check for the second loop operation. Change the values of the location Xj(ξ ) and size j(ξ ) for the jth TOD station, and repeat Steps 2.1–2.5. Check the resultant social welfare: if no better solution can be found, then stop and output SW(j) = max (SW(ξ ) ). Otherwise, set ξ = ξ + 1, and go to Step 2.1. Step 3. Convergence check for the first loop operation. Repeat Steps 2.0–2.6, and check the social welfare. If SW(j + 1) < SW(j) , then stop and output the optimal social welfare SW∗ = SW(j) . Otherwise, set j = j + 1, and go to Step 2. It should be pointed out that during the iterations, once constraints (44) and/or (45) are violated, the headway H and/or the TOD’s radius s are then set at the corresponding bounds. The proposed heuristic solution algorithm can arrive at a solution by making a sequence of decisions, each of which looks the best at that moment (that is, each decision is locally optimal). Thus, a global optimum solution cannot be guaranteed. 5. Numerical study In this section, a numerical example is used to illustrate the applications of the proposed models and the contributions of this paper. The example is intended to compare the optimal solutions under different TOD investment regimes and to Please cite this article as: Y.-T. Peng et al., Transit-oriented development in an urban rail transportation corridor, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.03.011

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Table 3 The input parameters for the numerical example. Symbol

Definition

Baseline value

M D1

Total number of stations on the rail line Length of the rail line (km) Value of access time (RMB/h) Value of waiting time (RMB/h) Value of in-vehicle time (RMB/h) Average walking speed of passengers (km/h) Average train cruise speed (km/h) Average train dwell time at a rail station (h) Fixed component of railway fare (RMB) Variable component of railway fare (RMB/km) Average annual number of trips to the CBD per household Average number of daily trips per household Sensitivity parameter in elastic demand function inside and outside TOD zones Parameters in household utility function Parameters in amenity function Average annual income of households (RMB/year) Parameters in housing production function Interest rate Fixed and variable components of TOD investment cost (RMB/year, RMB/km/year) Total number of households in the city Opportunity cost of land at the city boundary (RMB/km/year) Fixed and variable components of annual train operating cost (million RMB/year, million RMB/vehicle/year) Number of terminal times on the line Constant terminal time on the circular line (h) Peak-hour factor, i.e., the ratio of peak-hour flow to daily average flow Capacity of vehicles (passengers/vehicle) Lower and upper bounds for the radius of TOD zones (km)

10 20 30 40 20 5.0 50 0.01 3.0 0.3 365 1.0 0.001, 0.08 15,0 0 0, 80,0 0 0 0.1, 0.001 60,0 0 0 0.04, 0.7 0.05 160,0 0 0, 0.9, 10 0,0 0 0 60 0,0 0 0 150,0 0 0 50, 20 2.0 0.1 0.1 1500 0.1, 1.0

τa τw τt Va Vt

κ0 f0 f

ρ η ω1, ω2 α, β a1, a2 Y

θ 1, θ 2 k

c0, c1, c2 N rA

λ0, λ1 ζ t0

μ KVEH

¯ , 

Note: “RMB” stands for Chinese currency and US$1.0 approximates RMB6.88 on 1 January 2017.

reveal the effects of introducing the TOD projects on urban spatial structure in terms of household’s residential distribution, housing space, housing rental price, capital investment density and passenger demand for rail service. The effects of the TOD investment cost and the total number of households in the city on the optimal TOD design schemes are also explored, together with the urban population thresholds for investing in a TOD project under the public and private regimes. The proposed solution algorithm is coded in MATLAB and run on a ThinkPad T410 computer with an Intel(R) Core(TM) i5 CPU (2.4 GHz) and 4GB of RAM. The pre-specified tolerance is set as 0.0 0 0 01. The numerical experiment takes about 40 min of CPU time. In the following analysis, unless specifically stated otherwise, the values of the input parameters are the same as those in Table 3.

5.1. Comparison of solutions under public and private TOD investment regimes Fig. 4 shows the changes of the annual social welfare with the TOD investment under the public and private TOD investment regimes and the resultant optimal TOD design schemes along the rail line corridor. It can be seen in Fig. 4a that as the number of the invested TOD stations in the corridor increases, the annual social welfares of the urban system under these two TOD investment regimes first increase and then decrease. The optimal social optimum solutions, respectively, occur at 5 and 8 TOD stations for the public and private investment regimes, with the maximum social welfares of RMB63.68 billion and RMB64.73 billion per year. Herein, “RMB” stands for Chinese currency and US$1.0 approximates RMB6.88 on 1 January 2017. The no TOD case leads to the lowest social welfare of RMB63.50 billion per year. This implies that the public and private TOD investment regimes can bring an increase in the social welfare by RMB0.18 billion and RMB1.23 billion per year, respectively. It can also be seen in Fig. 4a that the social welfare curve with the private TOD investment regime is always above that with the public TOD investment regime. This means that the private TOD investment regime is superior to the public TOD investment regime in terms of the social welfare. It is, therefore, more appropriate for the authority to confer the development rights of the TOD projects to the private property developers from the society’s perspective of social welfare maximization. Fig. 4b and c depict the optimal TOD design schemes along the rail line corridor under the public and private TOD investment regimes, respectively. It can be noted that the number and size of the TOD zones and their investment importance rankings are different for the two regimes. Specifically, under the public regime (see Fig. 4b), 5 TOD zones are required to construct along the rail line, and the first TOD zone to be constructed is at the second nearest railway station from the CBD with a radius of 230 m, and the size of the TOD zones along the rail line first increases and then decreases. However, under the private regime (see Fig. 4c), 8 TOD stations to be invested are in turn constructed from the urban central area to the suburban area, and the size of the TOD zones always decreases outwards along the rail line. Such changes are a result of Please cite this article as: Y.-T. Peng et al., Transit-oriented development in an urban rail transportation corridor, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.03.011

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Fig. 4. (a) Changes of annual social welfare with number of TOD stations under public and private TOD investment regimes; (b) optimal location and size (or radius) of TOD stations under public regime; and (c) optimal location and size (or radius) of TOD stations under private regime.

the trade-off between the accessibility to the CBD and the TOD investment cost which is related to the land value and the amenity level. 5.2. Effects of TOD investment on urban system Fig. 5 displays the equilibrium household residential density, housing space, housing rental price, capital investment density, land value and daily passenger demand at each railway station under the public and private TOD investment regimes and the no TOD case. It can be seen in Fig. 5a that in contrast to the no TOD case, the TOD project investments under the public and private regimes lead to a higher residential density inside the TOD zones but a lower residential density outside the TOD zones. This is because the TOD zones can provide residents with a better accessibility to the railway station and the TOD amenities, such as the opportunities of entertainment and shopping. As a result, some residents would like to move into the inside of the TOD zones from the outside of the TOD zones. Consequently, the housing rental price and the capital investment density inside the TOD zones increase, while those outside the TOD zones decrease, as shown in Fig. 5c and d, respectively. Thus, the housing space inside the TOD zones decreases and that outside the TOD zones increases, as shown in Fig. 5b. It can also be seen in Fig. 5e that compared to the no TOD case, the TOD investment under the public regime causes an increase in the land value inside the TOD zones, but a decrease in the land value outside the TOD zones. However, under the private TOD investment the land value across the city always decreases. This is because in the public regime, the government invests in the amenities inside the TOD zones and the private property developers invest in the housing. This leads to a competition for the land between them. As a result, the land value inside the TOD zones increases. However, in the private regime, the private property developers invest in both the housing and the amenities, thus causing a lower land value. These observations are consistent with the results in Proposition 1. Fig. 5f depicts the changes of the passenger demand at each railway station under the scenarios with and without TOD project investment. It can be noted that as the distance from the CBD increases, the rail passenger demand decreases. In contrast with the no TOD case, the TOD investment can cause an increase in the passenger demand close to the CBD area Please cite this article as: Y.-T. Peng et al., Transit-oriented development in an urban rail transportation corridor, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.03.011

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Fig. 5. (a)–(f) represent household residential density, housing space per household, housing rental price, capital investment density, land value, and daily passenger demand at railway station with and without TOD projects, respectively.

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Table 4 Equilibrium solutions of urban system with and without TOD investment.

Solution

No TOD project

Number of TOD stations Total catchment area of all the TOD zones (km) Total number of households inside the TOD zones Total number of households outside the TOD zones Train headway (min) Length of corridor (km) Household utility level (RMB/year) Average household residential density (households/km) Inside TOD zones Outside TOD zones Average housing space (m2 /household) Inside TOD zones Outside TOD zones Average housing rental price (RMB/m2 /year) Inside TOD zones Outside TOD zones Average capital investment intensity (million RMB/km) Inside TOD zones Outside TOD zones Average land value (million RMB/km) Inside TOD zones Outside TOD zones Total daily passenger demand (million passengers/day) Annual social welfare (billion RMB/year)

– – – 60 0,0 0 0 3.27 25.68 101,140 23,358 – – 71.89 – – 208.66 – – 4905.26 – – 105.11 – – 0.18 63.50

Public investment regime

Private investment regime

5 8 1.66 6.18 277,634 450,282 322,366 149,718 2.50 1.90 25.23 24.86 103,328 105,099 23,783 24,139 167,250 72,861 13,677 8015 66.31 67.06 45.28 57.30 84.43 96.42 226.21 223.68 331.30 261.78 177.67 155.56 4994.40 5069.10 35,122.41 15,300.86 2872.17 1683.16 107.02 34.94 752.62 31.52 36.07 61.55 0.36 0.47 63.68 64.73 B Note: Average household residential density = N/citysize, Average housing space = 0 h (S (x ))dx/N, Average housing B B B B price = 0 p(x )h (S (x ) )dx/ 0 h (S (x ) )dx, Average capital investment intensity = 0 S (x )dx/city size, Average land value = 0 r (x )dx/city size.

rental

but a decrease in the passenger demand in the suburb. That is, there is a critical location such that the passenger demand increases on its left-hand side but decreases on its right-hand side due to the TOD investment. Specifically, the critical locations for the public and private investment regimes are 12 km and 18 km, respectively. Table 4 further summaries the effects of the TOD investment on the urban system under different investment regimes. Table 4 shows that compared to the no TOD case, the (public or private) TOD investment can cause an increase in the household utility level, average household residential density, average housing rental price, average capital investment intensity, total passenger demand and the annual total social welfare of the urban system, but a decrease in the city size and average housing space. Specifically, there are 277,634 and 450,282 households to move into the TOD zones under the public and private TOD investment regimes, respectively. Consequently, the city size decreases by 0.45 km (from 25.68 km to 25.23 km) for the public regime and by 0.82 km (from 25.68 km to 24.86 km) for the private regime, implying that the TOD investments lead to a more compact city and can thus restrain the urban sprawl. The public and private TOD investments, respectively, cause an increase in the rail passenger demand by 0.18 million (from 0.18 million to 0.36 million) and 0.29 million (from 0.18 million to 0.47 million). As a result, the train headway decreases (or equivalently, the train frequency increases) by 46 s and 82 s under the public and private regimes, respectively. In addition, in contrast with the no TOD case, the household utility level and the social welfare, respectively, increase by RMB2188 per year per household and RMB0.18 billion per year for the public investment regime, and by RMB3959 per year per household and RMB1.23 billion per year for the private investment regime. This implies that the TOD investment along the rail line benefits both households and the society. Table 4 also shows that in comparison with the public regime, the total catchment area of the TOD zones under the private regime increases by 4.52 km from 1.66 km to 6.18 km. This is because the private investment regime decreases the land value inside the TOD zones (see e.g., Fig. 5e) and thus the TOD investment cost, which leads to a larger total catchment area of the TOD zones. It can also be seen in Table 4 that in contrast to the public investment regime, the number of households inside the TOD zones, average household utility level, total daily passenger demand and social welfare under the private regime increase by 172,648 households, RMB1771 per year, 0.11 million passengers per day, and RMB1.05 billion per year, respectively. However, the average land value decreases by RMB72.08 million per kilometer. 5.3. Effects of TOD investment cost on TOD design schemes under public and private regimes To look at the effects of the TOD investment cost on the optimal TOD design scheme, we conduct numerical experiments by scaling the basic value of c0 in the TOD investment cost function (22) 0.8 time down and 1.2 times up for the public and private TOD investment regimes. Fig. 6 shows that for the public investment regime, the TOD investment begins with the second nearest station from the CBD, and the size of the TOD zones along the rail line first increases and then decreases. However, the TOD investment importance rankings under the private regime are in turn from the urban central area to the Please cite this article as: Y.-T. Peng et al., Transit-oriented development in an urban rail transportation corridor, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.03.011

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CBD

2nd

230m

1st

3rd

4th

260m 200m 160m

5th

6th

120m 100m

(a) Fixed cost of TOD investment = 0.8c0 CBD

2nd

1st

3rd

4th

5th

200m

230m

170m

130m

100m

(b) Fixed cost of TOD investment = c0 CBD

2nd

170m

1st

3rd

190m 140m

4th

5th

110m

100m

(c) Fixed cost of TOD investment = 1.2c0 Fig. 6. Optimal TOD design scheme versus TOD investment cost under public regime (c0 = 160 0 0 0).

Fig. 7. Optimal TOD design scheme versus TOD investment cost under private regime (c0 = 160 0 0 0).

suburban area, and the size of the TOD zones along the rail line always decreases, as indicated in Fig. 7. Figs. 6 and 7 also show that as the TOD investment cost increases from 0.8c0 to 1.2c0 , the optimal number of the TOD zones, respectively, decreases from 6 to 5 for the public regime, and from 9 to 8 for the private regime. These results mean that the TOD investment cost significantly affects the TOD investment decisions under different regimes. Table 5 further summarizes the effects of the TOD investment cost c0 on the urban system performance under the public and private investment regimes. It can be seen in Table 5 that for any of the both regimes, the increase of the TOD investment cost from 0.8c0 to 1.2c0 leads to a decrease in the total catchment area of the TOD zones, and thus in the total number of households inside the TOD zones, household utility level, total passenger demand and annual social welfare. As a result of decreased number of households inside the TOD zones, the total number of households outside the TOD zones and thus the train headway increase, leading to a growth in the city’s size. Please cite this article as: Y.-T. Peng et al., Transit-oriented development in an urban rail transportation corridor, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.03.011

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Table 5 Effects of TOD investment cost on urban system performance (c0 = 160 0 0 0).

Public regime

Private regime

Urban system performance

Fixed cost of TOD investment (RMB/year) 0.8c0

c0

1.2c0

Number of TOD stations Total catchment area of all the TOD zones (km) Total number of households inside the TOD zones Total number of households outside the TOD zones Train headway (min) Length of corridor (km) Household utility level (RMB/year) Total daily passenger demand (million passengers/day) Annual social welfare (billion RMB/year) Number of TOD stations Total catchment area of all the TOD zones (km) Total number of households inside the TOD zones Total number of households outside the TOD zones Train headway (min) Length of corridor (km) Household utility level (RMB/year) Total daily passenger demand (million passengers/day) Annual social welfare (billion RMB/year)

6 2.14 307,760 292,240 2.37 25.16 103,631 0.38 63.73 9 7.06 463,738 136,262 1.87 24.82 105,251 0.48 64.83

5 1.66 277,634 322,366 2.50 25.23 103,328 0.36 63.68 8 6.18 450,282 149,718 1.90 24.86 105,099 0.47 64.73

5 1.42 254,139 345,861 2.61 25.27 103,106 0.34 63.64 8 5.62 439,239 160,761 1.93 24.88 104,982 0.46 64.65

Fig. 8. Optimal TOD design scheme versus total number of households under public regime (N = 60 0 0 0 0).

5.4. Effects of number of households on TOD design schemes under public and private regimes Figs. 8 and 9 show the optimal TOD design schemes with different numbers of households under the public and private TOD investment regimes, respectively. It can be noted that the total number of households in the city significantly affects the number and size of the TOD zones. As the total number of households increases, the total number of the TOD zones and their associated sizes generally increase for each regime. The investment order of the TOD neighborhoods under the private regime is successively from the urban central area to the suburban area. However, it is not so for the public regime. Again, this is a result of the trade-off between the accessibility to the CBD and the TOD investment cost. Table 6 further indicates the effects of the total number of households on the urban system under the public and private investment regimes. It can be seen that for each investment regime, the increase in the total number of households leads to an increase in the total catchment area of all the TODs, the numbers of households inside and outside the TOD zones, city size, total daily passenger demand, and social welfare, but a decrease in the household utility level and the train headway. Fig. 10 depicts the changes of the net profit of the TOD investment projects with the total number of households in the city under the public and private TOD investment regimes. The net profit of the TOD investment projects is defined as the difference of the social welfare with optimal TOD investment scheme minus the social welfare without TOD investment projects. Fig. 10 shows that as the total number of households increases from 50,0 0 0 to 450,0 0 0, the net profit for each of the two regimes increases from a negative value to a positive value. The break-even point (i.e., the point with zero net ∗ profit) is 330,0 0 0 households for the public regime (i.e., NPUB = 330 0 0 0) and 70,0 0 0 households for the private regime (i.e., Please cite this article as: Y.-T. Peng et al., Transit-oriented development in an urban rail transportation corridor, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.03.011

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Fig. 9. Optimal TOD design scheme versus total number of households under private regime (N = 60 0 0 0 0). Table 6 Effects of total number of households on urban system performance (N = 60 0 0 0 0).

Public regime

Private regime

Urban system performance

Total number of households 0.8N

N

1.2N

Number of TOD stations Total catchment area of all the TOD zones (km) Total number of households inside the TOD zones Total number of households outside the TOD zones Train headway (min) Length of corridor (km) Household utility level (RMB/year) Total daily passenger demand (million passengers/day) Annual social welfare (billion RMB/year) Number of TOD stations Total catchment area of all the TOD zones (km) Total number of households inside the TOD zones Total number of households outside the TOD zones Train headway (min) Length of corridor (km) Household utility level (RMB/year) Total daily passenger demand (million passengers/day) Annual social welfare (billion RMB/year)

4 1.14 184,298 295,702 3.46 25.08 103,747 0.26 51.28 8 5.24 343,995 136,005 2.45 24.67 105,790 0.37 52.06

5 1.66 277,634 322,366 2.50 25.23 103,328 0.36 63.68 8 6.18 450,282 149,718 1.90 24.86 105,099 0.47 64.73

6 2.20 373,616 346,384 1.95 25.35 102,922 0.46 75.99 9 7.26 560,580 159,420 1.55 25.00 104,531 0.58 77.32

∗ = 70 0 0 0). This means that the population threshold for investing in a TOD project under the public regime is larger NPRI than that under the private regime. Thus, the public investment regime can cause a late investment in the TOD projects.

6. Conclusion and further studies This paper presented an analytical modeling approach to address the TOD investment issues in a rail transportation corridor. The TOD system, as a public facility, was assumed to be designed by the government, aiming to maximize the total social welfare of the urban system. The amenities inside the TOD zones can be invested by the government or the private property developers, leading to a public and a private investment regime, respectively. On the basis of an analysis of the urban system equilibrium with the TOD investment, social welfare maximization models under the public and private investment regimes were proposed to simultaneously determine the optimal location, number and size of the TOD zones and the optimal train headway on the rail line. In the proposed models, the effects of the TOD investments on the urban spatial structure in terms of households’ residential location choices and housing market were explicitly taken into account. Please cite this article as: Y.-T. Peng et al., Transit-oriented development in an urban rail transportation corridor, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.03.011

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Fig. 10. Changes of net profit of TOD investment projects with total number of households under public and private regimes.

Some important findings and new insights are obtained. First, the TOD investment can lead to a more compact city and thus effectively control the urban sprawl through changing households’ residential location choices and housing market. It can also lead to an increase in the total rail passenger demand (and thus an increase in the train frequency), household utility level, and social welfare. The TOD investment can thus benefit both households and the society. Second, the private investment regime outperforms the public regime in terms of total social welfare of the urban system. Thereby, in practice it is more appropriate for the government to confer the development rights of the TOD projects to the private property developers from the society’s perspective of social welfare maximization. In contrast to the private investment regime, the public investment regime can cause a late investment in the TOD projects. Third, the TOD investment cost and the total number of households in the city can significantly affect the TOD investment decisions in terms of the number and size of TOD neighborhoods. The proposed models can serve as a useful tool for addressing TOD investment issues from an economic viable and cost-effective perspective. Although the proposed models in this paper provide new revenues for examining the effects of the TOD investment on the urban system, some further extensions should be made. 1. The city concerned in this paper was assumed to be monocentric. However, modern cities generally have multiple business and commercial centers. Therefore, there is a need to extend the proposed models to consider a polycentric urban structure. 2. It is assumed in this paper that all households were homogeneous in the value of time and income level. However, in reality, people with different income levels have different attitudes/preferences towards residential location choices (Li and Peng, 2016). Therefore, it is worthwhile to investigate the effects of heterogeneous commuters with different income levels on the TOD design scheme. 3. The proposed models in this paper assumed that the rail line was operated by the government, with the objective of maximizing the total social welfare of urban system. Recently, the Chinese government has tried to encourage private sectors to take part in the public infrastructure investment projects through offering various confessional terms, such as Build-Operate-Transfer and Public-Private-Partnership. The objective of the private sectors is however to maximize their own profit. It is, thus, worthwhile to extend the proposed models to consider the interests of the private and public investors with different objectives. 4. The governmental regulations, such as regulations on urban population density, building height, city size, and floor area ratio, have a direct effect on the housing supply and households’ housing floor space (Brueckner, 2007; Kono et al., 2012) and thus on the TOD investment schemes. Therefore, it is meaningful to extend the proposed models to examine the effects of the governmental regulations on the urban spatial structure and the TOD investment decisions. 5. This paper focused mainly on rail mode, and thus the competition and substitution effects between private transportation and transit modes was ignored. It is thus meaningful to extend the proposed models to consider different modes of transportation and to address the effects of TOD investments on a multi-modal transportation system (Li et al., 2012a; Ma and Lo, 2013). 6. In the proposed models, rail line configuration (e.g., rail line length, station location and spacing) was assumed to be pre-given and fixed. However, in reality the TOD investment scheme affects the residential distribution along the rail line and thus the passenger demand for the rail transit service, which in turn affects the layout of the rail transit line configuration. Therefore, it is meaningful to investigate the integrated optimization issue of the rail transit line configuration and the TOD development scheme. Please cite this article as: Y.-T. Peng et al., Transit-oriented development in an urban rail transportation corridor, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.03.011

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7. The proposed heuristic solution algorithm in this paper cannot guarantee to find a global optimum solution of the models proposed in this paper. It is, thus, meaningful to develop an efficient solution algorithm for solving the models in a further study.

Acknowledgments We are grateful to Professor Hai Yang, Professor William H.K. Lam, four anonymous referees, and participants at the 8th International Workshop on Computational Transportation Science for their helpful comments and suggestions on earlier versions of the paper. The work described in this paper was jointly supported by grants from the National Natural Science Foundation of China (71525003, 71222107), the Major Program of National Social Science Foundation of China (13&ZD175), Huazhong University of Science and Technology (50 0130 0 0 01), and the National Research Foundation of Korea (NRF) Grant funded by the Korea government ( MSIP ) (NRF -2010-0029446). Appendix A. Proof of Proposition 1 According to Eqs. (27) and (28), one can obtain

  θ2 / ( 1 − θ 2 ) ∂ rPUB = θ1 pθ1 θ2 k−1 , ∂p    θ / ( 1 − θ2 ) θ1 pθ1 θ2 k−1 2 /(1 + c1 ), ∂ rPRI =   θ / 1 − θ ( ) 2 ∂p θ pθ θ k−1 2 , 1

(A.1)

∀x ∈ J,

(A.2)

∀x ∈ J¯,

1 2

1 ∂ rPRI =− , ∀x ∈ J, ∂ c0 1 + c1

(A.3)

∂ rPRI (x ) =− , ∀x ∈ J. ∂ c2 ( 1 + c1 )

(A.4)

Since θ 1 , θ 2 , k and c1 are positive parameters and (x) > 0 holds, we have 0, x ∈ J.

∂ rPUB ∂ rPRI ∂ rPRI ∂ rPRI ∂ p > 0, ∂ p > 0, ∂ c0 < 0, x ∈ J and ∂ c2 <

Appendix B. Proof of Proposition 2 According to Eqs. (29) and (30) and J ∪ J¯ = B, we obtain

σe

N=

σ

=

−

u−Y +α α ( 1 − θ2 )

u−Y +α − e α ( 1 − θ2 )

 

ϕ (x )

J¯

− e α (1−θ2 ) dx + B

0

 e J

 ϕ (x ) − e α (1−θ2 ) dx +

B

0

ϕ (x )

− e α (1−θ2 ) dx =



LM

0

ϕ (x )

− e α (1−θ2 ) dx +



ϕ (x )−β log (x ) α ( 1 − θ2 ) dx

 e

J

From Fig. 1, we have



−

−

ϕ (x )−β log (x ) ϕ (x ) − α ( 1 − θ2 ) − e α ( 1 − θ2 )

 M

s=1

Ds Ls

ϕ (x )

− e α (1−θ2 ) dx +



Ls−1

Ds

dx .

ϕ (x )

(B.1)



− e α (1−θ2 ) dx ,

(B.2)

where Ls is given by Eq. (9), and L0 = B. According to Eq. (7), one obtains



LM

0

ϕ (x ) e α (1−θ2 ) dx = −

 ϕ ( LM ) α (1 − θ2 )Va − α 1 − θ ( ) 2 1−e . 2ρτa

(B.3)

For any s = 1, 2, …, M, we have



Ds

Ls

and



Ls−1

Ds

ϕ (x )

− e α (1−θ2 ) dx =

ϕ (x )

 ϕ ( Ls ) α (1 − θ2 )Va − αϕ(1(−Dsθ)2 ) − e − e α ( 1 − θ2 ) , 2ρτa

− e α (1−θ2 ) dx =

 ϕ (Ls−1 ) α (1 − θ2 )Va − αϕ(1(−Dsθ)2 ) − e − e α ( 1 − θ2 ) . 2ρτa

(B.4)

(B.5)

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Substituting Eqs. (B.3)–(B.5) into Eq. (B.2) yields



B 0

ϕ (x ) e α (1−θ2 ) dx = −

   M M ϕ ( LM ) ϕ ( Ds ) ϕ ( Ls ) ϕ (B )

− ϕ (Ls−1 ) α (1 − θ2 )Va − − − − 1 − e α ( 1 − θ2 ) + 2 e α ( 1 − θ2 ) − e α ( 1 − θ2 ) − e α ( 1 − θ2 ) − e α ( 1 − θ2 ) . 2ρτa s=1 s=2

(B.6)

Substituting Eqs. (32), (35), (36) and (B.6) into Eq. (B.1) yields Eq. (33). From Eqs. (32) and (33), we obtain Eq. (34). This completes the proof of Proposition 2. Appendix C. Proof of Proposition 3 The partial derivatives of u and B with regard to N, Dj and j can be obtained as follows:

2ρτa α (1 − θ2 ) ∂u =− , ∂N 2ρτa N + Va rA

(C.1)

  ∂ c j (x ) ϕ (x ) β log (x ) exp −1 dx, α ( 1 − θ2 ) α ( 1 − θ2 ) ∂Dj D j − j    β log (D j +  j ) ϕ (D j +  j ) ∂u 4ρτa α (1 − θ2 ) = exp − exp −1 , ∂  j α (1 − θ2 )Va 1 + 2ρτa 2 α ( 1 − θ2 ) α ( 1 − θ2 ) ∂B Va α (1 − θ2 ) = , ∂ N 2ρτa N + Va rA     D j + j ∂ c j (x ) ∂B 2ρVa ϕ (x ) β log (x ) = exp − exp −1 dx, ∂ D j α (1 − θ2 )Va 1 + 2ρτa 2 D j − j α ( 1 − θ2 ) α ( 1 − θ2 ) ∂Dj    β log (D j +  j ) ϕ (D j +  j ) 2α (1 − θ2 )Va ∂B =− exp − exp −1 . ∂j α (1 − θ2 )Va 1 + 2ρτa 2 α ( 1 − θ2 ) α ( 1 − θ2 ) 4 ρ 2 τa ∂u =− ∂Dj α (1 − θ2 )Va 1 + 2ρτa 2



D j + j



exp −

(C.2) (C.3) (C.4) (C.5) (C.6)

∂ c (x )

Noted that 0 < θ 2 < 1 and ∂ jD > 0 hold. Therefore, we can easily obtain j

∂u ∂u ∂u < 0, < 0, > 0, ∂N ∂Dj ∂j

∂B > 0, ∂N

∂B ∂B > 0, and < 0. ∂Dj ∂j

(C.7)

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Please cite this article as: Y.-T. Peng et al., Transit-oriented development in an urban rail transportation corridor, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.03.011