The welfare effects of nationalization in a mixed duopoly public transport market

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1 Dublin Institute of Technology Aticles National Institute fo Tanspot and Logistics The welfae effects of nationalization in a mixed duopoly public tanspot maket Feifei Qin Soochow Univesity, meyqff@hotmail.com Xiaoning Zhang Tongji Univesity, cexzhang@tongji.edu.cn Eoin Plant Dublin Institute of Technology, Eoin.Plant@dit.ie Follow this and additional woks at: Pat of the Business Administation, Management, and Opeations Commons, and the Civil and Envionmental Engineeing Commons Recommended Citation Qin, F., Zhang, X. and Plant, E. (2016) The welfae effects of nationalization in a mixed duopoly public tanspot maket. Opeational Reseach Online 22nd. Mach This Aticle is bought to you fo fee and open access by the National Institute fo Tanspot and Logistics at ARROW@DIT. It has been accepted fo inclusion in Aticles by an authoized administato of ARROW@DIT. Fo moe infomation, please contact yvonne.desmond@dit.ie, aow.admin@dit.ie, bian.widdis@dit.ie. This wok is licensed unde a Ceative Commons Attibution- Noncommecial-Shae Alike 3.0 License

2 Ope Res Int J DOI /s ORIGINAL PAPER The welfae effects of nationalization in a mixed duopoly public tanspot maket Feifei Qin 1 Xiaoning Zhang 2 Eoin Plant 3 Received: 26 Mach 2015 / Revised: 5 Mach 2016 / Accepted: 11 Mach 2016 Spinge-Velag Belin Heidelbeg 2016 Abstact Recently, many cities have launched new ail tansit lines. Once these new ail tansit lines stat commecial opeation, they will play impotant oles as competitos to conventional bus sevices. In this pape, the effects of nationalization on equilibia have been studied in a mixed duopoly public tanspot maket, in which one publicly-owned ail tansit opeato competes with one pivate bus opeato. Two numeical case studies show the nationalization of a ail tansit company is always socially desiable both in Betand picing game in the shot un and in Nash non-coopeative game in the long un. Keywods Mixed duopoly tansit maket Nash non-coopeative competition Degee of nationalization & Xiaoning Zhang cexzhang@tongji.edu.cn Feifei Qin meyqff@hotmail.com Eoin Plant eoin.plant@dit.ie School of Uban Rail Tanspotation, Soochow Univesity, No. 1 Shizi Steet, Suzhou , China School of Economics and Management, Tongji Univesity, Siping Road 9, Shanghai , China School of Tanspot Engineeing, Envionment and Planning, National Institute fo Tanspot and Logistics, Dublin Institute of Technology, Bolton Steet, Dublin 1, Ieland

3 F. Qin et al. 1 Intoduction Since the 1990s, in paallel with the tend towads deegulation and pivatization, the oganizations of public tanspot in many cities have been undegoing a adical tansfomation. As fo the bus maket, it exhibits a tansition fom a state-owned monopolistic fom to a pivate competition egime in some developing egions and tansition counties, such as China. Unde fanchise ageements, pivate bus companies can feely povide local sevices to esidents. Howeve, due to the huge capital equiement and substantial economies of scale, most ail tansit systems aound the wold ae typically opeated by state-owned entepises (SOEs). Since the 2000s, in ode to tansfom SOEs fom cost centes to economic entities esponsible fo cetain pofit tagets, tansit authoities have begun to estuctue these SOEs into limited liability companies o joint stock companies, which can be temed as copoatization o patial pivatization. Consequently, the pevailing oganization in uban public tanspot maket pesents such a stuctue that one (semi-) public ail tansit opeato competes with one o seveal pivate bus companies, which suggests the pesence of a mixed oligopoly. 1 As an example, Shanghai-one of the lagest cities in China, has moe than 1000 bus lines, seved by moe than 10 pivate bus companies. Howeve, all ail tansit lines ae only opeated by one company Shengtong Meto Company, which was oiginally a complete public entity. In 2000, it was efomed as a semi-public coopeation with % of its shae holding by Shanghai municipality. Mixed oligopolies ae also common in some developing counties cities (such as Santiago in Chile, Kuala Lumpu in Malaysia), in which multiple opeatos with diffeent owneship ae vying fo passenges. Even in the developed counties, such as Noway, Sweden, the public govenments also hold some degee of owneship in tanspot fims seving local makets (Jøgensen and Santos 2014). While the coexistence of public and pivate opeatos become pevailing in the local tanspot secto, oveall it is still a new competition fom. In such a mixed oligopoly, public and pivate opeatos have a diffeent management goal, which aises impotant issues on picing and sevice level decisions. On the sevice level aspect, pivate opeatos aim to maximize its pofit, so both pivate monopoly and oligopoly competition intend to povide much less sevice fequency than the socially desied level. In some Chinese cities, it is fequently epoted public tanspot passenges often suffe fom long waiting times due to inadequate sevice levels affoded by pivate opeatos. On the pice aspect, pivate opeatos may chage high fae which eventually huts the whole welfae. Inefficient sevice supplies of pivate monopoly and oligopoly may oveshadow the possible benefits of pivate owneship, motivating the enty of welfae-maximizing public opeatos to compete against the pofit-maximizing pivate fims. Howeve, thee is still a debate as to whethe the mixed oligopoly competition can contibute to adjusting the maket failues aising fom pivate oligopoly. Being awae of these issues, the 1 In the field of industial economics, mixed oligopoly efes to the competition between public fims, which ae instucted to maximize thei contibutions to welfae, and pofit-maximization pivate companies.

4 The welfae effects of nationalization in a mixed duopoly objective of this pape is to investigate the maket outcomes and welfae effects of the mixed oligopoly competition in an uban tanspot maket, which is necessay and timely given that the mixed oligopoly is pesent in a boad ange of counties. Moe specifically, we conside the stategic inteactions between one public ail tansit company and one pivate bus company and attempt to answe the following questions: if one ail tansit opeato chooses to patially maximize consume suplus athe than to solely maximize its own pofit, how will this affect equilibia and, most impotantly, could it eally contibute to inceasing social welfae as its oganizational objective concens? The deegulation and pivatization in the uban tanspot maket have led to extensive eseach on the stategic behavios of oligopolistic competition duing ecent decades, using eithe Counot (e.g., Viton 1981; Oldfield and Emmeson 1986; Williams and Abdulaal 1993) o Betand (e.g., Baid 1986; De Palma and Leuth 1989; Wang and Yang 2005; Wichiensin et al. 2007) games. These studies povide consideable insights into pivate oligopolistic competitions. Moeove, mixed competition between public and pivate fims is also pevalent in a divese ange of industies, leading to extensive eseach on the mixed oligopoly (MO) in economic theoy (See, De Faja and Delbono 1990; Ceme et al. 1991; Andeson et al. 1997; Matsumua 1998; Fujiwaa 2007). While thee is a huge liteatue on mixed oligopoly in some industial sectos, such as telecommunications and banking sectos, only a little attention has been paid to modelling mixed oligopolistic competitions in the tanspot maket. Fo the seapot industy, Czeny et al. (2014) exploed the effect of pivatization a pot fo a local maket. Matsushima and Takauchi (2014) investigated how pot pivatization affects pot chages, pofits and welfae elements fom an intenational pespective. In the aviation secto, Matsumua and Matsushima (2012) and Mantin (2012) investigated behavioual pattens of aipots fo intenational ai tanspot makets, whee one domestic publicly-owned aipot competes against one pivate aipot in anothe county. In addition to these inta-mode competition models, Yang and Zhang (2012) investigated the effect of inte-modal competition between ai tanspot and high speed ail (HSR) in a mixed duopoly famewok. Vey ecently, D Alfonso et al. (2014) analyzed the impacts of competition between one pivate ailine and one public HSR on the envionment and social welfae. In the uban tanspot maket, Cantos-Sánchez and Mone-Colonques (2006) exploed fequency and picing competition between one pivate bus and one public ail tansit using a quadatic addess model. Employing a quadatic utility function, Clak et al. (2009) compaed equilibium esults that aise fom collusion, applying Counot and Betand competitions fo a mixed duopoly bus maket. The eview of the above-mentioned papes eveals thee common featues. Fist, the above studies only focus on full nationalization. In eality, howeve, the tanspot maket is likely to have patially nationalized tanspot opeatos, especially in the uban tanspot maket. Fo example, in China, Hangzhou govenment holds 51 % of the shae in Hangzhou Meto Company; the Nowegian Tansit Authoity holds the majoity of shaes in 36 bus companies. Despite these pevailing eal wold cases, the implications of patial pivatization fo fequency and picing choices have not been investigated in the tanspot secto. Ou eseach

5 F. Qin et al. intends to fill in this gap. Second, all these papes adopt linea demand functions, which can easily obtain analytical solutions. But, when exploing competitions among fims, an attaction model with nonlinea fom is supeio to linea one since it can not only epesent eality moe closely but also can eflect the nonlinea effects aising in competitive phenomena (Huang et al. 2013). To this end, we adopt a non-linea demand function (that is, a Logit model) in this pape to obtain moe geneal and pactical conclusions. Finally, some studies assumed the aggegate demand fo the whole maket is inelastic with espect to fae o full cost. Actually, this is a vey stict assumption. In this pape, a moe ealistic case will be developed in which total local tanspot demand is elastic with espect to the combination of fae and fequency. The contibutions of this pape can be summaized as follows: (1) given the gowing impotance of patial nationalization in the tanspot secto, this pape fills the eseach gap by analyzing the effects of patially nationalizing a ail tansit company that competes with a pivate bus sevice. (2) in contast to most mixed oligopoly studies with linea demand function, we employed an attaction demand function with non-linea fom to povide moe pactical and geneal insights; (3) Using a Chinese case study, two time hoizons ae analyzed fo this vetically and hoizontally diffeentiated mixed duopoly competition: a shot time hoizon in which opeatos compete only in faes, and a long time hoizon in which they compete in fequency along with the pice. The est of pape is oganized as follows. Section 2 povides a model using a Nash non-coopeative game in a mixed duopoly public tanspot maket. In Sect. 3, to evaluate the effects of nationalization on equilibium esults, two numeical cases fo shot-un and long-un competitions ae conducted by using Suzhou taffic data. Additionally, sensitivity analysis has been caied out to genealize the esults. Main findings and ecommendations fo futhe eseach ae epoted in Sect Modeling famewok 2.1 Basic setting of Nash non-coopeative game This study focuses on a mixed duopoly public tanspot maket, in which one semipublic ail tansit opeato,which is fully o patially concened with uses benefits, competes with a pivate bus company, which only consides its own pofits. Since each playe is tying to optimize its objective function without pio knowledge of othe playes functions (Evans 1992), these two playes make thei decisions simultaneously without any collusion and eceive payoffs depending on the sevice levels they offe. In this setting, the Nash non-coopeative Game in the context of mixed duopoly has the following featues: Playes Nowadays, although multiple pivate bus companies pevail in some cities, few ovelapping opeating situations lead us to view them as one vitual opeato. Thus, we theoetically assume only one bus opeato competes with one (semi-) public ail tansit opeato on an isolated oute duing a given peiod.

6 The welfae effects of nationalization in a mixed duopoly Stategies In the field of tanspot modelling, the stategic inteaction occus eithe in the pice dimension (Betand) o in the quantity dimension (Counot). What type of competition is appopiate not only depends on the stategic vaiables employed but also lagely depends on the poduction technology and the time hoizon (Keps and Scheinkman 1983). Counot competition may be applicable when the capacity is difficult to adjust (e.g., ail, aipot) and poductions ae peceived as pefect substitutes. But, when capacities ae flexible (e.g., bus) and poducts ae moe diffeentiated, a Betand competition would be moe appopiate (Quinet and Vickeman 2004). In the case of a competition between bus and ail tansit, it is not evident which type of competition is moe pope based on the ease o difficulty of capacity adjustment. Bus and ail sevices ae not only hoizontally diffeentiated in tems of thei distinct mode featues but also vetically diffeentiated in tems of diffeent sevice qualities. This implies that the substitution between these two modes is impefect and the poducts ae diffeentiated. As a esult, we believe a diffeentiated Betand competition may be easonable fo the shot un competition. Besides pice and quantity, fequency is anothe impotant stategic device. In the shot un, since the inceasing numbe of depatues could equie additional fleets, fequency is difficult to change. Thus, opeatos only compete ove pice, assuming fequency is exogenously given. But, in the long un, opeatos may compete ove fequency along with fae since opeatos can elocate stops and buying moe fleet. Payoff In the field of mixed oligopoly, the conventional objective function of one public company is to maximize social welfae, which is the sum of consume suplus, its own pofit and the pofits of pivate companies. Howeve, in the context of competition, it is somewhat uneasonable to assume the ail tansit company competing against one bus company also needs to conside its ival s pofit. Theefoe, we assume, in addition to the ail company s pofit, the ail company takes consumes inteests into account when it decides fae and fequency. The pofit of the bus company is out of its consideation. This kind of teatment is adopted by much ecent liteatue (Clak et al. 2009; Jøgensen and Santos 2014). It is easonable to assume one pivate bus company is only concened with maximizing its pofit. 2.2 Demand functions and opeating cost In view of the mixed duopoly competition between one public ail tansit opeato and one pivate bus opeato, this sub-section begins by constucting a demand function fo opeato i (Heeafte, the subscipt will be used to denote ail tansit and the subscipt b fo bus), which conventionally takes the following fom: Q i ¼ Q fn; ccgm i i ¼ b; ð1þ whee, Q i is the numbe of passenges selecting public tanspot sevice i. Q*{n, cc} epesents total taffic demand of public tanspot secto, which is the function of composite costs (cc) and one positive paamete n (n [ 0). M i denotes the maket shae of public tanspot sevice affoded by opeato i.

7 F. Qin et al. The total demand fo local public tanspot is assumed to be an exponential function of composite costs (cc) 2 : Q ¼ Q exp½nðcc ccþ ð2þ The composite cost (cc) epesents the expected disutility of using public tanspot modes. A ba ove a vaiable denotes its value in a efeence case befoe competition takes place. Thus, Q is the initial total public tanspot demand in a benchmak situation and cc means the composite cost of a efeence case. Total demand fo local public tanspot is elastic, because any changes in composite cost will shift the tavelling demand between public tanspot modes and pivate tanspot modes (such as pivate cas, motobikes, etc.). To eflect the elasticity of total demand change with espect to tavelles composite costs, we futhe specify the fomulation of composite costs as: cc ¼ 1 h ln X expðhc i Þ ð3þ C i denotes the genealized cost of tavelling by public tanspot mode i. h is a positive dispesion paamete. In this pape, individual tavelles mode choices ae assumed to be based on minimizing thei genealized cost pe tip (C i ), which equals the sum of the monetay cost (fae) and the tavel time weighted by value of time. C i ¼ P i þ a 1 qt i þ a 2 qð1=2f i Þ whee P i is the fae chaged by public tanspot opeato i. T i is the aveage invehicle tavelling time using public tanspot sevice i, which can be computed based on tavelling distance and speed. Nomally, duing ush hou, the public tanspot sevices ae much moe fequent. Thus, the aveage waiting time can be oughly estimated fom one-half the invese of fequency (f i ). q is the value of time (VOT). a 1 and a 2 ae paametes that measue diffeent weights that passenges put on in-vehicle tavelling time and waiting time. To focus on the pincipal aspects, this pape confines itself to the case of one homogeneous passenge goup, which indicates all passenges ae identical in egad to the value of time. Futhemoe, fo both bus and ail tansit sevices, since the station location and numbe of stops have been decided in the planning stage, accessing time costs have not been included in the genealized cost function. Tuning to the specifications of tavelling mode choices, the maket shae of opeato i is given by the Binay Logit fom: M i ¼ P expðhc iþ i expðhc ð5þ iþ Inseting Eqs. (5), (2) and (3) into Eq. (1), afte some manipulation, the numbe of passenges that choose public tanspot sevice i is: ð4þ 2 With this exponential function, the elasticity of total public tanspot demand will be popotional to composite cost ðe cc ¼ ðoq =occþðcc=q Þ¼ðnÞccÞ. This is a popety of public tanspot demand which has been found in many studies (see Johansen et al. 2001; Feanley et al. 2004).

8 The welfae effects of nationalization in a mixed duopoly Q i ¼ Q " # n X i expðh C i Þ h expðhc i Þ X i " #n h 1 expðhc i Þ ð6þ whee C i denotes the genealized cost of mode i in efeence case. To show the impact of pice and fequency on tavel demand, we evaluate patial deivatives of Eq. (6) afte substituting into (4) fom (4) and (5): oq i ¼Q i nm i þ hm j \0 i; j 2 fb; g i 6¼ j ð7þ op i oq i op j ¼ðh nþq i M j [ 0 i 6¼ j ð8þ oq i of i oq i of j ¼ a 2q 2fi 2 Q i ðnm i þ hm j Þ [ 0 ¼ðnhÞ a 2q 2fj 2 Q i M j \0 ð9þ ð10þ Vaious studies have shown that (h n) [ 0 in the uban public tanspot maket (Williams and Abdulaal 1993). Fom the above equations, it is clea that a high fae of one public tanspot sevice will educe its own demand and incease the demand of its competito. An incease of fequency will boost its own demand ate and simultaneously educe the demand of its competito. The vaiable opeating costs of public tanspot sevice i (OC i ) is the sum of maginal passenge cost (c i0 ) plus the vaiable cost (c i1 ) associated with sevice fequencies: OC i ¼ c i0 Q i þ c i1 f i ð11þ 2.3 Pice competition in the shot un In a mixed oligopoly maket, the bus opeato is viewed as a pue commecially oiented company, which is only concened with maximizing its pofit. The pofit of pivate bus company (p b ) is defined as the fae-box evenue minus the costs associated with bus opeation (OC b ): OF b ¼ p b ¼ P b Q b OC b ð12þ As mentioned ealie, the public ail tansit opeato maximizes a weighted combination of its pofit and consume suplus of the whole public tanspot maket. Given the poposed exponential demand function, the explicit fom of consume suplus can be simply expessed as: CS ¼ Q =n ð13þ

9 F. Qin et al. This simple expession fo consume suplus was deived by Evans (1992), and has since been used extensively by many studies (Williams and Abdulaal 1993; Wichiensin et al. 2007). The paamete n can be intepeted fom two pespectives. Fistly, n quantifies the sensitivity of the total local tanspot demand with espect to composite costs. The second intepetation is that the invese of n is the aveage peceived benefit expeienced by one epesentative tavele. Accodingly, the ail opeato s objective function can be specified as: OF ¼ CS þ p ¼ ðq =nþþðp Q OC Þ ð14þ The continuous paamete, which anges fom 0 to 1, can be efeed as the weight attached to consume suplus. It gives a measuement of nationalization degee. = 0 signifies the ail company is solely concened with its pofits. As ises, the weight on consume suplus becomes heavie in ail tansit fim s objective function. When = 1, the ail tansit opeato is fully nationalized and aims to maximize the sum of consume suplus and its own pofit. In this mixed duopoly maket, each opeato attempt to maximize its distinct objective function with espect to stategic valuables, subject to a capacity constaint. Max OF i P i ;f i ð15þ s:t: Q i K i f i K i is the designed vehicle capacity of public tanspot sevices i. In the shot un, since the expansion of olling stocks is stongly limited, the sevice fequency is assumed to be exogenously given. To this end, competition between two public tanspot modes is in pice only. By setting the patial deivatives of payoff functions with espect to fae equal to NE; zeo, and pefoming some manipulation, Nash equilibium pices (P i the supescipt NE denotes equilibium) fo opeato i can be calculated. Details of the deivation can be efeed to Appendix 1 : P NE i ¼ ð1 Þd þ c i0 ð16þ hm j þ nm i The indicato vaiable d takes on the value of one fo ail tansit and 0 fo bus. Next, we futhe patially diffeentiate Eq. (16) with espect to its ival s pice (P j ) o 2 OF i ¼ opne i op i op j op j ¼ M j 1 M j ðh nþ ð17þ A close look at the above mathematical expessions (16) fo equilibium faes gives ise to some inteesting insights. Fist, in the absence of capacity constaints, the equilibium pice is equal to maginal passenge cost of public tanspot sevice i (c i0 ) plus a mak-up, which elates to the maket shae of public tanspot sevices. Second, when ¼ 1 (that is, the public ail tansit company consides whole

10 The welfae effects of nationalization in a mixed duopoly consume suplus), the ail company consequently sets its fae equal to maginal cost (c i0 ). Finally, since we have assumed that (h - n) [ 0 in uban public tanspot maket, we have o 2 OF op i op j [ 0, which implies one opeato s maginal pofit inceases as its competito s fae incease. That is, the pice set by the bus company decease (incease) in esponse to the pice eduction (incease) of the ail tansit company op NE i opj [ 0. This popety matches the well-known definition of the stategic complement in oligopolies. To test the effect of patial nationalization () on equilibium faes, we invoke the techniques of monotone compaative statics fo the following analytical deivations and obtain: dp NE ðþ ¼ o2 OF op o d o 2 \0 ð18þ OF op 2 Fom Eq. (18), it is clea that with an incease of, the equilibium pice falls fo ail tansit. The explanation could be, an incease in povides moe incentives to ail tansit company to educe its fae fo impoving consume suplus. Futhemoe, since the equilibium faes between bus and ail tansit ae stategic complements, the bus fae coespondingly falls to compete fo passenges. So we can conclude in the mixed duopoly maket, as the publicly-owned ail tansit opeato pays moe attention to consume suplus elative to its own pofit, equilibium faes of bus and ail tansit both deceases. 2.4 Fequency-pice competition in the long un In the long un, since the fequency can be easily adjusted, it is easonably assumed that public tanspot opeatos compete in both pice and fequency. To deive the eaction functions fo sevice fequency, a simila pocedue is made by patially diffeentiating the payoff function of opeato i with espect to its fequency. Afte some algebaic manipulation, the equilibium sevice fequency is given as (Details of the deivation can be efeed to Appendix 2 ): f NE i ffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 q ¼ Q i 2c i1 ð19þ Obviously, the Nash Equilibium fequency follows a squae oot fomula. By analogy, the mathematical expession of equilibium fequency can be futhe diffeentiated with espect to the fequency of its competito: ffiffiffiffiffiffiffiffi a 2 q 1 of NE i of j ¼ 2c i1 2 Q1 2 i oq i of j ð20þ Fom Eq. (10), we have oq i ofj \0. Hence, the above equation takes a negative sign, which indicates a maginal incease (decease) in the fequency of ail tansit causes the bus opeato to decease (incease) its sevice fequency. This concept of

11 F. Qin et al. stategic substitution clealy addesses the elationship of equilibium fequencies between bus and ail tansit in this mixed oligopoly model. 3 Summaizing the above analysis, we obtain Poposition 1: Poposition 1 In the mixed duopoly maket involving one pivate bus opeato and one public ail tansit company, Nash Equilibium faes ae stategic complements. Nash Equilibium fequencies ae stategic substitutes. Poposition 1 offes an impotant extension, that is, the conventional concept of stategic substitutes and complements in the pivate oligopoly can also appopiately fit to the context of a mixed oligopoly model. It can contibute to explaining the stategic behavios (such as fae and fequency competitions) between the welfaemaximizing public and pofit-maximizing pivate opeatos. Next, we investigate the impact of nationalization degee on equilibium fequency. Given the mathematical specification of equilibium fequency fo ail tansit in Eq. (20), we patially diffeentiate it with espect to : of NE ffiffiffiffiffiffiffiffi o ¼ a 2 q 1 1 2c i1 2 QNE 2 oq NE op NE op NE ð21þ o Fom Eqs. (7) and (21), we have oq NE op NE\0 and op NE o\0, thus we can confim of NE o [ 0. This means the equilibium fequency of ail tansit inceases when ail opeato places moe weight on consume suplus. Since the equilibium fequencies between bus and ail ae stategic substitutes, an incease in will decease the equilibium fequency fo bus ðfb NE Þ. Poposition 2 summaizes the impact of nationalization degee () on equilibium faes and sevice fequencies. Poposition 2 In line with the initial objective of inceasing consume suplus, a lowe degee of nationalization leads to (1) lowe equilibium faes fo both ail tansit and bus; (2) highe equilibium fequency fo ail tansit and lowe equilibium fequency of bus. As noted in Poposition 2, as the ail tansit opeato concens consume suplus moe, its fae deceases. Since pices ae stategic complements, lowe ail tansit fae will put downwad pessue on bus fae. As a esult, bus fae will, in equilibium, fall as well. On the othe hand, the best espond functions of fequencies ae downwad sloping, indicating fequencies ae stategic substitutes. Moe concens about consume suplus leads to an incease in ail tansit s equilibium fequency and this occus at the expense of a eduction of bus fequency. 3 As pointed out by an anonymous efeee, the equilibium esult of Counot competition is equivalent to the outcome of a two-stage game, whee thee is a simultaneous capacity choice afte which pice competition occus (Keps and Scheinkman 1983). In the mixed monopoly, this equivalence found by Keps and Scheinkman (1983) is still valid, if assumptions of L-shaped maginal cost function and efficient ationing ule hold (a technical poof is available upon equest fom the authos). Howeve, if two assumptions ae violated, the K S esult might not emain valid. Extending the analysis to compae the equilibium esults of Counot competition with outcomes of two-stage game would be a useful futue study, if the demand function can be explicitly conveted to invese demand function.

12 The welfae effects of nationalization in a mixed duopoly Next, we analyze how the change of nationalization degee affects passenge demand, consume suplus and poduce suplus. All esults ae summaized in the following popositions. Poposition 3 An incease in the level of nationalization (1) inceases the total demand of local public tanspot in the equilibium oq NE o [ 0 ; (2) inceases the ideship of ail tansit unde the equilibium, but educes the ideship fo bus, oq NE o [ 0 and oq NE b o\0. Poof See Appendix 3. The insights behind Poposition 3 ae as follows: when the ail opeato attaches a high weight to consume suplus, this welfae concen puts a downwad pessue on ail fae and an upwad incentive on aising ail fequency. The combination of lowe fae and highe fequency makes ail tansit sevices moe attactive than befoe. Thus, its ideship is boosted along with its stengthened maket shae. Howeve, only concening its pofit, the bus sevice becomes pogessively less attactive, although it also educes its faes to compete against ail tansit. Poposition 4 Since tavelles pay less in the move fom a standad duopoly (SD) to a mixed duopoly (MD), the consume suplus necessaily ises alongside the boost in total public tanspot demand. At the same time, thee is a eduction in poduce suplus when we move away fom a standad pivate duopoly (SD) to a mixed duopoly (MD). Poof See the Appendix 4. Whilst the effects of nationalization on consume suplus and poduce suplus ae staightfowad now, we ae not cetain whethe the incease in consume suplus can compensate fo the loss of poduce suplus as the ail tansit opeato puts moe weight on consume suplus. Theefoe, the impact of nationalization on social welfae is still ambiguous. So we tun to a numeical study based pimaily on a mixed public tanspot maket in a Chinese city. 2.5 An extension to a lage-scale tansit netwok The above analytical modelling povides some useful and inteesting insights into the mixed oligopolistic competition. Nonetheless, the above analysis involves only single-path model. In eality, most cities commonly chaacteized with a lage-scale tanspot netwok with multiple paths and OD pais. Thus, it is meaningful to exploe the stategic inteactions and maket equilibium in a complex tansit netwok of a city. Consideing an uban aea, thee aleady exists one public tanspot netwok G = (N, A) whee, N is set of tansit stations (nodes) and A is the set of tansit links with the link a (a [ A). Within a W set of OD pai, bus and ail tansit companies opeate paallel sevices between a paticula OD pai, w (w [ W). The notation used fo this complex netwok is given as follows:

13 F. Qin et al. R w Set of tansit outes between OD pai w v a Total passenge flow on tansit link a; h w Passenge flow on path, [ R w ; i M a Pobability of passenges on link a choosing public tanspot mode i; i f a Sevice fequency of public tanspot mode i on link a; i P a The fae of public tanspot mode i on link a; q w Passenge flow between OD pai w; -1 D w Invese demand function between OD pai w; cc w Composite cost on oute between OD pai w; i OC a Opeating cost of unning tansit sevices though link a fo opeato i; With the above notion, the issue of inteest is a mixed oligopolistic equilibium subject to a stochastic taffic assignment. The poblem can be conveted as bi-level tansit equilibium model, which involve equilibium at both lowe (Stochastic Use Equilibium, SUE) and uppe levels (Nash Equilibium in the mixed oligopoly maket) The uppe level mixed oligopolistic equilibium MaxOF b ¼ X X P b a v ama b OCb a ð22þ w2w a2a MaxOF ¼ X X ðv a =nþ P a v ama OC a ð23þ w2w a2a Fo each opeato i, a level of fequency (f i a ) and pice chaged (P i a ) ae the only two vaiables chosen by each fim subject to the esulting passenge flow being in a stochastic use equilibium (SUE). The link flow (v a ) can be detemined by the following lowe poblem The lowe level Stochastic use equilibium Fo a given fae and sevice fequency, passenge will set thei tansit outes with the SUE manne. As poved by Lam and Zhou (1999), Lam et al. (2002), the SUE assignment poblem is equivalent to the following minimization poblem: Min 1 X X h w ðln h w 1Þ 1 X q w ðln q w 1Þ h w2w 2R w h w2w þ X cc a v a X a2a w2w Subject to v a ¼ P w2w P Z qw 0 D 1 w ðyþdy ð24þ 2R w s a h w s a ¼ 1 if link a lies on path ; 0 othewise ð25þ q w ¼ X 2R w d a h w d w ¼ 1 if path connects OD pai w; 0 othewise ð26þ

14 The welfae effects of nationalization in a mixed duopoly The pesented bi-level tansit netwok equilibium poblem follows simila modeling pinciples so that the esults do not qualitatively change with espect to the sign-path analytical model pesented in the pevious section. Nevetheless, with moe paths, moe links and moe OD pais, competition intensity inceases, thus eventually lowe the benefits of having a public tanspot opeato. The bi-level netwok equilibium model can be solved by a heuistic algoithm. Since the poblem becomes quite complex in a lage-scale netwok, up to now we cannot give an efficient algoithm to solve this bi-level netwok equilibium model. In the futue, it is much impeative to develop efficient solution algoithms applicable to geneal netwok. Moeove, due to the lack of data fo a lage-scale tanspot netwok, the numeical evaluation of the poposed bi-level tansit equilibium model fo a eal tanspotation netwok is also on open issue fo the futue eseach. 3 Numeical case studies To shed light on the impacts of patial nationalization on equilibium outputs unde a mixed duopoly maket, this section begins with intoducing Betand picing game in Case 1, in which sevice fequency is less flexible in the shot un. In Case 2, adding fequency as anothe stategic device, we illustate the long-un impact of nationalization on Nash equilibium solutions. To establish odes of magnitude fo key stategic vaiables and associated welfae elements, Suzhou taffic data ae used to gauge the esults. Table 1 summaizes all taffic data and paametes used. In the following numeical calculations, the taffic data and paametes, such as aveage faes, geneal opeating chaacteistics of public tanspot, and value of time Table 1 Taffic data and paametes Desciption Measuement Rail tansit Bus Aveage unning speed (V i ) Km/hou In-vehicle tavelling time (T i ) Hou Vehicle capacity (K mi ) Passenge/vehicle Vaiable unning costs (c i1 ) CNY/vehicle-km Maginal passenge cost (c io ) CNY/passenge Fae (P i0 ) CNY Fequency (f i ) Vehicle/hou 12 8 Maket shae (M i0 ) Pecent (%) % % Patonage of sevice i (Q i0 ) Passenges/hou Total public tanspot demand (Q ) Passenges/hou 5000 Value of time (q) CNY/hou 10.1 Scale paamete (h) 1 Weights of in-vehicle time (a 1 ) 1 Weights of waiting time (a 2 ) 2 Paamete in total tansit demand (n) 0.2

15 F. Qin et al. (VOT), wee souced fom the Annual Repot of Suzhou Uban Tanspotation (2008). Futhemoe, concening opeating costs of bus, the infomation stems fom Suzhou Bus Goup Annual Repot (2008). This epot illustates the vaiable opeating costs of unning an exta bus kilomete ae 8.9 CNY and the maginal cost pe passenge jouney CNY. In tems of ail tansit, the figues of vaiable opeating costs (22 CNY pe kilomete and CNY pe passenge) ae estimated by efeing to the accounting epot of Suzhou Meto Company. Hong and Zuo (2006) estimated the sensitivity of tansit passenges to changes of taveling costs in seveal Chinese cities. The esults indicate that fo a 1 % eduction in taveling costs, thee is a % incease in public tanspot patonage. In ode to make the following cases moe illustative, opeational featues of bus and ail tansit, such as thei aveage unning speed, vehicle capacity, ae also listed in Table 1. Setting the cuent situation as a efeence case, we can compae how faes, fequencies, maket shaes, social welfae and its constituent pats change with the degee of nationalization. With 5000 commutes pe moning peak hou, the ail tansit line catches a significantly lage maket shae (84.46 %) than the bus (15.54 %). 3.1 Case one: the effect of nationalization () on equiliba in the shot un In the shot un, public tanspot opeatos cannot affod moe fequent sevices due to the additional cost of inceasing thei fleet. Thus, both opeatos compete puely on pice, assuming sevice fequencies ae exogenously given. In Case one, to assess the impact of the level of nationalization () on equilibium configuations, the analysis begins with the move fom standad pivate duopoly (SD, = 0) toa mixed duopoly (MD, [ 0) case. As mentioned ealie, the paamete, is Table 2 Effect of nationalization () on equilibium solutions in the shot un = 0 = 0.2 = 0.4 = 0.6 = 0.8 = 1 Bus fae (P b ) Rail tansit fae (P ) Maket shae of bus (M b ) % % % 9.04 % 5.17 % 2.39 % Maket shae of ail tansit (M ) % % % % % % Total public tanspot demand (Q*) Passenge demand of bus (Q b ) Passenge demand of ail tansit (Q ) Poduce suplus (PS) D Pofit of bus (p b ) D Pofit of ail tansit (p ) Consume suplus (CS) 27,289 28,768 30,726 33,505 37,732 44,250 Social welfae (SW) 37,742 38,267 39,299 40,374 41,355 41,773 PS is the poduce suplus, which is the sum of bus s pofit (p b ) and ail s pofit (p ) SW is the social welfae, which is the sum of poduce suplus (PS) and consume suplus (CS)

16 The welfae effects of nationalization in a mixed duopoly Table 3 Effect of nationalization () on equilibium solutions in the long un = 0 = 0.1 = 0.3 = 0.5 = 0.7 = 0.9 Bus fae (P b ) Rail tansit fae (P ) Bus fequency (f b ) Rail tansit fequency (f ) Maket shae of bus (M b ) % % % % 4.37 % 0.75 % Maket shae of ail tansit (M ) % % % % % % Total public tanspot demand (Q*) Passenge demand of bus (Q b ) Passenge demand of ail tansit (Q ) Poduce suplus (PS) D Pofit of bus (p b ) D Pofit of ail tansit (p ) Consume suplus (CS) 29,287 29,933 31,472 33,542 36,599 41,637 Social welfae (SW) 38,022 38,489 39,465 40, continuous. But fo the sake of exposition, in Tables 2 and 3, we only list numeical equilibium solutions with vaying in steps of 0.2. The fist two ows of Table 2 show that, as expected, the equilibium pices of bus and ail tansit decease when ail opeato places moe weight on consume suplus. Thus, fo diffeent values of, the pai of equilibium faes ae highest in standad pivate duopoly with CNY fo bus and CNY fo ail tansit, and lowest in the mixed duopoly situation ( = 1) with only CNY fo bus and CNY fo ail tansit. Moeove, the synchonous decease in bus and ail tansit faes indicates the pices ae stategic complements, which means the pices set by bus company decease in esponse to the pice eduction of Rail Company. Although both equilibium faes move in the same diection (downwad), the ate of decease of ail tansit fae is much geate. When the nationalization degee exceeds 0.8, ail tansit has a competitive advantage ove bus in tems of its elatively low fae. Fom Fig. 1a, it can be obseved that the total demand of local public tanspot (Q*) and ail tansit (Q ) incease gadually with the incease in. On the contay, since ail tansit attacts inceasing numbes of tavelles fom buses due to its elatively low genealized cost, the maket shae of the bus quickly shinks fom to 2.39 %. The logic behind the esult can be explained as follows. The oveall impact of nationalization on patonage is the combination of two effects: the fist is inceasing total demand fo public tanspot, and the second is the shifting of maket shae between modes. Regading ail tansit, the moe weight the ail tansit fim places on consume suplus, the geate the aggegate demand and ail opeato s maket shae will be. Convesely, since the deceasing maket shae dominates the incease of total public tanspot demand, the equilibium patonage of bus falls with the inceasing degee of nationalization. Consequently, a futhe decease in both fims pices tiggeed by nationalization boosts the total public

17 F. Qin et al. Fig. 1 The impacts of patial nationalization on equlibia in shot un. a The effect of on passenge demand, b the effect of on opeato s pofits tanspot demand. In such a case, the ail tansit significantly eodes the maket shae of bus and the bus opeato gadually loses its competitive advantage. As illustated in Fig. 1b, although the ideship of ail tansit ises, the shap eduction in faes cause the ail tansit opeato s pofits to apidly tail off. Due to the ail tansit opeato s elatively high opeating cost, the low fae evenue cannot cove its vaiable opeating costs when the degee of nationalization exceeds 0.9. When eaches 1, it would be difficult fo ail tansit opeato to emain pofitable, the ail tansit opeato would equie subsidies fom local govenment. On the contay, although the combined effect of fae eduction and shinking ideship esults in a consideable decease in the pofits of the bus, the elatively lowe opeating cost makes the bus opeato pofitable when the value of does not exceed 0.9. In the exteme case of the ail tansit opeato fully consideing consume suplus ( = 1), the bus opeato cannot beak even. As the pofit is squeezed, the bus opeato might dop out of business. In the mixed duopoly, lowe equilibium pices and highe total tansit demand have positive impacts on consume suplus, implying consume suplus ises by a consideable amount. Although the poduce suplus falls with the incease in the degee of nationalization, these losses ae elatively small compaed with the gain in consume suplus. Consequently, total social welfae inceases slightly, this demonstates the desiability of nationalizing ail tansit opeatos. The numeical esults ae consistent with the analytical esults obtained in pevious section and can be summaized in Remak 1: Remak 1 An incease in the level of nationalization causes equilibium pices to fall fo both bus and ail tansit. Moeove, thee is a ise in social welfae when we move away fom a standad pivate duopoly (SD) to a mixed duopoly (MD). The above emak states that, in the pocess of nationalization, as the ail tansit opeato puts moe weights on consume suplus elative to its pofit, the pice

18 The welfae effects of nationalization in a mixed duopoly competition in the mixed duopoly maket become fiece. Since tavelles pay less in the move fom a standad duopoly (SD) to a mixed duopoly (MD), consume suplus, theeby boosting total public tanspot demand. Also, since the incease in consume suplus can compensate fo the loss of poduce suplus, patial nationalization is socially pefeable in tems of welfae impovement. 3.2 The effect of nationalization () on equilibia in the long un In the long un, since the opeato has the ability to adjust its fleet size to match fiece maket competitions, sevice fequency might be anothe stategic device. In this numeical case, we tun to Nash pice fequency competition and attempt to demonstate how the opeatos espond to the pogess of nationalization. The numeical equilibium solutions fo this Nash non-coopeative game ae summaized in Table 3. As Table 3 shows, the motive of inceasing consume suplus induces the ail tansit opeato to incease its fequencies and educe faes. Since pices ae stategic complements in this pice-fequency game, the gadually deceasing ail tansit faes povides an incentive to bus opeato to defend its maket shae by esponding with a fae eduction. On the contay, in this case with endogenous fequency, it emeges that sevice fequencies ae stategic substitutes. Thus, esponding to the incease in ail tansit fequency, the bus opeato is foced to educe its sevice fequency to pevent futhe deteioation of pofitability (see Fig. 2a). In compaison to Case 1, the most likely effect of adding sevice fequency as anothe stategic vaiable is that total public tanspot ideship has been geatly boosted, because the lowe composite costs esulting fom additional competitions ove fequency incease the attactiveness of public tanspot. Similaly, when ail opeato attaches a high weight to consume suplus and a coespondingly low Fig. 2 The impacts of patial nationalization on equlibia in long un. a The effect of on sevice, b the effect of on consume suplus and welfae

19 F. Qin et al. weight on pofit, its maket shae will be stengthened. On the othe hand, conventional bus sevices become pogessively wose off since its sevice levels ae not as attactive as befoe. The ideship of buses diminishes fast. Then, in the case of full nationalization, a cone solution can be obtained in which the ail opeato satisfies all demand. As fa as opeatos pofits ae concened, both opeatos expeience financial losses in the pocess of nationalization. Once the value of exceeds 0.5, the bus loses the commecial feasibility of opeating sevices due to the negative pofits. Howeve, the welfae compaisons show that the incease in consume suplus offsets the losses in opeatos pofits, which suppots the desiability of a mixed oligopoly (see Fig. 2b). The main finding of Case 2 is summaized in Remak 2. Remak 2 In Nash non-coopeative game, with the motive of inceasing consume suplus (1) the ail tansit opeato intends to educe fae and inceases its sevice fequency. The bus opeato stategically chooses to educe both fae and fequency (2) the incease in consume suplus offsets the losses in poduce suplus, leading to the incease of social welfae. The studies of mixed duopoly in industial oganization suggests that without othe egulations, nationalization will heighten social welfae if thee ae elatively few pivate fims in the maket and will degade social welfae if thee ae elatively many pivate fims, assuming Counot quantity-setting competition with an inelastic and linea demand function (Andeson et al. 1997; De Faja and Delbono 1990). Fom the above numeical case studies, it can be concluded that full nationalization yields the highest level of social welfae in the public tanspot maket. Thus, the numeical cases povide suppot fo pevious findings on Nash non-coopeative game in the context of an elastic and non-linea demand function. The above numeical esults ely heavily on specific dataset fom Suzhou city. Hence, the geneality of esults is esticted to some extent. To detemine whethe the esults ae univesal of specific, we cay out the following sensitivity analysis. 3.3 Sensitivity analysis The numeical esults of equilibium configuations ae dependent on a vaiety of system paametes. Among these paametes, the chaacteistics of tanspot demand and supply especially diffe fom one city to anothe. To this end, we focus on two paametes, namely, the uses cost peception (the demand paamete, n) and the poduces opeating cost (opeating cost coefficients of ail c 1 ), which ae impotant vaiations among cities. In what follows, we investigate the sensitivity of equilibia, in paticula, the opeatos faes and fequencies unde changes in these two paametes Sensitivity analysis with espect to uses cost peception (n) Chaacteizing with diffeent income level and diffeent infomation technology, passenges in diffeent cities may have diffeent peceptions on tavel cost vaiations. Fo example, when passenges in one city can achieve geate o moe accuate tavel infomation, they will have highe value of n. To investigate the

20 The welfae effects of nationalization in a mixed duopoly Fig. 3 Sensitivity of equilibium faes and fequencies w..p to changes in n. a The ail tansit fae change w..p n, b the bus fae change w..p n, c the ail tansit fequency change w..p n, d the bus fequency change w..p n likely esponse of equilibium pices and fequencies to the change of demand paamete (n), the value of n vaies fom 0.1 to 0.5 with steps of 0.1. In Fig. 3, we display the vaiations in the equilibium fae and fequency unde changes of demand paamete (n). Regading the sensitivity of equilibium esults to paamete n, the following can be obseved fom Fig. 3. Fist, when esidents of one city may have a highe accuacy in peceiving the tavel costs (a highe value of n), the equilibium faes fo both bus and ail tansit go down. Futhemoe, the fae diffeence between bus and ail tansit becomes smalle with the incease of value n, suggesting intense pice competition. Second, when the taveles have highe cost sensitivities n, the ail tansit opeato intends to povide moe fequent sevices; the bus opeato s opeating behavios in espond to changes of n becomes complicated. Finally, making tavelling demand moe cost sensitive does not affect the qualitative esults obtained in Remak 1 and Remak 2.

21 F. Qin et al Sensitivity analysis with espect to supplies opeating cost (c i1 ) In tems of opeating costs, we obseve that the cost efficiency of one tansit opeato has been affected by many factos, such as opeational scales, oganization stuctue, potential demand ate, egulatoy egime and so on. Thus, the vaious opeating envionments in diffeent cities do have geat impacts on the opeating efficiencies of opeatos. Since the slopes of fequency eaction functions ae diectly affected by the vaiable opeating cost, they ae likely to be impotant in detemining the equilibium esults. To show how the opeating cost diffeence affect the Nash equilibium fequencies and pices, we keep the opeating cost of bus fixed but the costs of opeating ail tansit sevices ae changed fo this 9 km tanspot link. With the incease of ail tansit opeating costs (fom 40 CNY to 198 CNY), the cost advantage of bus becomes stong (Fig. 4). The main insight fom this execise is that: an incease in the opeating cost of ail tansit educes the Nash equilibium fae and sevice fequency fo ail tansit sevices. Howeve, since the cost diffeence between bus and ail tansit becomes Fig. 4 Sensitivity of equilibium fae and fequency with espect to changes in (c 1 ). a The ail tansit fae change w..p c 1, b the bus fae change w..p c 1, c the ail tansit fequency change w..p c 1, d the bus fequency change w..p c 1

22 The welfae effects of nationalization in a mixed duopoly lage, the bus opeato has a stonge cost advantage ove ail tansit. Thus, the bus opeato can affod to un moe fequent sevices and chage highe fae than befoe. Not supisingly, we found that the magnitudes of equilibium outcomes ae somewhat sensitive to the demand paamete and opeating cost coefficient. Howeve, the qualitative conclusions about the effect of nationalization degee on equilibium esults in Remak 1 and Remak 2 ae found to be unaffected. 4 Conclusions In this pape, the duopolistic inteactions between one public ail tansit opeato and one pivate bus opeato ae pesented as a Nash non-coopeative game. To investigate the effects of nationalization on equilibium faes, sevice fequencies and associated welfae elements, two numeical cases based on Suzhou taffic data ae analyzed and the main insights ae pesented as follows. Fistly, given the less flexible sevice fequency in the shot un, the pesence of one publicly-owned ail tansit opeato is a useful measue fo appoaching social optima. The moe weight the ail opeato attached to consume suplus, the lowe the equilibium fae, leading to highe consume suplus and social welfae. Secondly, in Nash noncoopeative game with pice and fequency competition, pices ae a stategic complement. Due to this, when the ail opeato educes (o inceases) its fae level, it foces the bus opeato to do the same. Meanwhile, a maginal fequency incease (decease) in ail tansit sevice causes bus opeato to decease (incease) its fequency, indicating sevice fequencies ae stategic substitutes. Finally, if no additional egulations ae implemented, nationalizing one ail tansit company is socially desiable both in Betand picing game in the shot-un and in Nash noncoopeative game in the long-un. This esult is in line with the pevious studies esults on mixed oligopolies. That is, when the numbe of pivate companies is elatively small, pivatizing one public fim degades social welfae. The analysis pefomed in the pape captues the pesent status of uban public tanspot maket in many Chinese cities whee newly intoduced state-owned ail tansit sevices compete with extant pivate bus companies. Although the above model is simply based on a duopolistic setup, the esults have geneated many manageial implications fo the tanspot authoity, public and pivate tansit opeatos. Regading the tanspot authoity, it can use the nationalization as a mean to egulate the competitive uban tanspot maket. Moeove, in both shot and long tem, the authoity should caefully conside the amount of emphasis a public ail opeato should place on consume suplus compaed to its own pofit. The Suzhou case indicates that too much weight on consume suplus may cause the pivate bus opeato to educe its sevice fequencies and consequently lose incentive to emain the maket. On the othe hand, too little emphasis on consume suplus leads to highe faes, though it may yield highe fequent bus sevices. Thus a pope policy may be to find a happy compomise which will not only encouage pivate bus fims, ende public ail self-suppoting, but also yield adequate consume suplus in the total social welfae. This pape has aised these impotant issues to undestand such a compomise decision. As fa as Chinese uban tanspot policy is concened,