Ttle Optmal densty of radal major roads n a two-dmensonal monocentrc cty wth endogenous resdental dstrbuton and housng prces Author(s) L, ZC; Chen, YJ; Wang, YD; Lam, WHK; Wong, SC Ctaton Regonal Scence and Urban Economcs, 2013, v. 43 n. 6, p. 927-937 Issued Date 2013 URL http://hdl.handle.net/10722/192481 Rghts NOTICE: ths s the author s verson of a work that was accepted for publcaton n Regonal Scence and Urban Economcs. Changes resultng from the publshng process, such as peer revew, edtng, correctons, structural formattng, and other qualty control mechansms may not be reflected n ths document. Changes may have been made to ths work snce t was submtted for publcaton. A defntve verson was subsequently publshed n Regonal Scence and Urban Economcs, 2013, v. 43 n. 6, p. 927-937. DOI: 10.1016/j.regscurbeco.2013.09.010
Optmal densty of radal major roads n a two-dmensonal monocentrc cty wth endogenous resdental dstrbuton and housng prces Zh-Chun L a,*, Ya-Juan Chen a, Ya-Dong Wang a, Wllam H. K. Lam b,c, S. C. Wong d a School of Management, Huazhong Unversty of Scence and Technology, Wuhan 430074, Chna b Department of Cvl and Structural Engneerng, The Hong Kong Polytechnc Unversty, Kowloon, Hong Kong, Chna c School of Traffc & Transportaton, Bejng Jaotong Unversty, Bejng 100044, Chna d Department of Cvl Engneerng, The Unversty of Hong Kong, Pokfulam Road, Hong Kong, Chna Abstract Ths paper proposes an analytcal urban system equlbrum model for optmzng the densty of radal major roads n a two-dmensonal monocentrc cty. The proposed model nvolves four types of agents: local authortes, property developers, households and household workers (.e. commuters). The local authortes am to maxmze the total socal welfare of the urban system by determnng the optmal densty of radal major roads n the cty. The property developers seek to determne the ntensty of ther captal nvestment n the land market to maxmze the net proft generated from the housng supply. The households choose resdental locatons that maxmze ther utlty wthn a budget constrant, and the commuters choose the radal major roads that mnmze ther ndvdual costs of travel between home and workplace. A heurstc soluton procedure s developed to fnd the urban system equlbrum soluton. A system optmum model s also proposed to optmze the densty of radal major roads that maxmzes the socal welfare of the urban system. The proposed model can endogenously determne household resdental dstrbuton and land values across the cty, along wth the housng market structure n terms of housng prces and space. Numercal comparatve statc analyses of congeston prcng and road nfrastructure nvestment (addng a new radal major road) are carred out together wth evaluaton of the effects of the servce level of radal major roads, urban populaton sze, and household ncome level on the urban economy. Keywords: Two-dmensonal monocentrc cty; densty of radal major roads; urban system equlbrum; household resdental locaton choce; housng market; congeston prcng. * Correspondng author. Tel.: +86-27-8754-6073; Fax: +86-27-8755-6437. E-mal address: smzcl@gmal.com (Z.-C. L).
1. Introducton Many of the large modernzed ctes n Chna such as Bejng, Shangha and Wuhan have dramatcally ncreased n sze n recent years as a result of Chna s rapd urbanzaton and economc growth. Ths rapd urban expanson has led to a more decentralzed urban structure, longer average journey length, ncreased use of prvate cars, and hgher level of traffc congeston, partcularly wthn metropoltan areas. In response, local authortes have launched a number of transportaton nfrastructure mprovement projects to ncrease accessblty to the cty center and allevate urban traffc congeston, ncludng the constructon of radal major roads desgned to connect the cty s central busness dstrct (CBD) to ts suburban areas, as shown n Fg. 1. For example, the Wuhan muncpal government recently approved a proposal to buld a new radal major road wth a total length of 15 klometers to connect Gedan new town to Guanshan (Wuhan s CBD). Constructon for ths road project began n January 2012 and wll be completed at the end of 2013. The ntroducton of a new radal major road s expected to ncrease the capacty of a cty s transportaton system and mprove the accessblty of ts central area. Accordngly, the new radal major road can nfluence commuters road choces, whch affect the traffc flows and traffc congeston level on the major roads. However, the constructon of a new major road also nvolves a huge captal cost. The plannng and desgn of cty s radal major roads thus nvolve mportant welfare consderatons. Specfcally, an oversupply of major roads can lead to wastage of resources due to an neffcent use of road space. An undersupply of roads, n contrast, can result n traffc jams generated by a lack of suffcent road capacty to accommodate travel demand. Ths rases an nterestng queston: what s the optmal densty (or number) of radal major roads n terms of the total socal welfare (or surplus) of an urban system, partcularly when the cty boundary s expandng? The answer has long-term mplcatons for sustanable urban development, especally n rapdly expandng Chnese ctes such as Bejng, Shangha and Wuhan. To address ths mportant urban development ssue, advanced urban spatal models must be developed to help explan and evaluate the effects of dfferent denstes of radal major roads on the urban economy. In the classcal urban models that were developed by Alonso (1964), Mlls (1972) and Muth (1969), the urban transportaton network was assumed to be a 2
perfectly dvsble dense system consstng of an nfnte number of radal roads. These tradtonal urban models cannot be used to nvestgate the desgn ssue of the densty (or number) of urban radal major roads and evaluate the mpact of addng a new radal major road on the urban economy. One has thus to turn to a model that consders the urban structure as a sparse system of dscrete radal roads. In ths regard, Anas and Moses (1979) conducted a poneerng work. In ther work, t was assumed that commuters ether travel straght along dense surface streets to the cty center or travel along crcular dense surface streets to a radal road (.e. an expressway) and then travel along the radal road to get to the cty center (.e. a crcumferental-radal or rng-radal travel method). It was also assumed that the urban form was symmetrc (.e. the radal roads were evenly dstrbuted along the crcular cty) and there was no traffc congeston on the radal roads. The rng-radal travel method that they proposed s useful for modelng a dscrete-contnuous urban system (e.g., see Baum-Snow, 2007). However, the symmetry and congeston-free assumptons may cause a sgnfcant bas n the predcton capablty of the model, and thus restrct ts applcatons n practce. For example, congeston prcng ssues cannot be addressed under the congeston-free assumpton. D Este (1987) relaxed the assumptons of a symmetrc cty that s free of congeston, and proposed a trp assgnment model usng the rng-radal travel method of Anas and Moses (1979) to model commuters radal major road choces n a two-dmensonal monocentrc cty wth a small number of radal major roads. Wong (1994) reformulated D Este s trp assgnment model as a mathematcal programmng problem. These two studes assumed that the dstrbuton of resdental areas n the cty was exogenously gven and fxed. However, studes have shown that the mproved accessblty to road travel resultng from transportaton nfrastructure mprovements (e.g. ntroducng a new radal major road) can nduce changes n urban land-use patterns, land values and the housng market (n terms of housng prces and space) (e.g., see McDonald and Osuj, 1995; Henneberry, 1998; O Sullvan, 2000; Mkelbank, 2004; Ho and Wong, 2007; L et al., 2012b). Accordngly, ntroducton of a new major road may change a cty s household resdental locaton choces and resdental dstrbuton, whch can n turn affect the travel demand on the urban road network and thus the local authortes decsons regardng whether to ntroduce new major roads nto the system. Therefore, the effects that a new transportaton nfrastructure wll have on a cty s household resdental locaton choces, land values and housng market must be consdered n the plannng and desgn of new radal major roads of urban systems. 3
To account for these effects n the urban plannng, ths paper proposes an analytcal model for nvestgatng the desgn ssue of the densty (or number) of radal major roads n a two-dmensonal monocentrc cty (not necessarly symmetrc), as shown n Fg. 1. Ths paper makes two major extensons to the related lterature. Frst, an urban system equlbrum problem s formulated to model the nteractons among the followng four types of agents: local authortes, property developers, households and household workers (.e., commuters). The traffc congeston effects on the radal major roads are explctly consdered, and the household resdental dstrbuton and land values across the cty along wth the structure of the housng market n terms of housng prces and space can endogenously be determned. A heurstc soluton procedure s also presented to solve the urban system equlbrum problem. Second, a system optmum model for optmzng the densty of radal major roads s proposed to maxmze the urban system s total socal welfare. Wth use of the proposed model, numercal comparatve statc analyses of congeston prcng and road nfrastructure nvestment (.e. addng a new radal major road) are conducted. In addton, the effects of road level of servce (LOS), urban populaton sze and household ncome level on the urban system are also examned. The remander of ths paper s organzed as follows. The next secton descrbes the model s basc assumptons. Secton 3 formulates the equlbrum of the urban system, whch ncludes the commuters route choce equlbrum, the household resdental locaton choce equlbrum and the housng demand-supply equlbrum. A heurstc soluton approach s developed to solve the urban system s equlbrum problem. Secton 4 proposes a system optmum model for optmzng the densty of radal major roads n a two-dmensonal monocentrc cty. Secton 5 uses a numercal example to llustrate the applcatons of the proposed model. In partcular, comparatve statc analyses of some polcy parameters are conducted. Fnally, Secton 6 concludes the paper and provdes recommendatons for further study. 2. Basc assumptons To facltate the presentaton of essental deas wthout loss of generalty, ths paper makes the followng basc assumptons. 4
A1 The urban system s assumed to be a radal, closed and monocentrc cty, n whch the total populaton or number of households s exogenously gven and fxed and all job opportuntes are located n a hghly compact cty center or CBD. It s further assumed that all of the land wthn the cty boundary s owned by absentee landlords and that the value of the land at/beyond the cty boundary s equal to the agrcultural rent or opportunty cost of the land. These assumptons have been wdely adopted n the feld of urban economcs (e.g., Alonso, 1964; Muth, 1969; Mlls, 1972; Fujta, 1989; O Sullvan, 2000; Kraus, 2006; McDonald, 2009). A2 There are four types of agents n the urban economy: local authortes, property developers, households and commuters. Local authortes am to optmze the densty (or number) of radal major roads n the cty to maxmze the urban system s total socal welfare. The property developers determne the ntensty of ther captal nvestment n the land market to maxmze the net proft generated by the supply of housng. Each property developer s assumed to adopt Cobb-Douglas behavor for the housng producton functon (e.g., Beckmann, 1974; Qugley, 1984; Brueckner, 2007). A3 All households are assumed to be homogeneous, mplyng that the ncome level and utlty functon are dentcal for all households. Each household has a Cobb-Douglas utlty functon and the household ncome s spent on transportaton, housng and non-housng goods. The objectve of each household s to maxmze ts own utlty by choosng the optmum resdental locaton, area of housng space and amount of other goods wthn the household s budget constrants (e.g., Solow, 1972, 1973; Beckmann, 1969, 1974; Anas, 1982; Fujta, 1989). A4 Smlar to Anas and Moses (1979), D Este (1987) and Wong (1994), t s assumed that commuters travelng from ther home locatons to workplaces n the CBD frst travel along the mnor rng road that passes ther partcular home locatons to reach a radal major road, and then proceed along the major road to reach the cty center, whch consttutes a rng-radal routng system (see Fg. 1). It s also assumed that each commuter chooses the rng-radal route that mnmzes hs/her travel cost, thereby leadng to a Wardrop s user equlbrum for route choce. A5 Ths paper manly focuses on the commuters journeys between ther homes and ther 5
workplaces, whch are compulsory (or oblgatory) trps. The average number of commuters (or workers) per household s assumed to be exogenously gven and fxed, and s represented as. Every day, each worker completes one two-way commutng journey between hs/her place of resdence and a workplace located n the CBD. Thus, the average daly number of trps to the CBD per household s. For example, 1 ndcates that each household makes an average of one trp to the CBD per day. The assumpton that there s one commuter per household has also been adopted n prevous related studes (e.g., Anas and Xu, 1999; Song and Zenou, 2006; L et al., 2012a,b,c). 3. Equlbrum of the urban system Accordng to A2, there are four types of agents n the urban system: local authortes, property developers, households and commuters. The nteractons among these agents lead to a number of nterrelated equlbra: the commuters rng-radal route choce equlbrum, the household resdental locaton choce equlbrum and the housng demand-supply equlbrum. We formulate these respectve equlbra n the followng subsectons. 3.1. Commuters route choce equlbrum 3.1.1. Travel cost As shown n Fg. 1, represents the th radal major road and M s the total number of radal major roads n the cty. Let x be the radal dstance of a locaton from the CBD and the angle of a locaton from the nearest radal road. The travel cost from locaton ( x, ) to the CBD va radal major road s defned as ( x, ), whch conssts of the tme and monetary costs of travel on the rng-radal routng system,.e., ( x, ) T( x, ) C ( x, ), (1) where T( x, ) and C ( x, ) are the travel tme and the monetary cost from locaton ( x, ) to the CBD va radal major road, respectvely. s the value of travel tme, whch s used to convert the travel tme nto equvalent monetary unts. 6
Accordng to A4, the travel tme T( x, ) conssts of the tme taken to travel dstance x from the CBD along radal major road and the tme taken to travel a crcular arc wth a length of x along the mnor rng road at dstance x from the CBD. Ths s represented as x T( x, ) T( x), (2) V 0 where T ( x ) s the tme requred to travel dstance x from the CBD on radal major road and V 0 s the average vehcle travel speed on the mnor rng road. Travel tme ( ) T x depends on the level of traffc congeston on radal road. Let t Q( x ) be the travel tme per unt of dstance around locaton x on radal road, where Q ( x ) s the traffc volume (.e., the number of commuters) at locaton x on radal road. ( ) t Q x s assumed to be a strctly ncreasng functon of traffc volume Q ( x ) at locaton x and can be estmated usng the followng Bureau of Publc Roads (BPR) functon Q ( x) tq( x) t0 1.0a1 K a2, (3) where t 0 s the free-flow travel tme per unt of dstance on radal road, of radal road and a 1 and a 2 are postve parameters. K s the capacty Hence, T ( x ) can be expressed as x T( x) t Q( w) dw 0, (4) where t ( ) can be calculated usng Eq. (3). The monetary cost C ( x, ) s assumed to be a lnear functon of the dstance traveled, as assumed n Wang et al. (2004) and Lu et al. (2009),.e., C ( x, ) c c xc x, (5) 0 1 2 where c 0 s the fxed cost of travel (e.g., the parkng charge n the CBD per work trp) and c 1 and 2 c are the varable costs (e.g., fuel cost per unt of dstance) of travel on the radal and rng roads, respectvely. 7
Remark 1. (Propertes of the travel cost functon). The frst-order partal dervatves of ( x, ) wth regard to x and, respectvely, are gven by ( x, ) T( x, ) C( x, ) tq( x) c1c2 0 x x x V0 and (6) ( x, ) x c 2x0. (7) V 0 Therefore, for a gven, ( x, ) s a strctly ncreasng functon of the dstance between x and the CBD. For a gven x, ( x, ) s a strctly ncreasng functon of. Ths mples that the travel cost ncreases outwardly n the drecton of the radal major road extensons and away from the radal major roads along the mnor rng roads. 3.1.2. Boundary contour As Fg. 1 shows, two adjacent alternatve radal major roads n the two-dmensonal monocentrc cty, and +1, compete for travel demand. A watershed boundary exsts that dvdes the area between these adjacent radal roads nto two sub-areas. The watershed boundary s also called market area boundary n Anas and Moses (1979). For presentaton purposes, we defne B as the boundary contour, whch dvdes the travel demand between and +1 nto those who use and those who use +1, as shown n Fg. 1. Accordng to A4, when an equlbrum state s reached no commuter n the cty has an ncentve to change hs/her choce of radal major road when travelng to the CBD. Ths means that the cost of travelng from pont B on the boundary contour to the CBD usng radal roads and +1 s dentcal. The travel choce equlbrum can be mathematcally expressed as ( x, ˆ ) ( x, ˆ ), (8) 1 where ˆ ( x) s the angle between major road and boundary contour B at dstance x from the CBD and s the angle between radal major roads and +1. From Eqs. (1) and (8), one obtans xˆ ( x ) ˆ x ( ) 2 ( ) 1( ) ˆ ( ) 2 ˆ T x c x x T x x c x ( x) V. (9) 0 V0 Thus, ˆ ( x) can be expressed as 8
ˆ V ( x ) T ( x ) T ( x ), (10) 2 2x 2 0 0 1 c V where T ( x ) can be determned by Eq. (4). The boundary contours B1, B2,..., B M can be determned by Eq. (10). The catchment area of radal major road s the sector bounded by B 1 and B. Remark 2. (Property of angle ˆ ). Suppose that the varable travel cost c 2 on any rng road s a constant (.e. c 2 c2 n Eq. (5)), we then have sdes of Eq. (10) from = 1 to M yelds ˆ V ( x ) T ( x ) T ( x ) M M 1 1 2 2x 2 0 0 M1 1 cv M 1 ˆ ( x ). In fact, summng both. (11) Note that TM 1( x) T1( x) holds for a crcular cty. Therefore, Eq. (11) can be reduced to M Note that M 1 M ˆ ( ) x 2. (12) 1 1 ˆ ( x ). M 1 2. Substtutng t nto the above equaton, one mmedately obtans Remark 2 reveals that the traffc flows on the radal major roads are nterdependent; that s, an ncrease n the flow on one radal road leads to a decrease n the flow on another. It should be ponted out that the free-flow travel tmes and capactes (.e., t 0 and K n Eq. (3)) may change across the radal major roads due to the roads varyng geometres, sght dstances and servce levels. Thereby, ˆ ( x) 2 cannot hold. However, when a cty s served by evenly spaced radal roads wth the same free-flow travel tmes and capactes, as assumed n D Este (1987), then the cty s structure s entrely symmetrcal. 2 Remark 3. If the radal major roads n a cty are evenly dstrbuted (.e., 1, M 1,2,..., M ) and have dentcal travel tme functons (.e., t 0 t ( 1)0 and K K 1, 1,2,..., M ), then ˆ ( x). 2 M 9
3.1.3. Travel demand for each radal major road Let qx (, ) be the hourly densty of travel demand (.e., the number of commuters per unt of land area) at locaton ( x, ), be the average number of daly trps to the CBD per household and be the peak-hour factor (.e., the rato of peak-hour flow to the average daly flow), whch s used to convert the travel demand from a daly to an hourly bass. qx (, ) can then be defned as qx (, ) nx (, ) nx (, ), (13) where ( ) s the average number of peak hour trps to the CBD per household and nx (, ) s the household resdental densty at locaton ( x, ), whch s defned later. The hourly travel demand Q ( x ) at locaton x on radal major road can thus be gven by x ˆ ( ) ˆ w x ( w) Q ( x ) q ( w, ) wddw n ( w, ) wddw, (14) x ( ˆ ˆ 11( w)) x ( 11( w)) where x s the dstance from the cty boundary to the CBD (or the cty length) along radal road and ˆ s gven by Eq. (10). The frst-order dervatve of Q ( x ) wth regard to x s dq ( x) xn( x, ) d. (15) ˆ ( x ) ( ˆ dx 11( x)) dq ( x) Ths mples that 0. The travel demand functon Q ( x ) s therefore a decreasng dx functon of dstance x from the CBD. Remark 4. In vew of the above, once the household resdental densty nx (, ) s known, one can then determne the travel demand Q ( x ) by Eq. (14) or (15), the travel tme T ( x ) by Eq. (4), the travel cost ( x, ) by Eq. (1) and the boundary contour set M or, equvalently, the crtcal angle set ˆ ( x ), 1,2,..., M B, 1, 2,..., (10). n terms of Eq. 10
3.2. Housng market equlbrum 3.2.1. Household resdental locaton choce behavor To model household resdental locaton choce behavor, we frst defne the household utlty functon. In ths paper, the followng Cobb-Douglas form of the household utlty functon s adopted U( x, ) z( x, ) g( x, ),, 0, 1, (16) where U( x, ) s the utlty functon of households at locaton ( x, ), and are postve constants, zx (, ) s the composte non-housng goods consumed per household at locaton ( x, ), for whch the prce s normalzed to 1, and gx (, ) s the average consumpton of housng per household at locaton ( x, ), whch s measured n square meters of floor space. Accordng to A3, t s assumed that each household chooses the resdental locaton that maxmzes ts own utlty, subject to budget constrants. The household utlty maxmzaton problem can be represented as max U( x, ) z( x, ) g( x, ), (17) zg, s.t. zx (, ) px (, ) gx (, ) YEx (, ), (18) where px (, ) s the average annual rental prce per unt of housng floor area at locaton ( x, ), Y s the average annual household ncome and Ex (, ) s the average annual travel cost (or expendture) from locaton ( x, ) to the CBD along the rng-radal routng system, whch can be measured by Ex (, ) 2 ( x, ), (19) where the 2 denotes a round-trp journey between locaton ( x, ) and the CBD and s the average annual number of trps to the CBD per household. The (one-way) average travel cost, ( x, ), from locaton ( x, ) to the CBD can be determned by Eq. (1). Substtutng zx (, ) n Eq. (18) nto Eq. (17) and settng the dervatve of U ( ) wth regard to g equal to zero (.e., du dg 0 ), one obtans 11
Y E( x, ) gx (, ) px (, ). (20) When the household resdental locaton choce equlbrum state s reached, all households n the cty have the same utlty level regardless of ther resdental locatons. Let u be the common utlty level. The followng equaton then holds u z( x, ) g( x, ) Y E( x, ) p( x, ) g( x, ) g( x, ), (21) where g() and p() can be derved from Eqs. (20) and (21) as functons of the common utlty level u, as follows: 1 gx (,, u) YEx (, ) u and (22) 1 1 p( x,, u) Y E( x, ) u. (23) Eqs. (22) and (23) descrbe the equlbrum amount of housng floor space per household and the equlbrum prce per unt of housng floor space, respectvely, at locaton ( x, ). These equatons show that for a gven level of utlty, the average housng prce decreases and the average housng floor space per household ncreases as the travel cost ncreases, and vce versa. 3.2.2. Property developers housng producton behavor Ths subsecton focuses on the supply sde of the housng market. In ths paper, property developers are assumed to behave n keepng wth the followng Cobb-Douglas form of the housng producton functon S x h S( x, ) (, ) b, 0 b 1, (24) where (, ) h S x s the housng supply per unt of land area at locaton ( x, ), S( x, ) s the captal nvestment per unt of land area at locaton ( x, ) (also referred to as captal nvestment ntensty) and and b are postve parameters. Let rx (, ) be the rent or value per unt of land area at locaton ( x, ) and k be the prce of captal (.e., the nterest rate). The net proft per unt of land area, ( x, ), at locaton ( x, ) can then be gven by 12
( x, ) p( x, ) h S( x, ) r( x, ) ks( x, ), (25) where the prce per unt of housng floor space p( ) s gven by Eq. (23). The frst term on the rght-hand sde of Eq. (25) s the total revenue from housng leases. The fnal two terms are the land rent cost and the captal cost, respectvely. From A2, each property developer n the housng market ams to maxmze hs/her own net proft by determnng the optmal captal nvestment ntensty, whch s expressed as b max ( x, ) px (, ) S rx (, ) ks. (26) S The frst-order optmalty condton of the maxmzaton problem (26) yelds px bs k S b1 (, ) 0. (27) Substtutng p( ) n Eq. (23) nto Eq. (27) produces the captal nvestment ntensty as a functon of the utlty level u,.e., 1 1 1 1 1 b Sx (,, u) YEx (, ) u bk. (28) The household resdental densty nx (, ) at locaton ( x, ) can thus be calculated by 1(1 b) 1 1 ( b) h S( x,, u) nx (, ) bk u YEx (, ) gx (,, u) b ( b) ( b). (29) Note that under perfect competton, the property developers earn zero proft, thus b rx (, ) px (, ) Sx (, ) ksx (, ). (30) Substtutng Eqs. (23) and (28) nto Eq. (30) yelds 1 1 1 1 1 1 rx (,, u) k 1 YEx (, ) u bk b b. (31) Eqs. (28), (29) and (31) defne the equlbrum captal nvestment ntensty, resdental densty and land value at any locaton wthn the cty, respectvely. It can be seen that gven the utlty level u, the captal nvestment ntensty, resdental densty and land value all decrease as ether the travel cost or the nterest rate ncreases, and vce versa. 13
3.2.3. Housng demand-supply equlbrum Balancng the housng supply and demand requres that all households ft exactly nsde the cty boundary,.e., M 1 ˆ x( ) nx (, ) xdxd N, (32) ( ˆ 11) 0 where N s the total number of households n the cty. The left-hand sde of Eq. (32) s the sum of the number of households n the catchment areas for all of the radal major roads wthn a cty. The equlbrum rent per unt of land area devoted to housng at the cty frnge ( x, ) equals the exogenous agrcultural rent or the opportunty cost of the land; that s, rx ( ( ),, u) r, (33) where r a s a constant agrcultural rent. a Usng Eq. (31), Eq. (33) can be rewrtten as 1 1 1 1 1 1 1 (, ) b k Y E x u bk ra. (34) b Remark 5. Eqs. (32) and (34), together wth Eqs. (22), (23), (28), (29) and (31), determne the housng market equlbrum. Gven the travel cost Ex (, ) or ( x, ) (see Eq. (19)), one can determne the values of household utlty level u and the cty boundary x by solvng the system of Eqs. (32) and (34) and thus the followng functons: gx (, ), px (, ), S( x, ), nx (, ) and rx (, ). 3.3. Calculatng the equlbrum soluton for the urban system In ths subsecton, a procedure for calculatng the equlbrum soluton of the urban system s presented. Gven the household resdental densty nx (, ), the commuters route choce equlbrum problem can determne the functons Q ( x ), T ( x ) and ( x, ) n addton to the boundary contours B, 1, 2,..., M. In contrast, gven the travel cost ( x, ), the 14
housng market equlbrum problem can determne the functons nx (, ), gx (, ), px (, ), S( x, ) and rx (, ) and the constants u and x. The nterdependence of the nterrelated equlbra leads to a statonary soluton wth regard to resdental densty and travel costs, as follows: n (0) ψ (0) n (1) ψ (1) n *, ψ *, (35) where the bolded symbols represent the vectors of the correspondng varables; that s, ψ ( x, ) and nx (, ) n. The step-by-step procedure for determnng the (statonary) equlbrum soluton of the urban system s as follows. Step 1. Choose ntal values for the household resdental densty travel demand vector (0) Q (0) ( ) x vector (0) T (0) ( ) x (0) n, then determne the Q n terms of Eqs. (13) and (14), the travel tme T by Eq. (4), the travel cost vector crtcal angle vector ˆ (0) ˆ (0) ( ) x (0) ψ by Eq. (1) and the θ accordng to Eq. (10). Set the teraton counter to l 1. Step 2. Solvng the system of Eqs. (32) and (34) yelds the values of () l () u and x l. The values of the vectors g () l, substtutng () l () u and l () l p, () l S, () l n and () l r can then be obtaned by x nto Eqs. (22), (23), (28), (29) and (31). Step 3. Determne the travel demand vector () l Q n terms of Eqs. (13) and (14) and the travel tme vector () l T accordng to Eq. (4). The auxlary travel cost vector ψ () l can then be obtaned by Eq. (1) and thus, the crtcal angle vector l. ( Step 4. Update the travel cost accordng to l 1) ( ) ( ) ( ) ψ ψ l ψ l ψ l ˆ () θ l by Eq. (10). Step 5. If the relatve gap ( 1) ( ) ( ) l l / l ψ ψ ψ s smaller than a pre-specfed tolerance, then stop. Otherwse, set l l 1 and go to Step 2. In Step 1, the ntal household resdental densty can be assumed to be unform across ths cty. In Step 2, the system of Eqs. (32) and (34) can be solved usng an teratve method n whch the decson varables x and u are sequentally updated whle holdng the value of 15
the other varable fxed. Specfcally, one chooses ntal values for x, then solves Eq. (32) to obtan the value of u. The values of x can then be updated by solvng Eq. (34) usng the b-secton method for the value of u that has just been obtaned. Repeatng the teratve process yelds the solutons for x and u. 4. Model for optmzng the densty of radal major roads As prevously stated, local authortes am to maxmze the total socal welfare (SW) of the urban system by optmzng the densty of the radal major roads wthn the two-dmensonal monocentrc cty. The socal welfare, whch takes nto account the benefts of all partes n the urban system, s defned as the total utlty of all households plus the aggregate rent receved by the absentee landlords mnus the total road nvestment cost. The socal welfare maxmzaton problem can be formulated as M ˆ x( ) M ( ˆ 1 1) 0 a, (36) 1 1 max SW= un + r( x, ) r xddx M where s a parameter that converts the utlty level of households nto the equvalent monetary unts. rx (, ) can be determned usng the equlbrum formulaton of the urban system proposed n the prevous secton. represents the constructon cost of radal major road. The frst term on the rght-hand sde of Eq. (36) denotes the total utlty of all households, the second term denotes the aggregate rent receved by the absentee landlords, and the thrd term denotes the total constructon cost of all radal major roads. It s assumed that the constructon cost, to the length, x, of radal road and ts capacty, of radal major road s a functon wth respect K, as follows: xk, (37) where s a postve constant. Eq. (37) ndcates that the road constructon cost s proportonal to the road length and wdth. Note that the model (36) s an nteger programmng problem wth the number of radal major roads as the decson varable. In general, ths type of problem s dffcult to solve (L et al., 2011, 2012c). Fortunately, the number of radal major roads n a cty s fnte. Therefore, a 16
smple approach to determne the optmal densty of radal major roads s to compare the resultant values of objectve functon (36) usng dfferent numbers of radal major roads. 5. Model applcatons In ths secton, a numercal example s gven to llustrate the applcatons of the proposed model and the contrbutons of ths paper. The numercal example s ntended to ascertan the effects of road level of servce (LOS), urban populaton sze (number of households) and household ncome level on the optmal densty of the radal major roads and the performance of the urban system. It s also used to nvestgate the effects of varous urban polces, such as congeston prcng and road nfrastructure nvestment (ntroducng a new radal major road) on the urban economy. In order to quanttatvely evaluate the effects of congeston prcng scheme and addng a new radal major road, some performance measures are defned as follows: Cty area (or sze) = ˆ M x( ) xdxd 1 ( ˆ 11), (38) 0 Average household resdental densty = N cty area, (39) Average housng space per famly = ˆ x( ) M gx (, ) nx (, ) xdxd N, (40) 1 ( ˆ 11) 0 Average housng prce = ˆ x( ) M px (, ) xdxd cty area, (41) 1 ( ˆ 11) 0 Average land value = ˆ x( ) M rx (, ) xdxd cty area, (42) 1 ( ˆ 11) 0 Average captal nvestment ntensty = ˆ x( ) M S( x, ) xdxd cty area. (43) 1 ( ˆ 11) 0 For presentaton purposes, n the numercal experment, all of the radal major roads n the cty are assumed to have dentcal free-flow travel tmes and capactes. The baselne values (.e. base case) for the model s nput parameters are shown n Table 1. In the followng analyss, unless specfcally stated otherwse, the nput data are dentcal to those of the base case. The computer code for the numercal experment (wrtten n programmng language C) s not shown n the paper so as to save space, but avalable from the authors on request. 5.1. Effects of road LOS, urban populaton sze, and household ncome level on densty of 17
radal major roads We frst nvestgate the effects of the road LOS, number of households (or populaton sze), and household ncome level on the optmal densty of the radal major roads. Fg. 2 plots the contour of the annual total socal welfare of the urban system n the space of the road LOS (.e. road capacty) and the number (or densty) of radal major roads. It can be seen that for a gven road capacty, as the densty of the major roads ncreases, the annual total socal welfare of the urban system frst ncreases and then decreases. The maxmum socal welfare for a gven road capacty (or LOS) occurs at the pont of tangency between the horzontal lne passng through the gven road capacty and the socal welfare contour. For example, the horzontal lne wth the road capacty of 4950 vehcles per hour touches a socal welfare contour at pont F, as shown n Fg. 2. Ths means that when the road capacty s 4950 vehcles per hour, the optmal number of the radal major roads s 7. Any pont on the dotted curve n Fg. 2 represents the maxmum socal welfare for a gven road LOS. It can also be seen that as the road capacty (or LOS) ncreases, the optmal number of the radal major roads decreases, and vce versa. Fg. 3 dsplays the change of the annual total socal welfare of the urban system wth the total number of households and the number (or densty) of the radal major roads. It can be seen that for a gven number of households, the assocated horzontal lne touches a socal welfare contour, mplyng that there s an optmal number of the radal roads. For nstance, when the number of households s 300,000, the optmal number of the radal roads s 8. The dotted curve n Fg. 3 defnes the maxmum socal welfare levels for dfferent numbers of households. It shows that as the number of households (or populaton sze) grows, the optmal number of the radal roads ncreases. Fg. 4 ndcates the change of the annual total socal welfare of the urban system wth the household ncome level and the number of radal major roads. It can be observed that the optmal number of the radal major roads would be 7 regardless of the household ncome level. Ths mples that the household ncome level has no apparent effect on the optmal densty of the radal major roads. 5.2. Effects of optmal congeston prcng on densty of radal major roads 18
In the lterature, congeston prcng schemes generally nclude frst-best scheme, cordon-based scheme, and dstance-based scheme (see Mtchell et al., 2005; L et al., 2012a). The frst-best and cordon-based schemes have been wdely explored n the feld of urban economcs (for more detals, see Mun et al., 2003; Verhoef, 2005; Anas and Rhee, 2006; L et al., 2012a; De Lara et al., 2013). Here, we nvestgate the effects of the dstance-based congeston prcng scheme on the optmal densty of the radal major roads. The dstance-based prcng scheme mples that the toll s proportonal to the dstance traveled, that s, the product of the travel dstance and the toll per unt of dstance. Fg. 5 depcts the contour of the annual total socal welfare of the urban system n the toll-road densty space. It can be seen that the maxmum socal welfare occurs at the toll level of $0.36 per klometer and 6 major roads, causng a socal welfare of $10.389 bllon per year (see Table 2). Table 2 summarzes the effects of the congeston prcng scheme on the urban system. It can be seen that mplementaton of the congeston prcng scheme causes a decrease n the cty sze (or cty area) by 234 square klometers (from 1261.1 to 1027.0 square klometers). Ths s because the addtonal tolls under the dstance-based congeston prcng scheme ncrease the annual transportaton expendtures of households and thus lead them to move towards the cty center. As a result, the average resdental densty, average housng prce, average captal nvestment ntensty, and average land value ncrease by 90 households per square klometer, $92.3 per square meter, $25.4 mllon per square klometer, and $0.6 mllon per square klometer, respectvely. However, the average housng space per household decreases by 0.6 square meter. In addton, t can also be seen that the congeston prcng scheme leads to a decrease n the household utlty level by 4.6 utlty unts due to an ncreased travel cost, and n road constructon cost by $55.8 mllon due to a reduced cty sze, but the aggregate land rent ncreases by $62.8 mllon due to an ncreased land value. As a result, the annual socal welfare of the urban system ncreases by $18.8 mllon (from $10370.2 to $10389.0 mllon). 5.3. Effects of a new addtonal radal major road on urban system In ths subsecton, we address the followng two problems: (1) where s optmal locaton of a new addtonal radal major road? (2) What are the effects of a new addtonal radal major road on the urban system? For llustraton purposes, we consder a cty wth fve exstng radal major roads, whch are numbered as 1, 2, 3, 4, and 5, respectvely, as shown n bolded lnes n Fg. 6. The angles among the fve radal major roads are 120, 40, 50, 60, and 90, 19
respectvely. 5.3.1. Optmal locaton of a new addtonal radal road In order to determne the optmal locaton of a new addtonal radal major road n the cty shown n Fg. 6, Fg. 7 depcts the annual socal welfare curves when the new addtonal radal road s located between any two exstng radal roads, respectvely. In Fg. 7, represents to nsert the new addtonal radal road n between roads and j. It can be seen n ths fgure that the optmal locaton of the new addtonal radal road should be located between radal roads 1 and 2 wth 0.92 radan (.e. 52.7 ) from the radal road 1 (shown n dotted lne n Fg. 6), whch results n the maxmum socal welfare level of $10.359 bllon per year. R j 5.3.2. Comparson of urban system performances before and after addng a new radal road Fnally, we examne the effects of an addtonal radal road (shown n Fg. 6) on the urban system. Fgs. 8a-d and 8a'-d' plot the contours of the household resdental densty, housng prce, housng space per household, and captal nvestment ntensty across the cty before and after ntroducng a new radal road, respectvely. It can be seen that ntroducton of a new radal road leads to a change n the urban shape from a pentagram (.e. a fve-ponted star) to a hexagram (.e. a sx-ponted star). In the meantme, t also leads to a decrease n the household resdental densty, the housng prce, and the captal nvestment ntensty, but an ncrease n the housng space per household. Specfcally, the household resdental denstes on the cty boundary and n the cty center decrease by 0.5 (from 61 to 60.5) and 648 (from 3992 to 3344) households per square klometer respectvely, as shown n Fgs. 8a and a'. The housng prces on the cty boundary and n the cty center (shown n Fgs. 8b and b') decrease by $3 (from $1248 to $1245) and $250 (from $4836 to $4586) per square meter, respectvely. The housng spaces per household on the cty boundary and n the cty center (see Fgs. 8c and c') ncrease by 0.18 (from 13.24 to 13.42) and 0.26 (from 4.79 to 5.05) square meter, respectvely. The captal nvestment ntenstes on the cty boundary and n the cty center decrease by $0.1 (from $14.2 to $14.1) and $211 (from $1295 to $1084) mllon per square klometer respectvely, as shown n Fgs. 8d and d'. 20
Table 3 further summarzes the effects of addng a new radal road on the performance of the urban system. It shows that ntroducton of a new radal road ncreases the sze of the cty by 87.8 square klometers and the average housng space per household by 0.2 square meter. However, t also decreases the average household resdental densty from 432.9 to 402.3 households per square klometer, the average housng prce from $2094.8 to $2067.7 per square klometer, the average land value from $2.6 to $2.5 mllon per square klometer, and the average captal nvestment ntensty from $122.7 to $114.7 mllon per square klometer, respectvely. As a result, the household utlty level ncreases 2.7 utlty unts, the total land rent decreases by $8.3 mllon, the road constructon cost ncreases by $70 mllon, and the annual socal welfare ncreases by $29 mllon. 6. Concluson and further studes Ths paper nvestgated the desgn problem of the densty of radal major roads n a two-dmensonal monocentrc cty. The urban system concerned ncluded four types of agents: local authortes, property developers, households and commuters. The nterdependence among these agents nduces a number of nterrelated equlbra; namely, the housng demand-supply equlbrum, household resdental locaton choce equlbrum and commuters rng-radal route choce equlbrum. Based on the urban system equlbrum analyss, a heurstc soluton approach was developed to solve the urban system s equlbrum problem and a system optmum model was proposed to determne the optmal densty of the radal major roads so as to maxmze the urban system s socal welfare. In the proposed model, the household resdental dstrbuton, captal nvestment ntensty, land values, housng prces and housng space can all be endogenously determned. An llustratve numercal example was gven to assess the effects of road level of servce (LOS), urban populaton sze and household ncome level on the optmal densty of the radal major roads. The comparatve statc analyses of the congeston prcng and road nfrastructure nvestment were also carred out. The proposed model can be served as a useful tool for the long-term strategc plannng of sustanable urban developments and for evaluaton of varous transportaton, land-use and housng polces. The proposed model offers some new nsghts and nterestng fndngs. Frst, the road capacty (or LOS) and the number of households (or populaton sze) have a sgnfcant effect 21
on the optmal densty (or number) of the radal major roads. Specfcally, the optmal number of the radal major roads ncreases wth the ncrease of the number of households but decreases wth the ncrease of the road LOS. However, the household ncome level may have lttle mpact on the optmal densty of the radal roads. Second, congeston prcng scheme can decrease the sze of cty and thus lead to a more compact cty. It can also ncrease the socal welfare of the urban system and save the government s nvestment on road nfrastructure projects at the expense of the household utlty. Thrd, the ntroducton of an addtonal radal major road can ncrease the sze of cty and thus cause a more decentralzed cty. Households and the whole urban system can beneft from the road nfrastructure nvestment that s funded by the government. Although the modelng methodology proposed n ths paper provdes a new avenue for modelng the nteractons among land use, housng market and transportaton nfrastructure mprovements, further extensons are necessary. Frst, the cty of nterest was assumed to be monocentrc n ths paper. However, modern ctes generally have multple busness and commercal centers. Therefore, further studes could mprove the model by shftng from a monocentrc cty to a polycentrc structure (Wong, 1998; Kloosterman and Musterd, 2001; Yn et al., 2013; Ho et al., 2013). Second, ths paper focused manly on the auto mode, thus gnorng the competton and substtuton effects between auto and transt modes. To ncorporate the nteracton and competton between dfferent transportaton modes, the sngle-mode system should be extended nto a mult-modal system (Capozza, 1976; Anas and Moses, 1979; L et al., 2012a). Thrd, all of the households n ths paper were assumed to be homogenous. However, an emprcal study by Kwon (2003) showed that household ncome level may affect resdental locaton choce. Therefore, future models should be extended to consder the choce behavor of households wth dfferent ncome levels and demographc characterstcs. Acknowledgments The authors would lke to thank the edtor, Professor Yves Zenou, and two anonymous referees for ther helpful comments and suggestons on an earler draft of the paper. The work descrbed n ths paper was jontly supported by grants from the Natonal Natural Scence Foundaton of Chna (71222107, 71171013), the Research Foundaton for the Authors of 22
Natonal Excellent Doctoral Dssertatons (Chna) (200963), the Doctoral Fund of Mnstry of Educaton of Chna (20120142110044), and the Fok Yng Tung Educaton Foundaton (132015). References Alonso, W., 1964. Locaton and land use: toward a general theory of land rent. Massachusetts: Harvard Unversty Press, Cambrdge. Anas, A., 1982. Resdental locaton markets and urban transportaton. New York: Academc Press. Anas, A., Moses, L.N., 1979. Mode choce, transport structure and urban land use. Journal of Urban Economcs, 6 (2), 228-246. Anas, A., Rhee, H.J., 2006. Curbng excess sprawl wth congeston tolls and urban boundares. Regonal Scence and Urban Economcs, 36 (4), 510-541. Anas, A., Xu, R., 1999. Congeston, land use, and job dsperson: a general equlbrum model. Journal of Urban Economcs, 45 (3), 451-473. Baum-Snow, N., 2007. Suburbanzaton and transportaton n the monocentrc model. Journal of Urban Economcs, 62 (3), 405-423. Beckmann, M.J., 1969. On the dstrbuton of urban rent and resdental densty. Journal of Economc Theory, 1 (1), 60-67. Beckmann, M.J., 1974. Spatal equlbrum n housng market. Journal of Urban Economcs, 1 (1), 99-107. Brueckner, J.K., 2007. Urban growth boundares: an effectve second-best remedy for unprced traffc congeston? Journal of Housng Economcs, 16 (3-4), 263-273. Capozza, D.R., 1976. Land use n a cty wth two modes of transportaton. Southern Economc Journal, 42 (3), 442-450. De Lara, M., de Palma, A., Klan, M., Pperno, S., 2013. Congeston prcng and long term urban form: Applcaton to Pars regon. Regonal Scence and Urban Economcs, 43 (2), 282-295. D Este, G., 1987. Trp assgnment to radal major roads. Transportaton Research Part B, 21 (6), 433-442. Fujta, M., 1989. Urban economc theory. Cambrdge: Cambrdge Unversty Press. Henneberry, J., 1998. Transport nvestment and house prces. Journal of Property Valuaton 23
and Investment, 16 (2), 144-158. Ho, H.W., Wong, S.C., 2007. Housng allocaton problem n a contnuum transportaton system. Transportmetrca, 3 (1), 21-39. Ho H.W., Wong S.C., Sumalee, A., 2013. A congeston-prcng problem wth a polycentrc regon and mult-class users: a contnuum modelng approach. Transportmetrca, n press. DOI:10.1080/18128602.2011.621652. Kloosterman, R.C., Musterd, S., 2001. The polycentrc urban regon: towards a research agenda. Urban Studes, 38 (4), 623-633. Kraus, M., 2006. Monocentrc ctes. In Arnott, R. J., McMllan, D. P. (Eds.), A Companon to urban economcs (pp. 96-108). Oxford: Blackwell Publshng. Kwon, Y., 2003. The effect of a change n wages on welfare n a two-class monocentrc cty. Journal of Regonal Scence, 43 (1), 63-72. L, Z.C., Lam, W.H.K., Wong, S.C., 2011. On the allocaton of new lnes n a compettve transt network wth uncertan demand and scale economes. Journal of Advanced Transportaton, 45 (4), 233-251. L, Z.C., Lam, W.H.K., Wong, S.C., 2012a. Modelng ntermodal equlbrum for bmodal transportaton system desgn problems n a lnear monocentrc cty. Transportaton Research Part B, 46 (1), 30-49. L, Z.C., Lam, W.H.K., Wong, S.C., Cho, K., 2012b. Modelng the effects of ntegrated ral and property development on the desgn of ral lne servces n a lnear monocentrc cty. Transportaton Research Part B, 46 (6), 710-728. L, Z.C., Lam, W.H.K., Wong, S.C., Sumalee, A., 2012c. Desgn of a ral transt lne for proft maxmzaton n a lnear transportaton corrdor. Transportaton Research Part E, 48 (1), 50-70. Lu, T.L., Huang, H.J., Yang, H., Zhang, X., 2009. Contnuum modelng of park-and-rde servces n a lnear monocentrc cty wth determnstc mode choce. Transportaton Research Part B, 43 (6), 692-707. McDonald, J.F., 2009. Calbraton of a monocentrc cty model wth mxed land use and congeston. Regonal Scence and Urban Economcs, 39 (1), 90-96. McDonald, J.F., Osuj, C.I., 1995. The effect of antcpated transportaton mprovement on resdental land values. Regonal Scence and Urban Economcs, 25 (3), 261-278. Mkelbank, B.A., 2004. Spatal analyss of the relatonshp between housng values and nvestments n transportaton nfrastructure. Annals of Regonal Scence, 38 (4), 705-726. Mlls, E.S., 1972. Urban economcs. Illnos: Scott Foresman, Glenvew. 24
Mtchell, G., Namdeo, A., Mlne, D., 2005. The ar qualty mpact of cordon and dstance based road user chargng: an emprcal study of Leeds, UK. Atmospherc Envronment, 39 (33), 6231-6242. Muth, R.F., 1969. Ctes and housng. Chcago: Unversty of Chcago Press. Mun, S.I., Konsh, K.J., Yoshkawa, K., 2003. Optmal cordon prcng. Journal of Urban Economcs, 54 (1), 21-38. O Sullvan, A., 2000. Urban economcs. Boston: Irwn/McGraw-Hll. Qugley, J.M., 1984. The producton of housng servces and the derved demand for resdental energy. RAND Journal of Economcs, 15 (4), 555-567. Solow, R.M., 1972. Congeston, densty and the use of land n transportaton. Swedsh Journal of Economcs, 74 (1), 161-173. Solow, R.M., 1973. Congeston cost and the use of land for streets. Bell Journal of Economcs and Management Scence, 4 (2), 602-618. Song, Y., Zenou, Y., 2006. Property tax and urban sprawl: theory and mplcatons for US ctes. Journal of Urban Economcs, 60 (3), 519-534. Verhoef, E.T., 2005. Second-best congeston prcng schemes n the monocentrc cty. Journal of Urban Economcs, 58 (3), 367-388. Wang, J.Y.T., Yang, H., Lndsey, R., 2004. Locatng and prcng park-and-rde facltes n a lnear monocentrc cty wth determnstc mode choce. Transportaton Research Part B, 38 (8), 709-731. Wong, S.C., 1994. An alternatve formulaton of D Este s trp assgnment model. Transportaton Research Part B, 28 (3), 187-196. Wong, S.C., 1998. Mult-commodty traffc assgnment by contnuum approxmaton of network flow wth varable demand. Transportaton Research Part B, 32 (8), 567-581. Yn, J., Wong, S.C., Sze, N.N., Ho, H.W., 2013. A contnuum model for housng allocaton and transportaton emsson problems n a polycentrc cty. Internatonal Journal of Sustanable Transportaton, 7 (4), 275-298. 25