Allocating fixed costs in the postal sector in the presence of changing letter and parcel volumes: applied in outdoor delivery

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1 IDEI- 764 February 3 Allocatng ed costs n the postal sector n the presence o changng letter and parcel volumes: appled n outdoor delvery P.De Donder H.remer P.Dudley and F.Rodrguez

2 Allocatng ed costs n the postal sector n the presence o changng letter and parcel volumes: appled n outdoor delvery. Introducton P De Donder H remer 3 P Dudley 4 F Rodrguez 5 he m o volumes handled by the Unversal Servce Provder (USP) s changng. Letter mal volumes are n declne and parcel volumes are growng. hese trends are epected to contnue wth the e-substtuton o letter mal and the e-commerce o parcels. urrent costng methods are based on a letter mals market and not set up to take account o ths change. Wth the current method the market trends lead to the transer o common and jont costs to parcels and consequently an ever ncreasng allocaton o ully allocated costs per tem wthn parcels. he purpose o ths paper s to look at the propertes o the cost uncton that would provde a more ntutve and potentally more approprate cost allocaton n the changng crcumstances. ost allocaton s ncluded as part o the European Postal Servces Drectve Artcle 4.3. hs sets out that costs whch can be drectly assgned to a partcular servce or product shall be so assgned ; and common costs that s costs whch cannot be drectly assgned to a partcular servce or product shall be allocated n the ollowng manner () whenever possble common costs shall be allocated on the bass o drect analyss o the orgn o the costs themselves; () when drect analyss s not possble common cost categores shall be allocated on the bass o an ndrect lnkage to another cost category or group o cost categores or whch a drect assgnment or allocaton s possble; the ndrect lnkage shall be based on comparable cost structures; () when nether drect nor ndrect measures o cost allocaton can be ound the cost category shall be allocated on the bass o a general allocator computed by usng the rato o all epenses drectly or ndrectly assgned or allocated on the one hand to each o the unversal servces and on the other hand to the other servces; (v) common costs whch are necessary or the provson o both unversal servces and non-unversal servces shall be allocated approprately; the same cost drvers must be appled to both unversal servces and non-unversal servces. Wthn the postal sector Robnson and McMurde (9) developed an operatonal model and method or the allocaton o cost usng the drvers o class and products wthn the unversal servce; ther approach allocated proportonately more cost to hgher class and Unversal Servce Oblgaton (USO) products. De Donder et al () modelled parcels wthn the USP n the contet o competton wthn the delvery to rural areas; n subsequent papers by De Donder et al (8) the USP s charactersed as havng ed costs and margnal costs or the provson o ts servces. Mautno et al (3) developed an approach to estmate long run average ncremental costs or the postal sector. hs ncluded an eplanaton o the economes o scale and scope ound wthn outdoor collecton and delvery; there are jont costs n outdoor delvery whch mean he analyss contaned n ths paper relects the vew o the authors and may not necessarly be those o the nsttutons that employ them. orrespondng author oulouse School o Economcs (GREMAQ-NRS and IDEI) phlppe.dedonder@tse-r.eu 3 oulouse School o Economcs (GREMAQ Insttut unverstare de France and IDEI). 4 Head o Economcs Royal Mal Group. 5 Assocate Oera.

3 that the ncremental cost o an addtonal tem s oten assocated wth the lkelhood o havng to delver and thereore walk up the path to an address; the ncremental cost o delverng an tem reduces wth ncreased volume as the proporton o addresses vsted ncreases at a decreasng rate. hs paper develops a model wth welare mamsaton by the USP wth separate goods subject to a budget constrant where there are margnal jont and ed costs whch take account o the economes o scale and scope n delvery. Under the Postal Servces Drectve rst drect costs are dented or goods second ndrect attrbutable costs are allocated and nally ndrect nonattrbutable costs are to be apportoned usng EPMU. In the contet o the model the drect costs are the margnal costs the ndrect attrbutable costs are the jont costs and addtonal unt costs and the ndrect non-attrbutable costs are the resdual ed costs (.e. the ed costs less the margnal and jont (unt) costs). o ocus on the prncples o cost allocaton the model has two goods o letters and parcels n outdoor delvery; the same model could be etended to nclude more dmensons but the outdoor delvery s a sgncant component o the USP and s known to have sgncant economes o scale and scope. In the rst two stages o analyss the cost uncton ncludes the margnal and ed costs or letters and parcels and n the nal stage addtonally a jont cost component. he margnal costs ncrease at a decreasng rate and the jont cost s such that an ncrease n volume o one good ncreases the cost o the other good but at a decreasng rate. Hence both margnal and jont costs relect the presence o economes o scale. More speccally the ntal analyss assumes that the margnal costs or letters and parcels are separate and decreasng n volume wth the same mode o delvery method. hs s sucent to obtan an outcome whereby the margnal cost o letters ncreases and that or parcels reduces as letter volumes reduce and parcels ncrease such that the ed cost (allocated under constant demand elastcty assumptons or equ-proportonal mark up (EPMU)) leads to the unt cost o letters rsng more than parcels. hs s llustrated by a numercal eample. he llustraton s then etended to consder the case where parcels are more demand elastc than letters such that under Ramsey rules the movements n unt costs or letters and parcels are urther enhanced. he second analyss takes account o two delvery methods or letters and parcels one o ed capacty trolleys known as Hgh apacty rolleys (Hs) that are used on walks to carry letters and some parcels and one o a hgher capacty vans whch lend themselves more to parcels delvery. he letters are assumed to have prorty use o the Hs and the resdual space on a H s used or parcels wth resdual parcels then delvered by van. When the cost unctonal orm s obb Douglas or a sngle route only the results have smlar propertes to those n the ntal analyss. he thrd analyss develops the second analyss urther to nclude multple routes where there s a dstrbuton or the porton o H ll by letters by route. When the cost unctonal orm s obb Douglas assumng multple routes the results agan can have smlar propertes to those n the ntal analyss but n ths nstance there are not only margnal and ed costs but also jont costs wth the cost uncton havng cross elastctes o ts costs. he analytcal model s set out and developed n Secton. he llustratons are ncluded n Secton 3. Secton 4 concludes.

4 . Models. Frst model For the model there s one rm and two products and. One could be packets and the other one letters. he demands or the two products are ndependent o each other and gven by p and p where denotes the quantty and p the prce o good. hese demands are obtaned when consumers mamze ther utlty U p p U or gven p and p wth respect to and. he cost uncton o the rm has both varable and ed components and components that can be attrbuted to one good versus overheads. Wth no jont cost and decreasng margnal costs the total cost uncton o the rm s gven by: wth decreasng margnal costs: c F c F F c or =. he rm mamses welare subject to some prot constrant where prot gven by has to be larger than some mnmum amount. he Lagrangan epresson o ths problem s L U When mamzng L wth respect to p p U p p. p the ollowng Ramsey epresson or good s obtaned p / p where s the (absolute value o the) own-prce elastcty o the demand or good and where snce margnal cost s decreasng. c c c Observe that snce margnal cost s decreasng the Ramsey ormula wth generates prces (equal to margnal costs) that do not even cover varable costs. 3

5 I the volume o good decreases or an eogenous reason (meanng or any gven prce the demand s now lower) whle the demand or good ncreases than margnal costs wll ncrease or good whle they wll decrease or good meanng a larger prce o good and a smaller prce or good.. Second model he rst model s developed to account or two modes o delvery. Parcels are delvered jontly wth letters usng Hgh apacty rolleys (H). When the capacty constrant o the H s bndng the parcels that cannot t are delvered wth a van. For smplcty t s assumed that there s a sngle aggregate route that the H capacty s large enough to delver all letters but not large enough to accommodate all parcels. he volume o parcels delvered wth H s denoted by denoted by wth. 4 and the volume o parcels delvered by van s he capacty per tem taken by a letter n the H by a and the capacty per tem taken by a parcel by b wth b a. he capacty constrant o the H s a b and t s assumed that ths constrant s bndng (.e. holds wth equalty). he ollowng epresson s then obtaned and he cost uncton or vans s denoted by K K a () b K a () b c F and t s assumed that there are ncreasng returns to scale so that margnal costs are decreasng: he cost uncton or Hs s When combned wth () ths yelds c. F

6 ( K a ) b F / Observe that ths cost s only a uncton o wth and ( K a ) b K a / b K ( K a ) / b b so that margnal costs are decreasng. Hence ths has smlar propertes to the rst model wth no jont costs but decreasng margnal costs. he role played by the number o letters s more comple because havng more letters shts parcels rom H to vans. he parameters used nclude: K a and b (the capacty o a H n terms o both parcels and letters) the shape o c (the varable cost uncton or vans); and F and F (the ed costs o vans and trolleys)..3 hrd model In a urther development o the second model t s assumed that some Hs on some routes are not ull so that the total volume o parcels delvered by H ncreases wth the total amount o parcels delvered. As volumes ncrease the capacty constrant on Hs becomes bndng or an ncreasng racton o routes so that the relatonshp between and s concave and gven by the uncton wth and. he ntuton or ths assumpton runs as ollows. When s very small there s enough space on Hs on every route to accommodate parcels so that and. As parcels volumes ncrease the capacty constrant or Hs becomes bndng on an ncreasng number o routes so that n the aggregate and. At the lmt when volumes are very large Hs are saturated on all routes and nk (where K s the capacty constrant o an ndvdual H and where n s the number o routes) so that. It s assumed that the operaton o volumes s such that the capacty constrant o the Hs s bndng on some but not all routes so that and ". Usng the same ormulaton as n the Secton. or the cost unctons the ollowng s obtaned: Whle we smply the van delvery cost to have F 5

7 6 F c Where c s the constant margnal delvery cost wth vans. Denotng total cost by the ollowng s obtaned / ) ( so that a jont cost uncton s obtaned wth postve cross dervatve. In addton ) ( so that the total cost s ncreasng and concave n the number o letters. As or the mpact o the number o parcels we obtan ) ( c where the rst term s the margnal H cost and the second term the margnal van cost and where both terms are postve snce havng more parcels ncreases the number o parcels delvered both wth Hs and wth vans. onsequently ) " ( " c where the rst term pertans to the H cost uncton and s negatve (so that ths part o the uncton s concave) whle the second term pertans to the van cost uncton and s postve (so that ths part o the cost uncton s conve). he aggregate cost uncton s then concave n the number o parcels provded that the second term s not too postve (.e. that c s not too large). In that case the aggregate cost uncton s ncreasng and concave separately n both the number o parcels and o letters and ehbts a postve rst order cross-dervatve. hese are the same propertes as those o a smple obb-douglas uncton..

8 3. Illustraton For the rst model analyss n Secton. a numercal eample s developed assumng an so-elastc demand uncton p k p r and where costs are gven by so that c / and c / / / Good s twce more epensve than good or any margnal costs are decreasng but total costs are ncreasng. he same elastcty s assumed or both goods: r r. 3. he parameters k n the demand unctons are trend parameters. hese aect the sze o the demand wthout changng the senstvty o the demand unctons to prce (the demand elastctes). For smplcty t s assumed that F F F. able shows the optmal values o p and p when k k when (so that ) and when s obtaned endogenously so that. able : optmal values o p and p when k k when (so that ) and when s obtaned endogenously so that. = p.44 p As eplaned above even when there s no ed costs when s set eogenously at. When s set endogenously so that prces ncrease and we have p p snce good s more epensve. Observe that p s more than twce p (even though t costs eactly twce as much or the same quantty) because hgher prces translate nto lower quanttes and thus nto larger margnal costs. When k =.7 and k =. the demand elastctes are unchanged but the ncrease n the quantty o good s smaller than the decrease n the quantty demanded o good or any prce p p p. he results are reported n able. 7

9 able : optmal values o p and p when k. 7 and k. when (so that ) and when s obtaned endogenously so that. = p.55.3 p ompared to able p ncreases whle p decreases (both wth and wth endogenous ) because lower quanttes ncrease margnal costs or good whle the opposte occurs or good. In contrast t s assumed that parcels are more prce elastc than letters such that r. 3 and r the equvalent results to ables and are shown n ables 3 and 4 respectvely. able 3: optmal values o p and p when k k when (so that ) and when s obtaned endogenously so that wth more elastc demand or parcels. = p p able 4: optmal values o p and p when k. 7 and k. when (so that ) and when s obtaned endogenously so that wth more elastc demand or parcels. = p p Increasng r does not change the equlbrum prces wth when k k but slghtly decreases p when k k. Observe that there s margnal cost prcng when. he reason why the equlbrum prce changes only margnally s because the margnal cost s a uncton o quantty and the quantty demanded s aected by a varaton o r ecept n the specal case where p whch happens when k k. 8

10 When ncreasng r ncreases the prce o the prce-nelastc good (letter) and decreases the prce o the prce-elastc good (parcel). Hence p p even though good s twce more costly to delver than good when he varaton o k s has the same qualtatve mpact on equlbrum prces or the two conguratons o demand elastctes. Observe that the Ramsey prces wth endogenous reported n ables and are also EPMU prces here because the mark-up over margnal costs s proportonal to the same elastcty (wth r r ). able 5 compares the EPMU prces wth the Ramsey prces when the elastctes der. Wth EPMU the mark-up over margnal cost s eactly % or both goods. Good s twce as cheap to delver as good (when ). Wth p p ths cost derence s urther magned because the margnal costs decrease wth volume (when ) such that p 4 p under EPMU. Hence an EPMU rule or the recovery and allocaton o ed costs leads to a sgncantly hgher cost allocaton nto good n able 5 when compared to the results shown n the prevous tables. 4. oncluson able 5: recovery o ed costs under EMPU optmal values o p and p wth more elastc demand or parcels. EPMU p p 4 - k k r.3 r he postal market s changng such that letter volumes are n declne and parcel volumes ncreasng. urrent costng methods may not take account o ths change and lead unt costs or parcels ncreasng dsproportonately. As parcel volume ncreases and letter volume declnes t s possble that the porton o ed cost attrbuted to parcels ncreases stays the same or decreases dependng on the specc cost uncton and ts calbraton. he analyss shows how ths may be addressed through the dentcaton and applcaton o approprate cost unctons. he rst analyss assumes that the margnal costs o letters and packets are separate and decreasng n volume. Wth letter volumes decreasng and parcel volumes ncreasng ths unctonalty would ncrease the unt cost or letters relatve to parcels. However ths s a smple analyss. he second analyss ntroduces two delvery methods (e.g. Hs and vans) or letters and parcels and a obb Douglas unctonal orm wth prorty or Hs gven to letters. For a sngle route the results have smlar propertes whereby wth letter volumes decreasng and parcel volumes ncreasng the same unctonal orm or costs would ncrease the unt cost or letters relatvely more than parcels. However ths analyss assumes a sngle route. Hence the thrd analyss ntroduces multple routes and varaton n the share o H capacty dedcated to letters wthn a obb Douglas unctonal orm. Whle the analytc calculatons are more comple and ntroduce cross cost elastctes and thereore jont costs the result agan can be obtaned whereby 9

11 wth letter volumes decreasng and parcel volumes ncreasng the unctonal orm would ncrease the unt cost or letters relatvely more than parcels. Hence n summary our analyss dentes the unctonal orm or supply sde cost unctons that has the property o allocatng proportonally more o the ed cost to letters and less to parcels market trends contnue. hs outcome occurs smply through the supply sde cost uncton that s when the demand elastctes or letters and parcels are assumed to be the same or there s an EPMU mark up on margnal & jont costs. he drecton o the cost allocaton could be urther strengthened rom the demand sde under Ramsey rules where the demand elastcty or parcels s greater than that or letters as llustrated n ths paper. A obb Douglas cost uncton s a partcular orm that assumes constant cost elastctes or the two goods and a partcular relatonshp between those two elastctes. It may be approprate through emprcal analyss to assess whether the cost unctons have a obb Douglas unctonal orm and what ts parameter values look lke n practce. Publshed econometrc models n delvery to-date have tended to ocus on orms wth constant cost elastctes n volume but have not consdered ths specc pont. Rather than provde numercal estmates ths paper assumes values to llustrate through smple eamples or the rst analyss and dentes the parameters that would need to be valued rom emprcal analyss. hat s to say t asssts n dentyng what to look or wthn the emprcal analyss or the cost uncton to have the sad propertes. In terms o the uture development and applcaton o ths type o model ts ocus here has been on two goods and two delvery methods n outdoor delvery. Outdoor delvery comprses a sgncant porton o overall USP costs ts daly provson s a sgncant component wthn the unversal servce provson and there s known to be consderable economes o scale and scope. he model could be epanded to consder more goods and delvery methods. Further consderaton could also be gven to consder whether obb Douglas cost unctons apply n other parts o the ppelne. Moreover consderaton also could be gven as to whether the USP s goods contnue to recover ther margnal and jont costs n the presence o competton to the USP rom alternatve supplers. Reerences De Donder P. remer H. and Rodrguez F. () Access Prcng and Parcels Delvery n Postal and Delvery Servces: delverng on competton ed. rew and Klendorer Kluwer Academc Publshers. De Donder P. remer H. Dudley P. Rodrguez F. (8) Prcng welare and organzatonal constrants or postal operators n ompetton and Regulaton n the Postal and Delvery Sector ed. rew and Klendorer Edward Edgar. Mautno L. Dudley P. Prettyman J. and Heagney F. (3) Estmatng long run ncremental costs n the postal sector: A UK perspectve n Reormng the Postal Sector n the Face o Electronc ompetton ed. rew and Klendorer Edward Edgar. Robnson R. and McMurde J. (9) Postal costng beyond AB: estmatng the economc cost o mal servces n Progress n the ompettve Agenda n the Postal and Delvery Sector ed. M rew and P R Klendorer Edward Edgar.