1. 1 Competitive equilibrium in an Exchange Economy with Indivisibilities. By:Sushil Bikhchandani and John W.Mamer University of California. presented by: Meir Bing

2. 2 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.

3. 3 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.

4. 4 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.

5. 5 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.

6. 6 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 U i(*). Ui(S,W)=Vi(S)+W.

7. 7 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

8. 8 (S1’,…,Sm’) is an efficient allocation,if it is feasible allocation and if for every other feasible allocation (S1,…,Sm)

9. 9 mc”p are prices, one for each commodity at which there is no excess demand for any commodity.

10. 10 (S1,…,Sm) is said to be a market allocation which is supported by prices p 1,…,p n. 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:

11. 11 Lemma 1 Suppose that then prices (p 1,…,p n ) support a feasible allocation in the economy E1={N,(V i,W i ),i=1,..,m} if and only if (p 1,…,p n ) support the same feasible allocation in the economy E1’={N,(V i,W i ’),i=1,..,m} so now we can write E1={N,(V i ),i=1,..,m}

12. 12 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} V A (S) 0 0 0 3 3 3 4 V B (S) 0 0 0 3 3 3 4 efficient allocations are : S A ={1,2,3},S B =0 or S B ={1,2,3},S A =0. Any prices that support the first allocation most satisfy :

13. 13 Example con.

14. 14 Proposition 1 If mc”p exist in an economy E1, then the marcet allocation must be an efficient allocation. Proof: suppose that p 1,…,p n are mc”p and that (S 1 ’,…,S m ’) is the marcet allocation supported by these prices. Let (S 1,…,S m ) be any other allocation. Condition 1 implies that:

15. 15 Proof con.

16. 16 A necessary and sufficient condition Let E1={N,(V i ),i=1,…,m} be an economy with indivisible commodities. We define a divisible transformation E D (N,(V i )) of E1 as follows. Let be an enumeration of all the subsets of N. The agent i’s divisible allocation is:

17. 17 Let A be a matrix such that if object k is in subset The reservation value of agent i in E D is:

18. 18 E1 to E D We interpret W i (Y i ), Y i =(Y i1,…,Y in ) as a reservation value of agent i in E D over the divisible commodity bundle Y i. E D (N,(V i ))={N,(W i ),i=1,…,m} The utility function is : U i (Y i, w i )= W i (Y i )+w i. The endowments in E D are identical to those in E1.

19. 19 E1 to E D con. A feasible divisible allocation, An efficient divisible allocation is a feasible divisible allocation, Y 1 ’,…,Y m ’,such that for any other feasible divisible allocation, Y 1,…,Y m

20. 20 Proposition 2 Mc”p exsit in an indivisible economy E1={N,(V i )} if and only if an efficient allocation in E1 induces an efficient allocation in E D (N(W i )). Integer Program(IP):

21. 21 IP and LPR The optimal solution to IP is the set of efficient allocation in E1. Linear Programming Relaxation(LPR):

22. 22 DLPR Dual of LRP(DLRP): Let M IP,M LPR,M DLPR denote the value of an optimal solutions to IP, LPR and DLPR respectively.

23. 23 Thus, Lemma: Let (Y1’,…,Ym’) be an efficient divisible allocation in E D (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’=(x ij ) be an optimal solution to LRP, and be an optimal solution to DLRP, Proposition 2

24. 24 Proof (from the duality) The complementary slackness condition are:

25. 25 Proof con.

26. 26 Sufficiency This, together with, implies that the prices P support the allocation X. To prove sufficiency,suppose that M IP =M LPR so there exists a solution X’=x ij, which is feasible and optimal for both IP and LPR. Moreover X’ is efficient allocation in E1. The DLPR optimal variables P’=p i are prices which support X’ in E1.

27. 27 Necessity Suppose that, are mc”p which support (S j1,S j2,…,S jm ) a feasible allocation in E1. From Proposition 1 we know that (S j1,S j2,…,S jm ) is an efficient allocation. Define As the prices support (S j1,S j2,…,S jm ), we have are dual feasible.

28. 28 Necessity con. M LPR =M DLPR

29. 29 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.

30. 30 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 E D. This condition is satisfied when agents’ reservation value are additive.

31. 31 Assumption on agents’ preferences

32. 32 Superadditivity and supermodularity is not sufficient S {1} {2} {3} {1,2} {2,3} {1,3} {1,2,3} V A (S) 1 1 1 30 3 3 40 V B (S) 1 1 1 3 30 3 40 V C (S) 1 1 1 3 3 30 40 an efficient indivisible allocation is S A ={1,2,3}, S B =S C = -->M IP =40 where the efficient divisible allocation is S A =1/2{1,2}, S B = 1/2{2,3},S C =1/2{1,3} -->M LP =45>M IP -->mc”p do not exist.

33. 33 Supermodular preferences Proposition 3: Suppose there are two types of agents in an indivisible economy E1.type i agents with reservation value function V i..Further, suppose that V i are strictly supermodular and strictly increasing. Then mc”p exist.

34. 34 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.

35. 35 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.

36. 36 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.

37. 37 when mc”p do not exist con. An alternative approach is to set prices to some bundles( say those with 2-3 objects).

38. 38 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 M IP =M LPR. 2) we do not know how to find the mc”p even if we know that it exist.