Mathematical Social Sciences 41 2001 51–68 www.elsevier.nl locate econbase
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
: jlh.omcbs.dk J.L. Hougaard. 0165-4896 01 – see front matter
2001 Elsevier Science B.V. All rights reserved.
P I I : S 0 1 6 5 - 4 8 9 6 0 0 0 0 0 5 0 - 0
52 J
.L. Hougaard, L. Thorlund-Petersen Mathematical Social Sciences 41 2001 51 –68
C and suppose that 3 agents demand quantities q q q . According to serial cost
1 2
3
sharing, the common costs C q 1 q 1 q will be shared as follows. The agent with the
1 2
3
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 q 1 2q . The last agent
1 1
2
pays the residual costs. That is, payments x , x , x are defined as
1 2
3
1 1
] ]
x 5 C3q ,
x 5 x 1 C q 1 2q 2 C3q ,
s d
1 1
2 1
1 2
1
3 2
x 5 C q 1 q 1 q 2 x 2 x . 1
3 1
2 3
1 2
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 equal
1 2
3
1 1
] ]
y 5 C3q , y 5 y 1
C2q 1 q 2 C3q ,
s d
3 3
2 3
2 3
3
3 2
y 5 C q 1 q 1 q 2 y 2 y . 2
1 1
2 3
2 3
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.
In Section 2 preliminary definitions and concepts are presented and in Section 3, monotonicity properties of payments are analysed and corresponding upper and lower
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
