COST ALLOCATION IN PUBLIC ENTERPRISES: THE CORE AND ISSUES OF CROSS-SUBSIDIZATION. Haralambos D Sourbis*

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1 COST ALLOCATION IN PUBLIC ENTERPRISES: THE CORE AND ISSUES OF CROSS-SUBSIDIZATION By Haralambos D Sourbs* Abstract Ths paper examnes the mplcatons of core allocatons on the provson of a servce to a communty and the correspondng cost allocaton schemes and ndustral structures. In the process t s shown that the exstence of such allocatons s nether a necessary nor a suffcent condton for the exstence of subsdy-free cost allocaton schemes. In addton, we propose a set of condtons that wll make any noton of cross-subsdzaton "precse" and we ntroduce such notons. May 1985 *Unversty of Florda. I wsh to thank the Publc Utlty Research Center, Unversty of Florda, for ts support of my research, and Sanford Berg for hs comments, and suggestons, and for some very valuable conversatons I had wth hm. I have also benefted from comments by Vrgna Wlcox-Gok on an earler draft of ths paper.

2 1. Introducton The provson of a servce to a gven communty rases a number of ssues ncludng (a) how the members of the communty wll share the relevant costs, and (b) how the provson of the servce wll be organzed. Large overhead costs, common costs, jont costs, and long-run average costs that decrease over a relatvely wde range of output, are some of the reasons often cted as part of the problem. Ths has prompted a number of authors to use a varety of soluton concepts to n-person cooperatve games, such as the Shapley value, the core, and the nucleolus, to study some of those ssues. 1 Whle a game-theoretc approach can provde, useful nsghts n addressng certan problems assocated wth the provson of a servce, t would seem that some soluton concepts do not necessarly have all those attrbutes reported n the lterature. A case n pont s the core. In an approprately defned cooperatve game, the core can reveal what arrangements (output of the servce, cost allocaton schemes, and ndustral structures), f any, can be consdered stable, n a specfc sense, for a gven communty wth certan characterstcs. But, does the core reveal whether the correspondng cost allocaton schemes are subsdy-free as, for example, Faulhaber (1975), Lttlechld (1975), and Sharkey (1982a,b) have clamed? Suppose, for example, that a communty conssts of members who are dental to each other n all respects~ the same quantty of the servce s provded to each member under dentcal cost arrangements, and the resultng allocaton s Pareto optmal for the communty. To conclude that the correspondng cost allocaton scheme s 1See, e.g., the relevant sectons n Schotter and Schwodauer (1980), Sandler and Tschrhart (1980), Sharkey (1982b), and the varous references theren. 2

3 subsdy-free only f ths s a core-allocaton, but not subsdy-free otherwse, t wll be equvalent to confusng stablty (n ths case, core-stablty) wth cross-subsdzaton. No matter how vague a noton of cross-subsdzaton s, one should expect that n a stuaton lke the above there s not cross-subsdzaton. That the core of an approprately defned cooperatve game may not be suffcent to determne whether, n the provson of a servce, cost allocaton schemes are subsdy-free, rases two mportant questons: (a) what are the mplcatons of core allocatons on () the provson of a servce, () cost allocaton schemes, and () ndustral structures? (b) what s cross-subsdzaton? To address these questons the next secton ntroduces a model that:. (a) takes nto account the ncome and preferences of each member n the conununty (thus, questons about "ablty-to-pay" (Lttlechld (1975)), or "wllngness to pay" (Sharkey (1982a)) can be addressed wthn the model); and (b) s general enough so that t can be appled to many stuatons (e.g., mult-product frms, publcly owned or regulated enterprses, etc.). Secton 3 consders the model as an n-person cooperatve game. In Secton 4, core-allocatons are characterzed relatve to ther mplcatons ~or e.ffcency and stablty of outputs, cost allocaton schemes and ndustral structures. The same secton addresses questons of ' 'sustanablty" of partcular market structures. Secton 5 utlzes an ntutve noton of cross...subsdzaton to show that the exstence of core-allocatons s nether a necessary nor a suffcent condton for the exstence of subsdy-free cost allocaton schemes. A defnton of crosssubsdzaton s proposed n Secton 6. In partcular, we use some of the prevous arguments (from Secton 5) to ntroduce two axoms that wll make 3

4 any noton of cross-subsdzaton "precse." Ths leads to a seres of condtons that any "precse" and "complete" noton of cross-subsdzaton should not volate. Fnally, n the same secton, we utlze the prncple that a cost allocaton scheme can be consdered subsdy-free only f, whenever possble, those who beneft from the servce cover ts costs, n order to set up one of the weakest tests of cross-subsdzaton that wll satsfy our condtons. 2. A Model for the Provson of a Servce to a C6IlIlunty We are nterested n a communty whose members wll be ndexed by the fnte set N = {l,,n}. N can be nterpreted, alternatvely, as the set of ndvduals, households, localtes, neghborhoods, etc., n the communty. Ths s a sngle-perod model, where I wll represent the nonnegatve ncome of the -th member n the communty, E N. Each member n ths communty derves utlty from the consumpton of two commodtes n quanttes x and y, respectvely. Commodty x s a composte commodty. We are not gong to specfy what commodty y s, but we shall refer to t, from now on, as a servce. One can thnk of t as representng~ (a) the servce of a "publc enterprse" such as electrcty, water, natural gas, and telephone~ or (b) an excludable publc good, or (c) a dfferentated product (dfferentated as to neghborhoods, localtes, groups of users, etc.), or (d) the product of a mult-product enterprse, or even (e) an ordnary prvate good. For each E N, we shall denote by u (x,y ) that members utlty functon, and we assume that u (.) s an ncreasng functon of x and a non-decreasng functon of y 4

5 Any quantty of the composte commodty can be purchased freely by each ndvdual member n the communty at the prce of one unt of ncome per unt of that commodty. However, ntally, n the communty that we examne, there are no enterprses that wll provde the servce on demand and specal arrangements have to be made by the communty for that purpose. Nonetheless, each group S c N, of sze s,.e., S = {l,,s}, can make ts own, separate, arrangements f or t h e provlslon.. f S 0 a vector y of t h e servlce. to lts. members. 2 1 s (y,, y ) Although a producton process for the servce could be specfed, here we shall deal drectly wth costs. In terms of unts of ncome, C(yS;S) wll denote the cost that must be borne by group S f that group, actng unlaterally, s to provde ts members wth the vector of the S 3 4 servce y. We make no assumpton about subaddtvty of costs. It s assumed that: (a) C(yS;S) > 0, V S c N, f ys s such that a postve quantty of the servce s provded to some member n S, and (b) C(yS;S) 0, V S ~ N, f ys = O. 2Ths assumpton can be nterpreted as sayng that each group S ~ N has the technology to produce vectors ys of the servce for ts members. Thus, the model can be seen as a model of "voluntary clubs." Alternatvely, we can follow Sharkey (1982b) and say that there s a "pool of potental frms" each of whch can be commssoned by a dfferent group n the communty to provde the members of the latter wth the servce under specfed terms. In turn, those terms may nclude a fxed rate of proft for those frms. 3The ncluson of the varable S n the cost functon ndcates that group composton (ncludng sze) may be an mportant determnant of costs. For example, n addton to costs and benefts that may result from group sze, some servces may nvolve costs or savngs relatve to locaton, dstances, etc., specfc to the partcular members ncluded n a group. 4A cost functon s subaddtve f and only f C(yS;S) + C(yS*;S*) ~ (S U S*) - C(y ;(S U S*», for any S, S* C N such that S l S* = 0, and for any ys and ys*. Although our model wll be useful only f there are some benefts from cooperaton, at least to some groups, these benefts do not have to extend to the unon of every dsjont par of groups n order for our analyss to hold. 5

6 For a group S S N that chooses to act ndependently, a consumpton vector, of the composte commodty and the servce, wrtten as a par (xs,ys), s attanable f and only f (1) x >0, Y ~ 0, VE S, and I E S I E S x, Let us observe that by not assumng subaddtvty of costs, what s attanable by the communty as a whole may not be the same as what s attanable by the communty actng as a sngle group for the provson of the servce. Thus, let us denote by the Greek letter T a partton of the communty nto dsjont groups, and let us denote by T the set of all such parttons. n th~ An allocaton of the composte commodty and the servce, 1 n communty, conssts of a par (x,y) where x = (x,,x ) and 1 n y = (y,,y). An allocaton (x,y) s feasble f and only f (2) x > 0, y > 0, V E N, and I S I > Mn { I C(y ;S)} + I x, E N T E T S E T E N where ys s the subvector of elements n y that correspond to the members of group S. Our prevous observaton, then, s equvalent to sayng that allocatons (x,y) that satsfy (2) wll not necessarly satsfy (1) wth respect to N. Let Y denote the set of all feasble allocatons for the communty. Then, Y*, where (3) Y* {(x,y) E Y: I I = Mn E N T E T { I SET C(yS;S)} + I x} E N wll represent the set of effcent allocatons. 6

7 The actual costs realzed by the entre communty for obtanng a vector of the servce y, ndependently of what arrangements have brought that vector, wll be denoted by C(y), C(y) > O. A cost allocaton scheme relatve to C(y), denoted by c(y), conssts of a dstrbuton of the total actual costs C(y) among the members of the communty. Thus, (4) 1 n c(y) = (c (y),,c (y» where (5) I E: N c (y) C(y). Assumng that the communty must fnance on ts own the costs of the servae, we can restrct our attenton to feasble allocatons (x,y) E: Y, and cost allocaton schemes c(y) such that (6) I E: N I E: N c (y) + I E: N x Furthermore, wthn these bounds of feasblty, and under the same assumpton, there s a one to one correspondence between an effcent allocaton (x*,y*) and a choce of (a) the vector of the servce y*, and (b) the cost allocaton scheme c*(y*) that lead to ths allocaton. In other words~ gven an effcent allocaton (x*,y*) there corresponds a unque cost allocaton scheme c* (y~'() consstent wth ths allocaton (.e., where the communty fnances on ts own the servce) gven by (7) c* (y*) * x, for each E: N. Conversely, 'gven y*, and gven that I > C(y*), the only cost E: N allocaton scheme that leads to an effcent allocaton (x*,y*) E: Y*, s 7

8 the one gven by (7). In both cases we can check that the respectve cost allocaton scheme c*(y*) must satsfy (8) Mn T E T { I SET S C(y* ;S)}. An mplcaton of these conclusons s the followng. For any allocaton (x*,y*) E y* we can always fnd out what the correspondng cost allocaton scheme s from (7). Conversely, gven y*, and gven a cost allocaton scheme c*(y*) that satsfes (8), and such that E N I c*\y*), we can always fnd the correspondng effcent allocaton E N (x*,y*), agan from (7). Another mplcaton of the above conclusons s that we can extend the nqton of effcency appled to allocatns n the set Y*, to hold for the correspondng cost allocaton schemes. Thus, for a gven vector of the servce y*, we wll call a cost allocaton scheme c*(y*) effcent f and only f the correspondng consumpton vector x* obtanable from (7) leads to an allocaton (x*,y*) E Y*. Let us observe now that an effcent cost allocaton scheme c*(y*) must satsfy (8) relatve to y*. Furthermore, n our model (see footnote 2) a partton T of the communty nto dsjont groups, for obtanng a vector y* of the servce, can be nterpreted as representng the ndustral structure that provdes that vector of the servce to the communty. In other words, gven T, we can dentfy each SET wth a frm that serves exclusvely the members of S. Therefore, we can refer to any partton T* that solves the mnmzaton problem, (9) Mnmze: I C(yS;S), gven y, T E T SET 8

9 as representng an effcent ndustral structure for the provson of the vector y of the servce to the communty.5 Assumng that a postve amount of the servce s provded to each member n the communty, there s a vector of mplct "prces" that corresponds to each cost allocaton scheme. 1 n For example, f y = (y,,y ) represents a vector of the servce, y > 0, VE N, and f c(y) s a cost allocaton scheme, then, p(y), where (10) p(y) 1 n (p (y).",p (y)) and where, (11) p (y) c (y) y for each E N, wll represent ths vector of mplct "prces." However, we should pont out that, n our model, a cost allocaton scheme may represent a vector of fxed changes determned through some barganng process among the members of the communty. Therefore, the term "prces" may be napproprate n our model. Only ex post we can nfer to a p(y) as prces. Elsewhere n the lterature (see, e.g., Faulhaber (1975)), a vector of mplct prces p(y) s called a "prce structure.' ' Because a vector of mplct prces s dervable drectly from a cost allocaton scheme~ we wll refer to a p(y)~ whenever defned~ as an effcent vector of mplct prces or an effcent 'prce structure, whenever the correspondng cost allocaton scheme c(y) s effcent. 5Note that a soluton to (9) depends on the partcular vector of the servce y. Therefore, t s not necessary that the same ndustral structure should be effcent ndependently of y. 9

10 Fnally, let us conclude ths secton by observng that an allocaton (x,y) s Pareto optmal for the communty f and only f: (a) (x,y) y,.e., t s feasble, and (b) there does not exst a feasble allocaton (x*,y*) c Y suc'h that (, *) ( ) u 0 ~ u xx,y ~ u x,y, T 1 E: N, and f or at 1east one j E: N, uj(x*j,y*j) > uj(xj,yj). However, each utlty functon u (.) has been assumed to be ncreasng n at least the varable x Therefore, any Pareto optmal allocaton s effcent, that s, t belongs to the set y-;'c. 3. The Model as an n-person Cooperatve Game The model presented n the'precedng secton s equvalent to an n-person cooperatve game where the set N represents the set of players, each &ubset 5 ~N represents a coalton, and an algnment of the members of the communty nto a set of dsjont groups T represents a coalton structure. In terms of consumpton vectors, what each coalton n ths game can attan for ts members, f t acts ndependently, can be represented by the set of such vectors that satsfy condton (1). We shall 5 denote ths set by Y, for each 5 ~ N. Thus, (12) S 5 {(x,y): condton (1) s satsfed}. Let us observe now that for each coalton 5 ~ N, and for each -S -s s -s consumpton vector (x,y ) = «x,y ),,(x,y» attanable by that h (-S -S) S h d f 0 0 coa1ltl0n, l.e., or eac x,y E: Y, t ere correspon s a vector of (-1 -s) Utl ltles u= u,,u, where, for each E: 5, u = u'(x,y ). Therefore, for each coalton S ~ N, there s a set of attanable utlty vectors assocated wth the set y S that we shall denote by V(S). Thus, (13) V(S) -S,-1 -s I' -s-s {u = (u,,u): (x,y) = «x,y ),,(x,y», -S-5 S (x,y ) Y, and u = u (x,y ), for each E: S}. 10

11 In game-theoretc terms, then, we can say that our model s equvalent to an n-person cooperatve game n characterstc functon form (N,V), where V s the set valued functon defned for each SeN, by (12), and t repre 6 sents the characterstc functon of the game. Soluton concepts for a game (N,V) are defned over the space of attanable utlty vectors for the set of players N, that s, the set of utlty vectors that correspond to the set of feasble allocatons Y. However, because our nterest here les wth the set of allocatons that correspond to such utlty vectors, we wll proceed to examne drectly such allocatons. In partcular, here we shall examne the set of allocatons correspondng to the core of the game (N,V), that s, the set of core allocatons for the communty. Th us, 1 et ( x,y - -) = «_I x,y _I),, (_n x,y_n» anx,y d (--) = «-1-1) x,y,, (-n x,y -n» be any two feasble allocatons n the set Y, and let (xs,ys) = «x 1,yl),,(XS,Ys» be the subvector of elements of (x,y) correspondng to the members of some coalton SeN. We shall say that the allocaton {x,y) domnates the allocaton (x,y) va coalton S, and we shall wrte (x,y) dom S (x,y), f and only f (a) S ~ 0, (b) (xs,ys) E ys,.e., (xs,ys) s -- attanable for coalton S, (c) u(x,y ) ~ u (x,y ), VE S,.e., the allocaton (x,y) provdes each member of coalton S at least the same level of utlty as the allocaton (x,y), and (c) uj(xj,yj) > ujexj,yj), for some j E S,.e., at least one member.:j,of. coalton S s better 6In the game (N,V) t s not assumed that utlty s transferable. However, a cooperatve game wth transferable utlty could have been obtaned from our model, say the game (N,v), by postulaton that, for each S d N, v(s), where v(s) = Max {I u }, represents the value - us E V(S) E S of that coalton. In general, these two games wll not be equvalent because utlty vectors that are attanable under the game (N,v) may not be attanable under the game (N,V). 11

12 off wth the consumpton bundle (xj,yj) than what he s wth the consumpton bundle (xj,yj). We shall say that the allocaton (x,y) domnates the allocaton (,y), (wthout reference to any specfc coalton), and we shall wrte (x,y) dam (,y), f and only f there exsts a non-empty coalton SeN such that (x,y) dams (,y). The set of core-allocatons for the communty, denoted by the letter K, conssts of all feasble allocatons that are not domnated by other feasble allocatons,.e., (14) K { (x,y) E: Y: It (x, y) E: Y, 1- : (x, y) dom (x, y) } In dfferent terms we can say that the set of core-allocatons for the communty conssts of all those feasble allocatons such that no group n the communty, actng unlateraly, can attan an alternatve consumpton vector that wll make at least one of the members better off whle all other members of the group reman at least ndfferent. In ths sense then, we can say that a core allocaton s stable because no coalton of players has: the ncentve to depart from t. From the defnton of the bnary relatonshp dam, above, and the defnton of Pareto optmalty n the precedng secton, t s obvous that core allocatons, f any, are Pareto optmal, and thus, effcent. Therefore, K c Y*. 4~ Core Allocatons and the Provsonof'aServce Assumng that for the communty under consderaton the set of core allocatons K s not empty (somethng that s not always guaranteed) we can draw several conclusons about the provson of the correspondng vectors of the servce. That s, conclusons about cost allocaton schemes, prce structures, and ndustral structures. 12

13 Because core-allocatons are effcent,.e., K=Y*, we can utlze (7) to fnd the correspondng, to each element n K, cost allocaton scheme for the servce. Let, (15) KG 1 n {c(y) = (c (y),,c (y)): (x,y) E K, and c(y) = I - x, for each EN}. Then, KG represents all cost allocaton schemes for the servce correspondng to the set of core-allocatons K. From now on, we shall refer to KG as the set of core cost allocaton schemes for the servce n the communty. In a smlar manner, and provded that for each allocaton (x,y) E K, y > 0, VE N, we can utlze (11) to obtan the set of core prce structures for the servce, KP, where, (16) KP 1 n = {p (y) = ( p (y),, p (y): and p(y) = c ~y), for each 1: y c(y) E E N}. KG, What does t mean for a cost allocaton scheme c(y) to belong to the set KG KP)? (or the correspondng prce structure p(y) to belong to the set In the frst place, t means that the correspondng vector of the servce y has been obtaned by the communty effcently, that s at the mnmum total cost possble, whch follows from the effcency of core allocatons. Thus, n the termnology of Secton 2, the elements of the set KG are effcent cost allocaton schemes, and the elements of the set KP are effcent prce structures. Furthermore, ths noton of effcency extends to the coalton structures formed by the communty n obtanng the servce. Thus, for each cost allocaton scheme c(y) E KG, there exsts some coalton structure T* that solves the mnmzaton problem n (9), that represents the ndustral structure through whch the communty can obtan the correspondng vector of the servce y effcently. 13

14 However, we should pont out that the exstence of core-allocatons s not a necessary condton for obtanng effcency of (a) cost allocaton schemes, (b) prce structures, and (c) ndustral structures. One could obtan such effcency drectly from an allocaton n the set Y*. It s the stablty propertes of core-allocatons, then, that s more mportant, In partcular, because, as we have seen n the precedng secton, no coalton has the ncentve to move away from a coreallocaton (x,y) s K, the correspondng cost allocaton scheme c(y) s KG (or prce vector p(y) s KP) wll also be stable. In the context of our model, ths means that no group S S N, actng unlateraly, can obtan ts own consumpton vector ys, correspondng to a vector of the servce y such that c(y) s KG, at a lower cost (or a lower prce). To prove ths last statement, let c(y) s K~, and let (x,y) be the correspondng allocaton n the set K. A group n the communty, S ~ N, S S # 0, that acts unlateraly n obtanng y must bear a cost equal to S S G(y ;S). If the group could obtan y, on ts own, at "a lower cost than that correspondng to c(y), then, S c (y) > G(y ;5). Thus, ss ss S G(y ;S) > (I - c(y)), and group S could obtan a hgher consumpton ss of the composte commodty for at least some of ts members. Because utlty functons have been assumed to be ncreasng n x, the allocaton (x,y) wll be domnated va coalton S. A contradcton, snce (x,y) s K. An mportant mplcaton of ths result s the followng. For each c(y) s KG, (17) c (y), V S ~ N, 5 # 0. In smpler terms, (17) says that each core cost allocaton scheme s such that no, non-empty, coalton pays a hgher share for the cost of the 14

15 servce than what t would have to pay f t was to provde the same servce to ts members on ts own. However, we should pont out that the converse of ths statement s not necessarly true. In other words, a cost allocaton scheme c(y) that satsfes (17) 7 the set KC. for some y does not, necessarly, belong to Stablty of core allocatons and the correspondng core cost allocaton schemes (or core prce structures) has an mportant mplcaton about the stablty of the correspondng ndustral structures that can provde the servce to the communty. In partcular, let us say that, n our model, a coalton structure T E T represents a stable ndustral structure for the provson of a vector y of the servce to the communty f and only f no group n the communty has the ncentve to move away from that arrangement. It s obvous that any ndustral structure that can provde to the communty a vector of the servce y at a cost allocaton scheme c(y) such that the correspondng allocaton (x,y) s a core allocaton, and thus, c(y) E KC,.satsfes the above noton of stablty. Ths, n conjuncton wth our earler concluson about the correspondence of cost allocaton schemes c(y) E KC and coalton structures T* that solve (9), and thus, represent effcent ndustral structures, makes the proof of the followng proposton trval. Proposton 1: For each core allocaton (x,y) E K, any coalton structure T E T that solves the mnmzaton problem n (9), relatve to y, represents a stable ndustral structure for the provson of y. 7The set KC has been derved from the set of core-allocatons K and not from all those cost allocaton schemes that satsfy (17) for varous vectors of the servce y. Thus, a cost allocaton scheme could satsfy (17), relatve to some y, and yet the resultng allocaton mght not belong to the set K because t s not only the cost functon that determnes K, but preferences as well. However, as we shall see below, under some specal assumptons about utlty and cost functons, every cost allocaton scheme that satsfes (17) wll also belong to the set KC. 15

16 Let us observe that, lke wth the case of effcency, stablty of ndustral structures depends on the correspondng vector of the servce y to some allocaton (x,y) s K. In othe words, for dfferent allocatons n K (dfferent as to y) we could have dfferent coalton structures representng stable ndustral structures (n addton to possble varatons wth respect to a specfc (x,y) E K). Suppose, however, that, wthout gong nto the specfcs that ths holds, we were to assume the followng. Wthn the bounds of feasblty n (2), the soluton(s) to the mnmzaton problem n (9) s (are) ndependent of y. In other words, let us suppose that any coalton structure LET that represents an effcent ndustral structure for the provson of a vector y of the servce, does so wth respect to any effcent y. Then, Proposton 1 wll take the followng form. Proposton 2: If (9) has a soluton whch s ndependent of y, any coalton structure that solves (9) represents a stable ndustral structure for the provson of the respectve vector of the servce n any (x,y) E K. Proposton 2 brngs us to the queston of "sustanablty" of partcular market forms, such as monopoly, duopoly, olgopoly, etc. {On the subject, see, n partcular, Sharkey (1982b).) Wthn the framework of our model, a partcular market form of a gven composton wll be revealed through the coalton structures formed by the communty n obtanng the servce. In partcular then, such market forms, wll be sustanable f and only f the correspondng coalton structures represent a unque, stable, structure for the ndustry. Utlzng the stablty propertes of the core, we can obtan the followng corollary to Proposton 2. 16

17 Corollary 1: If K 1 0, a coalton structure T* E T represents a sustanable market form for the provson of the servce f and only f t s a unque soluton to (9), ndependently of y. Let us observe that a soluton to (9) should be ndependent of y s a necessary condton for sustanablty for the followng reason. The core may contan many allocatons and each one of those allocatons can be used as a, potental, stable outcome. Therefore, f a soluton to (9) s not ndependent of y, then, accordng to Proposton 1, dfferent coalton structures could represent stable ndustral structures for dfferent y's, and none of them wll be sustanable. Wth ths observaton n mnd the proof of Corollary 1 follows drectly from (a) the defnton of sustanablty, (b) the assumpton that the core s not empty, and (c) Proposton 2. As a consequence of Corollary 1, f the core s not empty, a natural monopoly wll be sustanable because the coalton structure T* {N} s the unque soluton to the mnmzaton problem n (9) ndependently of y, whch s what we mean by natural monopoly. Questons regardng the choce of the core as a crteron for stablty asde, there can be no doubt about the mplcatons of core allocatons, n the provson of a servce dscussed so far. 8 However, ths may not be the case regardng ssues of cross-subsdzaton and the relatonshp of the core to such ssues. Because the subject deserves specal attenton we examne t separately n the next secton. 8The results obtaned n ths secton wth respect to the core, wll hold equally well f we were to use a dfferent soluton concept to the respectve cooperatve game that selects effcent allocatons. The only dfference wll be on the sense of stablty utlzed n each case. 17

18 5. Core Allocatons and Issues of Cross-Subsdzaton Faulhaber (1975), n examnng the ssue of cross-subsdzaton n "publc enterprses," proposed that the core of an approprately defned cooperatve game should be utlzed as defnng "subsdy-free" prces for the respectve servces. Ths approach has been adopted by other authors as well (see, e.g., Sharkey (1982a,b)). Smply put, n our model, a cost allocaton scheme c(y), or the correspondng prce structure p(y), wll result n cross-subsdzaton f and only f (to paraphrase Faulhaber (p. 966)), t "unduly" favors one group n the communty at the expense of another group. Therefore, accordng to Faulhaber's proposal, cost allocaton schemes c(y) E KC are subsdy-free, thus, they do not unduly favor one group n the communty at the expense of another group. Furthermore, for any cost allocaton scheme c(y) KC, we should be able to fnd some group n the communty that t s unduly favored at the expense of another group, and thus, a case where the latter subsdzes the former. In essence, then, what Faulhaber's proposal amounts to, n our model, s the followng. The exstence of core cost allocaton schemes should be vewed as a necessary and suffcent condton for the exstence of subsdy-free cost allocaton schemes. Although an expresson such as "unduly favors" s vague and subject to a number of nterpretatons, there seems to be some confuson between stablty, on one hand, and cross-subsdzaton, on the other. The core does, n the sense explaned n the last secton, yeld stable cost allocaton schemes. However, t s doubtful whether all core cost allocaton schemes are subsdy-free. Conversely, there s no reason to beleve that a cost allocaton scheme must be stable n order to be subsdy-free. 18

19 Let us examne the second pont frst, by usng a smlar example to that of Faulhaber. Example 1: A communty consstng of three neghborhoods,.e., N = {1,2,3}, examnes the possblty of supplyng each neghborhood wth water. Any sngle neghborhood can make ts own arrangements for provdng tself wth up to 10,000 gallons of water at a fxed cost of $300. Any two neghborhoods can get together and obtan up to 20,000 gallons of water for common use at a fxed cost of $350. Fnally, all three neghborhoods, actng jontly can obtan up to 30,000 gallons of water for common use at a fxed cost of $600. Addtonal water can be obtaned through dental to the above arrangements and under the same condtons. It s assumed that all three neghborhoods are dental n all respects. In partcular, I = Y~ $300, Vs N, whle the representatve utlty functon for each neghborhood s N, between consumpton of water y and the composte commodty x takes the followng form: Y > 10, u (x,y ) = Xl + 2(y ) - O.OOOI(y ), f = x + 10,000, f x >0 and It s easy to check that for ths example, Pareto optmalty requres allocatons (x*,y*) such that: (18) , and (19) 9 y* = (10,000, 10,000, 10,000). 9If (18) s volated, and more than $600 are spent for water someone receves no utlty by consumng water above 10,000 gallons, and hs poston can be mproved by consumng more of the composte commodty and less water. The converse holds f less than $600 are spent for water. Wth (18) satsfed, (19) s obvous. 19

20 Because these allocatons can be obtaned only through the cooperaton of all neghborhoods, the grand coalton represents the most effcent coalton structure for provdng the vector of water y* n (19). Furthermore, wth the optmum supply of water for each neghborhood fxed at y*, the only queston that remans to be answered s how the communty wll dstrbute the cost of $600 among the three neghborhoods. Ths means that, here, we deal wth a specal case where the satsfacton of (17) by a cost allocaton scheme c(y*), that also satsfes (18) wll be a necessary and suffcent condton for the exstence of allocatons n the core K and core cost allocaton schemes n KC. It s easy to check that no cost allocaton scheme wll satsfy these condtons for ths partcular example, and thus K and KC are empty sets. As a consequence, here we deal wth a case very smlar to that examned by Faulhaber (1975, p.974), where an empty core s nterpreted to mply that any cost allocaton scheme assocated wth y* wll nvolve crosssubsdzaton. (See also a smlar example and nterpretaton n Sharkey (1982a, p. 58).) Suppose now that the communty was to select the vector y* and the cost allocaton scheme c(y*) = (200, 200, 200) whch satsfes (18). Although ths cost allocaton scheme does not satsfy the stablty propertes of the core, we cannot say that t "unduly favors one group n the communty at the expense of another group." Certanly, every 2-neghborhood coalton pays, at c(y*), $50 more than what t would have to pay f t was to make ts own arrangements for provdng 10,000 gallons of water to each of ts members. However, ths s true for every 2 neghborhood coalton, and not just, say, neghborhoods 1 and 2. On the other hand, f $50 s the amount by whch, say, neghborhood 3 s "unduly favored at the expense of neghborhoods 1 and 2," t s not just 20

21 neghborhood 3 that t s "favored" n ths way, but every neghborhood N. But ths s self-contradctory. Stated dfferently, no matter how vague an expresson lke "unduly favors" s, one should expect that, n a stuaton where everyone s dentcal to the others n all respects, an equal dstrbuton of costs for equal quanttes of the servce does not favor any group n the communty at the expense of another, partcularly f such cost allocaton scheme s effcent. What the above arguments amount to s a proof of the followng proposton. Proposton 3: In the provson of a servce to a gven communty, the exstence of the core s not a necessary condton for the exstence of subsqy-free cost allocaton schemes. Next, let us consder the clam that core cost allocaton schemes do not unduly favor one group n the communty at the expense of another. Example 2: A communty consstng of fve neghborhoods,.e., N = {1,2,3,4,5}, faces a smlar problem to that of the communty n Example 1. In fact, the utlty functon of each neghborhood n ths communty s dentcal to that of each neghborhood n Example 1, whle - I = I ~ $500, V N. However, the composton of each coalton n determnng the relevant costs n ths partcular communty s mportant. (For example, we can thnk of the locaton of water, ground consderatons, and dstances as playng some role n determnng costs.) ~1ore specfcally the communty can be devded nto two groups of neghborhoods, say, P and Q, such that cooperaton among neghborhoods can be beneftal only f they belong to dfferent groups. In partcular, we shall let P = {1,2}, and Q = {3,4,5}. Each neghborhood N can make ts own arrangements 21

22 - for provdng tself wth y gallons of water, up to 10,000, at a fxed cost of $500. Cooperaton, for obtanng the same quantty of water for each member of a coalton S, yelds the cost functon C(yS;S), gven by -S C(y ;S) 500s (mn{IS n P~ ls n QI }), V S ~ N, where, s denotes the number of neghborhoods n coalton S, and IS n pi, IS n QI, denote the number of neghborhoods n P and Qrespectvely, that also belong to coalton S. (In essence, the term 1000(mn{IS n pi, ls n QI }), n the above cost functon, represents the savngs from cooperaton for each coalton S.) As n Example 1, addtonal water can be obtaned through arrangements dentcal to the above and under the same condtons. The same arguments used n dscussng Example 1, can be utlzed here to show that Pareto optmalty requres allocatons (x*,y*) such that: (20) I E: N , and (21) y* (10,000, 10,000, 10,000, 10,000, 10,000). These allocatons are obtanable ether through the grand coalton N, or through coalton structures T* = {S*,S**}, where S* = {,j}, E: P, j E: Q, and S** (P-{}) U (Q-{j}). Wth the optmum supply of water fxed at y*, the only queston that remans to be answered s how the communty wll dstrbute the cost of $1,000 among the fve neghborhoods. Thus, here we deal agan wth the specal case where the satsfacton of (17) by a cost allocaton scheme c(y*), that also satsfes (20), wll be a necessary and suffcent condton for the exstence of the core K and core cost allocaton schemes n KC. 22

23 It s well known (see, e.g., Maschler (1976» that for a cooperatve game lke ths, the set KC s non-empty and t conssts of the sngle pont (500, 500, 0, 0, 0). Thus, the only core cost allocaton scheme s the one where 10,000 gallons of water ~re provded to each neghborhood at a total cost of $1,000 borne, at an equal share, by neghborhoods 1 and 2 only. The correspondng core allocaton s (x*,y*), where, y* s gven by (21), and x* s such that: x*l = x*2 = I - 500, whle x*3 = x*4 = x*5 = I. No matter how vague an expresson such as "unduly favors" s, t wll be very hard to convnce neghborhoods 1 and 2 that the above core cost allocaton scheme does not unduly favor neghborhoods 3, 4, and 5. It s a dfferent story to say that neghborhoods 1 and 2 do not have the power to move away from the above core cost allocaton scheme (and, thus, that t s stable) and a dfferent story to say that such a scheme does not unduly favor 3, 4, and 5. To take the argument one step further, why not call the alternatve cost allocaton scheme c(y*), where, (22) c(y*) (-250, -250, 500, 500, 500), "subsdy-free"? Ths cost allocaton scheme wll produce the Pareto optmal allocaton (x*,y*), where, y* s gven by (21), and x* s such that x* = x* = I + 250, whle x* = x* = x* = I Although (x*,y*) does not belong to the core, t represents the other extreme of the core allocaton (x*, y~) n the followng sense. Whle at (x*,y*) all benefts from cooperaton go to the set of neghborhoods Q, at (x*,y*) all benefts from cooperaton go to the set of neghborhoods P. But then, f ("X*,y*) unduly favors P, so does (~*,y*) wth respect to Q. In partcular, the neghborhoods n the set Q need the 23

24 neghborhoods n the set P n order to realze any benefts from cooperaton, as much as the former need the latter for the same purpose. What the above example llustrates, s the valdty of the followng proposton. Proposton 4: In the provson of a servce to a gven communty, the exstence of the core s not a suffcent condton for the relevant cost allocaton schemes to be subsdy-free. In addton to llustratng the valdty of Proposton 4, Example 2 s nterestng for the followng reason. Even f the core s not empty, we can stll fnd allocatons, not belongng to the core, such that, the correspondng cost allocaton schemes do not "unduly favor" one group n the communty at the expense of another. For example, the cost allocaton scheme c(y*) = (200, 200, 200, 200, 200), where y* s gven by (21), yelds the Pareto optmal allocaton (x*,y*) where x* s such that x~ = I - 200, VE N. The only coaltons that volate (17) relatve to c(y*), are three-neghborhood coaltons S** consstng of one neghborhood from the set P and two neghborhoods from the set Q. But, as wth the symmetrc allocaton n Example 1, ths s true for every such coalton. Thus, we cannot make a case that a coalton S**, as above, subsdzes ts complement S* = N - S**~ at c(y*). As.a consequence~ t s possble~ n a stuaton lke that of Example 2, that: (a) the core s not empty but core cost allocaton schemes are not subsdy-free, and (b) there exst allocatons outsde the core such that no vald case about cross-subsdzaton can be made for the correspondng cost allocaton schemes. We have seen so far n ths secton that core stablty and absence of cross-subsdzaton do not necessarly concde. Now, one may argue tha Faulhaber's proposal to use the core as defnng subsdy-free prce 24

25 structures (or cost allocaton schemes) has been based on arguments dfferent from stablty. Thus, that core-stablty s concdental to ths defnton. There are three cases n Faulhaber's paper where ths could be the case, but later on the relevant argument s abandoned and thus, t s core stablty that he uses n defnng subsdy-free prce structures. The frst case s where Faulhaber gves the followng defnton, as a "frst approxmaton," for subsdy-free prce structures: "If the provson of any commodty (or group of commodtes) by a multcommodty enterprse subject to a proft constrant leads to prces for the other commodtes no hgher than they would pay by themselves, then the prce structure s subsdy-free" (1975, p. 966). Applyng ths defnton to Example 2, we can see that both the core cost allocaton scheme (500, 500, 0, 0, 0), and the cost allocaton scheme c(y*) gven by (22) qualfy as "subsdy-free" schemes. For example, c(y*) = (-250, -250, 500, 500, 500) could result as follows. Intally, an enterprse comes nto agreement wth the neghborhoods n the set Qto provde each of them wth 10,000 gallons of water wth a zero proft constrant. Ths constrant wll not be volated later on f the same enterprse comes nto agreement wth the neghborhoods n the set P as well, provde each wth 10,000 gallons of water and pass the savngs of $500 to them. (Presumably, the enterprse saves ths amount of money by, say, utlzng the locaton of these neghborhoods.) Although both of the above cost allocaton schemes satsfy the above defnton".only the core cost allocaton scheme s consdered subsdy~ free by Faulhaber. To examne the second case, let us consder cost allocatn schemes that satsfy (8) and the followng condton. 25

26 S (23) I c (y) > Hn { I C(y ;S)} 8 S T 8 T S 8 T -Mn{ I S 8 ( T-5) S C(y ;S): T 8 T, S 8 T}, V S c:::: N. In our model, (23) represents a general form of the generalzed ncremental cost test (see, Schotter and Schwodauer (1980, pp ) who attrbute ths term to Faulhaber and Zajac). Now, t so happens that, n certan cases, cost allocaton schemes c(y) that satsfy (8) and (17) also satsfy (23). Arguments about ts valdty as a test defnng subsdy-free cost allocaton schemes asde, f (a) (17) s a necessary and suffcent condton for a cost allocaton scheme to belong to the set KC, and (b) every cost allocaton scheme that satsfes (8) and (17) also satsfes (23), one can argue that core cost allocaton schemes are subsdy-free because they satsfy the above generalzed ncremental cost test. However, even ths test, whch s more general and ndependent of stablty, s rejected by Faulhaber as defnng subsdy-free cost allocaton schemes n favor of core-stablty. As he states, " there may be prces whch pass an ncremental cost test and yet nvolve cross-subsdy" (1975, p. 976). Fnally, the thrd case s very smlar to the above, and nvolves condton (17). By tself, condton (17) does not defne the set of core cost allocaton schemes (as we have noted n Secton 4). However~ t represents the generalzed stand above test (see, the same reference as for generalzed ncremental cost test) for cost allocaton schemes, n that each coalton should not have to pay a hgher cost for ts own consumpton vector of the servce than what that coalton must pay f t was to act alone. Notce, though, that, n certan cases, ths test can be dentcal to the generalzed ncremental cost test n (23), whch, 26

27 as we have noted, t has been rejected by Faulhaber as a general test defnng subsdy-free prce structures, n favor of the core. Thus, (17) s not suffcent, accordng to these arguments, for the defnton of subsdy-free prce structures ether. What these conclusons ndcate s that f we are to follow Faulhaber's approach we must dentfy absence of cross-subsdzaton wth stablty, and n partcular wth core-stablty. However, as we have seen n dervng Propostons 3 and 4, ths s not necessarly the case. 6. What s Cross-Subsdzaton? Propostons 3 and 4, and the arguments used to demonstrate ther valdty leave open the queston of how one can defne, n a precse way, what Qross-subsdzaton s. In partcular, f core-stablty s not suffcent to determne whether a cost allocaton scheme s subsdy-free, and f an expresson lke "unduly favors" s vague, what s the alternatve? In ths secton we shall approach the subject from a dfferent perspectve. Frst of all we should make clear that our purpose here s not to fnd out whether the communty wll choose one allocaton over another. Nether to say that one allocaton s "better" than another. Our task s to characterze, n as a precse way as possble, dfferent cost allocaton schemes assocated wth a vector of the servce from the perspectve of whether or not they nvolve some form of cross-subsdzaton. Thus, our task should not be seen much dfferent than, say, someone tryng to determne whether a cost allocaton scheme s effcent. Gven a vector of the servce y, let us consder an arbtrary cost allocaton scheme C(Y)4 llow can one go ahead and determne whether c(y) s or s not subsdy-free? Well, we can start out by tryng to classfy 27

28 each member of the communty n accordance to whether that member s the recpent of a subsdy, a subsdzer, or nether. In partcular, let NR(c(y», Np(c(y», and NO(C(y» denote these three categores. Then, n the set NR(c(y» we should nclude every member of the communty that, relatve to c(y) and our noton of what cross-subsdzaton s, receves a subsdy. Lkewse, we should nclude n the set Np(c(y» every member of the communty who s a subsdzer. Fnally, NO(c(y» should contan all members of the communty that we cannot classfy ether as subsdzers or as the recpents of a subsdy. Now, any noton of cross-subsdzaton that s not precse wll not enable us to make the above classfcaton. For example, f we cannot decde whether a member of the communty s a subsdzer or the recpent of a subsdy, relatve to a cost allocaton scheme c(y), how are we to decde whether c(y) s subsdy-free? Therefore, any precse noton of what cross-subsdzaton s should always enable us to make the above classfcaton. Two condtons seem essental for ths purpose. The frst s a separaton axom (Axom 1, below) that classfes each member of the communty n one and only one of the above three categores. The second s a rason d'etre axom (Axom 2, below) that requres the exstence of subsdzers f there exst recpents of a subsdy and vce versa. Axom 1: The three sets NR(c(y», Np(c(y», and NO(c(y» are parwse dsjont for any cost allocaton scheme c(y). Axom 2: (NR(c(y» # 0) <=> (Np(c(y» #0), for any cost allocaton scheme c(y). 28

29 There are two mmedate mplcatons that follow from these axoms. The frst one s that NR(c(y)) ~ N, and Np(c(y)) ~ N, for any cost allocaton scheme c(y). Ths means that no precse noton of cross-subsdzaton should classfy every member of the communty as the recpent of a subsdy, and, lkewse, t should not classfy every member of the communty as a subsdzer. The second mplcaton s the followng defnton. A cost allocaton scheme c(y) s subsdy-free f and only f every member of the communty can be classfed as belongng to the set NO(c(y)). It s worth notng here that ths defnton s ndependent of any specfc noton of what cross-subsdzaton s. In other words, as long as a noton of what cross-subsdzaton s satsfes the two axoms (and thus, accordng to our termnology, s precse) there s only one way to determne when a cost allocaton scheme s subsdy-free, and that s through the condton NO(c(y)) = N. As a consequence of ths, t may be possble to come up wth a varety of notons about cross-subsdzaton. In other words, certan cost allocaton schemes can be characterzed subsdy-free n one sense but not under a dfferent sense. Although both notons can be precse. The natural queston to ask here s whether any of the classcal tests satsfy the above axoms, and f not whether there exst alternatve tests that do. Proposton 3 suggests that core-stablty, the generalzed stand alone test n (17), and the generalzed ncremental cost test n (23), wll all volate at least one of the two axoms. The problem wth these tests s that they fal to take nto account that n certan stuatons, t wll be mpossble to separate the subsdzers from the recpents of a subsdy. A case n pont s the symmetrc cost allocaton scheme ~(y*) = (200, 200, 200) n Example 1. In ths stuaton, all members of the communty are dentcal 29

30 n all respects, they receve the same quantty of the servce, they share equally the costs of obtanng effcently that servce, the resultng allocaton s Pareto optmal, and yet c(y*) would not be consdered subsdyfree by any of the three tests mentoned above. But as we have argued n Secton 5, t s mpossble to separate the subsdzers from the recpents of a subsdy n the above stuaton. What ths means s that any noton of cross-subsdzaton whch s expected to satsfy Axoms 1 and 2 must nclude some "symmetry condton." That s, a condton whch (a) wll classfy "smlar" members of the communty as belongng to the same category,.e., subsdzers, recpents of a subsdy, or nether one of these, and (b) wll enable us n stuatons lke the above to classfy every member of the communty as belongng to the set NO(C(y*)) relatve to c(y*). Althouth some symmetry condton seems necessary f Axoms 1 and 2 are to be satsfed, t s not suffcent for ths purpose. A vector of the servce y that s not obtaned effcently, can lead to cost allocaton schemes wth the followng characterstc. Some members of the communty are classfed as subsdzers wthout beng able to dentfy the recpents of such subsdy. Thus, Axoms 1 and 2 could be volated not because there s cross-subsdzaton but due to neffency. What ths concluson mples s that even the smple "stand along test" wll volate Axom 2 f we do not requre that y s obtaned effcently. A smlar stuaton to ths, but n reverse, can arse f we do not requre that the communty fnances on ts own the provson of any vector of the servce y. In other words, wthout ths requrement we may run nto a stuaton where everyone receves a subsdy wthout beng able to denfty 30

31 the members of the communty who pay that subsdy. Thus, Axom 2 could, agan, be volated. As a consequence, n a model lke the one n Secton 2, the satsfacton of Axoms 1 and 2 by any precse noton of what crosssubsdzaton s, requres not only that a "symmetry" condton should not be volated, but that the "effcency" and "self-fnance" condtons should not be volated ether. Although the above three condtons could make precse a rule for testng cost allocaton schemes about cross-subsdzaton, they are not suffcent for makng that rule complete. For example, the smple stand along test n (24) restrcted to apply to cost allocaton schemes c(y) that correspond to effcent allocatons (x,y) E Y*, wll not volate any of these condtons. However, as we have seen n Secton 5, a cost allocaton scheme can satsfy the so-modfed smple stand alone test and stll may not, necessarly, be subsdy-free. (As we have argued n dervng Proposton 4, the core cost allocaton scheme of Example 2 s not subsdyfree. However, t satsfes the modfed smple stand alone test.) What the above condtons do not provde s a fundamental prncple that can be used as the bass n any test about cross-subsdzaton. To get an dea of what ths prncple mght be, let us recall that we have declared the core cost allocaton scheme of Example 2 as not beng subsdy-free because one group of the communty, the group Q, benefts from the servce and they contrbute nothng towards the costs of obtanng t. Furthermore, the same members of the communty could be contrbutng somethng towards the costs of the servce, and stll they could fnd the new arrangements preferable to ether the non-provson, or the stuaton where they would have to obtan the servce on ther own. For example, as we have suggested n Secton 5, the cost allocaton scheme c(y*) = (200, 200, 200, 200, 200) represents such alternatve. What ths suggests s 31