- Email: [email protected]
Production costs, congestion, scope and scale economies in urban bus transportation corridors
Transportation Research Part A 39 (2005) 383–403 www.elsevier.com/locate/tra
Production costs, congestion, scope and scale economies in urban bus transportation corridors J. Enrique Ferna´ndez L. a
a,*
, Joaquı´n de Cea Ch. a, Louis de Grange C.
b,1
Departamento de Ingenierı´a de Transporte, Pontificia Universidad Cato´lica de Chile, Casilla 306, Santiago 22, Chile b Ferna´ndez y De Cea Ingenieros, Apoquindo 3650, Of. 902, Santiago, Chile Received 1 October 2002; received in revised form 31 March 2004; accepted 10 August 2004
Abstract A microeconomic model is developed to study the main characteristics of production costs in urban bus corridors. A multiproduct formulation is used, considering trips during peak and off peak periods as different products. The influence of the demand structure and congestion in the production of trips are considered in the analysis. Production and cost functions are specified using a fix proportion technology. The characteristics of period specific economies of scale, ray scale economies and transray convexity are studied with and without congestion. A numerical example is presented to illustrate the results obtained. 2004 Elsevier Ltd. All rights reserved. Keywords: Bus costs; Scope economies; Scale economies; Transray convexity; Bus congestion
1. Introduction In the transport economics literature most of the authors indicate that scale economies exist in the production of urban bus transportation services when, following Mohring, 1972 users travel time is considered as a production factor (see Mohring, 1972; Jansson, 1980; Kerin, 1992). *
1
Corresponding author. Tel.: +56 2 686 4270; fax: +56 2 553 0281/686 4818. E-mail addresses: [email protected] (J.E. Ferna´ndez L.), [email protected] (L. de Grange C.). Fax: +56 2 435 0099.
0965-8564/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.tra.2004.08.003
384
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
When only factors provided by the bus operator are considered, there is less agreement with respect to scale economies. Tauchen et al. (1983) conclude that scale economies exits only for small firms and low production levels. Then, Wabe and Coles (1975), show that total production costs increase proportionally more than to the firm size. Xu et al. (1994), using road network variables in the definition of the product conclude that there are scale diseconomies for urban bus firms. Recently important modifications to the operation of urban transportation systems are being implemented in various cities of the developing world.2 All of them are based in the implementation of a backbone network of segregated bus corridors. Production cost characteristics and in particular the degree of scale and scope economies are of importance for the design, definition of the production structure and correct regulation of these bus corridors. In this work a microeconomic cost model considering only factors provided by bus operators is developed for the production of trips in bus corridors. Our approach presents some important differences with previous works: we develop multiproduct production and cost models, considering trips during peak and off peak periods as different products. The model considers the influence of the structure of demand over the corridors production capacity. The possibility of congestion in the production of trips is considered for each period. The model is used to analyze the degree of scale and scope economies with and without congestion in the corridor. The influence of the structure of demand served is also analyzed. In Section 2, the main operational characteristics of the service offered and the main variables used in the model are defined. In Section 3 the production function is formulated. Section 4 defines the expenditure function. Section 5, specifies long and short run cost functions and analyzes scale and scope economies with and without congestion. A numerical example is developed in Section 6. The main conclusions are presented in Section 7.
2. Transportation service definition It is considered a bus corridor of length L, with n + 1 bus stops (n transit arcs, see Fig. 1). A fleet of buses operates over the corridor providing transportation services between all pairs of stops. The following variables and parameters are used to represent the service offered by the bus corridor: Y: total number of trips made in the corridor in the analysis period. TV: total travel time of a bus cycle excluding time spent at bus stops. t: time spent by a passenger while boarding and alighting a bus. tc: total time spent by a bus in performing a cycle (bus cycle time). l: average distance traveled by passengers using the corridor. K: bus capacity (maximum number of passengers per bus). k: bus occupancy (number of passengers carried per bus). k 6 K.
2
Curitiba, Bogota, Lima, Quito and Santiago—Chile are some examples.
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403 1
2
n-1
385
n
L
Fig. 1. Bus corridor.
B: total number of buses operating over the corridor (fleet size). f: service frequency (vehicles per unit of time). Given that a bus cycle time tc is equal to the sum of the vehicle travel time over the arcs of the corridor, plus the time spent at the bus stops, it can be written: tY ð1Þ tc ¼ T V þ 0 f where f 0 = B/tc is the frequency offered by a fleet of B buses, operating over the corridor. The frequency demanded over the corridor if Y trips are demanded per period is given by f d = aY/k (see, Appendix A) where a is a parameter representing the demand structure over the corridor. Its value is between 0 and 1.3 Notice that both the values of a and Y are necessary to describe the demand and output of the bus corridor. Using (1) with the definition of the frequency offered it is obtained that: f0 ¼
ðB tY Þ TV
ð2Þ
3. Corridor production function In Ferna´ndez et al. (1999) a production function is derived for abus corridor. Making the fre B aY quency offered equal to the frequency demanded over the corridor ¼ the following basic tc k production relation is obtained for the bus corridor (see Appendix A): Bk ð3Þ Y ¼ aT V þ kt Eq. (3) is similar to relations obtained by other authors but includes parameter a, which reflects the influence of the structure of demand over the corridors production capacity. It expresses the number of trips that will be produced over the corridor depending on the values taken by basic operational variables (B, k, TV, t) and the demand structure (given by a). A more heterogeneous demand will be represented by a higher value of a, which reduces the effectiveness of a fleet of B vehicles to produce trips over the corridor. In other words, to satisfy a more heterogeneous demand a higher fleet is needed. Additionally, from (3) it is obtained that larger vehicles allow 3 In Ferna´ndez et al. (1999) it is shown that a takes the values 0.5 when demands are homogeneous, and 1 when for feeder routes demand (see Appendices A and B).
386
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
carrying more passengers over each bus, which has a positive effect for the number of trips produced. However, at the same time larger buses require more time to board and alight at bus stops, increasing cycle time and reducing the frequency offered. One effect tends to cancel the other and the final result depends on the range of values considered. It is reasonable assuming that a direct relation exists between fleet size (B), number of drivers (H), and terminals size (M) required. This is given by M = m B, where m is the amount of terminal square meters necessary per bus, and H = gB, with g representing the number of drivers needed per bus. Therefore, the production process can be described by a fix coefficients production function (Leontief, 1946; Ferna´ndez et al., 1999). BK HK MK ; ; ð4Þ Y ¼ min aT V þ Kt gðaT V þ KtÞ mðaT V þ KtÞ This function has a zero elasticity of substitution between factors. It is well known that in this case the optimum combination of factors (that minimizes production spending) is the same for any level of production, therefore the production expansion path will be a ray through the origin, in the production factors space: B H M ð5Þ Y ¼ ¼ ¼ a b c Notice that a, b and c, represent the amount of factors (buses, drivers and terminals) necessary to produce a unit of Y (an average trip). Therefore, vehicles, bus drivers and terminals are perfectly complementary factors. If demands for trips are different for different periods during the day, both a and Y will take different values for each period (at, Yt) t = 1, . . . , T. Then the corridor output is given by the vector parameters ^a ¼ ða1 ; . . . ; aT Þ and of trips Yb ¼ ðY 1 ; . . . ; Y T Þ which is directly related to the vector of P the total number of trips produced considering all periods is Y ¼ t Y t . However, fleet and terminals size (an also number of drivers most of the cases) will be determined by the number of trips demanded during the peak period (Y = YP). Considering trips produced in each period as different products, the corridor is multi-product. Notice that in each period it is possible to use a different number of buses and drivers. Therefore, operating conditions can be different for each period and a different travel time can be experienced (T tV ). Finally, if the same buses are used, different bus occupancies (kt) can be experienced in different periods (kt 6 K). 4. Expenditure function To define the total expenditure of the firm operating the bus corridor defined above, the following expenditure items are considered: 1. Management expenses (A). Corresponds to resources spent in the administration of the firm that manages the bus corridor. It includes the salaries and participations of executives and managers and salaries paid to clerks, accountants and auxiliary personnel, general expenses related to buildings used and materials necessary for clerical work. It also includes, vehicles licenses,
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
387
vehicles and drivers insurance and other special permits. Finally, for simplicity, also infrastructure expenditures, or payments for its use are included. Units used are ($/day). These are fix expenditures in the short run. 2. Terminal expenses (pm Æ M). Here are included the expenses necessary to operate the terminal (of size M in square meters) where vehicles are kept and maintained and where the corridor operation is run. It includes expenditures (with generalized price pm, in $/(m2 day)) related to the land and buildings used, salaries of personal dedicated to the maintenance of vehicles and the control of operations and other related general expenditures. It corresponds to fix expenditures in the short run. 3. Vehicles capital expenditures (pb Æ B). Corresponds to the amortization of vehicles and related equipment and its fix maintenance (pb represents bus amortization in $/veh day). They are fix in the short run. 4. Operational expenditures. It includes the expenses in fuel, lubrication, tires and variable maintenance operations (C kb in $/(veh km) for a vehicle of capacity k, with ðoC kb =okÞ > 0) It also includes fuel and maintenance per unit of time spent while vehicles are at bus stops (Cr in $/ veh h). 5. Drivers (ph Æ H). Includes drivers salaries and other expenses relate to drivers contracts and labor legislation; ph is expressed in ($/driver day). In some cases, drivers receive a variable remuneration defined as a given amount p per passenger carried. The expenditure function can be analytically expressed as follows (in $/day): X C ktb Bt Lt N t þ p Y þ C r t Y GTOT ¼ A þ pm M þ pb B þ ph H þ
ð6Þ
t
where Nt is the number of round trips made by each bus during period t. Notice that corridor length Lt is considered to depend on t, because the round trip made by the buses may change for different periods.
5. Bus corridor cost function Following the usual microeconomic procedure, the corridor cost function can be obtained, by minimizing the expenditure necessary to produce a given output. Then the cost function is defined by the following minimization problem: Min :
GTOT ¼ A þ pm M þ pb B þ ph H þ
Yp ¼
C ktb Bt Lt N t þ p Y þ C r t Y
) Bt k t H t kt Mk t ; ; ; Y t ¼ min at T tV þ k t t gðat T tV þ k t tÞ m at T tV þ k t t BK ; þK t
ap T pV
ð7Þ
t
(
st :
X
Bt 6 B
8t 6¼ p
ð8Þ
ð9Þ
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
388
M ¼ m B;
H t ¼ g Bt ;
k t 6 K;
Y ¼
X
Yt
ð10Þ
t
Using the fix relations between factors M and B, and between H and B the expenditure function (7) can be written as X C ktb Bt Lt N t þ ðC r t þ pÞ Y ð11Þ GTOT ¼ A þ ðpm m þ g ph þ pb Þ B þ t
5.1. Short run cost function In the short run expenditures related to administration and terminals, amortization of vehicles and basic remuneration of drivers are fixed. Therefore, short run fix costs are given by F ¼ A þ ðpm m þ g ph þ pb Þ B Notice that the fix expenditure per bus is given by ~pb ¼ ðpm m þ g ph þ pb Þ. The short run expenditure function can be written as X GSR Bt C ktb Lt N t þ ðC r t þ pÞ Y ð12Þ TOT ¼ F þ t
Variable costs correspond to buses operating costs, and drivers variable remuneration. Using the basic production relation (3) to replace Bt the following short run total cost function is obtained: X at T t SR V C TOT ðB; H; M; Yb Þ ¼ F þ Yt þ t C ktb Lt N t þ ðC r t þ pÞ Y ð13Þ k t t Differentiating (13) with respect to Yt4 it is obtained the short run marginal cost per trip of period t: t kt at T V C b Lt N t t þ p þ tðC ktb Lt N t þ C r Þ ð14Þ SRMgC y ¼ kt Notice that Eq. (14) is valid only if Bt 6 B. In general it is assumed that B = Bp that is the fleet size necessary for the peak period t = p. Therefore, if Bt 6 B, there is not capacity to increase the number of trips produced and SRMgC ty ¼ 1. Differentiating (13) with respect to Nt the short run marginal cost per round trip of period t is obtained: t at T V oY t at T tV oY t kt þ t C b Lt þ þ t C ktb Lt N t þ ðC r t þ pÞ ð15Þ SRMgC N ¼ Y t oN t kt oN t kt But
4
oY t ¼ oN , therefore: t t oY t at T V oY t t SRMgC N ¼ Y t þ Nt þ t C ktb Lt þ ðC r t þ pÞ oN t kt oN t
oY oN t
Notice that oY =oY t ¼ 1 because Y ¼
P
Y t.
ð16Þ
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
389
The value of (oYt/oNt) must be estimated experimentally. Given the multiproduct formulation the average cost is not well defined, because different products Yt are obtained in each period t using the same fleet B, drivers H and terminals M. One alternative is to calculate average incremental costs (Baumol et al., 1982) for each period (AICt). AICt ¼
C TOT ðB; H; M; Y 1 ; . . . ; Y t ; . . . Y T Þ C TOT ðB; H ; M; Y 1 ; . . . ; 0; . . . ; Y T Þ Yt
and replacing the expression of CTOT from (13): t at T V AICt ¼ þ t C ktb Lt N t þ ðC r t þ pÞ kt
ð17Þ
ð18Þ
Then, comparing (18) with (14) it is obtained that AICt = SRMgCt. Therefore, there are constant product specific returns to scale to trips produced in period t, St = 1, "t (constant returns). This means that given the fleet of buses (B) the drivers (H) and the terminal (M), trips in period t can be produced incurring only in the corresponding short run marginal costs. A ray average cost (RAC) (Baumol et al., 1982) can also be calculated. For that, a unitary bundle of trips-periods must be defined as unity product. For instance the bundle o1 be arbitrarily assigned the unity value. In general any vector Yb o ¼ ð1=T ; . . . ; 1=T ; . . . ; 1=T Þ could P Yb ¼ ðY o1 ; . . . ; Y ot ; . . . ; Y oT Þ, such that t Y ot ¼ 1 could be used. Then, any amount of output can be o represented making Yb ¼ r Yb where r is the number of units in the bundle. This allows to define the average cost of the composite good Yb as o C TOT ðB; H ; M; Yb Þ C TOT ðB; H; M; r Yb Þ P ¼ RAC ¼ Y tY t
ð19Þ
Notice that given the definition of the unity product bundle Y ¼ r. For the special case in which o1 the unity bundle is Yb , ray average cost is given by
F 1 X a T t o1 t V b RAC r Y þ t C ktb Lt N t þ ðC r t þ pÞ ð20Þ ¼ þ kt Y T t Then ray scale economies can be calculated as C TOT ðB; H; M; Yb Þ A þ ðp m þ g ph þ pb Þ B >1 ¼ 1 þ X t m ST ¼ P 1 at T V Y SRMgC kt t t t Y þ t C b Lt N t þ C r t þ p t T kt
ð21Þ
Notice that in the short run there are ray scale economies, that are produced by the fix costs F. For a given fleet B, ray economies of scale decrease as the total number of trips produced increases and increase as the number. 5.2. Long run cost function In the long run all factors are variable and therefore, B, M, and H are optimized to produce each production vector Yb . Then considering that all factors can be adapted to Yb , we obtain the following expression for the long run total cost function:
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
390
C LR TOT
X a t T t ap T PV V þ t Yp þ ¼ AS ðY Þ þ ðpb þ m pm þ g ph Þ þ t C ktb Lt N t Y t K k t t þ ðC r t þ pÞ Y
ð22Þ
It is assumed that administration costs are a step function of Y . They are constant for some pb ¼ ðpb þ m pm þ g ph Þ will be used. ~pb corresponds to ranges of values of Y . In the notation: ~ a generalized price per bus. It is assumed that buses operate full during the peak period (kp = K). 5.2.1. Costs without congestion Then, if there is not congestion (T tV independent of Yt), differentiating (22) with respect to Yt, the following expression for the long run marginal cost is obtained for period t: ap T pV if t ¼ p ð23Þ þ t ð~ pb þ C kp LRMgC py ¼ b Lp N p Þ þ ðC r t þ pÞ; K t at T V t LRMgC y ¼ þ t C ktb Lt N t þ ðC r t þ pÞ; if t 6¼ p ð24Þ kt Using expressions (24) and (14), the following relations between long and short run marginal costs are obtained: SRMgC ty ¼ LRMgC ty ; LRMgC py
SRMgC py
if 6¼ p
¼
ap T pV þt ~ pb ; K
ð25Þ if t ¼ p
ð26Þ
The value of the difference in (26) is due to the fact that in the long run a marginal increase in the fleet size is necessary to accommodate the marginal increase in the number of trips produced during period p. Average incremental costs (Baumol et al., 1982) for each period (AICt) can be obtained as AICt ¼
C TOT ðY 1 ; . . . ; Y t ; . . . ; Y T Þ C TOT ðY 1 ; . . . ; 0; . . . ; Y T Þ Yt
Replacing the expression of CTOT from (22): ap T pV Y p Y ~t ap T pV AICp ¼ ~ pb L N þ C kp þt þ t þ ðC r t þ pÞ; p p b K Yp K t at T V AICt ¼ þ t C ktb Lt N t þ ðC r t þ pÞ; if t 6¼ p kt
ð27Þ
if t ¼ p
ð28Þ
ð29Þ
Comparing (28) with (23): ap T pV Y ~t þt pb AICp ¼ LRMgC p ~ K Yp
ð30Þ
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
391
Therefore, AICp < LRMgCp. In (30) Y ~t corresponds to the second largest demand period, that will become peak period after period p is eliminated. Comparing (29) with (24), AICt = LRMgCt, "t 5 p. Therefore, there are constant product specific returns to scale for trips produced in all non-peak periods t, St = 1, "t 5 p. This means that, given the fleet of buses (B), drivers (H) and terminal (M), required to operate peak period demands, trips in other period t can be produced incurring in a constant marginal cost. Notice that the average incremental cost for the peak period increases as the difference between peak and off peak demands increases. The minimum value is obtained when all periods have the same demand value (YP = Yt). Then: ap T pV AICp ¼ þ t C kp ð31Þ b Lp N p þ ðC r t þ pÞ; K In this limit case AICp = AICt = LRMgC. Peak period product specific returns to scale are given by5 ap T pV Yp Yt ap T pV ~ þ C kp L N pb þt þ t þ ðC r t þ pÞ p p b K Yp K Sp ¼ at oT tV t þ t ð~ pb þ C kp TV þ Yt b Lp N p Þ þ ðC r t þ pÞ kt oY t ap T pV Yt ~ þt pb K Y pp ð32Þ ¼1 ap T pV a pT V kp ~ þ t þ ðC b Lp N p Þ þ t þ ðC r t þ pÞ pb K K It can be noticed that the second term on the rhs of (32) is positive and less than one, which makes, 0 < Sp < 1. Therefore, there are product specific diseconomies for the peak period. Notice that by definition of peak period, Y p > Y ~t therefore as peak demand increases, keeping constant other periods demands, the degree of diseconomies is reduced, because the values of average and marginal costs become closer (see (30)) and Sp ! 1. The same happens as off peak demands are reduced. This is because the peak period average incremental cost, due to the need of additional buses, drivers and terminal space, increases when the difference between peak and off peak demands increases, making the value of AICp closer to the value of the LRMgCp. On the contrary, if all periods have the same demand, the size of the fleet, drivers and terminal space is equally justified by all the periods; then average incremental costs take their minimum value corresponding only to variable operating costs. the short run case, by defining Long run ray average cost (RACLR) can be obtained, similarly P to o o o o o b a unity product bundle Y ¼ ðY 1 ; . . . ; Y t ; . . . ; Y T Þ such that t Y t ¼ 1,
o b C r Y b TOT C TOT ð Y Þ ¼ RACLR ¼ P ð33Þ Y tY t 5 For the particular case in which all periods have the same demand this expression is not valid because there is not peak period. In such case all periods present constant product specific returns to scale.
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
392
o1 For the special case in which the unity bundle is Yb (see short run section) we obtain: " # X t
A ðY Þ 1 P o1 a T a T S p t kt V V pb þt þ þ t C b Lt N t þ ðC r t þ pÞ RACLR r Yb þ ~ ¼ T K kt Y t
ð34Þ Then long run ray scale economies will be given by As >1 X t S LR T ¼ 1þ P 1 ap T V at T V kt ~ Y þ t C b Lt N t þ C r t þ p p þt þ t T b K kt
ð35Þ
Therefore in the long run multi-product economies of scale are only generated by the fix administration expenses and should therefore be mild in magnitude.6 In addition it can be observed from (35) that: ray scale economies are reduced when the level of total production Y increases, and the contrary happens when the same total of trips production is distributed in a larger number of periods T. Non-uniform demand structures served by the corridor tend to reduce de degree of ray economies. For the case of uniform demands, scale economies increase when the trips produced are short with respect to the length of the corridor (l/L is small). If both peak period p and any other period t are considered, ray scale economies can be expressed as (Baumol et al., 1982; Jara-dı´az, 1983): ST ¼
/S p þ ð1 /ÞS t ; 1 SS
where S S ¼
CðY p Þ þ CðY t Þ CðY p ; Y t Þ CðY p ; Y t Þ
ð36Þ Y ðoCðY Þ=oY Þ
p p p corresponds to the degree of scope economies (Baumol et al., 1982), / ¼ Y t ðoCðY t Þ=oY t ÞþY p ðoCðY p Þ=oY p Þ
and Sp, St represent the economies of scale specific to periods p and t. Considering that, as was seen above, St = 1 and using Sp from (32) in (36), it is obtained that: 8 9 ap T pV Yt > > > > ~pb þt < = K Y p / 1 þ ð1 /Þ p ap T V > > > > : ; þt ~ L N þ ðC t þ pÞ pb þ C kp p p r b K ST ¼ ð37Þ 1 SS and rearranging (37): 8 9 ap T pV Yt > > > > ~ pb þt = 1 < K Yp 1 / SS ¼ 1 >0 > ap T pV ST > kp > > : þ t ð~ pb þ C b Lp N p Þ þ ðC r t þ pÞ; K
6
Cost studies made in Chile show that administration costs are around 10% of total costs.
ð38Þ
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
393
This indicates the existence of scope economies or sub-additivity if the second term in (38) is lower than 1. It is shipper to produce trips in different periods with the same fleet, than separating the same production in different fleets. Notice that the scope economies are consequence both of the ray scale economies (ST > 1), and the peak period diseconomies (SP < 1). However in (38) the effect of Sp is reduced by the presence of parameter /7 (notice that 0 < / < 1). If (Yt/Yp) ! 0, / ! 1 and SP ! 1 (see (32)); therefore, SS ! (1 1/ST) and scope economies are only related to existence of ray scale economies (ST > 1). On the contrary, if (Yt/Yp) ! 1, / < 1 and 1 1 S P ! S P < 1; then, S S ¼ 1 ½1 /ð1 S P Þ and economies of scope get their maximum value influenced both by ray scale economies and peak specific diseconomies. 5.2.2. Costs with congestion Assuming that some periods experience congestion (at least the peak period p) travel times will depend on the number of trips produced during the period: oT tV =oY t > 0, (see Appendix B). However notice that the expression for average incremental costs is the same as in the case without congestion (see (28) and (29)). The only difference is that the value taken by T tV will be higher if there is congestion. Notice that ray average costs have also the same expression as in the uncongested case (see (34)). Differentiating (22) with respect to Yt, the following expression for the long run marginal cost is obtained: ap oT pV p p þ t ð~pb þ C kp if t ¼ p ð39Þ LRMgCC y ¼ TV þ Yp b Lp N p Þ þ ðC r t þ pÞ; K oY p at oT tV t t LRMgCC y ¼ TV þ Yt ð40Þ þ t C ktb Lt N t þ ðC r t þ pÞ; if t 6¼ p kt oY t Then, period specific economies of scale with congestion are given by ap T pV Yp Yt ap T pV kp ~ þt þ t þ ðC r t þ pÞ þ C b Lp N p pb K Yp K ; S CP ¼ at oT tV kp t TV þ Yt þ t ð~ pb þ C b Lp N p Þ þ ðC r t þ pÞ kt oY t
if t ¼ p
ð41Þ
Comparing (41) with (32) it is clear that, as a result of including congestion ðoT PV =oY P > 0Þ peak specific diseconomies increase with congestion, S CP < S P < 1. For non-peak periods it is obtained: t at T V kt C b Lt N t þ t þ ðC r t þ pÞ kt S Ct ¼ < 1; if t 6¼ p; ð42Þ at oT tV kt t þ t þ ðC r t þ pÞ TV þ Yt C b Lt N t kt oY t In this case, as a consequence of congestion, non-peak periods also present period specific scale diseconomies S Ct < 1. As congestion increases, the denominators value also increases and period diseconomies become stronger (the values of Sp and St go to zero). 7
Expression (38) is equivalent to: S S ¼ 1 S1T ½1 /ð1 S P Þ .
394
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
Using (22), (39) and (40) in (21), the following expression for ray economies of scale is obtained: P at T tV ap T PV þ t Yp þ t AS ðY Þ þ ~ pb þ t C ktb Lt N t Y t þ ðC r t þ pÞY K k t S LR ð43Þ T ¼ P at T tV ap T PV kt þ t Yp þ t hðY Þ þ ~ pb þ t C b Lt N t Y t þ ðC r t þ pÞY K kt " # ap oT pV 2 X at oT tV kt 2 ð44Þ hðY Þ ¼ ~ Y þ C Lt N t Y t pb K oY p p k t oY t b t It can be noticed that hðY Þ takes into account the effect of congestion externalities and its value will be 0 without congestion, but will increase to any positive value depending on the level of congestion experienced. Therefore, congestion will have an important influence in the type of scale economies obtained. The following alternative cases can be experienced: S LR T > 1 () h < A
ð45Þ
S LR T ¼ 1 () h ¼ A
ð46Þ
S LR T < 1 () h > A
ð47Þ
Therefore, if buses experience a level of congestion such that h > A, there will be ray diseconoLR mies of scale,8 S LR T < 1. Ian indefinite increase of congestion will make that S T ! 0. Using (22) with the definition of economies of scope given in (36), the following expression is obtained, assuming only two periods t and p: t at T V AS ðY t Þ þ ðpb þ mpm þ gph Þ þ t Yt kt ð48Þ SS ¼ at T tV ap T PV P kt kp þ t Y p þ t C b Lt N t þ C r t þ p t AS ðY p Þ þ ~ p b þ C b Lp N p þ t Yt K kt From (48) it is easy to see that, independent of the levels of congestion experienced in the corridor, economies of scope will be always within the range: 0 < Ss < 1,. However, the value of Ss will change within the range indicated as a consequence of changes in the levels of congestion. First it can be observed that Ss ! 0 when Yt ! 0. Therefore in general, the magnitude of economies of scope will increase as the number of off peak trips increases as was also seen in the analysis of the uncongested case. An increase of congestion only during the peak period will reduce the degree of scope economies. On the contrary if congestion increases only during off peak period scope economies will increase. However the maximum, value that Ss can take will always be significantly lower than one, because T tV ¼ T tV ðY t Þ and by definition YP > Yt. Therefore, for a given value of YP the increase in the value of Yt is limited by YP. 8 This can explain why in developing countries where buses operate over congested infrastructure, operators are small. In the case of Santiago de Chile most bus operators own between one and two buses.
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
395
6. Numerical example A BPR type function is assumed to explain the congestion effect produced by the flow of buses over the travel time T tV ðY t Þ, mainly due to congestion in bus stops: c at Y t t TV ¼ T0 1 þ b ð49Þ k t CK where ft = (atYt/kt) is the flow of buses (equal to the frequency in (bus/h)) running over the corridor in period t, CK is the corridor nominal capacity in (bus/h), T0 is the travel time over the corridor under free flow conditions and b, c are calibration parameters. Parameter values assumed are given in Table 1. Basic relations for operational variables (fleet size (B), passengers demands (Y) and travel times (T)) are presented in Appendix B. Cost function parameters used are given in Table 2. Cost units considered are $US/month. Two different operating periods are considered: peak (p) and off-peak (t). Therefore, the product vector is Y = (Yp, Yt). Based on the values assumed and using expression (23) the multiproduct cost function shown in Fig. 2 is obtained. An specific ray ‘‘r’’ is represented in the figure to show ray costs and economies of scale.
Table 1 BPR and production function parameter value Parameter
Units
Value
t at, ap k t , kp , K T0 b CK c
s – Pas/Bus h – Bus/h –
8.0 0.5 25 1.0 1.0 300 5.0
Table 2 Cost function parameter values Parameter
Units
Value
A pm m pb ph H p kp kt C Pb , C b Lt N t Cr
$US/bus month $US/m2 month m2/bus $US/bus month $US/driver month drivers/bus $US/pass $US/bus-km km/month $US/bus h
373.0 8.4 24.0 741.0 127.4 2.0 0.14 0.23 9615.0 0.0
396
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403 C(yp,yt)
Yp1
n
Yp2
Yp
Yp
Yt1 r
Yt2 Yt n
Yt
Fig. 2. Long run total costs for the corridor.
Considering the parameter values assumed in Table 1, congestion starts in the corridor when the number of trips, increases over 8000 trips per hour,9 as can be observed in Fig. B.4 of Appendix B. The form of the cost surface shown in Fig. 2, suggests that there is transray convexity, which implies cost complementarity or economies of scope. In order to analyze in more detail the influence of congestion on these properties, some specific situations are analyzed in more detail. The demand values assumed for each of the six cases analyzed and the corresponding total cost values obtained are presented in Table 3. Two different production rays, identified by A and B in Table 3 and Fig. 3, are considered. Ray A is defined by a ratio (Yp/Yt = 3) and (Yp/Yt = 1) for ray B. For each ray, three levels of production (1, 2, 3) are considered, obtaining the six different cases defined in Table 3 and graphically represented in Fig. 3. Table 4 shows the corresponding values obtained for the index of economies of scale and scope for each of the cases analyzed. The results shown in Fig. 3 and Table 4 corroborate the results obtained in the previous section with respect to the influence of congestion on the degree of scale and scope economies. It can be observed that the values of ST decrease as congestion increases both for rays A and B; therefore an increase in the level of congestion always produces ray diseconomies of scale. However the values of SS decrease with congestion for ray A and increase for ray B. An increase of congestion 9 This value could change depending on the period if vehicle occupancy rate differs for peak and off peak periods. In this example we assume the same occupancy rate for both periods (see Table 1).
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
397
Table 3 Costs values $US/month Case
(Yp; 0)
(0; Yt)
(Y p ; Y t )
C(Yp; 0)
CðY p ; Y t Þ
C(0; Yt)
A-1 A-2 A-3 B-1 B-2 B-3
(4490; 0) (12,797; 0) (16,945; 0) (4490; 0) (12,797; 0) (16,945; 0)
(0; (0; (0; (0; (0; (0;
(2245.1; 748.3) (6398.3; 2132.7) (8472.5; 2,824.2) (2245; 2245) (6398; 6398) (8472; 8472)
364,900 1,474,246 3,728,291 364,900 1,474,246 3,728,291
216,685 625,045 837,463 287,350 826,432 1,104,134
82,619 235,438 311,710 364,900 1,474,246 3,728,291
C(0;Yt)
1497) 4266) 5648) 4490) 12,797) 16,945)
C(Yp;Yt )
C(Yp;0)
C(0;Yt)
C(Yp;Yt )
C(Yp;0)
3
2 3
2
1
1
Yt
Yt
3
(Yt;Yp ) 2
(Yt;Yp ) 1 3 2 1
Yp
Yp
A
B
Fig. 3. Transray convexity characteristics. Table 4 Economies level Ray
Case
Sp
St
ST
SS
A
1 2 3
0.66979046 0.27252797 0.16273242
0.99995054 0.99078760 0.96352638
0.69203244 0.28539983 0.17256263
0.0274 0.0203 0.0113
B
1 2 3
0.03190389 0.03155009 0.01283727
0.03190389 0.03155009 0.01283727
0.0433479 0.04292787 0.02208111
0.2640 0.2650 0.4186
398
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
only during the peak period reduces the degree of scope economies (ray A). On the contrary, if congestion increases also during off peak period scope economies increase (ray B). Therefore, an increase in congestion during off peak makes increase transray convexity or economies of scope.
7. Conclusions Multimodal bus corridor production and cost functions are developed and used to analyze the degree of scale and scope economies. An important feature of the cost functions is the multiproduct character of the trips production process because of different operating periods. Then, many of the corridor cost characteristics depend on the product expansion path in the trips-period space. This is related to the definition of the unitary production bundle. It was found that in the short run the existence of fixed costs produce ray scale economies. For a given fleet B he degree of these ray scale economies will increase with the magnitude of fixed costs F and will decrease when the level of total production Y increases. In the long run economies of scale are produced only by administration fix expenditures, which also can increase more than proportionally than production after some production level. Without congestion, there are constant product specific returns to scale for trips produced in all non-peak periods. This means that, given the fleet of buses (B), drivers (H) and terminal (M), required to operate peak period demands, trips in other period t can be produced incurring in a constant marginal cost. However, there are product specific diseconomies for the peak period. As peak demand increases, keeping constant other periods demands, the degree of diseconomies is reduced, because the values of average and marginal costs become closer. The same happens as off peak demands are reduced. This is because the amount of buses, drivers and terminal saved to serve off peak periods is reduced as the difference between peak and off peak demands increases. With respect to cost complementarity, there are always (with and without congestion) scope economies or sub-additivity in the production of trips for different periods. It is shipper to produce trips in different periods with the same fleet, than separating the same production in different fleets. These scope economies are consequence both of the ray scale economies and the peak period diseconomies. The degree of scope economies is reduced when the difference between trips produced during peak and non-peak period is large. On the contrary, when the number of trips produced in different periods is similar economies of scope increase. With congestion, period specific diseconomies increase and ray scale economies will turn into diseconomies, beyond some level of production. However, congestion can make increase or decrease the scope economies depending of the period in which it is experienced. Congestion increases during peak period makes economies of scope to decrease but congestion increases during off peak produce economies of scope increases. The value of parameters as length of the corridor, vehicles generalized price and demand structure directly influence the magnitude of diseconomies when congestion appears. As the value of this parameters increases the magnitude of the negative effects of congestion increases. On the contrary, an increase in the bus capacity reduces the negative effect of congestion; however this is counterbalanced by the increase in operating costs.
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
399
Appendix A. Influence of the demand structure on the production of trips Three different demand structures are considered, representing different trip distributions between transit stops over the route. P It is assumed that y is the umber of trips made between each pair of stops in the corridor ðY ¼ yÞ. • Uniform demand: corresponds to a set of trips that produces a uniform load of passenger flows over the route (Fig. A.1). This the simplest case usually assumed in this type of analysis. • Homogeneous demand: assumes a homogeneous distribution of trips among all pairs of transit stops (Fig. A.2). Some peripheral routes experience this type of demand in cities of developing countries. • Feeder routes demand: it is typical for feeder routes or routes going to CBD. Passengers boarding the service have a common destination at the end of the route (Fig. A.3). The number of passengers on the service increases as it approaches its final stop.
A.1. Uniform demand In this case, the passengers load is the same over all the corridor and equal to Y Ll . Therefore, the frequency required to transport all the demanded trips is equal to f d ¼ Yl=L . On the other hand, the k frequency that can be offered over the route served with the available fleet, is given by f 0 = B/tc, where the total cycle time tc is equal to the travel time necessary to cover the route, Tv, plus the time spent at the transit stops: tc = Tv + tY/f 0. Then, we can obtain the following expression for the total cycle time tc = BTv/(B tY). To obtain an equilibrium solution between service supply
1
3
2
y
y
n-1
y
y
n
y
Fig. A.1. Example of uniform demand.
y y 1 y
2
3
y
y
4 y y
y y
y
Fig. A.2. Example of homogeneous demand.
400
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403 y
1
3
2
4 y
y y
Fig. A.3. Example of feeder routes demand.
and demand we must equal the frequency supplied with the frequency demanded: f 0 = f d. Then, the number of trips that can be produced, per unit of time, under the above defined conditions is given by Y ¼ Bk= Ll T v þ tk with a ¼ Ll . This expression can be interpreted as the basic production relation for the operator of the service. It provides the total number of trips that can be produced with a fleet of B buses operating over a route of length L, each of which carry k passengers, with a travel time Tv and serving passengers with trip lengths l, when demand is uniformly distributed. A.2. Homogeneous demand In this case, the total number of trips demanded per unit of time is equal to the sum of trips demanded between each pair of stops (see Fig. A.2). Assuming that the number of trips demanded y. Given the homogeneous disbetween each pair (i, j) of stops is the same, we obtain Y ¼ nðnþ1Þ 2 tribution of demand in Fig. A.2, a different passenger load will be obtained for each link of the route. Therefore, the capacity constraint will be active only for the link (or links) with the maximum load of passengers, that in this case will be located at the center of the route. Imposing the constraint that the solution obtained must satisfy the capacity constraint of the service B/tc = T v Y/2k, and replacing the expression for tc it is obtained that: Y ¼ Bk= 2 þ kt where a ¼ 12. A.3. Feeder routes demand In this last case, the capacity constraint is active only in the last link of the route (see Fig. A.3). This will have a passengers load of ymax = ny. This value coincides also with the total number of trips demanded Y. Therefore, in a similar way we can obtain that the number of trips that can be produced is given by Y = Bk/(Tv + kt), where a = 1.
Appendix B. Basic operating relations for the bus corridor In this appendix the operational relations used to develop the numerical example included in the paper are presented. Replacing in (49) values given in Table 1 the relation shown in Fig. B.1 is obtained. It gives the values of travel time T tV (in h) on the corridor as a function of flow 5 ft (in buses/h) T tV ¼ 1 þ ðft =300Þ .
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
401
Fig. B.1. BPR function values for the corridor travel time.
Fig. B.2. Fleet size (B) vs. corridor flow (f).
Using equation (Bt = ft Æ tc) the relation between fleet size Bt and corridor flow ft shown in Fig. B.2, can be obtained. It can also be derived the relation between fleet size and travel time T tV shown in Fig. B.3. Using the previous relations it is obtained the relation between the trips produced over the corridor Yt and the travel time T tV , that is shown in Fig. B.4. This relation can be used to calculate the derivatives ðoT tV =oY t Þ in (42).
402
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
Fig. B.3. Fleet size (B) vs. travel time (Tv).
Fig. B.4. Travel time (Tv) vs. corridor trips produced (Y).
From the form of relations shown in Figs. B.3 and B.4 it is clear that o2 T tV > 0. oY 2
oT tV oBt
> 0,
oT tV oY t
> 0, and
t
References Baumol, W.J., Panzar, J.C., Willig, R.D., 1982. Contestable Markets and the Theory of Industry Structure. Harcourt Brace Jovanovich Inc.
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
403
Ferna´ndez, J.E., de Cea, J., de Grange, L., 1999. Caracterı´sticas de produccio´n de un servicio de transporte pu´blico de pasajeros. Actas del IX Congreso Chileno de Ingenierı´a de Transporte, pp. 431–448. Jansson, J.O., 1980. A simple bus line model for optimization of service frequency and bus size. Journal of Transportation Economic and Policy 14, 53–80. Jara-dı´az, S., 1983. Freight Transportation Multioutput Analysis. Transportation Research-A 17A, 429–438. Kerin, P., 1992. Efficient bus fares. Transport Reviews 12, 33–48. Leontief, W., 1946. The pure theory of the guaranteed annual wage contract. Journal of Political Economy. Mohring, H., 1972. Optimization and scale economies in urban bus transportation. American Economic Review 62, 591–604. Tauchen, H., Fravel, F., Gilbert, G., 1983. Cost structure of the intercity bus industry. Journal of Transportation Economic and Policy 17, 25–48. Wabe, S.J., Coles, O.B., 1975. The short and long run cost of bus transport in urban areas. Journal of Transportation Economic and Policy 9, 127–140. Xu, K., Windle, R., Grim, C., Corsi, T., 1994. Re-evaluting returns to scale in transport. Journal of Transportation Economic and Policy (September), 275–286.
Production costs, congestion, scope and scale economies in urban bus transportation corridors J. Enrique Ferna´ndez L. a
a,*
, Joaquı´n de Cea Ch. a, Louis de Grange C.
b,1
Departamento de Ingenierı´a de Transporte, Pontificia Universidad Cato´lica de Chile, Casilla 306, Santiago 22, Chile b Ferna´ndez y De Cea Ingenieros, Apoquindo 3650, Of. 902, Santiago, Chile Received 1 October 2002; received in revised form 31 March 2004; accepted 10 August 2004
Abstract A microeconomic model is developed to study the main characteristics of production costs in urban bus corridors. A multiproduct formulation is used, considering trips during peak and off peak periods as different products. The influence of the demand structure and congestion in the production of trips are considered in the analysis. Production and cost functions are specified using a fix proportion technology. The characteristics of period specific economies of scale, ray scale economies and transray convexity are studied with and without congestion. A numerical example is presented to illustrate the results obtained. 2004 Elsevier Ltd. All rights reserved. Keywords: Bus costs; Scope economies; Scale economies; Transray convexity; Bus congestion
1. Introduction In the transport economics literature most of the authors indicate that scale economies exist in the production of urban bus transportation services when, following Mohring, 1972 users travel time is considered as a production factor (see Mohring, 1972; Jansson, 1980; Kerin, 1992). *
1
Corresponding author. Tel.: +56 2 686 4270; fax: +56 2 553 0281/686 4818. E-mail addresses: [email protected] (J.E. Ferna´ndez L.), [email protected] (L. de Grange C.). Fax: +56 2 435 0099.
0965-8564/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.tra.2004.08.003
384
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
When only factors provided by the bus operator are considered, there is less agreement with respect to scale economies. Tauchen et al. (1983) conclude that scale economies exits only for small firms and low production levels. Then, Wabe and Coles (1975), show that total production costs increase proportionally more than to the firm size. Xu et al. (1994), using road network variables in the definition of the product conclude that there are scale diseconomies for urban bus firms. Recently important modifications to the operation of urban transportation systems are being implemented in various cities of the developing world.2 All of them are based in the implementation of a backbone network of segregated bus corridors. Production cost characteristics and in particular the degree of scale and scope economies are of importance for the design, definition of the production structure and correct regulation of these bus corridors. In this work a microeconomic cost model considering only factors provided by bus operators is developed for the production of trips in bus corridors. Our approach presents some important differences with previous works: we develop multiproduct production and cost models, considering trips during peak and off peak periods as different products. The model considers the influence of the structure of demand over the corridors production capacity. The possibility of congestion in the production of trips is considered for each period. The model is used to analyze the degree of scale and scope economies with and without congestion in the corridor. The influence of the structure of demand served is also analyzed. In Section 2, the main operational characteristics of the service offered and the main variables used in the model are defined. In Section 3 the production function is formulated. Section 4 defines the expenditure function. Section 5, specifies long and short run cost functions and analyzes scale and scope economies with and without congestion. A numerical example is developed in Section 6. The main conclusions are presented in Section 7.
2. Transportation service definition It is considered a bus corridor of length L, with n + 1 bus stops (n transit arcs, see Fig. 1). A fleet of buses operates over the corridor providing transportation services between all pairs of stops. The following variables and parameters are used to represent the service offered by the bus corridor: Y: total number of trips made in the corridor in the analysis period. TV: total travel time of a bus cycle excluding time spent at bus stops. t: time spent by a passenger while boarding and alighting a bus. tc: total time spent by a bus in performing a cycle (bus cycle time). l: average distance traveled by passengers using the corridor. K: bus capacity (maximum number of passengers per bus). k: bus occupancy (number of passengers carried per bus). k 6 K.
2
Curitiba, Bogota, Lima, Quito and Santiago—Chile are some examples.
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403 1
2
n-1
385
n
L
Fig. 1. Bus corridor.
B: total number of buses operating over the corridor (fleet size). f: service frequency (vehicles per unit of time). Given that a bus cycle time tc is equal to the sum of the vehicle travel time over the arcs of the corridor, plus the time spent at the bus stops, it can be written: tY ð1Þ tc ¼ T V þ 0 f where f 0 = B/tc is the frequency offered by a fleet of B buses, operating over the corridor. The frequency demanded over the corridor if Y trips are demanded per period is given by f d = aY/k (see, Appendix A) where a is a parameter representing the demand structure over the corridor. Its value is between 0 and 1.3 Notice that both the values of a and Y are necessary to describe the demand and output of the bus corridor. Using (1) with the definition of the frequency offered it is obtained that: f0 ¼
ðB tY Þ TV
ð2Þ
3. Corridor production function In Ferna´ndez et al. (1999) a production function is derived for abus corridor. Making the fre B aY quency offered equal to the frequency demanded over the corridor ¼ the following basic tc k production relation is obtained for the bus corridor (see Appendix A): Bk ð3Þ Y ¼ aT V þ kt Eq. (3) is similar to relations obtained by other authors but includes parameter a, which reflects the influence of the structure of demand over the corridors production capacity. It expresses the number of trips that will be produced over the corridor depending on the values taken by basic operational variables (B, k, TV, t) and the demand structure (given by a). A more heterogeneous demand will be represented by a higher value of a, which reduces the effectiveness of a fleet of B vehicles to produce trips over the corridor. In other words, to satisfy a more heterogeneous demand a higher fleet is needed. Additionally, from (3) it is obtained that larger vehicles allow 3 In Ferna´ndez et al. (1999) it is shown that a takes the values 0.5 when demands are homogeneous, and 1 when for feeder routes demand (see Appendices A and B).
386
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
carrying more passengers over each bus, which has a positive effect for the number of trips produced. However, at the same time larger buses require more time to board and alight at bus stops, increasing cycle time and reducing the frequency offered. One effect tends to cancel the other and the final result depends on the range of values considered. It is reasonable assuming that a direct relation exists between fleet size (B), number of drivers (H), and terminals size (M) required. This is given by M = m B, where m is the amount of terminal square meters necessary per bus, and H = gB, with g representing the number of drivers needed per bus. Therefore, the production process can be described by a fix coefficients production function (Leontief, 1946; Ferna´ndez et al., 1999). BK HK MK ; ; ð4Þ Y ¼ min aT V þ Kt gðaT V þ KtÞ mðaT V þ KtÞ This function has a zero elasticity of substitution between factors. It is well known that in this case the optimum combination of factors (that minimizes production spending) is the same for any level of production, therefore the production expansion path will be a ray through the origin, in the production factors space: B H M ð5Þ Y ¼ ¼ ¼ a b c Notice that a, b and c, represent the amount of factors (buses, drivers and terminals) necessary to produce a unit of Y (an average trip). Therefore, vehicles, bus drivers and terminals are perfectly complementary factors. If demands for trips are different for different periods during the day, both a and Y will take different values for each period (at, Yt) t = 1, . . . , T. Then the corridor output is given by the vector parameters ^a ¼ ða1 ; . . . ; aT Þ and of trips Yb ¼ ðY 1 ; . . . ; Y T Þ which is directly related to the vector of P the total number of trips produced considering all periods is Y ¼ t Y t . However, fleet and terminals size (an also number of drivers most of the cases) will be determined by the number of trips demanded during the peak period (Y = YP). Considering trips produced in each period as different products, the corridor is multi-product. Notice that in each period it is possible to use a different number of buses and drivers. Therefore, operating conditions can be different for each period and a different travel time can be experienced (T tV ). Finally, if the same buses are used, different bus occupancies (kt) can be experienced in different periods (kt 6 K). 4. Expenditure function To define the total expenditure of the firm operating the bus corridor defined above, the following expenditure items are considered: 1. Management expenses (A). Corresponds to resources spent in the administration of the firm that manages the bus corridor. It includes the salaries and participations of executives and managers and salaries paid to clerks, accountants and auxiliary personnel, general expenses related to buildings used and materials necessary for clerical work. It also includes, vehicles licenses,
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
387
vehicles and drivers insurance and other special permits. Finally, for simplicity, also infrastructure expenditures, or payments for its use are included. Units used are ($/day). These are fix expenditures in the short run. 2. Terminal expenses (pm Æ M). Here are included the expenses necessary to operate the terminal (of size M in square meters) where vehicles are kept and maintained and where the corridor operation is run. It includes expenditures (with generalized price pm, in $/(m2 day)) related to the land and buildings used, salaries of personal dedicated to the maintenance of vehicles and the control of operations and other related general expenditures. It corresponds to fix expenditures in the short run. 3. Vehicles capital expenditures (pb Æ B). Corresponds to the amortization of vehicles and related equipment and its fix maintenance (pb represents bus amortization in $/veh day). They are fix in the short run. 4. Operational expenditures. It includes the expenses in fuel, lubrication, tires and variable maintenance operations (C kb in $/(veh km) for a vehicle of capacity k, with ðoC kb =okÞ > 0) It also includes fuel and maintenance per unit of time spent while vehicles are at bus stops (Cr in $/ veh h). 5. Drivers (ph Æ H). Includes drivers salaries and other expenses relate to drivers contracts and labor legislation; ph is expressed in ($/driver day). In some cases, drivers receive a variable remuneration defined as a given amount p per passenger carried. The expenditure function can be analytically expressed as follows (in $/day): X C ktb Bt Lt N t þ p Y þ C r t Y GTOT ¼ A þ pm M þ pb B þ ph H þ
ð6Þ
t
where Nt is the number of round trips made by each bus during period t. Notice that corridor length Lt is considered to depend on t, because the round trip made by the buses may change for different periods.
5. Bus corridor cost function Following the usual microeconomic procedure, the corridor cost function can be obtained, by minimizing the expenditure necessary to produce a given output. Then the cost function is defined by the following minimization problem: Min :
GTOT ¼ A þ pm M þ pb B þ ph H þ
Yp ¼
C ktb Bt Lt N t þ p Y þ C r t Y
) Bt k t H t kt Mk t ; ; ; Y t ¼ min at T tV þ k t t gðat T tV þ k t tÞ m at T tV þ k t t BK ; þK t
ap T pV
ð7Þ
t
(
st :
X
Bt 6 B
8t 6¼ p
ð8Þ
ð9Þ
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
388
M ¼ m B;
H t ¼ g Bt ;
k t 6 K;
Y ¼
X
Yt
ð10Þ
t
Using the fix relations between factors M and B, and between H and B the expenditure function (7) can be written as X C ktb Bt Lt N t þ ðC r t þ pÞ Y ð11Þ GTOT ¼ A þ ðpm m þ g ph þ pb Þ B þ t
5.1. Short run cost function In the short run expenditures related to administration and terminals, amortization of vehicles and basic remuneration of drivers are fixed. Therefore, short run fix costs are given by F ¼ A þ ðpm m þ g ph þ pb Þ B Notice that the fix expenditure per bus is given by ~pb ¼ ðpm m þ g ph þ pb Þ. The short run expenditure function can be written as X GSR Bt C ktb Lt N t þ ðC r t þ pÞ Y ð12Þ TOT ¼ F þ t
Variable costs correspond to buses operating costs, and drivers variable remuneration. Using the basic production relation (3) to replace Bt the following short run total cost function is obtained: X at T t SR V C TOT ðB; H; M; Yb Þ ¼ F þ Yt þ t C ktb Lt N t þ ðC r t þ pÞ Y ð13Þ k t t Differentiating (13) with respect to Yt4 it is obtained the short run marginal cost per trip of period t: t kt at T V C b Lt N t t þ p þ tðC ktb Lt N t þ C r Þ ð14Þ SRMgC y ¼ kt Notice that Eq. (14) is valid only if Bt 6 B. In general it is assumed that B = Bp that is the fleet size necessary for the peak period t = p. Therefore, if Bt 6 B, there is not capacity to increase the number of trips produced and SRMgC ty ¼ 1. Differentiating (13) with respect to Nt the short run marginal cost per round trip of period t is obtained: t at T V oY t at T tV oY t kt þ t C b Lt þ þ t C ktb Lt N t þ ðC r t þ pÞ ð15Þ SRMgC N ¼ Y t oN t kt oN t kt But
4
oY t ¼ oN , therefore: t t oY t at T V oY t t SRMgC N ¼ Y t þ Nt þ t C ktb Lt þ ðC r t þ pÞ oN t kt oN t
oY oN t
Notice that oY =oY t ¼ 1 because Y ¼
P
Y t.
ð16Þ
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
389
The value of (oYt/oNt) must be estimated experimentally. Given the multiproduct formulation the average cost is not well defined, because different products Yt are obtained in each period t using the same fleet B, drivers H and terminals M. One alternative is to calculate average incremental costs (Baumol et al., 1982) for each period (AICt). AICt ¼
C TOT ðB; H; M; Y 1 ; . . . ; Y t ; . . . Y T Þ C TOT ðB; H ; M; Y 1 ; . . . ; 0; . . . ; Y T Þ Yt
and replacing the expression of CTOT from (13): t at T V AICt ¼ þ t C ktb Lt N t þ ðC r t þ pÞ kt
ð17Þ
ð18Þ
Then, comparing (18) with (14) it is obtained that AICt = SRMgCt. Therefore, there are constant product specific returns to scale to trips produced in period t, St = 1, "t (constant returns). This means that given the fleet of buses (B) the drivers (H) and the terminal (M), trips in period t can be produced incurring only in the corresponding short run marginal costs. A ray average cost (RAC) (Baumol et al., 1982) can also be calculated. For that, a unitary bundle of trips-periods must be defined as unity product. For instance the bundle o1 be arbitrarily assigned the unity value. In general any vector Yb o ¼ ð1=T ; . . . ; 1=T ; . . . ; 1=T Þ could P Yb ¼ ðY o1 ; . . . ; Y ot ; . . . ; Y oT Þ, such that t Y ot ¼ 1 could be used. Then, any amount of output can be o represented making Yb ¼ r Yb where r is the number of units in the bundle. This allows to define the average cost of the composite good Yb as o C TOT ðB; H ; M; Yb Þ C TOT ðB; H; M; r Yb Þ P ¼ RAC ¼ Y tY t
ð19Þ
Notice that given the definition of the unity product bundle Y ¼ r. For the special case in which o1 the unity bundle is Yb , ray average cost is given by
F 1 X a T t o1 t V b RAC r Y þ t C ktb Lt N t þ ðC r t þ pÞ ð20Þ ¼ þ kt Y T t Then ray scale economies can be calculated as C TOT ðB; H; M; Yb Þ A þ ðp m þ g ph þ pb Þ B >1 ¼ 1 þ X t m ST ¼ P 1 at T V Y SRMgC kt t t t Y þ t C b Lt N t þ C r t þ p t T kt
ð21Þ
Notice that in the short run there are ray scale economies, that are produced by the fix costs F. For a given fleet B, ray economies of scale decrease as the total number of trips produced increases and increase as the number. 5.2. Long run cost function In the long run all factors are variable and therefore, B, M, and H are optimized to produce each production vector Yb . Then considering that all factors can be adapted to Yb , we obtain the following expression for the long run total cost function:
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
390
C LR TOT
X a t T t ap T PV V þ t Yp þ ¼ AS ðY Þ þ ðpb þ m pm þ g ph Þ þ t C ktb Lt N t Y t K k t t þ ðC r t þ pÞ Y
ð22Þ
It is assumed that administration costs are a step function of Y . They are constant for some pb ¼ ðpb þ m pm þ g ph Þ will be used. ~pb corresponds to ranges of values of Y . In the notation: ~ a generalized price per bus. It is assumed that buses operate full during the peak period (kp = K). 5.2.1. Costs without congestion Then, if there is not congestion (T tV independent of Yt), differentiating (22) with respect to Yt, the following expression for the long run marginal cost is obtained for period t: ap T pV if t ¼ p ð23Þ þ t ð~ pb þ C kp LRMgC py ¼ b Lp N p Þ þ ðC r t þ pÞ; K t at T V t LRMgC y ¼ þ t C ktb Lt N t þ ðC r t þ pÞ; if t 6¼ p ð24Þ kt Using expressions (24) and (14), the following relations between long and short run marginal costs are obtained: SRMgC ty ¼ LRMgC ty ; LRMgC py
SRMgC py
if 6¼ p
¼
ap T pV þt ~ pb ; K
ð25Þ if t ¼ p
ð26Þ
The value of the difference in (26) is due to the fact that in the long run a marginal increase in the fleet size is necessary to accommodate the marginal increase in the number of trips produced during period p. Average incremental costs (Baumol et al., 1982) for each period (AICt) can be obtained as AICt ¼
C TOT ðY 1 ; . . . ; Y t ; . . . ; Y T Þ C TOT ðY 1 ; . . . ; 0; . . . ; Y T Þ Yt
Replacing the expression of CTOT from (22): ap T pV Y p Y ~t ap T pV AICp ¼ ~ pb L N þ C kp þt þ t þ ðC r t þ pÞ; p p b K Yp K t at T V AICt ¼ þ t C ktb Lt N t þ ðC r t þ pÞ; if t 6¼ p kt
ð27Þ
if t ¼ p
ð28Þ
ð29Þ
Comparing (28) with (23): ap T pV Y ~t þt pb AICp ¼ LRMgC p ~ K Yp
ð30Þ
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
391
Therefore, AICp < LRMgCp. In (30) Y ~t corresponds to the second largest demand period, that will become peak period after period p is eliminated. Comparing (29) with (24), AICt = LRMgCt, "t 5 p. Therefore, there are constant product specific returns to scale for trips produced in all non-peak periods t, St = 1, "t 5 p. This means that, given the fleet of buses (B), drivers (H) and terminal (M), required to operate peak period demands, trips in other period t can be produced incurring in a constant marginal cost. Notice that the average incremental cost for the peak period increases as the difference between peak and off peak demands increases. The minimum value is obtained when all periods have the same demand value (YP = Yt). Then: ap T pV AICp ¼ þ t C kp ð31Þ b Lp N p þ ðC r t þ pÞ; K In this limit case AICp = AICt = LRMgC. Peak period product specific returns to scale are given by5 ap T pV Yp Yt ap T pV ~ þ C kp L N pb þt þ t þ ðC r t þ pÞ p p b K Yp K Sp ¼ at oT tV t þ t ð~ pb þ C kp TV þ Yt b Lp N p Þ þ ðC r t þ pÞ kt oY t ap T pV Yt ~ þt pb K Y pp ð32Þ ¼1 ap T pV a pT V kp ~ þ t þ ðC b Lp N p Þ þ t þ ðC r t þ pÞ pb K K It can be noticed that the second term on the rhs of (32) is positive and less than one, which makes, 0 < Sp < 1. Therefore, there are product specific diseconomies for the peak period. Notice that by definition of peak period, Y p > Y ~t therefore as peak demand increases, keeping constant other periods demands, the degree of diseconomies is reduced, because the values of average and marginal costs become closer (see (30)) and Sp ! 1. The same happens as off peak demands are reduced. This is because the peak period average incremental cost, due to the need of additional buses, drivers and terminal space, increases when the difference between peak and off peak demands increases, making the value of AICp closer to the value of the LRMgCp. On the contrary, if all periods have the same demand, the size of the fleet, drivers and terminal space is equally justified by all the periods; then average incremental costs take their minimum value corresponding only to variable operating costs. the short run case, by defining Long run ray average cost (RACLR) can be obtained, similarly P to o o o o o b a unity product bundle Y ¼ ðY 1 ; . . . ; Y t ; . . . ; Y T Þ such that t Y t ¼ 1,
o b C r Y b TOT C TOT ð Y Þ ¼ RACLR ¼ P ð33Þ Y tY t 5 For the particular case in which all periods have the same demand this expression is not valid because there is not peak period. In such case all periods present constant product specific returns to scale.
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
392
o1 For the special case in which the unity bundle is Yb (see short run section) we obtain: " # X t
A ðY Þ 1 P o1 a T a T S p t kt V V pb þt þ þ t C b Lt N t þ ðC r t þ pÞ RACLR r Yb þ ~ ¼ T K kt Y t
ð34Þ Then long run ray scale economies will be given by As >1 X t S LR T ¼ 1þ P 1 ap T V at T V kt ~ Y þ t C b Lt N t þ C r t þ p p þt þ t T b K kt
ð35Þ
Therefore in the long run multi-product economies of scale are only generated by the fix administration expenses and should therefore be mild in magnitude.6 In addition it can be observed from (35) that: ray scale economies are reduced when the level of total production Y increases, and the contrary happens when the same total of trips production is distributed in a larger number of periods T. Non-uniform demand structures served by the corridor tend to reduce de degree of ray economies. For the case of uniform demands, scale economies increase when the trips produced are short with respect to the length of the corridor (l/L is small). If both peak period p and any other period t are considered, ray scale economies can be expressed as (Baumol et al., 1982; Jara-dı´az, 1983): ST ¼
/S p þ ð1 /ÞS t ; 1 SS
where S S ¼
CðY p Þ þ CðY t Þ CðY p ; Y t Þ CðY p ; Y t Þ
ð36Þ Y ðoCðY Þ=oY Þ
p p p corresponds to the degree of scope economies (Baumol et al., 1982), / ¼ Y t ðoCðY t Þ=oY t ÞþY p ðoCðY p Þ=oY p Þ
and Sp, St represent the economies of scale specific to periods p and t. Considering that, as was seen above, St = 1 and using Sp from (32) in (36), it is obtained that: 8 9 ap T pV Yt > > > > ~pb þt < = K Y p / 1 þ ð1 /Þ p ap T V > > > > : ; þt ~ L N þ ðC t þ pÞ pb þ C kp p p r b K ST ¼ ð37Þ 1 SS and rearranging (37): 8 9 ap T pV Yt > > > > ~ pb þt = 1 < K Yp 1 / SS ¼ 1 >0 > ap T pV ST > kp > > : þ t ð~ pb þ C b Lp N p Þ þ ðC r t þ pÞ; K
6
Cost studies made in Chile show that administration costs are around 10% of total costs.
ð38Þ
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
393
This indicates the existence of scope economies or sub-additivity if the second term in (38) is lower than 1. It is shipper to produce trips in different periods with the same fleet, than separating the same production in different fleets. Notice that the scope economies are consequence both of the ray scale economies (ST > 1), and the peak period diseconomies (SP < 1). However in (38) the effect of Sp is reduced by the presence of parameter /7 (notice that 0 < / < 1). If (Yt/Yp) ! 0, / ! 1 and SP ! 1 (see (32)); therefore, SS ! (1 1/ST) and scope economies are only related to existence of ray scale economies (ST > 1). On the contrary, if (Yt/Yp) ! 1, / < 1 and 1 1 S P ! S P < 1; then, S S ¼ 1 ½1 /ð1 S P Þ and economies of scope get their maximum value influenced both by ray scale economies and peak specific diseconomies. 5.2.2. Costs with congestion Assuming that some periods experience congestion (at least the peak period p) travel times will depend on the number of trips produced during the period: oT tV =oY t > 0, (see Appendix B). However notice that the expression for average incremental costs is the same as in the case without congestion (see (28) and (29)). The only difference is that the value taken by T tV will be higher if there is congestion. Notice that ray average costs have also the same expression as in the uncongested case (see (34)). Differentiating (22) with respect to Yt, the following expression for the long run marginal cost is obtained: ap oT pV p p þ t ð~pb þ C kp if t ¼ p ð39Þ LRMgCC y ¼ TV þ Yp b Lp N p Þ þ ðC r t þ pÞ; K oY p at oT tV t t LRMgCC y ¼ TV þ Yt ð40Þ þ t C ktb Lt N t þ ðC r t þ pÞ; if t 6¼ p kt oY t Then, period specific economies of scale with congestion are given by ap T pV Yp Yt ap T pV kp ~ þt þ t þ ðC r t þ pÞ þ C b Lp N p pb K Yp K ; S CP ¼ at oT tV kp t TV þ Yt þ t ð~ pb þ C b Lp N p Þ þ ðC r t þ pÞ kt oY t
if t ¼ p
ð41Þ
Comparing (41) with (32) it is clear that, as a result of including congestion ðoT PV =oY P > 0Þ peak specific diseconomies increase with congestion, S CP < S P < 1. For non-peak periods it is obtained: t at T V kt C b Lt N t þ t þ ðC r t þ pÞ kt S Ct ¼ < 1; if t 6¼ p; ð42Þ at oT tV kt t þ t þ ðC r t þ pÞ TV þ Yt C b Lt N t kt oY t In this case, as a consequence of congestion, non-peak periods also present period specific scale diseconomies S Ct < 1. As congestion increases, the denominators value also increases and period diseconomies become stronger (the values of Sp and St go to zero). 7
Expression (38) is equivalent to: S S ¼ 1 S1T ½1 /ð1 S P Þ .
394
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
Using (22), (39) and (40) in (21), the following expression for ray economies of scale is obtained: P at T tV ap T PV þ t Yp þ t AS ðY Þ þ ~ pb þ t C ktb Lt N t Y t þ ðC r t þ pÞY K k t S LR ð43Þ T ¼ P at T tV ap T PV kt þ t Yp þ t hðY Þ þ ~ pb þ t C b Lt N t Y t þ ðC r t þ pÞY K kt " # ap oT pV 2 X at oT tV kt 2 ð44Þ hðY Þ ¼ ~ Y þ C Lt N t Y t pb K oY p p k t oY t b t It can be noticed that hðY Þ takes into account the effect of congestion externalities and its value will be 0 without congestion, but will increase to any positive value depending on the level of congestion experienced. Therefore, congestion will have an important influence in the type of scale economies obtained. The following alternative cases can be experienced: S LR T > 1 () h < A
ð45Þ
S LR T ¼ 1 () h ¼ A
ð46Þ
S LR T < 1 () h > A
ð47Þ
Therefore, if buses experience a level of congestion such that h > A, there will be ray diseconoLR mies of scale,8 S LR T < 1. Ian indefinite increase of congestion will make that S T ! 0. Using (22) with the definition of economies of scope given in (36), the following expression is obtained, assuming only two periods t and p: t at T V AS ðY t Þ þ ðpb þ mpm þ gph Þ þ t Yt kt ð48Þ SS ¼ at T tV ap T PV P kt kp þ t Y p þ t C b Lt N t þ C r t þ p t AS ðY p Þ þ ~ p b þ C b Lp N p þ t Yt K kt From (48) it is easy to see that, independent of the levels of congestion experienced in the corridor, economies of scope will be always within the range: 0 < Ss < 1,. However, the value of Ss will change within the range indicated as a consequence of changes in the levels of congestion. First it can be observed that Ss ! 0 when Yt ! 0. Therefore in general, the magnitude of economies of scope will increase as the number of off peak trips increases as was also seen in the analysis of the uncongested case. An increase of congestion only during the peak period will reduce the degree of scope economies. On the contrary if congestion increases only during off peak period scope economies will increase. However the maximum, value that Ss can take will always be significantly lower than one, because T tV ¼ T tV ðY t Þ and by definition YP > Yt. Therefore, for a given value of YP the increase in the value of Yt is limited by YP. 8 This can explain why in developing countries where buses operate over congested infrastructure, operators are small. In the case of Santiago de Chile most bus operators own between one and two buses.
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
395
6. Numerical example A BPR type function is assumed to explain the congestion effect produced by the flow of buses over the travel time T tV ðY t Þ, mainly due to congestion in bus stops: c at Y t t TV ¼ T0 1 þ b ð49Þ k t CK where ft = (atYt/kt) is the flow of buses (equal to the frequency in (bus/h)) running over the corridor in period t, CK is the corridor nominal capacity in (bus/h), T0 is the travel time over the corridor under free flow conditions and b, c are calibration parameters. Parameter values assumed are given in Table 1. Basic relations for operational variables (fleet size (B), passengers demands (Y) and travel times (T)) are presented in Appendix B. Cost function parameters used are given in Table 2. Cost units considered are $US/month. Two different operating periods are considered: peak (p) and off-peak (t). Therefore, the product vector is Y = (Yp, Yt). Based on the values assumed and using expression (23) the multiproduct cost function shown in Fig. 2 is obtained. An specific ray ‘‘r’’ is represented in the figure to show ray costs and economies of scale.
Table 1 BPR and production function parameter value Parameter
Units
Value
t at, ap k t , kp , K T0 b CK c
s – Pas/Bus h – Bus/h –
8.0 0.5 25 1.0 1.0 300 5.0
Table 2 Cost function parameter values Parameter
Units
Value
A pm m pb ph H p kp kt C Pb , C b Lt N t Cr
$US/bus month $US/m2 month m2/bus $US/bus month $US/driver month drivers/bus $US/pass $US/bus-km km/month $US/bus h
373.0 8.4 24.0 741.0 127.4 2.0 0.14 0.23 9615.0 0.0
396
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403 C(yp,yt)
Yp1
n
Yp2
Yp
Yp
Yt1 r
Yt2 Yt n
Yt
Fig. 2. Long run total costs for the corridor.
Considering the parameter values assumed in Table 1, congestion starts in the corridor when the number of trips, increases over 8000 trips per hour,9 as can be observed in Fig. B.4 of Appendix B. The form of the cost surface shown in Fig. 2, suggests that there is transray convexity, which implies cost complementarity or economies of scope. In order to analyze in more detail the influence of congestion on these properties, some specific situations are analyzed in more detail. The demand values assumed for each of the six cases analyzed and the corresponding total cost values obtained are presented in Table 3. Two different production rays, identified by A and B in Table 3 and Fig. 3, are considered. Ray A is defined by a ratio (Yp/Yt = 3) and (Yp/Yt = 1) for ray B. For each ray, three levels of production (1, 2, 3) are considered, obtaining the six different cases defined in Table 3 and graphically represented in Fig. 3. Table 4 shows the corresponding values obtained for the index of economies of scale and scope for each of the cases analyzed. The results shown in Fig. 3 and Table 4 corroborate the results obtained in the previous section with respect to the influence of congestion on the degree of scale and scope economies. It can be observed that the values of ST decrease as congestion increases both for rays A and B; therefore an increase in the level of congestion always produces ray diseconomies of scale. However the values of SS decrease with congestion for ray A and increase for ray B. An increase of congestion 9 This value could change depending on the period if vehicle occupancy rate differs for peak and off peak periods. In this example we assume the same occupancy rate for both periods (see Table 1).
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
397
Table 3 Costs values $US/month Case
(Yp; 0)
(0; Yt)
(Y p ; Y t )
C(Yp; 0)
CðY p ; Y t Þ
C(0; Yt)
A-1 A-2 A-3 B-1 B-2 B-3
(4490; 0) (12,797; 0) (16,945; 0) (4490; 0) (12,797; 0) (16,945; 0)
(0; (0; (0; (0; (0; (0;
(2245.1; 748.3) (6398.3; 2132.7) (8472.5; 2,824.2) (2245; 2245) (6398; 6398) (8472; 8472)
364,900 1,474,246 3,728,291 364,900 1,474,246 3,728,291
216,685 625,045 837,463 287,350 826,432 1,104,134
82,619 235,438 311,710 364,900 1,474,246 3,728,291
C(0;Yt)
1497) 4266) 5648) 4490) 12,797) 16,945)
C(Yp;Yt )
C(Yp;0)
C(0;Yt)
C(Yp;Yt )
C(Yp;0)
3
2 3
2
1
1
Yt
Yt
3
(Yt;Yp ) 2
(Yt;Yp ) 1 3 2 1
Yp
Yp
A
B
Fig. 3. Transray convexity characteristics. Table 4 Economies level Ray
Case
Sp
St
ST
SS
A
1 2 3
0.66979046 0.27252797 0.16273242
0.99995054 0.99078760 0.96352638
0.69203244 0.28539983 0.17256263
0.0274 0.0203 0.0113
B
1 2 3
0.03190389 0.03155009 0.01283727
0.03190389 0.03155009 0.01283727
0.0433479 0.04292787 0.02208111
0.2640 0.2650 0.4186
398
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
only during the peak period reduces the degree of scope economies (ray A). On the contrary, if congestion increases also during off peak period scope economies increase (ray B). Therefore, an increase in congestion during off peak makes increase transray convexity or economies of scope.
7. Conclusions Multimodal bus corridor production and cost functions are developed and used to analyze the degree of scale and scope economies. An important feature of the cost functions is the multiproduct character of the trips production process because of different operating periods. Then, many of the corridor cost characteristics depend on the product expansion path in the trips-period space. This is related to the definition of the unitary production bundle. It was found that in the short run the existence of fixed costs produce ray scale economies. For a given fleet B he degree of these ray scale economies will increase with the magnitude of fixed costs F and will decrease when the level of total production Y increases. In the long run economies of scale are produced only by administration fix expenditures, which also can increase more than proportionally than production after some production level. Without congestion, there are constant product specific returns to scale for trips produced in all non-peak periods. This means that, given the fleet of buses (B), drivers (H) and terminal (M), required to operate peak period demands, trips in other period t can be produced incurring in a constant marginal cost. However, there are product specific diseconomies for the peak period. As peak demand increases, keeping constant other periods demands, the degree of diseconomies is reduced, because the values of average and marginal costs become closer. The same happens as off peak demands are reduced. This is because the amount of buses, drivers and terminal saved to serve off peak periods is reduced as the difference between peak and off peak demands increases. With respect to cost complementarity, there are always (with and without congestion) scope economies or sub-additivity in the production of trips for different periods. It is shipper to produce trips in different periods with the same fleet, than separating the same production in different fleets. These scope economies are consequence both of the ray scale economies and the peak period diseconomies. The degree of scope economies is reduced when the difference between trips produced during peak and non-peak period is large. On the contrary, when the number of trips produced in different periods is similar economies of scope increase. With congestion, period specific diseconomies increase and ray scale economies will turn into diseconomies, beyond some level of production. However, congestion can make increase or decrease the scope economies depending of the period in which it is experienced. Congestion increases during peak period makes economies of scope to decrease but congestion increases during off peak produce economies of scope increases. The value of parameters as length of the corridor, vehicles generalized price and demand structure directly influence the magnitude of diseconomies when congestion appears. As the value of this parameters increases the magnitude of the negative effects of congestion increases. On the contrary, an increase in the bus capacity reduces the negative effect of congestion; however this is counterbalanced by the increase in operating costs.
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
399
Appendix A. Influence of the demand structure on the production of trips Three different demand structures are considered, representing different trip distributions between transit stops over the route. P It is assumed that y is the umber of trips made between each pair of stops in the corridor ðY ¼ yÞ. • Uniform demand: corresponds to a set of trips that produces a uniform load of passenger flows over the route (Fig. A.1). This the simplest case usually assumed in this type of analysis. • Homogeneous demand: assumes a homogeneous distribution of trips among all pairs of transit stops (Fig. A.2). Some peripheral routes experience this type of demand in cities of developing countries. • Feeder routes demand: it is typical for feeder routes or routes going to CBD. Passengers boarding the service have a common destination at the end of the route (Fig. A.3). The number of passengers on the service increases as it approaches its final stop.
A.1. Uniform demand In this case, the passengers load is the same over all the corridor and equal to Y Ll . Therefore, the frequency required to transport all the demanded trips is equal to f d ¼ Yl=L . On the other hand, the k frequency that can be offered over the route served with the available fleet, is given by f 0 = B/tc, where the total cycle time tc is equal to the travel time necessary to cover the route, Tv, plus the time spent at the transit stops: tc = Tv + tY/f 0. Then, we can obtain the following expression for the total cycle time tc = BTv/(B tY). To obtain an equilibrium solution between service supply
1
3
2
y
y
n-1
y
y
n
y
Fig. A.1. Example of uniform demand.
y y 1 y
2
3
y
y
4 y y
y y
y
Fig. A.2. Example of homogeneous demand.
400
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403 y
1
3
2
4 y
y y
Fig. A.3. Example of feeder routes demand.
and demand we must equal the frequency supplied with the frequency demanded: f 0 = f d. Then, the number of trips that can be produced, per unit of time, under the above defined conditions is given by Y ¼ Bk= Ll T v þ tk with a ¼ Ll . This expression can be interpreted as the basic production relation for the operator of the service. It provides the total number of trips that can be produced with a fleet of B buses operating over a route of length L, each of which carry k passengers, with a travel time Tv and serving passengers with trip lengths l, when demand is uniformly distributed. A.2. Homogeneous demand In this case, the total number of trips demanded per unit of time is equal to the sum of trips demanded between each pair of stops (see Fig. A.2). Assuming that the number of trips demanded y. Given the homogeneous disbetween each pair (i, j) of stops is the same, we obtain Y ¼ nðnþ1Þ 2 tribution of demand in Fig. A.2, a different passenger load will be obtained for each link of the route. Therefore, the capacity constraint will be active only for the link (or links) with the maximum load of passengers, that in this case will be located at the center of the route. Imposing the constraint that the solution obtained must satisfy the capacity constraint of the service B/tc = T v Y/2k, and replacing the expression for tc it is obtained that: Y ¼ Bk= 2 þ kt where a ¼ 12. A.3. Feeder routes demand In this last case, the capacity constraint is active only in the last link of the route (see Fig. A.3). This will have a passengers load of ymax = ny. This value coincides also with the total number of trips demanded Y. Therefore, in a similar way we can obtain that the number of trips that can be produced is given by Y = Bk/(Tv + kt), where a = 1.
Appendix B. Basic operating relations for the bus corridor In this appendix the operational relations used to develop the numerical example included in the paper are presented. Replacing in (49) values given in Table 1 the relation shown in Fig. B.1 is obtained. It gives the values of travel time T tV (in h) on the corridor as a function of flow 5 ft (in buses/h) T tV ¼ 1 þ ðft =300Þ .
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
401
Fig. B.1. BPR function values for the corridor travel time.
Fig. B.2. Fleet size (B) vs. corridor flow (f).
Using equation (Bt = ft Æ tc) the relation between fleet size Bt and corridor flow ft shown in Fig. B.2, can be obtained. It can also be derived the relation between fleet size and travel time T tV shown in Fig. B.3. Using the previous relations it is obtained the relation between the trips produced over the corridor Yt and the travel time T tV , that is shown in Fig. B.4. This relation can be used to calculate the derivatives ðoT tV =oY t Þ in (42).
402
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
Fig. B.3. Fleet size (B) vs. travel time (Tv).
Fig. B.4. Travel time (Tv) vs. corridor trips produced (Y).
From the form of relations shown in Figs. B.3 and B.4 it is clear that o2 T tV > 0. oY 2
oT tV oBt
> 0,
oT tV oY t
> 0, and
t
References Baumol, W.J., Panzar, J.C., Willig, R.D., 1982. Contestable Markets and the Theory of Industry Structure. Harcourt Brace Jovanovich Inc.
J.E. Ferna´ndez L. et al. / Transportation Research Part A 39 (2005) 383–403
403
Ferna´ndez, J.E., de Cea, J., de Grange, L., 1999. Caracterı´sticas de produccio´n de un servicio de transporte pu´blico de pasajeros. Actas del IX Congreso Chileno de Ingenierı´a de Transporte, pp. 431–448. Jansson, J.O., 1980. A simple bus line model for optimization of service frequency and bus size. Journal of Transportation Economic and Policy 14, 53–80. Jara-dı´az, S., 1983. Freight Transportation Multioutput Analysis. Transportation Research-A 17A, 429–438. Kerin, P., 1992. Efficient bus fares. Transport Reviews 12, 33–48. Leontief, W., 1946. The pure theory of the guaranteed annual wage contract. Journal of Political Economy. Mohring, H., 1972. Optimization and scale economies in urban bus transportation. American Economic Review 62, 591–604. Tauchen, H., Fravel, F., Gilbert, G., 1983. Cost structure of the intercity bus industry. Journal of Transportation Economic and Policy 17, 25–48. Wabe, S.J., Coles, O.B., 1975. The short and long run cost of bus transport in urban areas. Journal of Transportation Economic and Policy 9, 127–140. Xu, K., Windle, R., Grim, C., Corsi, T., 1994. Re-evaluting returns to scale in transport. Journal of Transportation Economic and Policy (September), 275–286.