J .L. Hougaard, L. Thorlund-Petersen Mathematical Social Sciences 41 2001 51 –68
53
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 N 5 1, . . . , n of agents who share a common technology which h
j produces some homogeneous good. Each agent demands a quantity q 0 of the good.
i
Let q 5 q , . . . , q denote the vector of individual demands and let Q 5 q 1 ? ? ? 1
1 n
1
q denote total demand. The vector q is assumed to be increasingly ordered, q ? ? ?
n 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, `[
↑ ↓
such that C0 5 0 for all C [ D and let D and D
denote the convex and concave
↓
functions in D , respectively. A cost function F [ D which is constant for Q . 0 is
called a fixed-costs function. For any convex cone K D , a cost sharing mechanism y
n
on K is a mapping which for any C [ K and q [ R associates a vector of payments
1
v , . . . , v such that
1 n
i v 5 y C, q 0, i [ N;
ii v 1 ? ? ? 1 v 5 CQ . 3
i i
1 n
A mechanism y on K satisfies independence from above if for any i [
h1, . . . , n 2 1
j the cost share y C, q does not depend on q where j . i. Analogously, y satisfies
i j
independence from below if for any i [ h2, . . . , nj, y C, q does not depend on q where
i j
j , i. Finally call y additive if for all costs function C , C and any demand vector q,
1 2
yC 1 C , q 5 yC , q 1 yC , q.
1 2
1 2
For any vector of demands q 5 q , . . . , q define vectors r 5 r , . . . , r , s 5
1 n
1 n
s , . . . , s by the following linear transformations
1 n
n . . .
q r
1 1
1 n 2 1
. . . q
r
2 2
1 1
n 2 2 . . .
q r
5 ,
4
3 3
. . . :
:
3 43 4 3 4
1 1
1 . . .
1 q
r
n n
n . . .
q s
n 1
1 n 2 1
. . . q
s
n 21 2
1 1
n 2 2 . . .
q s
5 .
5
n 22 3
. . . :
:
3 43 4 3 4
1 1
1 . . .
1 q
s
1 n
54 J
.L. Hougaard, L. Thorlund-Petersen Mathematical Social Sciences 41 2001 51 –68
Notice that 5 is derived from 4 by reordering q inversely and that the n 3 n-matrix of 4 and 5 has row sums equal to n. Clearly, since q ? ? ? q , then
1 n
r ? ? ? r 5 Q 5 s ? ? ? s 6
1 n
n 1
and for i, j 5 2, . . . , n we have r 2 r
5 n 2 i 1 1q 2 q ,
s 2 s
5 j 2 1q 2 q .
7
i i 21
i i 21
n 2j 11 n 2j 12
j j 21
Following Moulin and Shenker 1992 and de Frutos 1998, we define for given n 2, C [ D , and q the following vectors of payments x 5 x , . . . , x , y 5 y , . . . , y by
1 n
1 n
i
Cr 2 Cr
k k 21
]]]]] x 5
O
, i 5 1, . . . , n,
8
i
n 1 1 2 k
k 51 j
Cs 2 Cs
k k 21
]]]]] y
5
O
, j 5 1, . . . , n,
9
n 2j 11
n 1 1 2 k
k 51
where by definition r 5 s 5 0. Clearly, if C is linear, then x 5 y. It is readily verified that for any C [ D , payments x and y satisfy the total-sum
condition corresponding to 3ii and the following inequalities hold: 0 x ? ? ? x ,
y ? ? ? y . 10
1 n
1 n
Nonnegativity of y does not follow for all C [ D . As an example, if n 5 2 and
1 ↓
C q 1 q 5 0 , C2q , then y 5 2 C2q 2 , 0. However, if C [ D , then pay-
1 2
2 1
2
ments are nonnegative as y Cnq n 0. The problem of determining bounds on
1 1
payments is further studied in Section 3.
↑ ↑
Define the increasing serial mechanism j on D by jC, q 5 x if C [ D , which on
↑
D is identical to the serial cost sharing mechanism in Moulin and Shenker 1992.
↓
Moreover, the decreasing serial mechanism h on D
is defined by hC, q 5 y if
↓
C [ D . It is easily verified, that mechanisms j and h satisfy independence from above
and from below, respectively. Furthermore, it follows from Moulin and Shenker 1992 and de Frutos 1998 that both mechanisms
j and h satisfy 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 q . 0 agent-specific prices
i
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 q . 0. If C is convex
, then x q ? ? ? x q and y q
1 1
1 n
n 1
1
? ? ? y q . If C is concave, then y q ? ? ? y q and x q ? ? ? x q .
n n
1 1
n n
1 1
n n
Proof. By definition, let q 5 0. If C is convex, then it follows from application of 8,
7, Jensen’s inequality, and finally 7, that for i 5 2, . . . , n we have
J .L. Hougaard, L. Thorlund-Petersen Mathematical Social Sciences 41 2001 51 –68
55
i 21 i 21
Cr 2 Cr Cr 2 Cr
k k 21
i i 21
]]]]] ]]]]]
x 5
O
q 2 q
O
q 2 q
i 21 k
k 21 k
k 21
r 2 r r 2 r
k k 21
i i 21
k 51 k 51
Cr 2 Cr Cr 2 Cr
x 2 x n 2 i 1 1
i i 21
i i 21
i i 21
]]]]] ]]]]] ]]]
]]] 5
q 5
q 5
q ,
i 21 i 21
i 21
r 2 r n 2 i 1 1
r 2 r q 2 q
i i 21
i i 21
i i 21
proving that x q ? ? ? x q . The remaining three sets of inequalities are proved in
1 1
n 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
