Competitive equilibrium in an
Exchange Economy with
Indivisibilities.
• By:Sushil Bikhchandani and John W.Mamer
• University of California.
• presented by: Meir Bing
1
• We analyze an exchange economy in which:
• all commodities except money are indivisible.
• agent’s preferences can be described by a
reservation value for each bundle of objects
• all agents are price takers.
• We will look for a necessary and sufficient
condition under which market clearing prices
(mc”p) exist.
2
• We saw already a good mechanism for one object
,but it is not known if a simple mechanism exist
for many objects where the buyer’s reservation
value for an object depends on which other objects
he obtains, it is called interdependent values.
• Example:FCC
• Vickrey auction in which bidders submit bids for
every bundle of objects assures an efficient
allocation, but it is too complex to implement.
3
• We ask at what condition there exist mc”p ?
• Market clearing prices (mc”p) are price for each
commodity , such that there is no excess demand
for any commodity.
• After we know that there are mc”p , the next step
is to investigate whether simple auction
procedures are capable of discovering the
competitive equilibrium prices.
4
• If mc”p are not exist then we believe it is unlikely
that any simple auction procedures will allocate
resources efficiently.
• This paper is also related to the matching literature
(we will not see it)
• we will see at what condition of the agents’
preferences will lead to mc”p.
5
• Consider an exchange economy with n indivisible
commodities and m agents.
• Each agent i=1,…,m has a reservation value
function :
Vi ( S ) : 2  
N
• this function is weakly increasing.
• Each agent i=1,…,m has a utility function Ui(*).
• Ui(S,W)=Vi(S)+W.
6
• where w is wealth level, and Wi’=is the initial
endowment of wealth of agent I.
• We assume that Wi’ Vi(N)
• E1={N,(Vi,Wi’) i=1,…,m}.
• A feasible allocation is an allocation in which no
object is assigned more than once.
• (S1,S2,…,Sm) denotes a feasible allocation ,where
agent i get Si and
7
S i  N , i  1,...m
S i  S j   far all i  j
• (S1’,…,Sm’) is an efficient allocation ,if it is
feasible allocation and if for every other feasible
allocation (S1,…,Sm)

m
Vi (S i ' )  i 1Vi (S i )
m
i 1
8
• mc”p are prices, one for each commodity at which
there is no excess demand for any commodity.
p k  0, k  1,...n are mc" p if there is a feasible
allocation (S1 ,..., S m ) such that :
1) Vi ( S i )   p k  Vi ( S )   p k ,
kS i
kS
S  N , i  1,..., m
n
m
2) p k    p k
k 1
i 1 kS i
9
• (S1,…,Sm) is said to be a market allocation which
is supported by prices p1,…,pn.
• We can see 2) at another way:
• 2) the price of any object which is unallocated at a
market allocation is zero:
pk  0
m
k  N \ ( S i ).
i 1
• agent i’s consumer surplus is: vi ( S i )   pk
kSi
10
Lemma 1
• Suppose that Wi  Vi ( N ) and Wi '  Vi ( N ) i then
prices (p1,…,pn) support a feasible allocation in the
economy E1={N,(Vi,Wi),i=1,..,m} if and only if
(p1,…,pn) support the same feasible allocation in
the economy E1’={N,(Vi,Wi’),i=1,..,m}
• so now we can write E1={N,(Vi),i=1,..,m}
11
Example where mc”p are not
exist
• There are two agents A,B ,and three objects 1,2
and 3. The reservation value function is:
• S
{1} {2} {3} {1,2} {2,3} {1,3} {1,2,3}
• VA(S) 0
0 0
3
3
3
4
• VB(S) 0
0
0
3
3
3
4
• efficient allocations are : SA={1,2,3} ,SB=0 or
SB={1,2,3} ,SA=0.
• Any prices that support the first allocation most
satisfy :
12
Example con.
p1  p 2  3, p1  p3  3, and p3  p 2  3, else B could
do better tha n buy nothing. but this implies that A will not
buy {1,2,3} at these prices as
p1  p 2  p3  4.5  4  V A ({1,2,3})
13
Proposition 1
• If mc”p exist in an economy E1, then the marcet
allocation must be an efficient allocation.
• Proof: suppose that p1,…,pn are mc”p and that
(S1’,…,Sm’) is the marcet allocation supported by
these prices. Let (S1,…,Sm) be any other
allocation. Condition 1 implies that:
Vi ( S i ' )  kS ' p k  Vi ( S i )  kS p k , i  1,..., m
i
i
Consequent ly,

m
i 1
Vi ( S i ' )  i 1
m

m
kSi '
m
p k   Vi ( S i )  
i 1
i 1

kSi
pk
14
Proof con.
m
m
i 1
i 1
Let S '   S i ' and S   Si
we know that p k  0 for all k  S \ S ' so we get that :
i1Vi (S i ' )  i1Vi (S i )   pk   pk
m
m
kS '
 i 1Vi ( S i ) 
m
 i 1Vi ( S i ) 
m
p
kS '\ S
k

p
kS \ S '
kS
k
 pk  i1Vi (S i )
m
kS '\ S
15
A necessary and sufficient
condition
• Let E1={N,(Vi),i=1,…,m} be an economy with
indivisible commodities. We define a divisible
transformation ED(N,(Vi)) of E1 as follows.
• Let S 0 , S1 ,..., S 2 1
be an enumeration of all the subsets of N.
• The agent i’s divisible allocation is:
X i  ( xi1 , xi 2 ,..., xi 2n 1 )
n
if x ij  f than agent i gets fraction
f of the jth subset.
16
n
n

(
2
 1) matrix such that if object k
• Let A be a
is in subset S j , j  1 then a kj  1 , otherwise a kj  0.
• The reservation value of agent i in ED is:
Define y ik  [0,1], k  1,..., n i  1,..., m
Wi ( y i1 ,..., y in )  max xi
2 n 1
s.t.
a
j 1
j 1
V (S
j 1
i
j
) xij
kj
xij  y ik , k  1,..., n
ij
1
2 n 1
x
2 n 1
x ij  0 ,
j.
17
E1 to ED
• We interpret Wi(Yi), Yi=(Yi1,…,Yin) as a
reservation value of agent i in ED over the
divisible commodity bundle Yi.
• ED(N,(Vi))={N,(Wi),i=1,…,m}
• The utility function is :
• Ui(Yi, wi)= Wi(Yi)+wi.
• The endowments in ED are identical to those in E1.
18
E1 to ED con.
• A feasible divisible allocation,
YI  ( yi1 , yi 2 ,..., yin ), i  1,..., m is one
m
which satisfies
y
i 1
ik
 1, k  1,..., n.
• An efficient divisible allocation is a feasible
divisible allocation, Y1’,…,Ym’,such that for any
other feasible divisible allocation, Y1,…,Ym
m
m
W (Y ' )  W (Y ).
i 1
i
i
i 1
i
i
19
Proposition 2
• Mc”p exsit in an indivisible economy
E1={N,(Vi)} if and only if an efficient allocation
in E1 induces an efficient allocation in ED(N(Wi)).
• Integer Program(IP):
m 2 n 1
max x1 , x2 ,..., xm
2 n 1
s.t.
 V (S
i 1 j 1
i
j
) xij
 akj i 1 xij  1 k  1,..., n
m
j 1
2 n 1
x
j 1
ij
1
i  1,..., m
x ij  0 or 1 i, j
20
IP and LPR
• The optimal solution to IP is the set of efficient
allocation in E1.
• Linear Programming Relaxation(LPR):
m 2 n 1
max x1 , x2 ,..., xm
 V (S
i 1 j 1
2 n 1
s.t.
a 
kj
j 1
2 n 1
x
j 1
ij
m
i 1
1
x ij  0
i
j
) xij
xij  1 k  1,..., n
i  1,..., m
i, j
21
DLPR
Dual of LRP(DLRP):
n
min
pk , i
m
 p  
k
k 1
n
s.t.
a
k 1
kj
i 1
i
pk   i  Vi ( S j ), i, j
pk  0  i  0
i, k
• Let MIP ,MLPR ,MDLPR denote the value of an
optimal solutions to IP, LPR and DLPR
respectively.
22
Proposition 2
• Thus, M DLPR  M LPR  M IP
• Lemma: Let (Y1’,…,Ym’) be an efficient divisible
allocation in ED(N,(Vi)).Then i Wi (Yi ' )  M LPR .
• Now we can write Proposition 2 in a new way:
• Lemma (Proposition 2): mc”p exist in E1 if and
only if MIP =MLPR.
• Proof: Let X’=(xij) i  1,..., m j  1,...,2  1 be an optimal
solution to LRP, and P'  ( p1 ' ,..., pn ' ) and  '  ( 1 ' ,...,  n ' )
be an optimal solution to DLRP,
n
23
Proof
m
2n
• V ( S
n
j
i
i 1 j 1
m
) xij '   pk '    i '
k 1
i 1
(from the duality)
• The complementary slackness condition are:
2n 1
m
j 1
i 1
1) [1   akj  xij '] pk '  0
2 n 1
2) [1   xij' ] i  0
k
i
j 1
n
3) [ akj pk ' i 'Vi ( S j )] xij '  0 i, j
k 1
24
Proof con.
2 n 1
1) implies that if
m
a  x '  1
kj
j 1
ij
i 1
then p k '  0.
3) implies that if x ij '  0 then
n
Vi (S j ) -  a kj pk '   i '
k 1
n
 Vi (S j' ) -  a kj' pk ' j'
k 1
25
Sufficiency
• This, together with  i  0 , implies that the prices
P support the allocation X.
• To prove sufficiency ,suppose that MIP=MLPR so
there exists a solution X’=xij, which is feasible and
optimal for both IP and LPR. Moreover X’ is
efficient allocation in E1. The DLPR optimal
variables P’=pi are prices which support X’ in E1.
26
Necessity
• Suppose that P'  ( p1 ' ,..., pn ' ) p k  0 , are mc”p
which support (Sj1,Sj2,…,Sjm) a feasible allocation
in E1. From Proposition 1 we know that
(Sj1,Sj2,…,Sjm) is an efficient allocation.
i,
• Define  i '  Vi (S ji )   pk '
kS
• As the prices P'  ( p1 ' ,..., pn ' ) support
(Sj1,Sj2,…,Sjm) , we have  i ' 0 and
ji
 i '  Vi ( S ji ) 
p
kS ji
k
'  Vi ( S j ) 
p
kS j
• thus p k ' and  i ' are dual feasible.
k
'
i, j
27
Necessity con.
• MLPR=MDLPR
m
n
i 1
k 1
   i '   pk '
m
  [Vi ( S ji ) 
i 1
p
kS ji
n
k
']   p k '
k 1
m
  Vi ( S ji )  M IP
i 1
28
Corollaries
• Corollary 1: If one efficient allocation in E1 is
supported by a price vector P, then all efficient
allocation in E1 are supported by P.
• Corollary 2: The set of mc”p in E1 is a closed,
bounded , convex (and possible empty) set.
• Corollary 3: If all agent have the same reservation
value function V( ), and if V( ) is balanced then
mc”p exist.
29
Extensions
• 1)if there are more than one unit from one or more
objects.( it is exactly the same)
• 2) we can limit the agents’ choices to S   .
• 3) we can exclude constraint  xij  1 from IP ,
j
then Proposition 2 is modified to:
mc”p, which give each agent zero consumer
surplus, exist if and only if an efficient allocation
in E1 induces an efficient divisible allocation in
ED. This condition is satisfied when agents’
reservation value are additive .
30
Assumption on agents’
preferences
V is superaddit ive if for all S, T  N, such that S  T  
V(S)  V(T)  V(S  T).
V is supermodul ar if for all S, T  N,
V(S)  V(T)  V(S  T)  V(S  T ).
another definition : if for all T1 , T2 , T3  N,
V(T1  T3 )  V (T1 )  V (T1  T2  T3 )  V (T1  T2 )
31
Superadditivity and
supermodularity is not sufficient
•
•
•
•
•
S
{1} {2} {3} {1,2} {2,3} {1,3} {1,2,3}
VA(S) 1
1
1
30
3
3
40
VB(S) 1
1
1
3
30
3
40
VC(S) 1
1
1
3
3
30
40
an efficient indivisible allocation is SA={1,2,3},
SB=SC=  -->MIP=40 where the efficient divisible
allocation is SA=1/2{1,2}, SB= 1/2{2,3}
,SC=1/2{1,3} -->MLP=45>MIP -->mc”p do not
exist.

32
Supermodular preferences
• Proposition 3: Suppose there are two types of
agents in an indivisible economy E1.type i agents
with reservation value function Vi..Further,
suppose that Vi are strictly supermodular and
strictly increasing. Then mc”p exist.
33
Implication?
• Do there exist simple market mechanisms (I.e.
mechanisms that assign a price to each object)
which efficiently allocate multiple indivisible
objects when mc”p exist?
• It is an open question.
• We have simple market mechanisms when:
• 1) agent want only one object.
• 2) reservation value function are additive .
34
Implication? Con.
• Other assumption under which simple market
mechanisms may be efficient are:
• 1) buyers have a common unknown balanced
reservation value function.
• 2)buyers’ preferences satisfy the hypothesis of
Proposition 3, with each buyer’s type being
private information.
35
when mc”p do not exist.
• Two implication for market mechanisms when
mc”p do not exist.
• First, nonexistence of mc”p implies that when
bidders value more than one object and have
interdependent values, then simultaneous oral
ascending price auction will not have the no regret
property.
• Second, bundling a few of the objects together
may lead to existence, with some loss of
efficiency.
36
when mc”p do not exist con.
• An alternative approach is to set prices to
some bundles( say those with 2-3 objects).
37
• 1) we saw a condition when mc”p exist.
• 2) we do not know a lot about the existence of
mc”p from the condition of the reservation value.
• Problem:
• 1) we do not know how to check if MIP=MLPR.
• 2) we do not know how to find the mc”p even if
we know that it exist.
38