´ C
. Bevia et al. Mathematical Social Sciences 37 1999 1 –23 9
Section 3. The analysis follows the route of the second Theorem of Welfare economics
M m
and shows that the prices p and p
as well as a convex, complete lattice of intermediate prices support the efficient allocations of the objects among the agents.
3. Equilibrium prices
M
3.1. Definition of p and p
m
The analysis of this section is made under the following set of assumptions on the utility functions which will not be repeated.
Assumption SC. For all i [I, the utility function Vi,? is submodular and satisfies the cardinality condition.
Let s be an efficient assignment of the objects V to the I agents. The goal of this section is to derive the prices supporting this allocation of the objects. In a model with
divisible goods and quasi-linear utilities, the prices supporting a Pareto optimal allocation are given by the multipliers associated to the scarcity constraints in the
program of maximization of the sum of the utilities social welfare subject to the feasibility constraints. The envelope theorem then permits interpreting the multiplier
associated to the scarcity constraint for a good let us say good a as the change in social welfare resulting from a marginal decrease or increase in the supply of this good.
Suppose now that good a is indivisible and exists in a single unit. If we proceed by analogy, there are two changes in the supply of a which play the role of a marginal
change in the supply of a when the good is divisible: the supply can be decreased by one unit by taking the good a out of the available supply of goods; or the supply can be
increased by one unit by adding a copy of a to the supply of available goods. These changes induce changes in social welfare analogous to the changes in social welfare
accompanying a marginal change in the supply of a divisible good. We will prove that these changes in social welfare define the maximum and minimum prices supporting the
efficient allocation s.
Let us thus define the social welfare created by a supply V of objects by U
V 8
O
Vi, si, for any efficient assignment s of V
i [I M
Define p as the change in social welfare when the object a is taken out of the
available objects, i.e.
M
p a 5 UV 2 UV•a, a [ V
or alternatively as the contribution of a to the social welfare. To define the minimum ˜
prices, for all a[V let a denote an exact copy of object a. To define the social welfare
˜ associated to V
a, we need either to extend the agents’ reservation value functions to ˜
˜ subsets containing both a and
a or to restrict ourselves to assignments of Va which ˜
assign a and a to different agents. We choose the second solution and define the set of
˜ ˜
feasible assignments S9I, V a as the sets of maps r from I to Va satisfying i
´ 10
C . Bevia et al. Mathematical Social Sciences 37 1999 1 –23
˜ ˜
ri5V a, ii i ±j
⇒ rir j5[, iii a[ri,
a [r j ⇒
i ±j. Then for all
i [I
˜ r[S9I,V
a , ri is in V so that Vi,ri is well defined. The social welfare ˜
associated to V a is then defined by
˜ ˜
U V a 5 max
O
Vi, ri
ur [ S 9I,V a
H J
i [I
˜ and a solution to the maximum problem is called an efficient allocation of V
a. The ˜
definition of S9I,V a is chosen in order to make it possible to consider assignments
which allocate the object a to the agent i who has it in an efficient assignment of V, while giving a copy of a to the agent who would most benefit from a after agent i. Thus
if we define
˜ p
a 5 UV a 2 UV , a [ V
m
we can interpret p a as the social value of a in its second best use see Remark 1 after
m
Lemma 3.3.
M M
M
Define p 5 p a and p 5 p a
. We first show that p a and p a give
a[V m
m a[V
m
respectively the highest and lowest possible equilibrium prices for object a.
Proposition 3.1. Let s be an efficient assignment of V , and suppose that there exists
uVu
˜ p [R
supporting the assignment s . Then for all a[V, UVa 2UVpa
1
UV2UV•a.
Proof. Since the vector p is supporting the assignment s, for all i [I, and for all A[PV, Vi,si 2psi Vi,A2pA. In particular, given a[V, let t be an
˜ efficient assignment of V•a among I, and let r be an efficient assignment of V
a. As noted above, assume w.l.o.g. that ri [V for all i. Then, Vi,si 2psi Vi,ti 2
pti and Vi,si 2psi Vi,ri 2pri for all i [I. Summing up these ˜
inequalities, we get UV2paUV•a and UVUV a 2pa. h
M
The next two lemmas will be frequently used in proving that p and p
are
m
equilibrium price vectors.
Lemma 3.2. Let a and b be two objects in V and let C and D be two subsets of V such that
ha,bjC5[ and ha,bjD5[. Then for all i[ I Vi,C
a 2Vi,C b 5Vi,D a 2Vi,D b
Proof. The property follows directly from the characterization of the cardinality condition in Proposition 2.5.
h
Lemma 3.3. Let s be an efficient assignment of V to the agents . For any a[V there is
˜ an efficient assignment t of V•a and an efficient assignment r of V
a such that utiuusiuuriu for all i [I. Moreover r can be constructed such that the agent who
has a in the assignment s also has it in the assignment r.
´ C
. Bevia et al. Mathematical Social Sciences 37 1999 1 –23 11
Proof. See Appendix A.
¯
Remark 1. The property that the agent i who is assigned a under s also receives a under r
, justifies the interpretation of p a as the value of a in its second best use: r
m
¯ attributes the object a to the agent i to whom it is efficient
under s to give it, and ¯
attributes the second copy of a to the agent who , after i, would benefit most of
consuming a — perhaps after a reallocation of the other objects .
M
We now prove that p and p
are equilibrium prices supporting the efficient
m
allocations of the objects V.
M
3.2. p supports the efficient assignments of
V to I
Lemma 3.4. Let s be an efficient assignment of V among I
. For all i [I
M
1. if a[si , then p aVi,si 2Vi,si •a
M
2. if b[ ⁄ si , then p bVi,si b2Vi,si .
Proof.
M
1. Let a[si , then UV•a o V j,s j1Vi,si•a. Since p a5UV2
j [I, j ±i
UV•a,
M
p a
O
V j, s j 2
O
V j, s j 2Vi,si•a
j [I j [I, j ±i
5Vi, si 2Vi,si•a
2. Let b[ ⁄ si , and suppose that
M
p b 5 UV 2 UV•b ,Vi,si b 2Vi,si
then, U
V , UV•b 1Vi,si b 2Vi,si By Lemma 3.3 there exists an efficient assignment, t, of V•b among I such that
utiuusiu, and by Assumption SC Vi,
si b 2Vi,si Vi,ti b 2Vi,ti Then,
U V , UV•b 1Vi,ti b 2Vi,ti UV
which is a contradiction. h
M
Lemma 3.5. For all i [I , there is A[PV such that
uAu5usiu and A[Di, p .
´ 12
C . Bevia et al. Mathematical Social Sciences 37 1999 1 –23
Proof.
M
1. Suppose that there is B [Di, p such that uBu,usiu. From part 1 of Lemma 3.4
M
we know that p aVi,si 2Vi,si •a for all a[si . Since uBu,usiu, if
M M
a[ ⁄ B, p aVi,B a2Vi,B so that B a[Di, p . Following this process, we
M
can add elements to B and obtain a set B such that uBu5usiu and B[Di, p .
M
2. Suppose that there is B [Di, p such that uBu.usiu. Then by part 2 of Lemma
M
3.4 we know that for all b[B such that b[ ⁄ si , p bVi,si b2Vi,si .
M M
This implies, since uBu.usiu, that p bVi,B2Vi,B•b. Since B [Di, p , it
M M
must be that p b5Vi,B 2Vi,B•b, so that B•b[Di, p . Following this process, we can subtract elements from B and obtain a set B such that
uBu5usiu
M
and B[Di, p . h
M
Theorem 3.6. If s is an efficient assignment of V among I , then p
supports s.
Proof. In order to prove the theorem, we must prove that if s is an efficient assignment
M M
of the objects, then for all i [I, si [Di, p . By Lemma 3.5, there exist A[Di, p such that
uAu5usiu. Suppose that A±si. Then there exists b[si such that b[⁄ A
M
and there is a[ A such that a[ ⁄ si . Let us show that Ab•a is also in Di, p .
M M
Suppose this is not true. Then it must be that Vi,A2p A.Vi,Ab•a2p A
M M
b•a, which is equivalent to p b2p a.Vi,Ab•a2Vi,A. By Lemma 3.2 with C 5 A•a and D 5si •b this is equivalent to
U V•a 2 UV•b .Vi,si 2Vi,si a•b
3.1 To show that this inequality is impossible, consider an efficient assignment t of V•a
such that ut juus jufor all j [I. By Lemma 3.3 such an assignment exists. In order to
contradict inequality 3.1 we construct an assignment of V•b from t by removing the object b from the agent who has it under t and ‘appropriately’ assigning the object a.
Consider the agent j who receives b under t. If j 5i, then take b from agent i and
1 1
replace it by a. If j ±i then since b[t j and b[ ⁄ s j , and since
ut j uus j u,
1 1
1 1
1
there is an object b in s j which is not in t j . If this object is either a or is such
1 1
1
that it belongs to ti , then the procedure stops: in the first case replace b by a in the assignment of agent j , in the second replace b by b for agent j and replace b by a
1 1
1 1
for agent i. If b cannot be either a or an object of ti , then there exists an agent j such
1 2
that b [t j . By the same reasoning, since b [ ⁄ s j there exists an object b in
1 2
1 2
2
s j which is not in t j . If either this object is a or if it belongs to ti, then
2 2
procedure stops by replacing b by b for agent j , and b by a for agent j in the first
1 1
1 2
case, by b in the second case and b by a for agent i; otherwise it continues. As long
2 2
as the procedure continues the objects b , b , . . . , can be chosen so as to be different
1 2
from each other since each time that an object in t j and not in s j , since ut ju
us ju, there is a corresponding object in s j. Since there is a finite number of objects the procedure must end by finding an agent j
such that there exists an object b in
m m
s j which is either a or is such that b [ti. In the first case consider the assignment
m m
of V•b such that b is replaced by b for agent j , b is replaced by b for agent
1 1
1 2
´ C
. Bevia et al. Mathematical Social Sciences 37 1999 1 –23 13
j , . . . , and b is replaced by a for agent j . In the second case b is replaced by b
2 m 21
m 1
for agent j , b is replaced by b for agent j , . . . , b is replaced by b for agent j
1 1
2 2
m 21 m
m
and b is replaced by a for agent i. Note that the agents j , . . . , j can be chosen so as
m 1
m
to be different from each other since, if the same agent j is chosen twice, i.e. if for some
,
, 1, r1, j 5j
, then the object b can directly be chosen in s j instead of b
, ,
1r ,
1r ,
,
the first time that agent j is selected. We now use the assignment just constructed to find
,
a bound on the difference UV•a2UV•b. Consider the first case where b 5a.
m
U V•a 2 UV•b V j ,t j 2V j ,t j •b b
1 1
1 1
1
1V j , t j 2V j ,t j •b b
2 2
2 2
1 2
1 . . . 1V j ,
t j 2V j ,t j •b a
m m
m m
m 21
By Lemma 3.2 with C 5t j •b and D 5s j •b for ,51,...,m
, ,
21 ,
,
U V•a 2 UV•b V j ,s j •b b 2V j ,s j
1 1
1 1
1
1V j , s j •b b 2V j ,s j
2 2
2 1
2 2
1 . . . 1V j ,
s j •a b 2 V j ,
s j
m m
m 21 m
m
By the efficiency of the assignment s Vi,
si 1V j ,s j 1 ? ? ? 1V j ,s j Vi,si a•b
1 1
m m
1V j , s j •b b 1 ? ? ?
1 1
1
1V j , s j •a b
m m
m 21
which, combined with the previous inequality implies U
V•a 2 UV•b Vi,si 2Vi,si a•b and contradicts 3.1. The proof for the case b [ti is similar and left to the reader.
m
Note that it covers, with m 50, the case where b[ti .
M
Thus inequality 3.1 is impossible so that if A[Di, p is different from si then each object of A which is not in si can be replaced by a corresponding object of si
and the new subset obtained in this way is still in the demand of A. After a finite number
M
of such replacements the subset si will be obtained, so that si [Di, p . h
Remark 2. Since efficient assignments exist , Theorem 3.6 proves the existence of a
competitive equilibrium for the economy e . The end of this section describes the
structure of the set of equilibrium prices .
´ 14
C . Bevia et al. Mathematical Social Sciences 37 1999 1 –23
3.3. p supports the efficient assignments of V to I
m
Lemma 3.7. Let s be an efficient assignment of V among I . For all i [I
1. if a[si , then p aVi,si 2Vi,si •a
m
2. if b[ ⁄ si , then p bVi,si b2Vi,si
m
Proof.
M
1. Since we have proved that p is an equilibrium price, Proposition 3.1 implies that
M M
p p . Since, by Lemma 3.4, p satisfies the inequality 1, so does p .
m m
˜ 2. If b[
⁄ si , adding b to the objects of i creates an assignment of V
b. Thus ˜
UV b UV2Vi,si1Vi,sib, which is equivalent to the inequality in
2. h
Lemma 3.8. For all i [I , there is A[PV such that
uAu5usiu and A[Di,p .
m
Proof. The proof is identical to the proof of Lemma 3.5. h
Theorem 3.9. If s is an efficient assignment of V among I , then p supports s.
m
Proof. We must prove that for all i, si [Di, p . By Lemma 3.8, there exists A in the
m
demand of agent i such that uAu5usiu. Let us show that if A±si, then every object a
in A and not in si can be replaced by an object b in si and not in A, so that Ab•a is in the demand of agent i. By the same reasoning than in the proof of
Theorem 3.6, if Ab•a were not in the demand of agent i, then the following inequality would have to hold
˜ ˜
U V b 2 UV a .Vi,si 2Vi,si a•b
3.2 ˜
To show that this equality is impossible, choose an efficient assignment r of V b
such that r j [V, ur juus ju, for all j [I and such that b[ri. By Lemma 3.3 such
an assignment exists. There are two possible cases. ˜
Case 1. r assigns b and not a to agent i. Then consider the assignment of V
a obtained in replacing b by a.
˜ ˜
U V a UV b 2Vi,ri 1Vi,ri•b a
˜ 5 U
V b 2Vi,si 1Vi,si•b a where the last equality follows from Lemma 3.2 with C 5ri •b, D 5si •b. This
contradicts inequality 3.2.
Case 2. r assigns b and a to agent i. Let j be the agent who receives a under s, and let k
´ C
. Bevia et al. Mathematical Social Sciences 37 1999 1 –23 15
˜ ˜
be the agent who receives the copy b of b under r. If j5k, then take b from agent j and
˜ replace it by
a. ˜
˜ U
V b 2 UV a V j,r j 2V j,r j a•b 5V j,
s j b•a 2V j,s j where the last equality follows from Lemma 3.2 with C 5r j •b and D 5s j •a. By
efficiency of the assignment s, V j,
s j b•a 2V j,s j Vi,si 2Vi,si a•b This contradicts inequality 3.2. If j ±k, then since a[s j , a[
⁄ r j , and
ur ju us ju, there is an object b in r j which is not in s j. If this object b belongs to sk
˜ or si , then the procedure stops: in the first case replace b by
a for agent j, and ˜
˜ replace
b by b for agent k. In the second case replace b by a for agent j, and replace b by b for agent i. If b is neither in sk nor in si, then there is an agent j such that
1
b [s j . By the same reasoning, since b [ ⁄ r j , and
ur j uus j u, there is an
1 1
1 1
object b in r j which is not in s j . If b belongs to sk or si , then the procedure
1 1
1 1
˜ ˜
stops by replacing b by a for agent j, b by b for agent j , and b by b for agent k in
1 1
1
the first case, and b by b for agent i in the second case; otherwise it continues. As long
1
as the procedure continues the objects b , b , . . . , can be chosen so as to be different
1 2
from each other since each time that an object in sl is not in rl , and since urluuslu, there is a corresponding object in rl. Since there is a finite number of
objects the procedure must end by finding an agent j such that b [r j , and
m m
m
b [ ⁄ s j , and such that b is in sk or si . In the first case consider the assignment
m m
m
˜ ˜
of V a such that b is replaced by a for agent j, b is replaced by b for agent j , b
1 1
2
˜ is replaced by b
for agent j , . . . , b is replaced by b
for agent j , and b is
1 2
m m 21
m
˜ replaced by b
for agent k. In the second case b is replaced by a for agent j, b is
m 1
replaced by b for agent j , b is replaced by b for agent j , . . . , b is replaced by
1 2
1 2
m
b for agent j , and b is replaced by b
for agent i. Note that, as in the proofs of
m 21 m
m
Lemma 3.3 and Theorem 3.6, we can assume w.l.o.g. that the agents j , . . . , j are all
1 m
different. We now use the assignment just constructed to find a bound on the difference
˜ ˜
UV b 2UVa . Consider the case where b [si.
m
˜ ˜
U V b 2 UV a V j,r j 2V j,r j a•b
1V j , r j 2V j ,r j b •b
1 1
1 1
1
1 ? ? ? 1V j ,
r j 2V j ,r j b •
b
m m
m m
m 21 m
1Vi, ri 2Vi,ri b •b
m
By Lemma 3.2 with C 5r j •b and D 5s j •b for l 50,1, . . . ,m where j 5j,
l l
l l 21
and b 5a, and C95ri•b, D95sia•
hb,b j
021 m
´ 16
C . Bevia et al. Mathematical Social Sciences 37 1999 1 –23
˜ ˜
U V b 2 UV a V j,s j b •a 2V j,s j
1V j , s j b •b 2V j ,s j
1 1
1 1
1
1 ? ? ? 1V j ,
s j b •b 2 V j ,
s j
m m
m m 21
m m
1Vi, si a•b 2Vi,si
m
1Vi, si 2Vi,si a•b
By the efficiency of the assignment s V j,
s j 1 ? ? ? 1V j ,s j 1Vi,si
m m
V j, s j b •a 1 ? ? ? 1V j ,s j b •b
1 Vi, si a•b
m m
m m 21
m
which, combined with the previous inequality implies ˜
˜ U
V b 2 UV a Vi,si 2Vi,si a•b and contradicts inequality 3.2. Thus 3.2 is impossible so that if A[Di, p is
m
different from si then each object of A which is not in si can be replaced by a corresponding object of si , and the new subset obtained in this way is still in the
demand of A. After a finite number of such replacements the subset si will be obtained, so that si [Di, p . The case where b belongs to sk is similar and left to
m m
the reader. h
3.4. The lattice structure of equilibrium prices
Theorem 3.10. The set of prices supporting the efficient assignments of V is a convex ,
complete lattice .
Proof. The set of prices supporting an efficient assignment s of V is the set of solutions to the linear inequalities Vi,si 2psi Vi,A2pA, ;A[V and is thus closed and
convex. To prove the theorem, we thus only need to prove that if p and p9 are two prices supporting an efficient assignment s of V then p
∧ p9 and p
∨ p9, defined by p
∧ p9a5
min h pa, p9aj and p
∨ p9a5max
h pa, p9aj for all a[V, also support s. This amounts to showing that si [Di, p
∧ p9 and si [Di, p
∨ p9 for all i [I. First note
M
that since, by Proposition 3.1, p p ∧
p9p ∨
p9p , the inequalities i and ii of
m
Lemma 3.4 or 3.7 are satisfied by p ∧
p9 and p ∨
p9. By the same reasoning as in Lemma
9
3.5, this implies that, for all i, there exists A in Di, p ∧
p9 and A in Di, p ∨
p9 such
i i
9
that uA u5uA u5usiu. Suppose that, for some agent i, A ±si. Then there exists a such
i i
i
that a[ A and a[ ⁄ si , and there exists b such that b[si and b[
⁄ A . Let us show
i i
that A •ab is also in Di, p ∧
p9. By Lemma 3.2 with C 5 A •a and D 5si •b
i 1
and the fact that p and p9 support s
´ C
. Bevia et al. Mathematical Social Sciences 37 1999 1 –23 17
Table 3
M
A price vector obtained from p and p which is not an equilibrium price vector
m
V \A
a b
g ab
ag bg
abg V1,A
8 9
8 16
15 16
22 V2,A
3 7
6 9
8 12
13 V3,A
5 4
7 8
11 10
13
Vi,A • a b 2Vi,A 5Vi,si 2Vi,si•b a
i i
max h pa 2 pb , p9a 2 p9b j
max p ∨
p9 a 2 p
∨ p9
b , p ∨
p9 a 2 p
∨ p9
b h
j where the last inequality can easily be checked case by case. Thus the objects of A
i
which are not in si can be replaced by objects of si , which proves that si [Di, p ∧
9
p9. The same reasoning applied to A shows that si [Di, p
∨ p9. Thus the set of
i
prices supporting the assignment s is a lattice, and being closed, it is complete. h
M
Note that choosing prices independently for each object a between p a and p a
m
does not generally lead to a vector of equilibrium prices, as shown by the following example.
Example 3.11. Let e [´ be such that I 5
h1,2,3j, V5ha,b,gj. The reservation values of the agents for the different subsets of objects are given in Table 3.
For this economy the efficient assignment is s15 habj, s25[, s35hgj. The
M
vectors p and p
are
m M
p 5 7,8,7, p 5 4,7,6
m
The price vector p 54,7,7 however is not an equilibrium price vector since at these prices agent 3 would demand object a and not object g. The prices of objects need to be
compatible: in particular the surplus of agent 3 on object g has to be as least as large as on object a. The set of equilibrium prices is
h41e,7, pgu6pgmin61e,7, 0´3
j.
4. Relation between the cardinality condition and gross substitutability
