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Competitive equilibrium in an Exchange Economy with

Indivisibilities.

• By:Sushil Bikhchandani and John W.Mamer

• University of California.

• presented by: Meir Bing

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• 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.

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• 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.

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• 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.

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• 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.

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• Consider an exchange economy with n indivisible commodities and m agents.

• Each agent i=1,…,m has a reservation value function :

• this function is weakly increasing.• Each agent i=1,…,m has a utility function Ui(*).

• Ui(S,W)=Vi(S)+W.

Ni SV 2:)(

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• 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

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• (S1’,…,Sm’) is an efficient allocation ,if it is feasible allocation and if for every other feasible allocation (S1,…,Sm)

ji allfar

,...1,

ji

i

SS

miNS

)()'(11 i

m

i ii

m

i i SVSV

9

• mc”p are prices, one for each commodity at which there is no excess demand for any commodity.

m

i Skk

n

kk

Ski

Skki

m

k

i

i

pp

SVpS

S

nkp

11

ki

1

)2

m1,...,i , NS

, p)()(V 1)

:such that ),...,(S allocation

feasible a is thereif pmc" are ,...1,0

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• (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:

• agent i’s consumer surplus is:

).(\ 01

m

iik SNkp

iSk

kii pSv )(

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Lemma 1

• Suppose that 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}

iNVWNVW iiii )(' and )(

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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 :

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Example con.

})3,2,1({45.4

as prices at these {1,2,3}buy

not A will that implies but this nothing.buy n better tha do

could B else ,3 and ,3,3

321

233121

AVppp

pppppp

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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:

ii

ii

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iii

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pSVpS

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ly,Consequent

m1,...,i , )()'(V

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Proof con.

m

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SS and ''Let

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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 be an enumeration of all the subsets of N.

• The agent i’s divisible allocation is:

1210 ,...,,n

SSS

subset. jth theof f

fraction gets iagent thanx if

),...,,(

ij

1221

f

xxxX niiii

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• Let A be a matrix such that if object k is in subset

• The reservation value of agent i in ED is:

)12( nn

.0a otherwise, 1a then 1, kjkj jS j

j. , 0 x

1

n1,...,k , y ..

)(max),...,(W

m1,...,i ,...,1 ],1,0[y Define

ij

12

1

ik

12

1

12

11i

ik

n

n

n

i

jij

jijkj

jijjixini

x

xats

xSVyy

nk

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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.

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E1 to ED con.

• A feasible divisible allocation,

• An efficient divisible allocation is a feasible divisible allocation, Y1’,…,Ym’,such that for any other feasible divisible allocation, Y1,…,Ym

.n1,...,k ,1 satisfies which

one is ,...,1 ),,...,,(m

1i

21

ik

iniiI

y

miyyyY

.)()'(11

m

iii

m

iii YWYW

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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):

ji, 1or 0 x

m1,...,i 1

n1,...,k 1 ..

)(max

ij

12

1

12

11

1

12

1,...,, 21

n

n

n

m

jij

jij

m

ikj

m

i jijjixxx

x

xats

xSV

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IP and LPR

• The optimal solution to IP is the set of efficient allocation in E1.

• Linear Programming Relaxation(LPR):

ji, 0 x

m1,...,i 1

n1,...,k 1 ..

)(max

ij

12

1

12

11

1

12

1,...,, 21

n

n

n

m

jij

jij

m

ikj

m

i jijjixxx

x

xats

xSV

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DLPR

Dual of LRP(DLRP):

• Let MIP ,MLPR ,MDLPR denote the value of an optimal solutions to IP, LPR and DLPR respectively.

ki, 0 0p

ji, ),( ..

min

ik

1

1 1,

n

kjiikkj

n

k

m

iik

SVpats

pikp

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• Thus,• Lemma: Let (Y1’,…,Ym’) be an efficient divisible

allocation in ED(N,(Vi)).Then• 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) be an optimal solution to LRP, and be an optimal solution to DLRP,

IPLPRDLPR MMM

.)'( LPRi ii MYW

)',...,'(' and )',...,'(' 11 nnppP

12,...,1 ,...,1 njmi

Proposition 2

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Proof

• (from the duality)

• The complementary slackness condition are:

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kkij

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pxSV1

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1 11

''')(

ji, 0')](''[ )3

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1

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1'

1

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1

ijjiik

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kkj

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pxa

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Proof con.

j' 'a-)(S

''a-)(S

then0'x if that implies 3)

0.'p then

1' if that implies 1)

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1kkj'j'

i

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1kkjj

ij

k

1

12

1

ki

ki

m

iij

jkj

pV

pV

xan

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Sufficiency Sufficiency

• This, together with , 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.

0i

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Necessity Necessity

• Suppose that , 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.

• Define • As the prices support (Sj1,Sj2,…,Sjm) ,

we have

• are dual feasible.

0p )',...,'(' k1 nppP

jiSk

kjiii pSV i, ')('

)',...,'(' 1 nppP

' and ' thus

i, ')( ')('

ik

Skkji

Skkjiii

p

jpSVpSVjji

and 0'i

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Necessity con.Necessity con.

• MLPR=MDLPR

IP

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ijii

n

kk

m

i Skkjii

n

kk

m

ii

MSV

ppSV

p

ji

1

11

11

)(

']')([

''

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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.

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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 .• 3) we can exclude constraint from IP , 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 .

S 1

jijx

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Assumption on agents’ preferences

)T()T()()V(T

N,,,T allfor if :definitionanother

).V(ST)V(SV(T)V(S)

N,TS, allfor ifar supermodul is V

T).V(SV(T)V(S)

TSsuch that N,TS, allfor if ivesuperaddit is V

21321131

321

TVTTVTVT

TT

T

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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.

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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.

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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 .

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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.

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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.

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when mc”p do not exist con.

• An alternative approach is to set prices to some bundles( say those with 2-3 objects).

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• 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.