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Mixed serial cost sharing
*
Jens Leth Hougaard , Lars Thorlund-Petersen
Copenhagen Business School, Solbjerg Pl. 3, DK-2000 Frederiksberg, Denmark
Received 11 February 1999; received in revised form 11 November 1999; accepted 20 December 1999
Abstract
A new serial cost sharing rule, called mixed serial cost sharing, is defined on the class of cost functions which equal a sum of an increasing convex and increasing concave function. This rule is based on a particular decomposition principle known as complementary-slackness decomposition and it coincides with the original serial rule of Moulin and Shenker (1992) [Moulin, H., Shenker, S., 1992. Serial cost sharing. Econometrica 60, 1009–1037] and the reversed serial rule of de Frutos (1998) [de Frutos, M.A., 1998. Decreasing serial cost sharing under economies of scale. Journal of Economic Theory 79, 245–275] if the cost function is convex or concave, respectively. The rule and its decomposition are characterized by three properties and the vector of payments is compared with existing cost sharing rules with respect to economic inequality measurement.
2001 Elsevier Science B.V. All rights reserved.
Keywords: Serial cost sharing; Complementary-slackness decomposition; Cost function
JEL classification: D63; D62
1. Introduction
Sharing a joint cost or surplus among a set of agents, such as projects, departments or members of a club, has always been a challenging task for economists and consequently various cost sharing schemes have emerged. Recently, in the particular case of a single homogeneous good new schemes have called for attention due to their nice axiomatical as well as strategical properties – the serial cost sharing schemes, as studied in Moulin and Shenker (1992, 1994) and in de Frutos (1998).
The intuition behind the serial cost sharing scheme in Moulin and Shenker (1992) is straightforward: Consider the production of some good under an increasing cost function
*Corresponding author. Tel.:145-3815-3523; fax:145-3815-2440. E-mail address: [email protected] (J.L. Hougaard).
C and suppose that 3 agents demand quantities q1#q2#q . According to serial cost3
sharing, the common costs C( q11q21q ) will be shared as follows. The agent with the3
smallest demand pays one-third of the total costs in case all agents had demanded q .1
The agent with the second smallest demand pays the same share as the first agent plus one-half of the incremental costs from a total demand of 3q to q1 112q . The last agent2
pays the residual costs. That is, payments x , x , x are defined as1 2 3
1 1
] ]
x153C(3q ),1 x25x112sC( q112q )2 2C(3q ) ,1 d
x35C( q11q21q )3 2x12x .2 (1)
Clearly, the payment of agent i is independent of the demand of any agent j.i. With a
convex cost function this seems a reasonable property since agents with modest demands are not penalized by the fact that agents with higher demands cause the common total costs to escalate.
Now, consider the reversed serial scheme in de Frutos (1998); using the serial principle as expressed in (1) but commencing with the highest rather than the lowest demand. Then payments y , y , y equal1 2 3
1 1
] ]
y353C(3q ), y3 25y312sC(2q21q )3 2C(3q ) ,3 d
y15C( q11q21q )3 2y22y .3 (2)
In (2) payments of agent i do not depend on payments of any agent j,i. When marginal
costs are decreasing, small demands are penalized under scheme (2), something which seems to be a natural mirror-image of the discussion in Moulin and Shenker (1992) for the case of increasing marginal costs. We consider the payment schemes (1) and (2) suitable on the domain of convex and concave cost functions, respectively. However, in numerous applications one encounters cost functions which do not belong to either of these domains; for example, in managerial economics one often considers cost functions having first decreasing and then increasing marginal costs (concave–convex cost functions.) Thus we define in this paper a payment scheme and its associated mechanism on a class of cost functions which comprises the concave–convex functions: The class of functions which equal a sum of an increasing convex and an increasing concave function. This new mechanism will be called mixed serial cost sharing.
Since the domain of mixed serial cost sharing equals the sum of all convex and concave cost functions, it seems natural to base its definition on some rule of decomposition of any given cost function into two such components. In the following, it will be argued that mixed serial cost sharing should be defined by the unique so-called complementary-slackness decomposition as in Thon and Thorlund-Petersen (1986). Due to the neat properties of this decomposition rule, mixed serial cost sharing prescribes that cost shares are computed by application of payment schemes x and y to the convex and concave component functions, respectively. Thus mixed serial cost sharing is an extension of both serial rules to a common, larger domain of cost functions.
bounds are given. Mixed serial cost sharing is defined in Section 4 and its properties are characterized in terms of the choice of decomposition rule. In Section 5 it is shown that the choice of the complementary-slackness decomposition for a given cost function, can be construed as choosing the unique payment vector which is maximal in the sense of economic income inequality. Furthermore, mixed serial cost sharing is compared in this respect with average and marginal cost pricing as well as with the original serial scheme in Moulin and Shenker (1992). Section 6 contains final remarks.
2. Increasing and decreasing serial cost sharing
Consider a finite set N5h1, . . . , n of agents who share a common technology whichj produces some homogeneous good. Each agent demands a quantity qi$0 of the good. Let q5( q , . . . , q ) denote the vector of (individual) demands and let Q1 n 5q11 ? ? ? 1
q denote total demand. The vector q is assumed to be increasingly ordered, qn 1# ? ? ? #q .n
Throughout, the cost of producing any quantity Q is determined by a cost function of the following kind. Let D denote the class of increasing functions on the interval [0,0 `[
↑ ↓
such that C(0)50 for all C[D and let D0 0 and D0 denote the convex and concave
↓
functions in D , respectively. A cost function F0 [D0 which is constant for Q.0 is called a fixed-costs function. For any convex cone K0#D , a cost sharing mechanism0 y
n
on K is a mapping which for any C0 [K and q0 [R1 associates a vector of payments (v , . . . , v ) such that
1 n
(i )v5y(C, q)$0, i [ N; (ii )v 1 ? ? ? 1v 5C(Q ). (3)
i i 1 n
A mechanism y on K satisfies independence from above if for any i0 [ h1, . . . ,
n21jthe cost shareyi(C, q) does not depend on q where jj .i. Analogously,ysatisfies
independence from below if for any i[h2, . . . , nj,yi(C, q) does not depend on q wherej
j,i. Finally call y additive if for all costs function C , C and any demand vector q,1 2
y(C11C , q)2 5y(C , q)1 1y(C , q).2
For any vector of demands q5( q , . . . , q ) define vectors r1 n 5(r , . . . , r ), s1 n 5 (s , . . . , s ) by the following linear transformations1 n
n 0 0 . . . 0 q1 r1
1 n21 0 . . . 0 q2 r2
1 1 n22 . . . 0 q3 5 r3 , (4)
. . . : :
3
1 1 1 . . . 143 4 3 4
qn rnn 0 0 . . . 0 qn s1
1 n21 0 . . . 0 qn21 s2
1 1 n22 . . . 0 qn22 5 s3 . (5)
. . . : :
Notice that (5) is derived from (4) by reordering q inversely and that the n3n-matrix of
(4) and (5) has row sums equal to n. Clearly, since q1# ? ? ? #q , thenn
r1# ? ? ? #rn5Q5sn# ? ? ? #s1 (6)
and for i, j52, . . . , n we have
ri2ri215(n2i11)( qi2qi21), sn2j112sn2j125( j21)( qj2qj21). (7) Following Moulin and Shenker (1992) and de Frutos (1998), we define for given n$2,
C[D , and q the following vectors of payments x0 5(x , . . . , x ), y1 n 5( y , . . . , y ) by1 n
i
C(r )k 2C(rk21) ]]]]]
xi5
O
n112k , i51, . . . , n, (8)k51 j
C(s )k 2C(sk21) ]]]]]
yn2j115
O
n112k , j51, . . . , n, (9)k51
where by definition r05s050. Clearly, if C is linear, then x5y.
It is readily verified that for any C[D , payments x and y satisfy the total-sum0
condition corresponding to (3(ii )) and the following inequalities hold:
0#x1# ? ? ? #x ,n y1 # ? ? ? # y .n (10)
Nonnegativity of y1 does not follow for all C[D . As an example, if n0 52 and
↓
C( q11q )2 50,C(2q ), then y2 15 2C(2q ) / 22 ,0. However, if C[D , then pay-0
ments are nonnegative as y1$C(nq ) /n1 $0. The problem of determining bounds on payments is further studied in Section 3.
↑ ↑
Define the increasing serial mechanismjon D by0 j(C, q)5x if C[D , which on0 ↑
D0 is identical to the serial cost sharing mechanism in Moulin and Shenker (1992).
↓
Moreover, the decreasing serial mechanism h on D0 is defined by h(C, q)5y if
↓
C[D . It is easily verified, that mechanisms0 j andh satisfy independence from above and from below, respectively. Furthermore, it follows from Moulin and Shenker (1992) and de Frutos (1998) that both mechanismsj andhsatisfy positive cross-monotonicity (as they are increasing in q), and that they are additive on their respective domains.
Finally, corresponding to any payment vector v, define for qi.0 agent-specific prices v/q . For the serial mechanisms, payment vectors x and y determine prices that are
i i
monotone with respect to index i:
Lemma 1. Suppose that q1.0. If C is convex, then x /q1 1# ? ? ? #x /q and y /qn n 1 1# ? ? ? #y /qn n. If C is concave, then y /q1 1$ ? ? ? $y /q and x /qn n 1 1$ ? ? ? $x /q .n n
i21 i21
C(r )k 2C(rk21) C(r )i 2C(ri21)
]]]]] ]]]]]
xi215
O
r 2r ( qk2qk21)#O
r 2r ( qk2qk21)k k21 i i21
k51 k51
C(r )i 2C(ri21) C(r )i 2C(ri21) n2i11 xi2xi21
]]]]] ]]]]] ]]] ]]]
5 r 2r qi215 n2i11 r 2r qi215q2q qi21,
i i21 i i21 i i21
proving that x /q1 1# ? ? ? #x /q . The remaining three sets of inequalities are proved inn n
a similar fashion. h
As a consequence of Lemma 1, one can determine bounds on agent-specific prices even in the more general case of cost functions considered in the following sections. Furthermore, Lemma 1 makes it possible to compare x and y with average and marginal cost pricing, see Section 5.
3. Upper and lower bounds on payments
In the literature, one has considered bounds on payments of agent i which depend only on q . It follows from Moulin and Shenker (1992) that if C is convex, theni
payments x are bounded from above by the unanimity cost C(nq ) /n and from below byi i
the stand-alone costs C( q ), thus (I) C( q )i i #xi#C(nq ) /n. Furthermore, it follows fromi
de Frutos (1998) and Moulin (1996) that if C is concave, then (II) C(nq ) /ni #y , (III)i
0#xi#C( q ). The bounds (I–III) cannot in general be improved if they are to dependi
on q only. However, the asymmetric nature of these bounds is a disadvantage from thei
point of view of mixed serial cost sharing as will become apparent in Section 4; for example, there is no upper bound for y in (II). In order to overcome this asymmetry and obtain stronger bounds, define for any vector of demands q the ith lower sum Q andi
i i
upper sum Q by Qi5q11 ? ? ? 1qi21if i$2, Q 5qi111 ? ? ? 1q if in #n21, and
n
Q15Q 50. In this section we establish upper and lower bounds on x and y which arei i i
stronger than (I–III) and depend only on q , Q , and Q .i i
If demand of agent i is fixed, then x depends only on demands q , . . . , qi 1 i21 and yi
depends only on demands qi11, . . . , q . For any index in 51, . . . , n and fixed demand
0
vector q define the sets
0 0 0
Vi( q )5
h
( q , . . . , q ) 01 nu
#q1# ? ? ? #qi21#q and qi j5q if jj $i ,j
i52, . . . , n,i 0 0 0
V( q )5
h
( q , . . . , q ) q1 nu
j5q if jj #i and qi #qi11 # ? ? ? # q ,nj
i51, . . . , n21,
0 n 0 0
and V1( q )5V ( q )5
h
qj
. It turns out that over these domains, ji and hi are order-preserving functions with respect to the partial ordering of majorization (see0
Appendix B). Thusji(C, q) is Schur-concave onVi( q ) andhi(C, q) is Schur-convex on
i 0
averaging their demands. Moreover, manipulation of h is possible by spreading out demands; for example, if qn22,qn21 and agents n21and n demand qn212´ and
qn1´, for´ .0 sufficiently small, then by Schur-convexity the payment of agent n22 will increase. Consequently, neither j norh are immune to reallocations of demand.
Now, consider some demand vector q and index i with qi.0. We define two vectors related to the pair ( q, i ) which are more spread out than q, and one vector related to ( q,
i )which is less spread out than q. Firstly, for i$2 let h[h1, . . . , i21 be the numberj '%i '%1
uniquely determined by (i2h21)qi,Qi# (i2h)q and define the vector q by qi 5q
and
' %i
q 5(0, . . . , 0, Qi2(i2h21)q , q , . . . ,q , qi i i i11, . . . ,q ),n i52, . . . , n. (11)
%
3 %%%3 %3 %%%3 %3 %%%%%3
h21 i2h n2i
%&i
Secondly, define the vector q by i5n2j11 and
%&n2j11 i
q 5(q , . . . , q1 n2j, qn2j11, qn2j11, . . . ,qn2j11, Q 2( j22)qn2j11),
%
3 %%%%%3 %3 %%%%%%%%%%%%%%%%%3
n2j j21
%&n
j52, . . . , n, and q 5q. (12)
i 1 1 1 n
¯ ¯ ¯
Finally, define the vector q by q 5( q , Q /(n1 21), . . . , Q /(n21)), q 5(Q /(nn 2 1), . . . , Q /(nn 21), q ), andn
i i i
¯q 5(Q /(ii 21), . . . , Q /(ii 21), q , Q /(ni 3%%%%%%%%%%%%32i ), . . . , Q /(n2i ) ),
%
3 %%%%%%%%%%%3
n2i i21
i52, . . . , n21.
Lemma 2. Let a denote the ordering of majorization, (see Appendix B). For any pair
'
%i %&i '%i i
¯
( q, i ), i51, . . . , n, one has q aqaq , q . In particular, q is a 2 maximal inV( q) i
%&i
i
and q is a 2 maximal inV( q). Furthermore, by replacing the last n2i or the first i
¯
i21 elements of q by those of q, one obtains the a2 minimal element inV( q) or i i
V( q), respectively.
Proof. This follows by application of the difference criterion, Appendix B, (B.2). h
One can now establish the following bounds.
Proposition 1. Consider a given demand vector q and index i with qi.0. If C is convex,
then
'
%i i
¯
ji(C, q )#ji(C, q)#ji(C, q ). (14)
In particular, for i52, . . . , n,
q C(r )i i 1 i21 Qi C(nq )i
]] ]]]
S
]]S
]]DD
]]C( q )i # r #xi#n2i11 C(r )i 2 n C ni21 # n . (15)
i
%&i i
¯
hi(C, q )#hi(C, q)#hi(C, q ), (16)
and for j52, . . . , n,
i
C(nqn2j11) 1 j21 Q
]]]n #]]]n2j11
S
C(s )j 2]]n CS DD
n]]j21 #yn2j11 iC(s )j C n(Q
s
2( j22)qn2j11)d
]] ]]]]]]]
# 2 . (17)
n21 n(n21)
Proof. Inequalities (14), (16) follow from Lemma 2, concavity, and Schur-convexity. The first and the last inequality of (15) follows from convexity of C. Furthermore, the second inequality of (14) is equivalent to the third inequality of (15). In order to prove the second inequality of (15), we proceed as follows.
'
% '% '%i
If i52, . . . , n, then payments x , . . . , x corresponding to q are determined by1 i
'
% '% '%
21
x , . . . , x1 h2150, xh5(n2h11) C (ns 2h11) Qs i2(i2h21)q , andidd
'
% '%
xh115 ? ? ? 5xi
1 1
]]
F
]]]G
5n2h C rs di 2n2h11C (ns 2h11) Qs i2(i2h21)qidd (18)
where ri5(n2i11)qi1Q . Note that ni 2h115(n2i12)1i2h21,(n2i1
2
2)1Q /q and definei i g 5i (i2h)qi2Qi2q /(ri i1q ). Then by (18) and convexity ofi
C,
C (ns 2i121Q /q ) Qi i s i2(i2h21)qidd
]]]]]]]]]]]] (n2h)xi$C rs di 2 n2i121Q /q
i i
r C(r )i i C(r )i 2C rs i2gi(ri1q ) /qi id r C(r )i i C(r )i
]] ]]]]]]]] ]] ]]
5 1gi $ 1gi
ri1qi gi(ri1q ) /qi i ri1qi ri
q C(r )i i
]]
5(n2h) . (19)
ri
By (19), the second inequality of (15) follows from the first one of (14).
Finally, consider inequality (17). Under mechanismh agents n and n21 each pays
%&i
yn5C(nq ) /n and yn %&i n215C(s ) /(n2 2%&i1)2C(s ) /n(n1 21). Thus the vector s corre-%&i
i i
sponding to q , (see (5)), satisfies s15n(Q 2( j22)qn2j11) and s25Q 1(n2j1 1)qn2j115s . Consequently,j
%& %&
i i
%& %& C(s )2 C(s )1
]] ]]]
yn2j11#yn2j115yn215n212n(n21)
i
C(s )j C n(Q
s
2( j22)qn2j11)d
]] ]]]]]]]
5 2 . (20)
n21 n(n21)
Then (17) follows by concavity of C, (16) and (20). h
The second inequality of (15) establishes a lower bound on x in terms of averagei
demand never pays a price below average costs; this inequality can also be derived from Lemma 1. Note that in (17) there is no corresponding upper bound for y in terms of average costs since y1.0 is possible even if q150. One further important asymmetry between (15) and (17) remains in the sense that in (17), there is no upper bound in terms of stand-alone costs C( q ). In order to establish such a bound one needs additionali
assumptions on C and q, see Hougaard and Thorlund-Petersen (2000).
4. Mixed serial cost sharing
As mentioned in Section 1, the mechanism of mixed serial cost sharing will be defined on the class of cost functions which equal a sum of a convex and a concave cost
1 ↑ ↓
function, denoted D0 5D01D . In general, we are interested in a cost sharing0
1 ↑ ↓
mechanism on D0 which is completely determined by its values on D0 and D .0
1 ↑ ↓
Therefore, a decomposition rule is defined as a mapping G: D0 →D03D0 where
C5R1S forG(C )5(R, S ) and normalized by the requirement that the right-derivative of R at Q50 equals zero. Furthermore, consider a pair of additive mechanismsr on R
1
ands on S. Then a corresponding decomposition mechanismyG;r,s is defined on D0 by yG;r,s(C, q)5r(R, q)1s(S, q) and G(C )5(R, S ). Clearly, the particular choice of decomposition ruleG is crucial for the allocation of costs via the mechanismyG;r,s. Note also that only additive mechanisms r and s are considered.
Mixed serial cost sharing z relates to the particular decomposition rule, in short denoted GCS, studied in Thon and Thorlund-Petersen (1986) and defined as follows.
1
Every function C[D0 has a right-derivative C9, possibly with C9(0)5 `, on [0,`[. For
2 21
´±0 define the quotient D´C(Q )5´ sC9(Q1´)2C9(Q ) for Qd 1´ $0. Then the
↑ ↓
CS-decomposition C5R1S, is determined by functions R[D , S0 [D0 such that
2 2
(i ) lim supD´R(Q )D´S(Q )50; (ii ) R9(0)50. (21)
´→0
If C5R1S is twice continuously differentiable on ]0,`], then (21(i )) is equivalent to
;Q.0: R0(Q )S0(Q )50. (22)
If C5R1S is piecewise affine with respect to a finite subdivision of [0,`[, then (21(i )) means that at any Q.0, at least one of the functions R and S is affine on an open neighbourhood of Q. It follows from Thon and Thorlund-Petersen (1986) that among all possible decompositions the CS-decomposition maximizes the concave component.
As a simple example, consider the following cost function. Suppose that a good is sold at a price of 1 dollar per unit and furthermore that a single bundle of 10 units is offered at a price of 80 cents per unit. Then the relevant cost function C is determined by C(Q )5Q for Q,8, C(Q )58 if 8#Q,10, and C(Q )5Q22 if 10#Q with
unique decomposition as a sum of a convex and concave cost function given by
C(Q )5R(Q )1S(Q )5max 0, Qh 210j1min Q, 8 . However, if a 10 cent excise taxh j
is added, then total costs equal C(Q )10.1Q with CS-decomposition max 0, Qh 210j1 min 1.1Q, 0.1Qh 18 , which is not the only possible decomposition as for examplej
1
Now, define mixed serial cost sharing as the mechanismz 5 yGCS;j,h on D , thus for0
1
any C[D0 with CS-decomposition C5R1S, one has
z(C, q)5j(R, q)1h(S, q). (23)
Therefore, if C is convex, then z(C, q)5j(C, q)5x and if C is concave, then
z(C, q)5h(C, q)5y. Note that even though payment scheme y may be negative for
non-concave C, the vector of payments under z, denoted z5z(C, q), is always nonnegative. Consequently, z defines a mechanism in the sense of (3).
In order to motivate the choice of the CS-decomposition consider some general decomposition mechanism yG;r,s which satisfies the following properties.
↑ ↓
(a) Extension Mechanism yG;r,s is identical toj on D0 and toh on D .0
(b) Independence of irrelevant cost levels Let q be given. If two cost functions C ,1
1
C2[D0 coincide on the interval [a, b],[0,`[ and a#nq1,nqn#b, thenyG;r,s(C ,1
q)5yG;r,s(C , q).2
1
(c) Fixed-costs additivity For any cost function C[D0 and fixed-costs function
1
F[D ,0 yG;r,s(C1F, q)5yG;r,s(C, q)1yG;r,s(F, q).
Property (a) reflects the basic idea behind mixed serial cost sharing as a mechanism which is an extension of bothjandh. Hence consider for someG the mechanismyG;j,h. Since the CS-decomposition prescribes that if the function C is concave, then it is decomposed according to C501C, mixed serial costs sharing clearly satisfies (a). On
the other hand, replacing the CS-decomposition by what would seem to be its natural counterpart, i.e. a decomposition which maximizes the convex component, leads to a mechanism which violates property (a); consider for example the concave function
C(Q )5min 2Q, Qh 11j5max 0, Qh 21j1min 2Q, 2 . Clearly, (b) holds for bothh j j andh and reflects a basic property of the serial method as only cost levels between nq1
and nq matter. Finally, property (c) appears to be a natural requirement of any costn
sharing mechanism on D . Note that the original serial scheme in Moulin and Shenker0
(1992) is additive and hence fixed-costs additive. Moreover, properties (a) and (c) jointly imply that if for example the cost function equals fixed costs plus a linear function, then even an agent having zero demand pays an equal share of fixed costs; this can be interpreted as an equal split of a common entrance fee.
1
It turns out that mixed serial cost sharing is the only decomposition mechanism on D0
which satisfies all three properties:
Theorem 1. Let G and r, s be a given decomposition rule and a pair of additive
mechanisms as above. Then the decomposition mechanism y satisfies properties
G;r,s
(a –c) if and only if it is identical to mixed serial cost sharing, i.e. G 5 GCS andr 5 j, s 5h.
Proof. It is easily verified thatz satisfies (a–c). Thus we prove the converse. It suffices to consider a piecewise affine cost function C which is determined by some subdivision
a,b
0,Q1,Q2, . . . of [0,`[. If C equals the convex angle function C (Q )5max 0,h aQ2bj,a,b .0, then C has no other decomposition than C5C10. Consequently,
a,b a,b
affine convex cost function equals a finite sum of convex angle functions on any bounded interval and by assumption, mechanismris additive over its domain; therefore r 5 j. In a similar fashion, consideration of concave angle functions Ca,b(Q )5minhaQ,
bjshows that s 5h.
1
Now, consider a cost function C[D0 decomposed by G(C )5(R, S ) and a demand vector q confined by the interval a, b in the sense that af g #nq1,nqn#b for some a,
∨
b.0. Firstly, if C is convex on a, b , then there exists a convex cost function C and af g
∨ ∨ ∨
fixed-costs function F such that C 1F coincides with C on fa, b . Hence, fromg
r 5 j,s 5h, (b); (c); (a) and q1.0, it follows that
∨ ∨ ∨ ∨
j(R, q)1h(S, q)5yG;j,h(C 1F , q)5yG;j,h(C , q)1yG;j,h(F , q)
∨ ∨ ∨ ∨
5yG;j,h(C , q)1h(F , q)5j(R1S2F , q)1h(F , q) 5j(R1S, q).
Thus asj(R, q)1h(S, q)5j(R1S, q) for any q being confined by a, b , S must bef g
affine on this interval.
∧
Secondly, if C is concave on a, b , then there exists concave cost function C and af g
∧ ∧ ∧
fixed-costs function F such that C1F coincides with C on a, b . Then again usingf g
∧ ∧ ∧
(a–c), one obtainsh(R1S1F , q)5yG;j,h(R1S, q)1h(F , q)5j(R, q)1h(S1F , q), and it follows that R must be affine on a, b .f g
Consequently, by Thon and Thorlund-Petersen (1986) the piecewise affine function C is decomposed according to the CS-decomposition, meaning that G 5 G . h
CS
It follows from (23) that mixed serial cost sharing z inherits positive cross-monotonicity from j andh on their respective domain. Furthermore, for C5R1S,
inserting C5R in (15) and C5S in (17) one obtains directly by additivity in the sense
of (23) upper and lower bounds for the mechanismz. For example, the payment of agent 1 satisfies C(nq ) /n1 #z1#R(nq ) /n1 1S(Q ) /(n21)2S n(Qs 2(n21)q ) /n(n1 d 21).
For given C and strictly positive q, consider corresponding agent-specific prices (z /q ), ii i 51, . . . , n. Such prices satisfy the following condition.
Proposition 2. Suppose that q1.0. Let i1,i2, ? ? ? ,i2m11 be an odd number of agents in N. Then the vector of prices (z /q , . . . , z /q ) satisfies
1 1 n n
z /qi2 i21z /qi4 i41 ? ? ? 1zi2mqi2m#z /qi1 i11z /qi3 i31 ? ? ? 1zi2m11/qi2m11 (24) Proof. This follows by Lemma 1 since the vector (z /q , . . . , z /q ) equals the sum of1 1 n n
two nonnegative vectors (x /q , . . . , x /q ) and ( y /q , . . . , y /q ), each of which are1 1 n n 1 1 n n
increasing and decreasing, respectively, in index i, and therefore satisfy (24). h
In particular, Proposition 2 implies that for all 1#h,i,k#n one has z /qi i#z /h
qh1z /q . Thus for ik k 52, . . . , n21, z /qi i#2 maxj±i
h
z /q .j jj
functions, then it must necessarily violate this principle because individual rationality is violated by the decreasing serial rule. Here, we take the view that in a number of cost sharing problems individual rationality is irrelevant as agents may be forced to cooperate due to regulatory economic or organizational constraints. Sufficient conditions for the decreasing serial rule to pass the stand-alone test are given in Hougaard and Thorlund-Petersen (2000).
5. Inequality comparisons of payment schemes
In this section, mixed serial cost sharing z is related, in terms of economic inequality, to the well-known schemes of average and marginal cost pricing as well as to the original serial scheme in Moulin and Shenker (1992). Moreover, in the same spirit, we compare the use of the CS-decomposition with any alternative decomposition.
Define the mechanism of average cost pricing a on D by the vector of payments0
1
a5sC(Q ) /Q q. Secondly, for Cd [D0 and q, define the vector of payments m by
1 ]
mi5q Ci 9(Q )1nsC(Q )2QC9(Q ) ,d i51, . . . , n. (25)
AC↓
Let D0 denote the functions in D for which average costs C(Q ) /Q are decreasing in0
Q.0. If
1 AC↓
C[D0 >D0 , (26)
then the vector m is nonnegative. Hence define the mechanism of marginal cost pricing m on the domain (26) by m(C,q)5m. As an example of (26), consider the concave–
convex function C(Q )5Q if Q,1, C(Q )51 if 1#Q,2, and C(Q )5(1 / 2)Q if
Q$2.
Our first result shows that under schemes x and y, payment vectors are more (less) spread out than the payment vector of average cost pricing if the cost function is convex (concave). Furthermore, marginal cost pricing yields payments which are more spread out than under increasing and decreasing serial cost sharing, respectively.
Lemma 3. Let a denote the ordering of majorization, see Appendix B. Consider a given demand vector q and cost function C[D . If C is convex, then aaxam and
0
aay. If C is concave, then yamaa and xaa.
Proof. Suppose that C is convex (or concave) and that q1.0. Then by Lemma 1, x /qi i
and y /q are increasing (or decreasing) in i. Hence the four majorizations involving x, ai i
or y, a follow from application of the quotient criterion, Appendix B, (B.3).
If C is convex and i51, . . . , n21, then by (6), (7), mi112mi5( qi112q )Ci 9(Q ) $(C(r )2C(r )) /(n2i )5x 2x , hence xam follows by the difference criterion,
i11 i i11 i
(7), and the difference criterion, it follows that yam. Finally, maa follows from the
quotient criterion. h
Lemma 3 shows that no definite conclusion can be drawn concerning majorization between the payment vectors of mixed serial cost sharing and of average cost pricing. With respect to marginal cost pricing, there is such a relation. Moreover, in order to compare mixed serial cost sharing zwith the original serial mechanism determined by x
1
on C[D0 one needs the following inequalities.
Lemma 4. Consider the payment schemes x and y of (8) and (9). For a given demand q
and index k[N, we have
↑
C[D implies (i ) x 1 ? ? ? 1x 1y 1 ? ? ? 1y $C(Q ), (ii ) xay, (27)
0 1 k k11 n
↓
C[D implies (i ) x 1 ? ? ? 1x 1y 1 ? ? ? 1y #C(Q ), (ii ) y a x. (28)
0 1 k k11 n
Proof. We prove (28), as (27) can be proved in a similar manner. Thus suppose throughout in this proof that C is concave. Then if k51, . . . , n, it follows from (A.2), (A.4), (A.5) of Appendix A that
C(r )k C(r )1 C(r )2
]]] ]]] ]]]]
x11 ? ? ? 1xk5n2k111(n2k)
S
n(n21)1(n21)(n22)1 ? ? ?C(rk21) ]]]]]]]
1
D
(n2k12)(n2k11)
C(r )k
]]]
# n2k111(n2k)(s 2 sn n2i11)
r1 r2
21
]]] ]]]]
3C
S
(s 2 sn n2i11)S
n(n21)1(n21)(n22)1 ? ? ?ri21
]]]]]]
1
DD
(n2i12)(n2i11)
q 1 ? ? ? 1q
1 (n2k)(k21) 1 k21
]]]
S
]]]]S
]]]]]D
D
5n2k11 C(r )k 1 n C n k21
k n
] ]]]] 5n
S
k(n2k11)C(r )kq 1 ? ? ? 1q
(n2k)(k21) 1 k21
]]]]
S
]]]]]D
1 C n
D
k21
k(n2k11)
k n n(n2k)
] ]]]] ]]]]
# nC
S
k(n2k11)rk1k(n2k11)sq11 ? ? ? 1qk21dD
k n
] ]
5nC
S
ksq11 ? ? ? 1qkdD
. (29)C(sn2k) ]]]
yk111 ? ? ? 1yn5 k11
C(s )1 C(s )2 C(sn2k21)
]]] ]]]] ]]]]
1k
S
1 1 ? ? ? 1D
n(n21) (n21)(n22) (k12)(k11)
C(sn2k) 21 s1
]]] ]]]
# k11 1k(s 2 sn n2j11)C
S
(s 2 sn n2j11)S
n(n21)sn
s2 sn2k21
]]]]] ]]]]
1 1 ? ? ? 1
DD
(n21)(n22)sn (k12)(k11)
q 1 ? ? ? 1q
1 k(n2k21) n k12
]]
S
]]]]S
]]]]]D
D
5k11 C(sn2k)1 n C n n2k21
q 1 ? ? ? 1q
n2k n k(n2k21) n k12
]] ]]]] ]]]]
S
]]]]]D
5 n
S
(n2k)(k11)C(sn2k)1(n2k)(k11)C n n2k21D
n2k n nk
]] ]]]] ]]]]
# n C
S
(n2k)(k11)sn2k1(n2k)(k11)sqn1 ? ? ? 1qk12dD
n2k n
]] ]]
5 n C
S
(n2k)sqn1 ? ? ? 1qk11dD
. (30)Adding (29) and (30) and using concavity yields (28(i )). Thus the pair of vectors x, y satisfies (B.1) of Appendix B thus (28(ii )) follows. h
Now the following result can be established.
1
Proposition 3. For demand q and cost function C[D , the paymentvector x is more
0
spread out than z, z(C, q)ax. Furthermore, if average costs are nonincreasing, (26),
then payments under marginal cost pricing are more spread out than under mixed serial cost sharing, z(C, q)am(C, q).
Proof. Since xax, this follows from Lemmas 3 and 4. h
Consider the CS-decomposition, C5R1S as used in mixed serial cost sharing. It
follows from Lemma 4 that on the class of concave cost functions, choosing the payment scheme y rather than x amounts to selecting the minimum with respect to a of the set x, y . Similarly, choosing payment scheme x over y on the class of convex functions is
h j
equivalent to selecting the a-minimal payment scheme. In other words, choosing
payment schemes x for increasing marginal costs and y for decreasing marginal costs is tantamount to choosing the most egalitarian payment scheme under the given circum-stances.
On the other hand, it turns out that the vector of payments z based on the CS-decomposition is maximal with respect to a in the convex set of n-vectors hj(C ,
1
↑ ↓
↑ ↓
Theorem 2. Consider functions R , R1 [D , S , S0 1 [D0 such that the cost function C5R 1S has CS-decomposition C5R1S. Then for anyvector of demands q,
1 1
j(R , q)1h(S , q)az(C, q), (31)
1 1
Proof. The function G5R12R is increasing convex as can rather easily be proved
from Thon and Thorlund-Petersen (1986). In particular, if C is twice continuously
99 99 99 99
differentiable, then one has 05R0s(R1 2R0)1(S1 2S0)d5R0(R1 2R0)1R0S , thus1
9 9
G0 $0. As G9 5R (0)1 2R9(0)5R (0)1 $0, the function G is increasing convex and Lemma 4 applies. One must prove that for any k51, . . . , n
k k
O
sji(R, q)1hi(S, q)d#O
sji(R , q)1 1hi(S , q) .1 d (32)i51 i51
Since G5R12R5S2S , (32) is equivalent to1
k k k
O
shi(S, q)2hi(S , q)1 d#O
sji(R , q)1 2ji(R, q)d5O
ji(G, q),i51 i51 i51
which follows from (27(ii )). h
At first sight the result of Theorem 2 is rather striking; as the CS-decomposition picks the maximal concave component, this method intuitively maximizes the role ofhwhich one might expect to lead to more equal payment vectors. On the other hand, it follows
1
from Proposition 3 that the use of payment scheme x on the entire domain D0 leads to a more unequal payment vector than what results from z. It remains an open question whether there exists a counterpart to Theorem 2 in terms of an alternative decomposition rule leading to a unique payment vector of minimal inequality. An obvious candidate would be a decomposition which maximizes the convex component. However, a cost sharing rule based on such a decomposition, will violate the condition of Extension defined in Section 4; consider, as in Section 4, the concave cost function C(Q )5 min 2Q, Qh 11 which has CS-decomposition Cj 501C as well as the decomposition C(Q )5max 0, Qh 21j1min 2Q, 2 .h j
6. Final remarks
Consider an example of a seller, such as an airline, offering more than a single bundle of tickets; sometimes airlines offer an arbitrary number of bundles and possibly even a
1
mix of different bundles. Then one ends up with a cost function outside the domain D .0
For example, with two bundles the cost function consists of five pieces of slope t1, . . . ,t5 where t 5 t 5 t .1 3 5 0 and t 5 t 52 4 0; by Thon and Thorlund-Petersen
1
(1986), the cost function is in D0 only if t # t 1 t3 2 4 and this condition is clearly violated.
cost function allow an additive decomposition into those two cases. Accordingly, such problems cannot readily be solved by mixed serial costs sharing. Therefore, extending our mechanism to these more general cases calls for further research.
Acknowledgements
´
The authors are grateful to Herve Moulin and an anonymous referee for many helpful comments.
Appendix A
Consider the matrix of (4), (5). It is simple to prove that
n 0 0 . . . 0
1 n21 0 . . . 0
1 1 1
]]] ]]]], , . . . ,], 0 1 1 n22 . . . 0
F
n(n21) (n21)(n22) 2G
3
. . .
4
1 1 1 . . . 1
5f1, 1, . . . , 1, 0 .g (A.1)
Using (A.1), (4), and (5), we obtain
r1
:
1 1 1 ri21 1
]]] ]]]], , . . . ,], 0 5]]]fq 1 ? ? ? 1q g,
F
n(n21) (n21)(n22) 2G
3 4
0 n2i11 1 i21:
0
(A.2)
s1
:
s
1 1 1 j21 1
]]] ]]]], , . . . ,], 0 5]]] q 1 ? ? ? 1q .
F
n(n21) (n21)(n22) 2G
n2j11f
n n2j12g
0
3 4
:0
(A.3)
1
1 1 1 1
]]] ]]]], , . . . ,], 0
F
n(n21) (n21)(n22) 2G
3 4
:1
n 0 0 . . . 0
1 /n 1 n21 0 . . . 0
1 1 1 1 /n
]]] ]]]] ]
5
F
, , . . . , , 0G
1 1 n22 . . . 0 : 2n(n21) (n21)(n22)
3
3 4
. . .
4
1 /n1 1 1 . . . 1
1 /n
n21 1 /n
F G
]] 5f1 1 . . . 1 0g : 5 ,n
3 4
1 /nwhere the second equality follows from (A.1). Accordingly, the sum sn defined by
1 1 1 1
]]] ]]]] ]] ]
s 5n n(n21)1(n21)(n22)1 ? ? ? 13?212 (A.4)
satisfiess 5n (n21) /n. Consequently,
s 2 sn n2i115(i21) /n(n2i11). (A.5)
Appendix B
If v and u are n-vectors with increasingly ordered components, u # ? ? ? #u ,
1 k
v # ? ? ? #v, then u majorizesv if for k51, . . . , n
1 k
u 1 ? ? ? 1u #v 1 ? ? ? 1v, (B.1)
1 k 1 k
with equality for k5n, in which case we write vau, see e.g. Marshall and Olkin (1979) and Thon and Thorlund-Petersen (1986). The partial ordering a is well known
for example from the economic literature on income-inequality measurement, where vau is interpreted as income distribution u being more spread out (or unequal) than distribution v.
Two criteria for majorization are particularly useful in this paper: if vectors u andvas above satisfy the difference- or the quotient criterion:
u 2v # ? ? ? #u 2v, (B.2)
1 1 n n
v .0 and u /v # . . . #u /v , (B.3)
1 1 1 n n
thenvau, see Marshall and Olkin (1979, chapter 5). The class of real functions which preserve the ordering a is often useful for identifying and proving inequalities. A real
n
function f defined on some subset V#h(u , . . . , u )1 n [R uu1# ? ? ? #unj is called
Schur-convex if u,v[V,vau implies that f(v)#f(u); f is called Schur-concave if 2f
n
≠f(u) ≠f(u) ≠f(u)
]] ]] ]]
0# # # ? ? ? # , (B.4)
f g ≠u ≠u ≠u
1 2 n
see Marshall and Olkin (1979, chapter 3).
In order to prove that for given qi.0, i$2, the function ji is Schur-concave in variables q , . . . , q1 i21 on the convex set V 5i h( q , . . . , q1 i21)u0#q1# ? ? ? #qi21#
qij, we apply (B4). For i52 this is trivial. Thus consider an index h such that 1#h,h11,i and suppose that q is interior to Vi. Furthermore, assume that C is continuously differentiable. Now,
≠ji C9(r )i C9(r )1 C9(r )2
]5]]]2
S
]]]1]]]]1 ? ? ?≠qh n2i11 n(n21) (n21)(n22)
C9(rh21) C9(r )h ≠ji
]]]]]]] ]] ]]
1
D
2≠q
(n2h12)(n2h11) (n2h) h11
C9(r )i C9(r )1 C9(r )2 C9(rh21)
]]] ]]] ]]]] ]]]]]]]
5 2
S
1 1 ? ? ? 1D
n2i11 n(n21) (n21)(n22) (n2h12)(n2h21)
C9(r )h C9(rh11)
]]]]] ]]]]
2 2
(n2h11)(n2h) (n2h21)
Consequently,
≠ji ≠ji C9(rh11)2C9(r )h
]≠ 2]]5]]]]]$0.
qh ≠qh11 (n2h21) Since ≠ji/≠qi5C9(r ), theni
≠ji ≠ C(r )i 2C(ri21) C(r )i
]]≠q 5]]≠q
S
xi211]]]]]n2i11D
5]]]n2i11$0,i21 i21
hence Schur-concavity follows by convexity of C. The proof can easily be generalized to a non-differentiable convex C. Note that increasingness of ji is equivalent to positive cross-monotonicity. In a similar fashion, it can be proved thathi is increasing Schur-convex in qi11, . . . , q .n
References
de Frutos, M.A., 1998. Decreasing serial cost sharing under economies of scale. Journal of Economic Theory 79, 245–275.
Hougaard, J. L., Thorlund-Petersen, L., 2000. The stand-alone test and decreasing serial cost sharing. To appear in Economic Theory.
Marshall, A., Olkin, I., 1979. Inequalities: Theory of Majorization and its Applications. Academic Press, New York.
Moulin, H., Shenker, S., 1992. Serial cost sharing. Econometrica 60, 1009–1037.
Moulin, H., 1996. Cost sharing under increasing returns: a comparison of simple mechanisms. Games and Economic Behavior 13, 225–251.
