Research in Transportation Economics

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1 Researc in Transportation Economics 3 (8) 3 4 Contents lists available at ScienceDirect Researc in Transportation Economics journal omepage: Optimal economic interventions in sceduled public transport q Kjell Jansson a, *, Harald Lang b, Dan Mattsson b a Royal Institute of Tecnology, Scool of Arcitecture and Built Environment, Division of Transport and Economics, Stockolm, Sweden b Royal Institute of Tecnology, Scool of Engineering Sciences, Division of Matematics, Stockolm, Sweden article info abstract Article istory: Available online Marc 9 Tis capter analyses appropriate regulatory instruments for public transport markets under monopoly and competition, respectively. For te monopoly case, te operator cooses too low supply, compared to welfare optimum. In contrast, for te competition case te operators coose too ig supply, at least for te competition model tat we ave considered most appropriate. It is found tat under monopoly a subsidy sould be applied, wile under competition taxation sould be applied. Ó 8 Elsevier Ltd. All rigts reserved.. Introduction Is public transport caracterised by market failure, so tere is a reason for government intervention? If so, does intervention mean subsidisation or taxation? We are in tis study concerned solely wit pure economic incentives to correct te beaviour of profit maximising operators. We disregard oter possible regulatory measures, tus leaving te operators maximum freedom. Muc concern as been dedicated to public transport services, bot from te general public and politicians in most countries. Sould tey be state owned or private? Sould tey be supported financially? In most industrialised countries, te responsibility for local and regional public transport gradually became in to te ands of a Public Transport Autority, were a driving force was growing car ownersip and consequently lower public transport demand and revenues. In many of tese countries, te autority is also still in carge of bot planning and operations of te services. During te last two decades, anoter trend as flourised: deregulation. In Western Europe, tis trend commenced witin local and regional public transport. Te privatisation of te Englis urban bus industry represents te full market solution, were bot supply, prices and te operation are in te ands of competing profit maximising firms. In Scandinavian countries and in London, and to some extent in oter countries, te decision over local and regional public transport supply and prices as been kept in te ands of a public q Tis work was financed by te Swedis Governmental Agency for Innovation Systems (VINNOVA). We are grateful for valuable comments by Roger Pyddoke and Reza Mortasawi. * Corresponding autor. address: kjell.jansson@infra.kt.se (K. Jansson). autority, wile te actual operation is left for competition troug tendering. Typically, tese services need local or central government grants for financing and tere is economic reason for subsidisation, partly due to te so-called Moring effect (See Moring, 97), i.e., positive external effects of public transport. Te approac by Moring as ten been followed by, e.g., Jansson (984) and Turvey and Moring (975) wo deal wit price and service frequency, assuming one passenger group. Nas (978) optimises price and output in terms of miles operated for frequent urban bus services, contrasting maximum profit and maximum welfare solutions and assuming demand in terms of passenger miles to be dependent on price and bus miles operated. Jansson (99) considers and contrasts frequent and infrequent services, and takes into account a variety of passenger groups. Panzar (979) analyses infrequent airline services, assuming demand to be dependent on price and service frequency and allowing for a distribution of ideal departure times. Te oter reason, for subsidisation, most relevant for urban public transport, is te second-best argument, i.e., tat public transport sould be subsidised wen motorists do not pay te full social costs. Te second-best argument is not discussed in tis capter. In many developing countries, te typical situation is tat urban public transport is bot planned and operated by profit maximising firms. For long-distance public transport, te circumstances and te economically efficient policy are less clear. Most long-distance public transport operators are commercial profit maximisers, private or state owned. Airlines, railway, coac services often compete in some ranges of distance. Is tere a cause for government intervention? Rail transport, urban and long-distance, as in most countries so far been left in te ands of governmentally controlled bodies, but two exceptions are Great Britain and Sweden. In bot countries, te railway as been split into a governmental autority in carge of /$ see front matter Ó 8 Elsevier Ltd. All rigts reserved. doi:.6/j.retrec.8..

2 K. Jansson et al. / Researc in Transportation Economics 3 (8) te infrastructure and one or more operating companies. Tese companies compete troug competitive tendering, in Sweden toug only for regional commuter train services. Te analysis and te results in tis capter are valid for all kinds of sceduled public transport. We analyse te scope for intervention for a monopolistic and a competitive situation, respectively. We ignore te income effect and excess burden of public financing. We find tat monopolies sould be subsidised and competitive operators be taxed, in order to acieve te social welfare optimum. Tis capter is organised as follows. In Section, we specify our basic modelling of utility and demand. In Section 3, we study a profit maximising public transport operator wo acts under monopoly, facing imperfect elastic demand. Tis section follows Jansson (993) in te basic modelling approac, were price, service frequency and ride time as a function on load factor are welfare optimised, but ere we also contrast welfare and profit maximisation in order to find te appropriate regulatory instrument. We find tat a profit maximising operator cooses a iger price and a lower seat capacity tan wat a welfare maximiser would do. We find tat te regulatory instrument for acieving a social optimum is to subsidise te profit maximising operator troug te price paid by passengers. Tis subsidy can acieve a social optimum for all te relevant variables: price, service frequency and transport unit size, wile a productionrelated subsidy cannot yield a social optimum. Some autors assume tat te appropriate regulatory instrument is a subsidy related to production, see for example, Carlquist () for applications in Norway. Larsen () finds tat bot a production related subsidy and a subsidy related to te passengers sould be applied. Else (985) and Wallis and Gale () on te oter and, argue tat te subsidy sould be related to te passengers only. In Section 4, we study two profit maximising public transport operators wo compete ( Bertrand competition ) wit eterogeneous transport modes. Heterogenity ere stems from te difference between randomly distributed ideal departure times and actual departure times for te competing routes or modes and from randomly distributed taste for te routes or modes. Since no analytical solution to te optimisation problem is available in tis case, we employ numerical calculations. Te conclusion in tis case is significantly different compared to te monopolistic case in Section 3. Under competition, we find tat welfare optimising frequencies are below te frequencies tat te operators would coose. Tat competition may imply too muc supply from a social welfare point of view is not surprising and found in oter studies; see for example, Jansson (997) wo employs a numerical example to demonstrate tis. Under competition, we find tat taxation instead of subsidies sould be applied in order to correct a profit maximising operator. However, we cannot rule out tat tis result is model dependent, but we believe tat te assumption of Bertrand competition is te most realistic. In Section 5, we discuss te results from a teoretical point of view and provide examples for possible real-world applications.. Utility and demand Witout loss of generality, we assume tat calculations of assignment, demand and consumer surplus refer to one passenger group in one origin destination pair. Tis group sould be as omogeneous as possible wit respect to valuation of time in relation to price. For real-world analyses and applications, it is evident tat passengers ave to be segmented according to valuations of time and te segment specific ticket price tey ave to pay. We ignore te income effect, wic is standard in transport analysis. All travel time components are expressed in terms of money or time by conversion wit values of time. Demand is specified for certain periods, suc as te average weekday afternoon peak our in wintertime, te average Saturday etc. Only one type of cargeda per-trip pricedis considered. Te operating firm reaces decisions about relevant inputs and prices well aead of implementation because of a necessary planning lag. All factors of production tat are variable between decision and implementation are, terefore, considered relevant for te joint decision on te magnitude of policy variables. Tese factors include, we assume, te frequency and te size of transport unit, including te personnel required. Te generalised cost, i.e., te sum of price and travel time components expressed in monetary terms or time, for a journey from door to door as one part G common to all passengers, and one idiosyncratic part e i specific for passenger i. Tus te generalised cost for traveller i is: G i ¼ G þ e i : Eac individual is assumed to ave a utility of travel from origin to destination, i.e., te utility of te journey itself, wic is denoted n i. Te net utility for individual i is: v i G i ¼ v i e i G: Let u i ¼ n i e i and denote by f[u] te density of u i. Individual i cooses to travel if u i G, so te aggregate demand, X, is te integral of f[u] above G: X½GŠ ¼ Z N G f ½uŠdu: (.) Note tat X is a function of G only, and tat X ½GŠ ¼ f ½GŠ < : (.) Te consumer surplus, S, is tus: S ¼ S½GŠ ¼ Z N G It follows tat vs ðg GÞf½GŠ vg ðu GÞf ½uŠdu: Z N 3. Public transport under monopoly 3.. Introduction G f ½uŠdu ¼ X½GŠ: (.3) Our aim in tis section is to find te appropriate regulatory instrument under monopoly. First, we compare optimal price and quality according to welfare maximisation optimum and profit maximisation optimum, respectively. Quality is ere understood as wait time and ride time as a function of te load factor (seats per departure). Te discrepancies between optima provide te ground for te basic aim: to find a regulatory instrument tat makes te profit maximising operator beave according to welfare optimum criteria. 3.. Notation and assumptions We introduce te following notation: p is te price for a trip. s is a subsidy paid to te operator per journey. X is te number of passengers during a period of time, tougt of as. F is frequency in number of departures per our.

3 3 K. Jansson et al. / Researc in Transportation Economics 3 (8) 3 4 N is te number of seats per departure. U is te product FN, i.e., te number of seats per our. Z is te load factor, Z ¼ X=U ¼ X=ðFNÞ: I is te variable infrastructure cost per departure. e is te external cost per departure. f is te fee paid by te operator per departure, including infrastructure costs, external costs and a possible production-related subsidy. c is a cost proportionate to number of passengers, mainly sales costs. r is te ride time cost; an affine or convex function of Z. T is te cost of frequency delay (wait time). C[N] is te variable cost per departure directly related to te size of te transport unit, assumed to be an affine or convex function of N. e p is te own-price elasticity of demand. vx=vpp=x; e F is te frequency elasticity of demand.vx=vff=x; e N is te elasticity of demand wit respect to number of carriages vx=vnn=x: Arguments of functions are delimited by [ ], wile expressions are grouped by parenteses ( ) Autorities Te Public Transport Autority is assumed to be welfare maximising and responsible for infrastructure, wic is subject to a user carge, f Te operating firms Te operating firms may be eiter welfare or profit maximising. In bot cases, it optimises prices, service frequency and te number of seats or carriages. We deal wit consumption efficiency related to determination of optimal prices, optimal frequency, optimal transport unit size and possible subsidies and infrastructure carges, assuming tat production is efficient, irrespective of weter te actual producer is welfare or profit maximising Te passengers Aggregate demand and consumers surplus are expressed as functions of te common generalised cost G ¼ (ticket price p) þ (time costs). Time costs are ere assumed to comprise ride time and frequency delay, wic is te time interval between ideal and actual departure time. Te cost of ride time is: X r ¼ r : FN Te cost of ride time, r, is tus assumed to depend on te load factor (occupancy rate,) Z ¼ X=ðFNÞ, were r is an increasing and convex function, i.e., r [Z] > and r [Z]. We ignore tat tere are several passenger groups. Tis simplification will substantially facilitate te expressions derived witout disturbing te purpose of te analysis (te optimal welfare maximisation conditions wen various distances travelled by different passenger groups are taken into account can be found in Jansson (99)). Te interval between departures is /F. Ideal departure times, t, are uniformly distributed witin tis interval. In tis section, we assume tat te delay cost is a function of frequency only, i.e., T ¼ T[F]. In Section 4, we will, owever, consider te individual variations of delay times. If p denotes te price, te generalised cost of travel for a group (index omitted) at time t is: X G½p; F; NŠp þ r þ T½FŠ: (3.) FN Note ere tat G is a function of occupancy rate, wic is a function of demand, X, wic in its turn is a function of frequency and price. Tat is, price affects demand and tus te riding time cost. We know tat own-price elasticities, denoted e p, are negative, e p <. We assume, based on solid empirical evidence for most situations, tat demand elasticities wit respect to frequency and transport unit size, denoted e F and e N, are suc tat < e F <, < e N <, implying tat vz=vf < ; vz=vn <. Tis means tat an increase in frequency or unit size will not generate so many passengers tat occupancy rate is uncanged or increases Welfare and profit optima Objective functions Below we present te objective functions of te welfare maximisation and te profit maximisation models. Te maximisation relates to one service during a period normalised to (). Te analysis may ten be repeated for oter periods and routes. Infrastructure costs of rail operation is I$F. Te operation gives rise to external costs ef. Te operators are supposed to pay te infrastructure carge ff, were not necessarily f ¼ I þ e. Fixed costs are not taken into account in te model since tey are not affected by te operation decisions. Te welfare objective function is expressed as: W ¼ S½GŠþðp cþx FðC½NŠþI þ eþ: Te objective function for profit maximisation includes only producer s surplus, taking into account te infrastructure fee, f: p ¼ðp cþx FC½NŠ ff: Relations between welfare and profit optima We will now examine te relations between optimal price, frequency and unit size for welfare and profit maximisation, respectively. In oter words, we want to know weter price, frequency and transportunitsizeislargerorsmallerforwelfareorprofitmaximisation, respectively. For tis purpose, we employ Topkis teorem. We differentiate te welfare and profit functions wit respect to te variables X, F and U FN; i.e., we transform te variables F and N into one, so tat U reflects te total capacity per our in terms of number of seats. Of course, X is endogenous, but in te calculations below we consider X, F and U to be exogenous, and p endogenous. Tus, wen we differentiate w.r.t. X, we keep U and F fixed, but let p vary. Similarly, wen we differentiate w.r.t. F, we keep X and U fixed, and let p vary (so as to keep X fixed.) Note tat wenwe differentiate w.r.t. Fand U,sinceX is fixed, also G is fixed (cf. Eq. (.);) so for instance v=v FS½GŠ ¼. Note tat wen employing Topkis teorem, we sould compare te derivatives of te objective functions, i.e., te comparison sould be at te same point (not teir respective optimum points). In te case of no subsidy, s ¼, we ave, employing Eqs. (.) and (.3): v v ðw pþ¼ vw vx ðs½gšþfðf I eþþ ¼ S ½GŠG ½XŠ¼ XG ½XŠ ¼ X >: (3.) X ½GŠ Te marginal effect on occupancy rate of, for example, te frequency F is vz=vf ¼ FvX=vF X=F N ¼ Xðe F Þ=F N; wic is negative only if e F <. Topkis teorem, in te form we use it ere, says tat if vf ½x Š =vx i vg½x Š =vx and eiter vf ½x Š=vx i is increasing in x j for all j s i, or te same is true for g[x], ten argmax f[x] argmax g[x].

4 K. Jansson et al. / Researc in Transportation Economics 3 (8) As long as te infrastructure carge, f, is equal to, or larger tan, te marginal infrastructure and external cost, we ave tat: v v ðw pþ ¼ ðs½gšþfðf I eþþ ¼ f I e : (3.3) vf vf Next, we differentiate wit respect to U: ¼ vw vp ¼ XvG vp þðp cþx ½GŠ vg vp þ X ¼ vw vf ¼ XvG vf þðp cþx ½GŠ vg ðc½nšþi þ eþ vf (3.5b) (3.5a) v ðw pþ ¼ v ðs½gšþfðf I eþþ ¼ : (3.4) vu vu We are now almost prepared to conclude tat at welfare optimum, X, F and U is greater (or possibly equal) to te situation wit profit maximisation. Te first condition in Topkis teorem is satisfied, as is seen from Eqs. (3.), (3.3), and (3.4). However, we must also ceck te cross effects for one of te objective functions, and we coose W. First, we compute vw=vf: ¼ vw vn ¼ XvG vn þðp cþx ½GŠ vg vn FC ½NŠ: (3.5c) Te first order conditions referring to price, frequency and unit size for a monopolist are, respectively, ¼ vp vp ¼ ðp cþx ½GŠ vg vp þ X (3.6a) vw vf ¼ vp vf X C½NŠþNC ½NŠ I e: In order to compute p F, we note tat by Eq. (3.): ¼ vp vf ¼ðp cþx ½GŠ vg C½NŠ f vf (3.6b) ¼ vg vf ¼ vp vf þ T ½FŠ i.e., vp vf ¼ T ½FŠ: Hence, vw vf ¼ XT ½FŠ C½NŠ þ NC ½NŠ I e: Now we can compute v W=vFvX and v W=vFvU: v W vfvx ¼ T ½FŠ > ; v W vfvu ¼ N F C ½NŠ : Te computation of vw=vu is similar to tat of vw=vf: ¼ vp vn ¼ðp cþx ½GŠ vg vn FC ½NŠ: (3.6c) Note tat it is impossible to acieve te welfare optimum by adjusting f away from I þ e. Tis is seen by a reductio ad absurdum: assume tat we can acieve welfare optimum by te profit maximiser by a suitable value of f. It ten follows tat Eqs. (3.5a) and (3.6a) are evaluated at te same values of p, F and N, and ence tat XG p ¼ at tis point, wic is clearly absurd. Hence, one cannot apply a subsidy only related to frequency in order to acieve a welfare optimum. If te monopolist enjoys a subsidy s of te price, is objective function is p ¼ðp þ s cþx FC½NŠ ff; ence, te first order conditions become ¼ vp vp ¼ðp þ s cþx ½GŠ vg vp þ X (3.7a) vw vu ¼ vp vu X C ½NŠ: Here vp=vu can be computed in te same way as vp=vf: ¼ vp vf ¼ðpþs cþx ½GŠ vg C½NŠ f vf (3.7b) ¼ vg vu ¼ vp X X r vu U U ; i.e., vp X X vu ¼ r U U ; ence, v W vuvx ¼ v vx! X X X X U U C ½NŠ ¼ r X X U U 3 þ r U U > : r To conclude: Te profit maximiser as too few passengers and too low frequency from a welfare point of view. He as also lower capacity FN tan te welfare maximiser; owever, given te fewer passengers, tis may or may not be sub-optimal Welfare and monopoly equilibria Te first order conditions referring to price, frequency and unit size for a welfare optimiser are, respectively, ¼ vp vn ¼ ðp þ s cþx ½GŠ vg vn FC ½NŠ: (3.7c) We see tat if te subsidy s is set to S ¼ X=X ½GŠ at te welfare optimum and f ¼ I þ e, ten te tree Eqs. (3.5a), (3.5b), and (3.5c) coincide wit Eqs. (3.7a), (3.7b), and (3.7c), and ence te profit optimum coincides wit te welfare optimum. Te passenger related subsidy can tus yield te social optimum wit respect to all te relevant policy variables: price, frequency and transport unit size. Te expression for te subsidy s can be expressed in various ways: s ¼ X X ½GŠ ¼ G e G ¼ p e p r ½ZŠZ were p is te price of te welfare optimiser. Te last equality follows from te fact tat vg=vp ¼ þ r ½ZZe p =p. If, for example, te elasticity wit respect to price would be around, te subsidy would be equal to te cosen price minus a term tat depends on te marginal congestion cost. Wallis and

5 34 K. Jansson et al. / Researc in Transportation Economics 3 (8) 3 4 Gale () find, like us, tat te passenger related subsidy is G=e G, but also tat te subsidy can be expressed as p=e p. Our result includes an additional term related to te marginal congestion cost, tis, since we explicitly take tis quality variable into account. Else (985) finds an optimal subsidy related to passengers, but wic differs from ours. 4. Public transport under competition Wen dealing wit competition between operators or modes a crucial issue is wat factors affect te passengers coice. Clearly travel time components and price matter, among oter factors, and time and price are not valued te same by all individuals. Tese are important facts tat sould not be ignored. Tere are various ways tat one can take care of variations of travel time and price, as well as oter factors: Apply randomness for taste variations witin eac group, segment passenger groups wit different values of travel time, i.e., take taste into account in tis way, apply randomness to te passengers different ideal departure or arrival times. A popular way to model taste variation is te so-called logit model (belonging to te extreme value family). Here we ave cosen a different approac toug, since we believe te logit model as sortcomings wen applied to public transport 3. We, basically, take into account randomness wit respect to te difference between passengers ideal departure or arrival times, bearing in mind tat te analysis can be repeated for a number of segments. 4.. Basic micro-economic model In Section 3, we defined generalised cost as: X G½p; F; NŠ ¼p þ r þ T½FŠ: FN In tis section, we ignore tat ride time cost may depend on inveicle congestion, and call p þ r travel cost R. Tat is, R ¼ p þ r.we also ignore te size of te veicle, N. Instead, we take into account tat te wait time is variable, i.e., dependent on t, te difference between ideal departure or arrival time and actual departure or arrival time. We model competition by using te fact tat te passengers will coose te competitor tat as te smallest total travel cost R þ t. We assume tat te individual differences between ideal and actual departure or arrival times are randomly, uniformly distributed. Eac travel alternative in a specific origin destination pair as a total travel cost and eac passenger cooses te alternative wit minimum total cost. In order to simplify notation and calculations, we assume tat tere are only two alternatives, and. Te total cost of alternative j (j ¼, ) for eac individual i is te sum of travel cost R j (including all travel time components plus price, except wait time) and a random variable, t j, te time between i ideal arrival time and actual arrival time. Wit a valuation of time, wait time, ride time, and transfer time can all be expressed in monetary units, or conversely monetary units can be tougt of as measures of time. In te following, all quantities are measures of time but we may, and often do, describe tem as costs. Wen eac individual cooses te alternative wit te minimum total cost, te effective cost becomes: 3 See Jansson et al. (8). i min R þ ti þ R þ ti : We now assume tat (t, t ) is uniformly distributed on [, H ] [, H ], were eadway H i ¼ /F i. Tis, since we ave no knowledge of te true distribution of ideal departure or arrival times for te period of time (peak ours or non-peak ours for example) we are analysing. 4 Notation H eadway of route. H eadway of route. R travel time (including price expressed in minutes) of route. R travel time (including price expressed in minutes) of route. t time to departure of route. t time to departure of route. Te probability for coosing alternative is ten: Pr½Š ¼ Z H H H Z H R R þ t t i dt dt (4.) were [s] is te Heaviside function, defined by: if s ½sŠ ¼ if s < : Te model tus assumes tat passengers know te timetable and coose route, stop and mode, taking into account all travel time components and price and ow well ideal departure times relate to actual departure times. Here, te expected wait time, T, can be expressed as: T ¼ Z H H H Z H R R þ t t i t t þ t dt dt : In general, if tere are k acceptable routes and te travel cost for route j is R j and te probability of coice of route j is denoted Pr[j], te expected travel time R, and te expected wait time T are given by: R ¼ Xk j ¼ Pr½jŠR j ; T ¼ Xk j ¼ Pr½jŠE½tjjŠ (4.) respectively, were E[$jj] denotes expectation conditioned on route number j being cosen. Te probability in expression Eq. (4.) and te conditional expectation E[t j jj] used in Eq. (4.) can be calculated explicitly. Tis is illustrated wen aving two alternative modes. Te first route is selected wen R þ t < R þ t, tat is wen t > t þ (R R ). For different values of d ¼ R R, te probabilities of t > t þ d and te conditional expectations are given by: (i) if d > H ten Pr½Š ¼ 4 Te result tat follows for coice probability and expected wait time as also been presented, witout specific derivation, by Hasselström (98). Te principles for taking into account randomness of ideal departure times ave been implementd in te public transport network softwares Vips and Visum.

6 i E t j is undefined i E t j ¼ H =; (ii) if H > d > max [H H, ] ten Pr½Š ¼ H d H H i E t j ¼ H d 3 K. Jansson et al. / Researc in Transportation Economics 3 (8) i E t j ¼ H þ d ; 3 (vi) if H > d ten Pr½Š ¼ i E t j ¼ H = i E t j is undefined: i E t j ¼ d 3 3d H H 3H H H ; 3 d dh H H þ H (iii) if max [H H,]> d > ten Pr½Š ¼ H = þ d =H i E t j ¼ 3dH þ H H 3H 3 d þ H H i 3d þ 3dH þ H E t j ¼ 3H ; þ 6d (iv) if > d > max [H H, ] ten Pr½Š¼ i E t j ¼ i E t j ¼ H = d =H 3d 3dH þ H 3 H d 3dH þ H H 3H ; 3 d þ H H (v) if max [H H,]> d > H ten Pr½Š ¼ i E t j ¼ H þ d H H d 3 þ 3d H H 3H H H 3 d þ dh H H þ H Below we provide an example concerning te coice between two alternatives. We assume tat te eadways for bot routes and are 5 min, i.e., H ¼ H ¼ 5, but te travel times for te routes differ; for route it is R ¼5 min, for route it is R ¼ min. Route is selected if R þ t R þ t, were t and t are independent and uniformly distributed on [, H ] and [, H ], respectively. Hence, if t 5 ten: Pr t i ¼ Pr t R R þ t fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} Furtermore, wen 5 t 5 ten: Pr t i ¼ Pr Tis gives, Pr½Š ¼ ¼ ¼ Z N N Z 5 t R R þ t fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} t 5 : Pr t i f, 5 dt þ t i dt ¼ Z 5 5 t t ¼ Pr t t i ¼ : t ¼ R R þ t H Z 5 Pr t i, 5 dt 5, 5 dt ¼ 3 þ 4 9 ¼ 7 9 : Fig. below illustrates points of time, eadway and travel times of te two routes. Te total bar lengts represent te maximal costs associated wit te routes. We assume tat te two routes arrive at te same time (). Te passengers ideal departure times are along te x-axis; te wait time for eac alternative is a uniformly distributed random variable over te ligt coloured parts of te bars. Te passengers cose te alternative were tis wait time variable is closest to te origin, aving smallest total cost. For ideal departure times between 5 and min only route is cosen. For oter times te passengers are split between te two alternatives according to te proportions Pr[jt ] and Pr[jt ]. Te conditional probability of selecting route is sown in Fig.. Let u denote te net utility, measured in pecuniary terms, of a journey for an individual. Te total consumer surplus S for travel is ten: S ¼ X H H ¼ X Z H Z H Z Z u min R þ t ; R þ t i dt dt max u R H s ; u R H s i ds ds : Since v=vx max½x; yš ¼ ½x yš we ave

7 36 K. Jansson et al. / Researc in Transportation Economics 3 (8) 3 4 Route Route Z vs vr ¼ X and Z ¼ X,Pr½Š ¼ X Z vs vh ¼ X Z ¼ i t H E j X : R þ H s R H s i ds ds s R þ H s R H s i ds ds Note tat E[t j] is sorter tan H =, since tere is anoter alternative! Hence, in order to compute te cange DS in consumer surplus due to a cange DR of R i, one can integrate te demand function X i for mode i: DS ¼ Z R i þdr R i X i dr i and similarly for a cange in eadway H i by DH: DS ¼ H i Z H i þdh H i R R i X i E t i ji dh i : (4.3) Note, owever, tat in Eq. (4.3) we must take into account tat E[t i ji] depends on wat oter alternative travel modes tere are, so te mode i cannot be considered in isolation Demand We consider two competing firms providing public transport meeting a fixed total demand. Te modes of transport differ in some respect, i.e., one firm may transport by air and anoter by railway. We denote te two transport firms and. Te basic model is now extended to include a taste variation. Eac traveller H Cost of travel for te routes Fig.. Travel time and eadway for routes and. For eac alternative te wait time is a random variable uniformly distributed over te yellow (ligt coloured) parts of te bars. Pr [ t ] /3 5 5 Wait time of route, t Fig.. Te conditional probability of selecting route as a function of t. H as a taste parameter q wic is interpreted as a disutility (measured in pecuniary measure, i.e., a cost ) for travel: q is te disutility for travel wit, and L q te disutility for travel wit firm. Te taste parameter q over individuals is assumed to be uniformly distributed in te interval q L. Eac traveller as also an ideal departure time, and te difference between te ideal departure (arrival) time and te most convenient actual departure (arrival) time of firm i is denoted t i. Furtermore, te travel time and ticket price for firm i are denoted r i and p i. Wit tese parameters, te total travel cost R i is given by R i ¼ r i þ p i. Te traveller will cose te firm offering te lowest generalised cost: e will cose firm if p þ r þ t þ q < p þ r þ t þ L q and firm oterwise. Te random variables t, t, and q are assumed to be uniformly distributed on [, H ] [, H ] [, L]. Here H j is te eadway of firm j. We can now express te demand for te two firms. Tere is a total coort of X individuals, all of wom will travel wit eiter firm. Te number of travellers coosing firm is Z L Z H Z H X X ¼ LH H p þ r þ t þ L q p þ r þ t þ q i dt dt dq were [x] is again te Heaviside function. Te demand for firm is obviously X ¼ X X Welfare In order to study te welfare effects, we ave to compute te welfare function. In tis case, wit inelastic demand, it is convenient to calculate welfare costs, i.e., we set te travellers utility of travel to zero (tis number will not affect welfare canges) and compute te negative of te welfare. Te total welfare costs for te travellers is: C c ¼ Z X L Z H LH H q i dt dt dq: Z H Te total welfare cost is ten min p þ r þ t þ q; p þ r þ t þ L W c ¼ C c profit of firm profit of firm þ external costs ðpollution; noise; etc:þ 4.. Competition Tere are tree common models of competition in te literature. Te Bertrand model, te Cournot model and te Stackelberg model. In te Bertrand model, eac firm takes te oter firms prices as given (fixed,) and maximise teir own profit under tat assumption by coosing te relevant price. Tat is, eac firm s price is optimal (i.e., profit maximising) given te oter firms prices. Te equilibrium outcome is te Bertrand equilibrium. Te Bertrand competition is very strong; if all firms produce te same good (perfect substitutes,) ten te price is driven down to marginal cost, even wit only two firms in te market. If firms ave different marginal costs, only te firm(s) wit minimum marginal cost(s) will stay in te market. Terefore, one usually assumes tat products are imperfect substitutes, so tat an equilibrium wit prices above marginal cost results. No model, including Stackelberg and Cournot, is perfectly convincing intuitivelydindeed, tey produce differing equilibria,

8 K. Jansson et al. / Researc in Transportation Economics 3 (8) so tey cannot all prevail at te same time. In te literature, it is our impression tat te Bertrand model is te most favoured wen products are differentiated (imperfect substitutes), and since transport modes are eterogeneous, tis is te model we coose in tis study. In order to compute te Bertrand equilibrium values of p, p, H and H, we first consider firm s decision problem, given some (arbitrarily cosen) values of p and H ; e maximises profit: argmax X p ; p ; H ; H i p c C H i H ;p were X is te demand for te given values of parameters, c is te cost per passenger, and C [H ] te cost associated wit te departure interval H. Te outcome produces firm s price and eadway as functions of te corresponding values cosen by firm : p ; H p ; H (4.4) Now te decision problem of firm is similar; e maximises is profit, given p and H : argmax X p ; p ; H ; H i p c C H i : H ;p Te outcome produces firm s price and eadway as functions of te corresponding values cosen by firm : p ; H p ; H : (4.5) Now we ave to find values of p, p, H and H, suc tat bot Eqs. (4.4) and (4.5) are satisfied simultaneously. Tecnically, we do tis by fixed point iteration: we start wit some initial values of p and H, compute te resulting values of p and H according to Eq. (4.4), use tese values in Eq. (4.5) to compute new values of p and H, use tese values in turn to compute new values of p and H employing Eq. (4.4), and so on, iteratively, until te values converge Numerical evaluation and results Tis section contains results obtained by numerical computation of Bertrand equilibrium for competing public transport services, as well as numerical values of minimal welfare cost for suc competitors. Several computations are performed to reflect ow te equilibrium and obtained welfare vary wen fundamental parameters cange. Te basic set of parameter values are cosen to resemble te situation for train and airline operators in Sweden, specifically on te route between Stockolm and Gotenburg, te capital and te second largest city. Te numbers seem reasonable according to information from te Swedis National Rail Administration (Banverket) and literature on airline costs. Te external costs are tose normally applied by Banverket and te Swedis Institute for Transport and Communications Analysis (SIKA). We also use te infrastructure carges applied in Sweden. Te total demand is 4 travellers per day. Te taste parameter (te disutility) is uniformly distributed on te interval (, 5) SEK, i.e., some passengers are willing to pay up to 5 SEK extra to avoid a particular operator. Te parameters of importance are presented in Table below. Table summarises te effect of different settings of te policy variables on te frequencies of te operators. From te table one finds tat a low tax (or ig subsidy) yields operating frequencies for bot operators tat are iger tan wat is optimal from a welfare perspective. Conversely, a ig tax yields too low frequencies. Te infrastructure carge beaves similarly. Te competition wit an airline operator made te train operate at a frequency iger tan optimal regardless of te value of travel time or wait time. For ig values of te travel/wait time te airline operated wit too low frequencies toug. Te basic results are as follows. Since we assume a constant total demand, te welfare is unaffected by te level of prices as long as teir difference is te same. We can tus compute te welfare optimising price difference, but not a level. We find tat under competition, tis price difference is te same as under welfare optimisation, but tat welfare optimising frequencies are below te frequencies tat te operators would coose. Te last result can be explained in terms of negative externalities: wen one firm increases its frequency, it imposes a negative externality on te oter firm, in tat it decreases tat oter firm s demand. Since tis cost is not internalised by te first firm, it cooses a iger frequency tan is optimal Te numerical procedure Te numerical computations are performed as follows. Wen calculating te Bertrand equilibrium, an operator s action and te competitor s reaction are calculated successively until te equilibrium is reaced. Te equilibrium tus obtained is a fix point. In eac step of te iteration, a competitor s reaction is determined by searcing a grid of price and interval pairs for te values tat maximise te profit of te operator. As te algoritm converges, te searc is made on a finer and finer grid, allowing te optimal price and interval to be accurately determined. Te minimal welfare cost is determined by an exaustive searc on a grid of distinct interval lengts for te operators and teir price difference. Te welfare optimum is te particular triple of interval lengts and price difference tat minimise te welfare cost. Te searc is ten performed repeatedly on a finer parameter grid, centered on te previously obtained parameter triple, until te optimum is determined wit sufficient accuracy. Table Assumed fixed and variable parameters. Fixed parameters Costs Distance (km) Travel time () External Costs (SEK/km) Operating (SEK/km) Passengers (SEK/pass) Rail Air Variable parameters Values of time Ride time weigt Infrastructure carge (SEK/km) Demand No of passengers Taste parameter Max (SEK/journey) Travel (SEK/) Wait (SEK/) Rail Air Total 4,

9 38 K. Jansson et al. / Researc in Transportation Economics 3 (8) 3 4 Table An overview of ow different settings of te policy variables affect te frequencies of te operators. Low Hig Train Air Train Air Tax þ þ Infrastructure carge þ þ Value of travel time þ þ þ Value of waiting time þ þ þ Demand þ þ þ Taste/disutility þ þ A þ -sign indicates tat te frequency is iger tan optimal for te operator and parameter setting, conversely a -sign indicates a too low frequency. In tese numerical computations, te eadways, H and H,are not allowed to be greater tan 6, tus bounding te maximal time interval between departures, or putting a lower bound on te frequency of departures. Wen a time interval for a competitor is set to 6 by te computer programme, it usually signifies tat te actual (optimal) interval is greater, possibly infinite Effects of tax/subsidy on departures A subsidy or tax on prices did noting but cange te ticket price wit te corresponding amount. Te operators did not cange teir beaviour and consequently did not come any closer to an optimal welfare cost. Tis is an artifact of two model specifications: () te total demand is inelastic and () te demand for te individual firms depend only on te price difference (Fig. 3). A lowered subsidy or increased tax on departures wen aving two equal (train) operators decreased te frequency but did not affect te ticket price (cf. subsidy or tax on prices). Te profits decreased as te taxes increased. Te optimal welfare cost is not affected by taxes but te obtained welfare is, and te least difference between obtained and optimal welfare costs is acieved wen te tax is SEK/km iger per departure tan te current tax level. For two different operators, a lowered subsidy or increased tax decreases te frequencies. Again, te least difference between obtained and optimal welfare costs is acieved wen te tax is increased by SEK/km per departure from te current tax level Infrastructure carge As an alternative to add taxes/subsidies relative to te current level of taxation, one may subject every mode to an infrastructure carge, equal to te external costs of te mode. Tis makes te modes internalise te external costs. In tis setting, te infrastructure carge is varied around te zero level were te external costs are fully compensated. Canging te infrastructure carge does not affect te ticket price wen aving two equal operators, similar to te beaviour wit tax on departures. Profits decrease wen te carge is increased. Te optimal welfare cost is unaffected by any suc carge, and te minimal difference between obtained and optimal welfare costs is acieved wen tere is a carge around SEK/km and departure. For two different operators, an increased carge decreased te frequencies. Again, te least difference between obtained and optimal welfare costs is acieved wen te carge is around SEK/km per departure Value of travel time Wen aving two equal operators, larger travel time values give iger welfare costs, bot under competition and optimality, but constant difference between tem. Te ticket price and frequency of operation were not affected by te canges in travel time values, and tus, te profits of te operators are constant. Te optimal welfare, aving bot an air and a train operator and travel time values above 6 SEK/, is obtained if te train does not operate. Under competition, te ticket price decreases for train but increases for air operators, as te travel time value increases. Te total cost of travel increases wit larger travel time values. Te difference in total cost of travel between operators is zero wen te travel time value is SEK/. Wen te travel time value is larger tan SEK/ ten te obtained total price difference between te operators is smaller tan te optimal price difference. Te frequency of trains decreases but for fligts increases, as te travel time value increases. Under competition te train operates more frequently tan wat is optimal, but te airline less, at least 9 Difference, SEK Tax per veicle km Fig. 3. Illustration of te difference between te welfare obtained by te profit maximiser and te welfare maximiser for various subsidy levels.

10 K. Jansson et al. / Researc in Transportation Economics 3 (8) for large values of te travel time value. For small travel time values, even te airline is too frequent. Te welfare cost decreases as te travel time value decreases. Te difference between obtained and optimal welfare cost is at a minimum wen te travel time value is 3 SEK/ Value of wait time Generally, wen te wait time value is small, te competitors operate wit a low frequency, i.e., tey ave long intervals between departures. As te wait time value increases, so does te frequency for bot operators. Te welfare cost increases wit te wait time value and also te difference between obtained welfare cost under competition and minimal welfare cost, determined by optimisation. We note tat te intervals obtained under competition are sorter tan te corresponding intervals under welfare optimisation. An analysis of a competition between two equal operators (train services) reveals tat wen te value of te wait time is large, i.e., it is expensive to wait, it is not optimal to ave two operators. Tat is, for wait time values larger tan 3 SEK/, say, it is better in a welfare sense to ave a single firm operate at, more or less, te double frequency. Te analysis of te competition between a train and an airline operator exibits similar beaviour. Wen aving large wait time values (5 SEK/ or more) it is optimal to ave only an airline operator. It is wort noting tat, under optimality, te difference in price between te operators is constantly 65 SEK, irrespective of te wait time value. Te 65 SEK is te difference in travel time value, meaning tat optimally te operators sould ave te same ticket price Demand Te ticket price for two competing train operators decreases as te demand increases, but never falls below 54 SEK. Te frequency increases as te demand increases, and te difference between obtained and optimal welfare cost decreases in absolute numbers. It is wort noting tat for very small demand it is probably not optimal to run any services at all. Wen aving two different operators and a demand in te range from 6 to travellers per day, it is optimal in te welfare cost sense to only run a single airline operator. Te ticket price for te train is constantly decreasing wile te ticket price for te airline company as a minimum wen te demand is around 3 passengers per day. As te demand increases, so does te frequency, and wen it is optimal to ave two operators, bot operate too frequent Taste/disutility Te welfare cost increases as passengers are more willing to pay for avoiding a particular operator. Te difference between obtained and optimal welfare cost decreases wit te willingness to pay. Two equal operators: te frequencies are not affected by a cange in te taste parameter, but te prices increase wen te willingness to pay increases, ence, te profits increase. If te willingness to pay is small, it is optimal to ave only a single operator. Two different operators: te prices (and profits) increase but te difference in frequency diminises; a train increases te number of departures, an airline lowers te frequency. Yet, wen te taste parameter is small te train runs too frequent and te airline too seldom Comfort A measure of comfort is te time value of a mode. Hig time values indicate tat passengers are willing to pay a lot for a reduction of time, i.e., te mode is uncomfortable. Te ratio of time values for airline and train is. in te standard setting; signifying tat travelling by airplane is % more uncomfortable tan by train, in a sense. Varying te ratio between te time values for airline and train sows tat small values (train uncomfortable compared to airplane) give a welfare optimum wit no train, and ig values a welfare optimum wit no airline. Only for values of te ratio in te range [.9,.4] optimality is obtained wit two operators. Wen te airplane is % more uncomfortable tan te train (ratio.), te operators split te demand in two equal parts. 5. Discussion We ave found tat profit maximisng public transport operators sould be subsidised if tey are monopolies, but tey sould be taxed if tey operate in competitive markets. Te fact tat a monopoly carges too ig a price comes as no surprise; maybe less obvious is te fact tat te regulatory instrument is a subsidy upon te pecuniary price te operator cooses, wic makes te operator coose welfare optimal price and frequency as well as transport unit size. We believe tat tis result is reasonably robust to model specifications. Te situation is different under competition, were we find tat supply tend be too ig. If one operator increases te frequency of departures, te competitor will suffer a loss as a consequence of fewer passengers, and tis negative externality causes te equilibrium to sow too ig frequencies. In order to come closer to welfare optimum, a tax per departure is suggested. It is, owever, a common experience tat comparative statics results in competition models can be very sensitive to model specifications, in particular to te mode of competition (competition in prices versus supply,) so furter investigation of models wit oter modes of competition is called for. In some cases, demand is too low to sustain more tan one operator. For instance, if travel time values are ig, people will prefer te fast mode, air instead of rail, for example, and tere may be no room for te slow mode. So, according to our modelling, wen a competitive situation sifts to monopolistic situation, also te optimal policy sifts from taxation to subsidisation. Wat about real-world examples and possible applications? For local and regional transport, te situation were a private profit maximising firm virtually as monopoly, like in many cities in Great Britain, a passenger related subsidy may be considered. On te oter and, in cities were tere is real on-road competition, like in many developing countries and in some places in Great Britain for example, taxation may be considered. Wit respect to long-distance public transport, we can give two examples from Scandinavia. Tere are many long-distance transport markets were competition is fierce. One example is Stockolm centre Gotenburg centre in Sweden were tere are tree airlines ( plus access time), one railway operator (3 ) and several coac operators (6.) Te supply ere may be too large and inefficiently costly. Taxation would reduce te number of departures and maybe also squeeze out some operators. One would acieve resource saving tat increases overall welfare. In many places in te sparsely populated Norway, for example, mountains, islands and fjords make rail-, coac- and car transport very expensive or impossible. Only air transport is efficient. In tese cases, most of te origin destinations ave only one operating airline. Te simple reason is tat more tan one operator could not survive due to te low demand. Consequently prices are very ig and sould be subsidised, wic tey actually also are to some extent. Wit respect to bot urban and long-distance transport were competitive tendering for operation under gross-cost contract is

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