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FROM ECONOMIES OF DENSITY AND NETWORK SCALE TO MULTIOUTPUT ECONOMIES OF SCALE AND SCOPE: A SYNTHESIS Leonardo J. Basso Sergo R. Jara-Díaz Unversdad de Chle. INTRODUCTION For more than twent ears, the cost structure of transport ndustres n general, and the arlne ndustr n partcular, has been analzed through the calculaton of two ndces: returns to denst (RTD) and returns to scale (RTS), whch were orgnall proposed b Caves et al. (984). Durng ths perod, there have been man observatons, amendments, caveats and correctons about ther calculaton (e.g. Gagne, 990, Antonou, 99; Yng, 992, Flppn and Magg, 992; Xu et al. 994). However, both RTD and RTS have become the textbook tools to examne emprcall, from a cost perspectve, mportant ssues such as mergng, or the optmal shape and sze of transport networks (see for nstance Small, 992; Braeutgam, 999; Pels and Retveld, 2000). The obectve of ths artcle s to organze ths dscusson, and to offer an nterpretatve snthess of a number of results on the subect that have been obtaned b the authors. The man concluson s that RTD and RTS should be replaced wth three concepts: a corrected verson of economes of denst (Jara-Díaz and Cortés, 996), the multoutput degree of economes of scale (Basso and Jara-Díaz, 2006a), and the degree of economes of spatal scope (Basso and Jara-Díaz, 2005). The artcled s organzed as follows. In the next Secton we present the concepts of scale and scope n general, and ther applcaton to transport ndustres n partcular. In Secton we explan what has been the usual wa to analze transport cost structure n the lterature; we show here what we thnk are the shortcomngs of RTD and RTS, and propose methodologes to emprcall estmate a corrected verson of economes of denst, economes of scale, and economes of spatal scope. In the fourth Secton, we snthesze the analtcal results and show, wth (rudmentar) emprcal evdence, how the proposed methodologes affect conclusons regardng the technolog and the expected structure of transport ndustres. Secton 5 concludes. 2. SCALE AND SCOPE IN TRANSPORT The output of a transport frm s a vector of flows between man orgns- kt Y =, dsaggregated b the tpe of cargo k destnaton pars, of the form { } and perod t (Jara-Díaz, 982). If we keep onl the spatal dmenson of output, transport product s a vector of components. In order to produce a flow vector Y, the frm has to take a number of decsons: number and capact of vehcles (fleet sze), desgn of the was (locaton, capact), desgn of termnals (locaton, loadng and unloadng capactes), frequences, and so on. Some decsons nvolve choces about the characterstc of nputs, whle others are related to ther use, that s, to the wa n whch nputs are combned n order to produce the flow vector; we shall call the latter tpe of decsons operaton rules. Snce transport takes place on a network, the frm also has also to choose a servce structure the generc wa n whch vehcles vst the nodes n order to produce the flows and a lnk sequence. Together, these two endogenous decsons defne a route structure, whch s to be chosen based on exogenous nformaton, namel the OD structure of demand (defned b vector Y), and the phscal network (Jara-Díaz and Basso, 200). It s mportant to note that, n the end, the route structure decson s a consequence of the spatal dmenson of transport output. To llustrate these concepts, consder a three nodes OD sstem as n Fgure.a, together wth a phscal network as n Fgure.b (a) Fgure. OD structure and phscal network 2 (b) For a gven vector{ }, the best combnaton of nputs and operaton rules wll depend on man factors. Three possble servce structures are shown n Fgure 2 (Jara-Díaz, 2000). Structure (a) corresponds to a general cclcal sstem (Gálvez, 978), structure (b) to three smple cclcal sstems (pont-topont servce), and structure (c) to hub-and spoke, where a dstrbuton node the hub has been created (ths structure s ver pervasve n the arlnes ndustr; the hub ma concde or not wth an orgn or a destnaton). H (a) (b) (c) Fgure 2. Servce structures Regardng the allocaton of vehcles to fleets one of the components of a servce structure, n case (a) there can onl be one fleet (and therefore onl one frequenc), whle n cases (b) and (c) there ma be up to three dfferent fleets. If the servce structure chosen s as n Fgure 2.a, a possble route structure s the one shown n Fgure. 2 Fgure. A route structure Conceptuall, the transformaton of nputs X nto outputs Y n ths case flows ma be represented through a transformaton functon F( X, Y ) 0 (see Jara-Díaz and Basso, 200), where equalt represents an effcent use of nputs. The multoutput degree of economes of scale, S, s defned as the maxmal equproportonal expanson of Y, λ s Y, that s possble after an equproportonal expanson of X to λx (Panzar and Wllg, 977). Analtcall, S ( X, Y ) = 0 F λ λ () A value of S larger, equal or smaller than mples ncreasng, constant or decreasng returns to scale respectvel. Hence, what s relevant n scale analss s the optmal combnaton of nputs when all components of the output vector ncrease b the same proporton. Under some smple regulart condtons, S can be calculated from the cost functon C(w,Y), whch represents the mnmum expendture necessar to produce Y at nput prces w. In partcular, ncreasng returns to scale mpl that an equproportonal expanson of Y wll nduce a less than proportonal ncrease n cost. In general, and omttng nput prces for notatonal smplct, S can be calculated from C as, S = C(Y) C = η, (2) where η s the elastct of C wth respect to the -th output. Applng ths to the case of transport, the degree of economes of scale wll depend on the operatonal re-organzaton that the frm ma acheve, after proportonal ncreases of the flow vector. For example, f the flows are small and smlar, the OD structure of Fgure.a ma be well served wth small vehcles and a servce structure as n Fgure 2.a. If flows were larger, servce structures such as (b) or (c) ma become more attractve. Hence, not onl the number and sze of nputs s relevant, but also the spatal re-desgn of servce. What s mportant to note here s that the frm chooses endogenousl ts route structure, whch s ndeed not assumed as fxed n equaton (2). On the other hand, economes of scope exst f SC A A B D C( Y ) + C( Y ) C( Y ) = SC B = () D C( Y ) s postve, where D s the set of all outputs, A B=D and A B= (.e. A and B are an orthogonal partton of D). Y A s vector Y D but wth = 0, A D; Y B s defned analogousl. Therefore, a negatve value for SC A ndcates that t s cheaper to have a second frm producng Y B, rather than to expand the producton lne of a frm alread producng Y A. If SC A s postve, then t s cheaper that a sngle frm produces everthng (Y D ). It s eas to verf that SC should le n the [-;] nterval. Snce the output n transport s a vector of OD flows, an expanson of the lne of producton necessarl mples servng new OD pars. Therefore, n transport, economes of spatal scope are analzed n a context n whch the sze of the network understood as the OD structure changes. Ths does not happen wth S. Hence, SC enables to examne whether t s cost convenent that a frm A, who serves PS A A A nodes and potentall PS ( PS ) OD flows, expands ts network to PS D D D A nodes servng PS ( PS ) PS ( PS A ) new flows, or f t s more cost convenent that other frm does t. If frm D produces on the OD structure of Fgure.a, a possble partton s the one show n Fgure 4. Note that analzng ths tpe of economes of (spatal) scope s equvalent to analze an ncrease n one node of frm A s network. [A] [B] [D] ; ;0;0;0;0} Y B = 0;0; ; ; ; D = ; ; ; ; ; } Y A = 2 } Y { 2 { 2 2 { Fgure 4. Varable network sze and spatal scope For snthess, and emphaszng the spatal dmenson of output n the transport case, wth S one analzes the behavour of costs after an equproportonal expanson of the OD flows keepng the number of OD pars constant, whle wth SC one analzes the behavour of cost when new OD flows are added. . RTD, RTS, AND EMPIRICAL METHODS FOR SCALE AND SCOPE The large sze that the output vector Y acheves n practce precludes ts drect use n emprcal work. It s a necesst, then, to estmate cost functons usng aggregate output descrptons, Y = { h }, whch represents outputs and attrbutes such as ton-klometers, seat-klometers, average dstance or load factor. When a network sze varable, N, s ncluded n the estmaton, emprcal studes of transport ndustres dstngush between two concepts of scale : returns to denst (RTD) and returns to scale (RTS). In the former t s assumed that the network s fxed when output ncreases; t s sad that traffc denst ncreases. In the latter, though, both output and network sze would ncrease, keepng traffc denst unchanged. RTD s calculated as the nverse of the sum of a subset of the cost-output elastctes. Ths subset vares from stud to stud, whch has become a source of ambgut. In RTS, the elastct of the network sze s also ncluded n the calculaton. Several emprcal studes on the arlne network (where the number of ponts served, PS, s usuall the network sze varable) have reported the exstence of ncreasng returns to denst (RTD ) and constant returns to scale ( RTS ). Ths would ndcate that there would be cost advantages f the denst of traffc s ncreased, but there would not be such advantages f frms operated larger networks. However, the observed behavour n the ndustr has been dfferent: after the deregulaton, n the US frst and then n the rest of the world, the concentraton of the ndustr and the sze of the networks have ncreased through mergers, acqustons and allances. 2 These efforts from the part of frms to ncrease ther network sze n seemng contradcton wth the constant returns to scale provoked a re-examnaton of the methods used to calculate economes of scale (e.g. Gagné, 990; Yng, 992; Xu et al., 994 and Oum and Zhang, 997). Jara-Díaz and Cortés (996, hereafter JDC) proposed a dfferent approach to stud economes of scale n transport. The noted that behnd the aggregates ncluded n the vector Y = { }, les the real output of a transport frm, that s, h the vector Y = { kt } of flows of tpe k between orgns and destnatons n perods t. JDC noted that the nablt of usng Y n the emprcal work does not mean that ts defnton should be abandoned when usng an estmated cost functons to make economc nferences. If the estmated functon represents well the real multoutput cost functon, then the characterstcs of the latter should be obtanable from the estmated parameters of the former. Let us take the case of economes of scale. Snce economes of scale analze the behavour of costs when the output vector ncreases equproportonall, a correct calculaton of economes of scale n transport would be related to an ncrease n the same proporton of all the flows n Y. Ths ma be analzed from an estmated cost functon C ( Y, N) f one examnes the behavour of aggregates h when Y vares. If the aggregates can be actuall descrbed as functons of Y,.e. ( Y h h ), then C ˆ ( Y ) C ( Y ( Y ), N ) can be consdered as an approxmaton of the cost functon n terms of Y. Calculatng from C ˆ ( Y ) C Y (Y ), N the elastctes of cost wth respect to the components of Y, ( ) JDC obtaned a method to calculate the degree of economes of scale. Frst, dsaggregate margnal costs can be calculated as: Cˆ = n = C The correspondng cost elastctes wth respect to are = = =. (4) n n n Cˆ C C ˆ η = = = = C C ε η (5) C where ε s the elastct of aggregate output the elastct of C wth respect to,.e. wth respect to, and η s ε = and C η =. (6) C In ths wa, the correct calculaton for an estmator of S, Ŝ, s = = ε S ˆ = ˆ η α η where α. (7) Note that α s the degree of homogenet of the -th aggregate wth respect to the dsaggregated flows, and that ts calculaton avods the dscusson regardng whch aggregate should be consdered n the calculaton of S. η s the elastct whch s obtaned drectl from the estmated cost functon. Snce the number of OD pars do not change when flows ncrease, JDC argued that the elastct of the network sze should never be ncluded n a scale calculaton. Recall that ths s also mposed n the calculaton of RTD. Does ths mean that RTD s actuall S, as suggested b Panzar (989)? Not exactl; although the man dea behnd RTD s to examne the behavor of costs when there s an output ncrease but the network sze does not change, Basso and Jara-Díaz (2006a) dentfed a second mplct condton behnd ts calculaton: the route structure also remans unchanged. Ths condton s requred because the dea of estmatng the degree of economes of denst s to analze whether the average costs of a drect connecton decreases wth proportonate ncreases n both flows on that connecton (Hendrcks et al., 995). Hence, a fxed route structure s needed to ensure that onl the exstng lnks handle the new traffc. If the route structure changes, some new lnks ma be added whle others ma dsappear. Ths condton, however, was not even mentoned n the strct calculaton of Ŝ sntheszed above. Along these lnes, Basso and Jara-Díaz (2006a) proposed to dstngush RTD from S, assumng the route structure fxed n the former, but varable n the latter. Obvousl, ths dstncton nduces dfferences n the applcaton of equaton (7), partcularl n the calculaton of the α. For example, n RTD, the α of the average dstance wll alwas be zero as flows grow b the same proporton holdng the route structure fxed; t could be dfferent from zero n S f the mnmum cost occurs for a dfferent rout structure after flows grow. We consder ths dstncton to be useful and relevant. Economes of denst wll be useful to know f, for example, there are economes of vehcle sze, that s, f larger flows n non-stop routes mpl decreasng average costs n that route because of larger vehcles. Hub-and-spoke networks would be strongl nfluenced b the exstence of economes of denst. On the other hand, economes of scale S, are mportant because, when traffc ncreases sgnfcantl, t ma not be effcent to further ncrease the sze of the vehcles, whle a frequenc ncrease ma be expensve because of congeston. Wth a reconfguraton of the route structure however, t ma happen that the ncreases n flows ma be handled wthout ncreasng costs too much; for example, through pont-to-pont servce n certan OD pars (phenomenon that has been observed; see Swan, 200). As explaned above, RTS s amed at analzng the behavour of costs when both traffc and network sze ncrease b the same proporton. As n RTD, the proportonal ncrease apples to the vector of aggregates Y = { h } (or a subvector), but n ths case the network sze varable N also ncreases. Although ths makes RTS look lke the defnton of scale, t s not the case because of two nterrelated structural reasons. Frst, as shown b Basso and Jara-Díaz (2006b), ths procedure, performed on the aggregates, mposes analtcal condtons on the OD flow vector whch seem to be ndefensble. Second, ncreasng N mples the varaton of the number of OD pars, that s, a varaton on the dmenson of Y, whch s somethng that should be examned wth a scope analss. As an example of the strange mplct condtons mposed b RTS on Y, consder the cost functon C = C( PK, ALT, PS), where PK s passengerklometres, ALT s average length of trp, and PS s the number of ponts served. If onl the expanson of PK and PS were consdered (onl two elastctes n the calculaton of RTS), then t s mposed that (Basso and Jara- Díaz, 2006b) ' PS =, (8) 2 PS where s the average of the new flows served, s the average of the flows served orgnall, and PS s evaluated at ts startng value. Hence, (8) shows that RTS mplctl examnes the addton of flows that are, on average, between a quarter and a half the orgnal ones, dependng on the orgnal number of ponts served. Note that the ver fact that the condton depends on the value of PS s an undesrable propert, because comparsons between frms, who are lkel to have dfferent number of ponts served, s a useless exercse. If both PK and ALT were consdered n the calculaton of RTS (plus the elastct of PS, whch s alwas consdered b defnton), the condtons on the flows would be even stronger. Denotng wth an astersk the value of the aggregates after the network expanson, t holds that PK * =λpk and * ALT = λalt (equproportonal expanson of both products ). However, snce ALT PK/P where P s the total number of passengers, then * * PK / P = λpk / P, whch leads to P * =P. That s, the total number of passengers, before and after the expanson of the number of ponts served, s the same. Hence, calculaton of RTS assumes that the new flows to be served after the network expanson wll be zero. And ths s ndependent of the network, the number of ponts served and the route structure: a new node s added, but nothng arrves to or departs from t. As stated above, the man problem wth RTS s that t was desgned as a scale ndex, but t examnes a problem that should be dealt wth as a scope problem: network sze. The emprcal problem s that a drect calculaton of SC usng Equaton () s seldom feasble (an example s Jara-Díaz, 988). However, the approach proposed b JDC delvers a wa to deal wth the problem: snce most aggregates h are mplct functons of Y, even though the (dsaggregate) output vectors Y A, Y B and Y D mght be unknown, SC mght A B be calculated correctl f the correspondng aggregate vectors. Y ( Y ), Y ( Y ) D e Y ( Y ) were known, and a cost functon C( Y, N) was avalable (Jara-Díaz, Cortés and Ponce, 200). Analtcall, and consderng PS as the network sze varable, scope could be calculated as SC A B A A B B D D ( ( Y ), PS ) + C( Y ( Y ), PS ) C( Y ( Y ), PS ) D D C( Y ( Y ), PS ) C Y = SC =. (9) Note that the arguments n C ( Y, PS) n Equaton (9) are lkel never evaluated at zero, as opposed to what happens b defnton wth C ( ) n (). Ths occurs because aggregates (such as total passengers or ton-klometres) do not go to zero when onl some OD flows are zero, as s the case wth Y A or Y B n Fgure 4. The problem s then reduced to stud the behavour of the aggregates under dfferent orthogonal parttons of Y, when possble. It s mportant to explan that the calculaton of equaton (9) can be seen from dfferent perspectves. For example, f one knows Y ( Y A, PS A ) and the functons ( Y h h ), the problem becomes dentfng Y D, Y B (whch must be orthogonal to Y A ), PS D and PS B, n order to generate Y ( Y B B B ), C ( Y ( Y ), PS ), ( D D D Y Y ) C ( Y ( Y ), PS ). Analtcall, the challenge s to fnd out a sstem of equatons that allows these calculatons. The prevous dscusson provdes a wa to face the problem of calculatng SC, but t does not solve everthng. Analses of several cases partcularl the one presented b Basso and Jara-Díaz (2005) have shown the necesst to assume, n occasons, condtons on the new flows that appear after the network expanson; for example, to sa somethng explct about how the nonl flows n Y B compare to those n Y A n Fgure 4. Ths s not somethng new, because n RTS there are related (but mplct) condtons through the constant denst mposton, as we have shown. When these sorts of condtons are needed, we have deemed reasonable to mpose that the new flows wll be, n average, smlar to the exstng ones. Ths s, precsel, what some authors (e.g. Braeutgam, 999) suggested t was mposed b RTS. For snthess, Table shows whch ndcator should be calculated wth the methods presented n ths artcle, and under what condtons. We beleve that these delver correct nferences about the technologcal characterstcs of transport ndustres from cost functons estmated wth aggregate product. Table. Condtons for the calculatons of cost structure ndces n transport Fxed route structure Varable route structure Fxed network sze Returns to denst (RTD) Multoutput degree of economes of scale (S) Varable network sze Economes of spatal scope (SC) 4. EMPIRICAL EVIDENCE AND POLICY CONCLUSIONS. We establshed n the prevous Secton that proper calculatons of RTD and S requre estmaton of the α coeffcents. In our analss of the US arlne ndustr (Basso and Jara-Díaz, 2006a), for the average length of trp, we used smple log-lnear equatons to estmate α ALT from annual nformaton of fve arlnes n the perod The coeffcents obtaned where used together
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