Stability and Competitive Equilibrium in Trading Networks

Stability and Competitive Equilibrium in Trading Networks∗ John William Hatfield† Alexandru Nichifor§

Scott Duke Kominers‡

Michael Ostrovsky¶

Alexander Westkampk

November 9, 2010

Abstract We introduce a model in which a network of agents trades via bilateral contracts. In contrast to previous results in settings without transferable utility, we find that when continuous transfers are allowed and utility is quasilinear, the full substitutability of preferences is on its own both sufficient and necessary for the guaranteed existence of stable outcomes. Furthermore, when transfers are possible and preferences are fully substitutable, the set of stable outcomes is equivalent to the set of competitive equilibria, and all stable outcomes are in the core and efficient. JEL classification: C78; C62; C71; D85; L14 Keywords: Matching; Networks; Stability; Competitive Equilibrium; Core; Efficiency; Submodularity; Lattice; Substitutes ∗

We thank Peter Cramton, C ¸ a˘ gatay Kayı, Paul Milgrom, Al Roth, and Glen Weyl for helpful comments and suggestions. Hatfield and Nichifor appreciate the hospitality of Harvard Business School, which hosted them during parts of this research. Kominers thanks the National Science Foundation and the Yahoo! Key Scientific Challenges Program for support. Nichifor thanks the Netherlands Organisation for Scientific Research (NWO) for its support under grant VIDI-452-06013. † Graduate School of Business, Stanford University; Stanford, CA 94305, U.S.A.; [email protected]. ‡ Department of Economics, Harvard University, and Harvard Business School; Wyss Hall, Harvard Business School, Soldiers Field, Boston, MA 02163, U.S.A.; [email protected]. § Department of Economics, Maastricht University; P.O. Box 616, 6200 MD Maastricht, The Netherlands; [email protected]. ¶ Graduate School of Business, Stanford University; Stanford, CA 94305, U.S.A.; [email protected]. k Economics Department, University of Bonn; Leneestr.37, 53113 Bonn, Germany; [email protected].

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1

Introduction

An extensive literature has studied stability in matching markets, leading to highprofile applications in entry-level labor markets such as the National Resident Matching Program and school choice programs in Boston (Abdulkadiroˇglu et al., 2005) and New York (Abdulkadiroˇglu et al., 2005, 2009).1 Another extensive literature has studied competitive equilibria in exchange economies with indivisibilities leading to high-profile applications in spectrum auctions (McMillan, 1994) and electricity markets (Wolfram, 1998; Fabra et al., 2006).2 In this paper we unify these literatures via a model in which a network of agents trade indivisible goods or services via bilateral contracts. We show that the preference substitutability conditions identified by the matching and competitive equilibrium literatures are equivalent in the presence of transferable utility. Furthermore, we show that, when preferences are substitutable, both stable outcomes and competitive equilibria exist and are efficient.3 In fact, all competitive equilibria induce stable outcomes, and, in the presence of substitutable preferences, the converse is also true. Our existence results for general trading networks are particularly unexpected, because in a similar setting without transfers (or with only discrete transfers), acyclicity of the contractual set is a necessary condition to guarantee existence of stable outcomes. In our model, contracts specify a buyer, a seller, provision of a good or service, and a monetary transfer. An agent may be involved in some contracts as a seller, and in other contracts as a buyer. Agents’ preferences are given by cardinal utility functions over sets of contracts and are quasilinear with respect to the numeraire.4 We say that preferences are fully substitutable if contracts are substitutes for each other in a generalized sense, i.e., whenever an agent gains a new purchase opportunity, he becomes both less willing to make other purchases, and more willing to make 1

Roth and Sotomayor (1990) and Roth (2008) survey the theory and practice of matching market design. 2 Milgrom (2004) surveys the theory and practice of auction design for the exchange of indivisible goods. 3 In the literature on matching with contracts, the term “allocation” has been used to refer to a set of contracts. Unfortunately, the term “allocation” is also used in the competitive equilibrium literature to denote an assignment of goods, without specifying transfers. For this reason, we distringuish between these two concepts by introducing the term “outcome” to refer to an assignment of goods and associated transfers. 4 We limit our exposition to the case of exchange economies, where agents trade goods, but our approach may also be used to consider production economies, where the act of production may be codified by differences in items that agents buy and sell.

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sales. We show that this intuitive notion of substitutability, which has appeared in the matching literature (Ostrovsky, 2008; Hatfield and Kominers, 2010b; Westkamp, 2010), is equivalent to the gross substitutes and complements condition introduced in the demand theory literature (Sun and Yang, 2006, 2009). Furthermore, both formulations of the full substitutability condition are equivalent to the formulation of substitutability put forth in the auction literature (Ausubel and Milgrom, 2002). We show that if preferences are fully substitutable, then competitive equilibria are guaranteed to exist. Moreover, there is a simple and natural mapping from the set of competitive equilibria to the set of stable outcomes, and thus fully substitutable preferences are sufficient for the existence of stable outcomes. We also show that full substitutability is in a sense necessary to guarantee the existence of stable outcomes, and hence is necessary to guarantee the existence of competitive equilibria: if any agent has preferences which are not fully substitutable, then fully substitutable preferences can be found for other agents such that no stable outcome exists. Substitutability has been identified in both the matching and exchange economy literatures as central to the existence of equilibria. In the setting of many-to-one matching with transferable utility, Kelso and Crawford (1982) showed that substitutability is sufficient for the existence of stable outcomes. 5 Gul and Stacchetti (1999) and Sun and Yang (2006) showed the sufficiency of substitutability for the existence of competitive equilibria in exchange economies with indivisibilities.6,7 Our results on existence extend those of Kelso and Crawford (1982) and Gul and Stacchetti (1999) to a more general network setting, and furthermore illustrate that they are closely linked to the classical matching literature, which has shown substitutability to be necessary and sufficient for the existence of stable outcomes in a variety of settings.8 5

Hatfield and Kojima (2008) showed the corresponding necessity result. Most of this literature assumes quasilinear utility (as do we); a recent literature (e.g., Alaei et al., 2010) extended the results from the quasilinear case to settings without quasilinear utility. 7 Gul and Stacchetti (1999) provide a corresponding necessity result. Additionally, Gul and Stacchetti (2000) and Sun and Yang (2009) provide a tâtonnement process to find the equilibria identified in their earilier works. 8 In the settings of: many-to-one matching, Roth (1984) proves sufficiency and Hatfield and Kojima (2008) prove necessity; many-to-many matching, Roth (1984) and Echenique and Oviedo (2006) prove sufficiency and necessity follows from the results of Hatfield and Kojima (2008); manyto-many matching with contracts, Klaus and Walzl (2009) and Hatfield and Kominers (2010a) prove sufficiency and Hatfield and Kominers (2010a) prove necessity; supply-chain matching, Ostrovsky (2008) and Hatfield and Kominers (2010b) prove sufficiency and Hatfield and Kominers (2010b) prove necessity. In the setting of many-to-one matching with contracts, substitutable preferences 6

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Further investigating the relationship between stable outcomes and competitive equilibria, we show that, not only do competitive equilibria induce stable outcomes, but, in the presence of full substitutability, all stable outcomes correspond to competitive equilibria. The intuition that stability may be related to price-theoretic solution concepts has been suggested by previous works. Kelso and Crawford (1982) showed that the gross substitutes condition is sufficient for the existence of a core outcome in two-sided matching markets allowing transfers. Hatfield and Milgrom (2005) showed a correspondence between solution concepts of two-sided matching theory and auction theory.9 Our work develops the relationship suggested by these works, fully characterizing the relationship between matching-theoretic stability and price-theoretic competitive equilibrium in our setting. We find that the set of competitive equilibrium price vectors forms a lattice, generalizing the results of Gul and Stacchetti (1999) and Sun and Yang (2006). Furthermore, any efficient allocation can be supported by any competitive equilibrium price vector. These results are also closely related to the well-known “polarization of interests” results in classical matching theory, which shows that the set of stable outcomes forms a lattice.10 Finally, we consider the relationship between the set of stable outcomes and other solution concepts introduced in the literature. Generalizing results of Shapley and Shubik (1971) and Sotomayor (2007), we show that all stable outcomes are in the core, and that the core may contain unstable outcomes. We then consider the notion of strong group stability, which is strictly stronger than both stability and the core in classical matching settings.11 We show that the set of stable outcomes is equal to the set of strongly group stable outcomes in our setting. We also show that the notion of chain stability introduced by Ostrovsky (2008) is equivalent in our setting to stability whenever preferences are fully substitutable.12 The remainder of this paper is organized as follows. In Section 2, we formalize our model. In Section 3, we introduce substitutability notions from matching theory, are sufficient for the existence of a stable outcome (Hatfield and Milgrom (2005)), but are not necessary (Hatfield and Kojima (2008, 2010)). 9 These issues are also discussed by Ashlagi et al. (2010). 10 See, e.g., the work of Roth (1984), Hatfield and Milgrom (2005), and Ostrovsky (2008). 11 Strong group stability is a strengthening of several solution concepts of the literature; see Section 5.1 for a discussion. 12 Hatfield and Kominers (2010b) show an analogous result in the setting of supply-chain matching, generalizing earlier results in matching theory; see Proposition 6.4 of Roth and Sotomayor (1990).

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demand theory, and auction theory, and prove they are equivalent. In Section 4, we present our main results. In Section 5, we consider the relationship among competitive equilibria, stable outcomes, and other solution concepts such as the core. We conclude in Section 6. All proofs are in the Appendix.

2

Model

There is a finite set of agents I in the economy. These agents can participate in bilateral trades. Each trade ω is associated with a seller s (ω) ∈ I and a buyer b (ω) ∈ I. For instance, a worker (seller) may be hired by a firm (buyer) in a variety of different capacities with a variety of job conditions and characteristics, and each possible type of job may be represented by a different trade. The set of possible trades Ω is finite and exogenously given. The set Ω may contain multiple trades that have the same buyer and the same seller; indeed, a pair of agents may simultaneously agree to multiple trades. We allow the set Ω to contain trades ω1 and ω2 such that s(ω1 ) = b(ω2 ) and s(ω2 ) = b(ω1 ). It is convenient to think of a trade as representing the “physical” aspects of a transaction between a seller and a buyer (although in principle it could include some “financial” terms and conditions as well). The purely financial aspect of a trade ω is represented by a price pω ; the complete price vector is denoted by p ∈ R|Ω| . Formally, a contract x is a pair (ω, pω ), with ω ∈ Ω denoting the trade and pω ∈ R denoting the price at which the trade occurs. The set of available contracts is X ≡ Ω × R. For any set of contracts Y , we denote by τ (Y ) the set of trades involved in contracts in Y , i.e. τ (Y ) ≡ {ω ∈ Ω : (ω, pω ) ∈ Y }. For a contract x = (ω, pω ), we will let s (x) = s (ω) and b (x) = b (ω), the seller and the buyer associated with the trade ω of contract x. Consider any set of contracts Y ⊆ X. We denote by Y→i the set of “upstream” contracts for i in Y , that is, the set of contracts in Y in which agent i is the buyer: Y→i ≡ {y ∈ Y : i = b (y)}. Similarly, we denote by Yi→ the set of “downstream” contracts for i in Y , that is, the set of contracts in Y in which agent i is the seller: Yi→ ≡ {y ∈ Y : i = s (y)}. We denote by Yi the set of contracts in Y in which agent i is involved either as the buyer or S S the seller: Yi ≡ Y→i ∪ Yi→ . We let b (Y ) ≡ y∈Y b (y) and s (Y ) ≡ y∈Y s (y). We use analogous notation to denote the sets of trades associated with agent i for sets of trades Ψ ⊆ Ω. 5

We say that the set of contracts Y is feasible if there is no trade ω and prices pω 6= pˆω such that both contracts (ω, pω ) and (ω, pˆω ) are in Y ; in other words, a set of contracts is feasible if each trade is associated with at most one contract. An outcome Y ⊆ X is a feasible set of contracts. Note that an outcome A only specifies prices associated with trades in τ (A). For any set of trades Ψ and price vector p, we let λ ([Ψ; p]) ≡ {(ψ, p¯ψ ) ∈ X : ψ ∈ Ψ and p¯ψ = pψ }. Note that λ ([Ψ; p]) is an outcome. Additionally, we have that τ (λ ([Ψ; p])) = Ψ.

2.1

Preferences

Each agent i has a quasilinear utility function over sets of trades involving that agent (Ψ ⊆ Ωi ) and associated vectors of prices (with element pψ denoting the price corresponding to trade ψ): Ui (Y ) ≡ ui (τ (Y )) +

X

X

pω −

(ω,pω )∈Yi→

pω .

(ω,pω )∈Y→i

For any [Ψ; p], we let Ui ([Ψ; p]) ≡ Ui (λ [Ψ; p]) = ui (Ψ) +

X

pψ −

ψ∈Ψi→

X

pψ .

ψ∈Ψ→i

We allow ui (Ψ) to take the value −∞ for some sets of trades Ψ, in order to incorporate various technological constraints. However, we also assume that for all i, the outside option is finite: ui (∅) ∈ R. That is, no agent is “forced” to sign any contracts at extremely unfavorable prices: he always has an outside option of completely withdrawing from the market at some potentially high but finite price. Finally, for many of our results, a critical assumption is the full substitutability of preferences. We present and discuss this assumption in Section 3. The utility function Ui gives rise to both demand and choice correspondences. The demand correspondence of agent i given price vector p ∈ R|Ω| is defined as the collection of the sets of trades maximizing agent i’s utility under prices p: Di (p) ≡ arg max Ui ([Ψ; p]) . Ψ⊆Ωi

The choice correspondence of agent i from the set of contracts Y ⊆ X is defined as

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the collection of the sets of contracts maximizing agent i’s utility: Ci (Y ) ≡

arg max

Ui (Z) .

Z⊆Yi ; Z is feasible

Note that while the demand correspondence always returns at least one (possibly empty) set of trades, the choice correspondence may be empty-valued (e.g., if Y consists of all contracts with prices strictly between 0 and 1).

2.2

Stability and Competitive Equilibrium

We study two notions of market equilibria, stability and competitive equilibrium. Each of these concepts specifies which trades transact in equilibrium and the associated transfers. Definition 1. An outcome A is stable if it is 1. Individually rational: Ai ∈ Ci (A) for all i; 2. Unblocked: There does not exist a nonempty blocking set Z ⊆ X such that (a) Z ∩ A = ∅, and (b) for all i ∈ b (Z) ∪ s (Z), for all of i’s choices Y ∈ Ci (Z ∪ A), we have Zi ⊆ Y . Individual rationality requires that no agent can become strictly better off by dropping some of the contracts that he is involved in. This is a standard requirement in the matching literature. The second condition states that there are no strict blocks: presented with a stable outcome A, one cannot propose a new set of contracts such that all the agents involved in these new contracts would strictly prefer to form all of them (and possibly drop some of their existing contracts in A) to sticking with their contracts in A. This requirement is a natural adaptation of the stability condition in Hatfield and Kominers (2010b) to the current setting, to take into account that the presence of transferable utility generates indifferences between choices, and we therefore need to be careful about strict versus weak preferences. It is strictly stronger than pairwise stability, which requires that the blocks Z must contain exactly one contract. It is also stronger than chain stability, which requires that the blocking set Z exhibit a special chain structure. We discuss this latter condition further in 7

Section 5.2, where we show that in the presence of fully substitutable preferences, chain stability is equivalent to our stability concept. In contrast to stable outcomes, a competitive equilibrium [Ψ; p] specifies prices for all trades in Ω. Definition 2. A price vector p ∈ R|Ω| and set of trades Ψ ∈ Ω form a competitive equilibrium if for all i ∈ I, Ψi ∈ Di (p) . This is the standard notion of competitive equilibrium, applied to the current setting: market-clearing is “built in,” since each trade in Ψ carries with it the corresponding buyer and the corresponding seller, and the condition is simply that each agent is optimizing given market prices. Note that we only require weak optimality: the agent may be indifferent between the assigned bundle and some other bundle of trades.

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Full Substitutability

In this section, we introduce full substitutability, a natural condition which is crucial for many of our results. We provide a brief description of this condition’s predecessors, and then present several equivalent characterizations of the condition.

3.1

Predecessors

Kelso and Crawford (1982) introduced the gross substitutes condition (GS), which requires that an increase in the price of some trades never causes an agent to drop other demanded trades for which the prices are invariant. In the terms of our model, (GS) means that each agent finds that choosing some contracts as a buyer makes choosing other contracts as a buyer less valuable. Kelso and Crawford (1982) use (GS) to establish the existence of a core outcome for many-to-one matching models in which complete heterogeneity prevails and in which prices are either discrete or continuous. Subsequent development of (GS) continued independently in matching and demand theory. In matching theory, Ostrovsky (2008) generalized (GS) to a combination of two assumptions: same-side substitutability (SSS) and cross-side complementarity (CSC) (see also Hatfield and Kominers, 2010b; Westkamp, 2010). In 8

demand theory, Sun and Yang (2006) introduced the analogous gross substitutes and complements (GSC) condition (see also Sun and Yang, 2009). Finally, in auction theory, Ausubel and Milgrom (2002) adapted the (GS) condition to auction theory (see also Milgrom (2004)). These conditions impose that two constraints: First, when an agent’s opportunity set on one side of the market expands, that agent does not choose any options previously rejected from that side of the market. Second, when an agent’s opportunity set on one side of the market expands, that agent does not reject any options previously chosen from the other side of the market. When contracts encode trades, these two conditions may be naturally reinterpted as a single notion of full substitutability:13 when presented with additional contractual options to purchase (sell), an agent both rejects any previously rejected purchase (sale) options, and continues to choose any previously chosen sale (purchase) options. Adapting or extending the current definitions from matching, demand, and auction theory, we extend all of this work via three distinct definitions of full substitutability. As we show, these definitions are equivalent.

3.2

Matching-Theory Full Substitutability

First, we define full substitutability in the language of matching theory. From this literature, we borrow and merge (SSS) and (CSC). Definition 3. The preferences of agent i are matching-theory fully substitutable (MFS) if: 1. for all sets of contracts Y, Z ⊆ Xi such that Yi→ = Zi→ and Y→i ⊇ Z→i , and for ∗ ∩ Z ⊆ Z ∗ and each Y ∗ ∈ Ci (Y ), there exists a set Z ∗ ∈ Ci (Z) such that Y→i ∗ ∗ Zi→ ⊆ Yi→ ; 2. for all sets of contracts Y, Z ⊆ Xi such that Y→i = Z→i and Yi→ ⊇ Zi→ , and for ∗ each Y ∗ ∈ Ci (Y ), there exists a set Z ∗ ∈ Ci (Z) such that Yi→ ∩ Z ⊆ Z ∗ and ∗ ∗ Z→i ⊆ Y→i . The choice correspondence Ci is fully substitutable if choosing from a smaller set of contracts on one side, agent i always chooses on this side all the contracts that 13

Since preferences are quasilinear in our settting, there is no distinction between gross and net substitutes. Therefore, we drop the “gross” specification.

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she selected from the larger set that are still available (SSS) and on the other side a subset of the contracts initially chosen (CSC).14

3.3

Demand-Theory Full Substitutability

Second, we define full substitutability in the language of demand-theory, by borrowing the gross substitutes and complements (GSC) condition introduced by Sun and Yang (2006, 2009). Definition 4. The preferences of agent i are demand-theory fully substitutable (DFS) if: 1. for all price vectors p, p0 ∈ R|Ω| such that pω = p0ω for each ω ∈ Ωi→ and pω ≤ p0ω for each ω ∈ Ω→i , for each Ψ ∈ Di (p), there exists Ψ0 ∈ Di (p0 ) such that {ω ∈ Ψ→i : pω = p0ω } ⊆ Ψ0→i and Ψ0i→ ⊆ Ψi→ ; 2. for all price vectors p, p0 ∈ R|Ω| such that pω = p0ω for each ω ∈ Ω→i and pω ≥ p0ω for each ω ∈ Ωi→ , for each Ψ ∈ Di (p), there exists Ψ0 ∈ Di (p0 ) such that {ω ∈ Ψi→ : pω = p0ω } ⊆ Ψ0i→ and Ψ0→i ⊆ Ψ→i .15 The demand correspondence Di is fully substitutable if, when the upstream (downstream) prices agent i faces increase (decrease), agent i always demands any upstream (downstream) trades initially demanded for which prices have not changed, and does not demand any new downstream (upstream) trades.16 14

Full substitutability is the natural strengthening of (SSS) and (CSC) when the choice is extended from functions to correspondences. It ensures that for each choice from the larger set, there is a consistent choice (from possibly several indifferent ones) from the smaller set. Without this strengthening, for some initial choice of the agent from the larger set, it could be that there are two distinct indifferent choices from the smaller set, each of them satisfying either (SSS) or (CSC), but neither of them satisfying both. For instance, consider the following example: Let Yi→ = {x1 , x2 }, Y→i = {x3 , x4 }, and X = Y = Yi→ ∪ Y→i . Assume Ci (Y ) = {Y ∗ , Y ∗∗ }, where Y ∗ = {x1 , x3 , x4 } and Y ∗∗ = {x2 , x3 }. Let Z = {x1 , x2 , x3 }. Assume Ci (Z) = {Z ∗ , Z ∗∗ }, where Z ∗ = {x2 , x3 } and Z ∗∗ = {x1 }. If we impose separately (SSS) and (CSC) on Y ∗ , then by (SSS), there exists Z ∗ ∈ Ci (Z) ∗ ∗∗ ∗ such that Y→i ∩ Z ⊆ Z ∗ ; by (CSC), there exists Z ∗∗ ∈ Ci (Z), such that Zi→ ⊆ Yi→ . However, for 0 0 0 ∗ ∗ ∗ Y , there is no Z ∈ Ci (Z) such that {x3 } = Y→i ∩ Z ⊆ Z and Zi→ ⊆ Yi→ = {x1 }. 15 We show in Lemma 4 that this definition is equivalent to the weaker condition that preferences are (DFS) for any prices such that the demand correspondence is single-valued, a simpler condition often used in auction setting: see for instance Milgrom (2004). 16 (DFS) coincides with that of (GSC) in the model of Sun and Yang (2006, 2009). To see this, note that from the perspective of an individual agent, our setting is exactly the same as theirs: There are two types of trades (upstream and downstream) and (DFS) restricts how choices vary with price changes. This implies in particular that the results of Sun and Yang (2006) on preferences of individual agents carry over to our setting.

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3.4

Auction-Theory Full Substitutability

Finally, we extend to correspondences the reformulation of demand-theory full substitutability in vector notation of Hatfield and Kominers (2010b) to underscore the similarities with the definition of substitutability that is widely used in the auction literature (see Ausubel and Milgrom (2002)). For each agent i, for any set of trades Ψ ⊆ Ωi , define the (generalized) indicator function e(Ψ) ∈ {−1, 0, 1}|Ωi | to be the vector with component 1 (−1) for each upstream (downstream) trades in Ψ, and component 0 for all trade not in Ψ. The convention here is that an agent “buys” a strictly positive (negative) amount of a good if he is the buyer (seller) in some trade. Definition 5. The preferences of agent i are auction-theory fully substitutable (AFS) if for all price vectors p, p0 ∈ R|Ω| such that pω ≤ p0ω for each ω ∈ Ωi , and for each Ψ ∈ Di (p), there exists Ψ0 ∈ Di (p0 ) such that for each ω ∈ Ωi with pω = p0ω , eω (Ψ0 ) ≥ eω (Ψ) .

3.5

Characterization of Fully Substitutable Preferences

All the full substitutability notions we have presented so far take the preferences of the agent as primitive. However, if the indirect utility of an agent is known, then his preferences can be derived. We introduce next a generalization of the earlier submodularity assumption on the indirect utility of an agent (Gul and Stacchetti, 1999; Ausubel and Milgrom, 2002). This notion leads to a characterization of full substitutability which extends that of Sun and Yang (2009) to our setting. For all price vectors p, p0 ∈ R|Ω| , let their join p ∨ p0 be defined as the pointwise maximum of the two price vectors and their meet p ∧ p0 be defined as the pointwise minimum of the two price vectors.17 Definition 6. The indirect utility function Vi (p) ≡ max Ui ([Ψ; p]) Ψ⊆Ωi

of agent i is submodular if, for all price vectors p, p0 ∈ R|Ω| , Vi (p ∧ p0 ) + Vi (p ∨ p0 ) ≤ Vi (p) + Vi (p0 ). 17

Since we use the convention that agents receive money for any downstream contracts they sign and pay money for any upstream contracts they sign, we can use the usual definition of join and meet instead of the “generalized join and meet” used by Sun and Yang (2006, 2009).

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The main result of this section shows equivalence between the three definitions of full substitutability, along with a characterization of fully substitutable preferences in terms of the submodularity of the indirect utility function.18 Subsequently, we will use the term full substitutability in place of (MFS), (DFS), and (AFS). Theorem 1. (MFS), (DFS), and (AFS) are all equivalent, and hold if and only if the indirect utility function is submodular.19

4

Main Results

We first show that competitive equilibria exist when preferences are fully substitutable and characterize the structure of the set of competitive equilibria. We then show that full substitutability implies that the set of competitive equilibria essentially coincides with the set of stable outcomes. Finally, we show that if preferences are not fully substitutable, then stable outcomes and competitive equilibria need not exist.

4.1

The Existence and Structure of Competitive Equilibria

Theorem 2. Suppose that preferences are fully substitutable. Then there exists a competitive equilibrium. We proceed in several steps. First, we associate to the original market a twosided many-to-one matching market with transferable utility, in which each agent corresponds to a “firm” and each trade corresponds to a “worker”. In this associated market, the utility of firm i from hiring a set of workers Ψ ⊆ Ωi is given by vi (Ψ) ≡ ui (Ψ→i ∪ (Ω − Ψ)i→ ) . 18 In settings of matching in networds without transferable utility, there is a similar characterization due to Hatfield and Kominers (2010b). However, because only ordinal preferences are available in that setting, the relevant concept is quasisubmodularity, rather than submodularity. 19 Full substitutability is also equivalent to the valuation function vi (Ψ) ≡ ui (Ψ→i ∪ (Ω − Ψ)i→ ) being M\ -concave, where v is M\ -concave if for all Φ, Ψ ⊆ Ω and all ϕ ∈ Φ−Ψ the function v satisfies v (Φ)+v (Ψ) ≤ max v (Φ − {ϕ}) + v (Ψ ∪ {ϕ}) , max {v (Φ ∪ {ψ} − {ϕ}) + v (Ψ ∪ {ϕ} − {ψ})} . ψ∈Ψ−Φ

This equivalence is shown by Reijnierse et al. (2002), Fujishige and Yang (2003), and Lien and Yan (2003). This is particularly useful in the computation and characterization of efficient allocations; see Murota (2003).

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Intuitively, we think of the firm as employing all of the workers associated with trades that the firm buys and with trades that the firm does not sell. We show that vi satisfies the gross substitutes condition (GS) of Kelso and Crawford (1982) so long as ui is fully substitutable. We then take the utilities of workers to be monotonically increasing in their wages; the workers are indifferent over firms. With these definitions, we have constructed a market of the type studied by Kelso and Crawford (1982). It is known that in such markets, a competitive equilibrium exists so long as preferences of the firms satisfy (GS) (Kelso and Crawford (1982)). We take the competitive equilibrium thus derived and transform it back into a set of allocations and prices in the original economy as follows: we let ω be part of the allocation in the original economy if and only if the worker ω was hired by the buyer in the associated market; we use the wages in the associated market as prices in the original market. We thus obtain a competitive equilibrium of the original economy: given the prices generated, agents demand optimal bundles, and a trade ω is demanded by its buyer if and only if it is demanded by its seller. This construction also provides an algorithm for finding a competitive equilibrium. For instance, once we have transformed the original economy into an associated market, we can use an ascending auction for workers to find the minimal-price competitive equilibrium of the associated market; we may then map that competitive equilibrium back to a competitive equilibrium of the original economy. We now consider the structure of the set of competitive equilibria. We first prove an analogue of the first welfare theorem in our economy. Theorem 3. Suppose [Ψ; p] is a competitive equilibrium. Then, Ψ is an efficient set of trades. This result, while important in and of itself, also allows us to characterize the set of competitive equilibria. In particular, the set of competitive equilibrium price vectors forms a lattice, and for any p in this set, and for any efficient allocation Ψ, [Ψ; p] is a competitive equilibrium. Theorem 4. Suppose that preferences are fully substitutable. Then, the set of competitive equilibrium price vectors forms a complete lattice. Furthermore, for any element of this lattice p, and any efficient allocation Ψ, [Ψ; p] is a competitive equilibrium. The lattice structure of the set of competitive equilibrium prices is analogous to the lattice structure of the set of stable allocations for economies without transferable 13

utility (see, e.g., Hatfield and Milgrom (2005)). In those models, there is a buyeroptimal and a seller-optimal stable allocation. These correspond to the lowest-price and highest-price competitive equilibria in our model, as if some agent were only a buyer, the lowest-price competitive equilibrium would be optimal from the perspective of that agent, and similarly, if some agent were only a seller, the highest-price competitive equilibrium would be optimal from the perspective of that agent.

4.2

The Relationship between Stability and Competitive Equilibrium

We now demonstrate an equivalence between stable outcomes and competitive equilibria. We first show that for every competitive equilibrium [Ψ; p], the associated outcome λ ([Ψ; p]) is stable. Theorem 5. Suppose that [Ψ; p] is a competitive equilibrium. Then, λ ([Ψ; p]) is a stable outcome. If for some competitive equilibrium [Ψ; p] the outcome λ ([Ψ; p]) is not stable, then either it is not individually rational or it is blocked. If it is not individually rational for some agent i, then λ ([Ψ; p])i ∈ / Ci (λ ([Ψ; p])). Hence, Ψi ∈ / Di (p), and so [Ψ; p] can not be a competitive equilibrium. If λ ([Ψ; p]) admits a blocking set Z, then all the agents with contracts in Z are strictly better off after the deviation. Hence, there must exist some agent i with a contract in Z such that i is strictly better off choosing an element of Ci (Z ∪ λ ([Ψ; p])) given the original price vector p; hence, Ψi ∈ / Di (p) and so [Ψ; p] is not a competitive equilibrium. Note that this result does not rely on substitutability. However, it is generally not true that all stable outcomes correspond to competitive equilibria. Consider the following example where Ω = {χ, ψ}, I = {i, j} and s (ψ) = s (χ) = i and b (ψ) = b (χ) = j, and ui ({χ, ψ}) = ui ({χ}) = ui ({ψ}) = −4, uj ({χ, ψ}) = uj ({χ}) = uj ({ψ}) = 3. In this case, ∅ is stable. Since ∅ is the only efficient allocation, by Theorem 4 any competitive equilibrium is of the form [∅; p]. However, the preferences of agent i imply that pχ + pψ ≤ 4, as otherwise i will choose to sell at least one of ψ or χ. 14

Moreover, the preferences of agent j imply that pχ , pψ ≥ 3, as otherwise j will buy at least one of ψ or χ. Clearly, all three inequalities can not jointly hold. Hence, while ∅ is stable, there is no corresponding competitive equilibrium. The key issue is that an outcome A need only specify prices for the trades in τ (A) while a competitive equilibrium, by contrast, must specify prices for all trades (including those trades that do not transact). Hence, it may be possible, as in the example, for an outcome A to be stable, but, because of preference complementarity over trades outside of τ (A), it may be impossible to find a price vector such that τ (A) satisfies demand. If, however, preferences are fully substitutable, then for any stable outcome A we can find a supporting set of prices p such that [τ (A) ; p] is a competitive equilibrium and the prices of trades that transact are the same as in the stable outcome. Theorem 6. Suppose that preferences are fully substitutable and A is a stable outcome. Then, there exists a price vector p ∈ R|Ω| such that [τ (A) ; p] is a competitive equilibrium and if (ω, p¯ω ) ∈ A, then pω = p¯ω . As demonstrated above, the key difficulty in constructing a competitive equilibrium from a stable outcome is to price the trades that are not part of the stable outcome, i.e. those trades ξ ∈ Ξ ≡ Ω − τ (A). To do this, we construct a pseudomarket where the set of trades available is Ξ. For each ω ∈ τ (A), let p¯ω ∈ R be the ˆ i in the price for which (ω, p¯ω ) ∈ A. Each agent i has a demand correspondence D pseudo-market given by ˆ i (pΞ ) = Φ ∩ Ξ : Φ ∈ arg max Ui Ψ; p¯τ (A) , pΞ D , Ψ⊆Ωi

which is agent i’s demand for trades in Ξ given that he has the option to take any contract ω that he obtains as part of the stable outcome A at the price p¯ω . This demand correspondence is fully substitutable and so, by Theorem 2, we have a competitive equilibrium in this pseudo-market. Furthermore, some competitive equilibrium of this pseudo-market must be of the form [∅; pΞ ]—otherwise, any competitive equilibrium would provide a strictly higher utility than no trade, yielding a blocking set for A. As [∅; pΞ ] is a competitive equilibrium of the pseudo-market, each agent i demands τ (A)i given the price vector p¯τ (A) , pΞ ; otherwise the original outcome A would not be individually rational for i. Hence, any price vector pΞ that supports 15

∅ as a competitive equilibrium of the pseudo-market can then be used to show that τ (A) ; p¯τ (A) , pΞ is a competitive equilibrium of the original market.

4.3

The Necessity of Full Substitutability for Existence

We now show that if the preferences of any one agent are not fully substitutable, then stable outcomes need not exist. In particular, we can construct simple preferences for other agents such that no stable outcome exists. By simple preferences, we mean preferences such that when faced with any two contracts, an agent considers them either perfect substitutes or values them independently. Definition 7. An agent i, with utility function over trades ui , has simple preferences if, for all ψ, ω ∈ Ωi and all Φ ⊆ Ωi − {ψ, ω} such that ui (Φ) > −∞: 1. For all trades ψ, ω such that i plays the same role, i.e. {ψ, ω} ⊆ Ω→i or {ψ, ω} ⊆ Ωi→ , either (a) ψ and ω are independent: ui ({ψ, ω} ∪ Φ) − ui ({ω} ∪ Φ) = ui ({ψ} ∪ Φ) − ui (Φ) or (b) ψ and ω are incompatible: ui ({ψ, ω} ∪ Φ) − ui (Φ) = −∞. 2. For all trades ψ, ω such that i plays different roles, i.e. ψ ∈ Ω→i or ω ∈ Ωi→ , either (a) ψ and ω are independent: ui ({ψ, ω} ∪ Φ) − ui ({ω} ∪ Φ) = ui ({ψ} ∪ Φ) − ui (Φ) or (b) ψ and ω are incompatible: either ui ({ω} ∪ Φ) = −∞ or ui ({ψ} ∪ Φ) = −∞. 16

Note that it is immediate that all simple preferences are substitutable. Simple preferences are a restrictive class of preferences that allow agents to be both buyers and sellers and to have multi-unit demand. When considering two trades ψ and ω as a buyer (or as a seller), the agent either considers them independently (in which case the marginal utility of performing both trades is just the sum of the marginal utilities of performing the individual trades), or can not sign both trades simultaneously (so that whenever he performs both trades he obtains a utility of −∞). The idea behind the second part of the definition is that each trade ω is associated with an object ω, so that if the trade is performed, b(ω) acquires the object ω from s(ω), and if the trade does not take place, s(ω) keeps the object ω. With this convention in mind, the reasoning behind the second part of the definition is completely analogous to that behind the first: In case (a), performing only ψ (in addition to trades in Φ) means that i acquires ψ and holds on to ω (since ω is not performed). The condition then says that the agent either considers the two underlying goods independently (in which case the marginal utility of having both goods, ui ({ψ} ∪ Φ) − ui ({ω} ∪ Φ), is just the sum of the marginal utility of keeping ω, ui (Φ) − ui ({ω} ∪ Φ), and the marginal utility of acquiring ψ, ui ({ω, ψ} ∪ Φ) − ui ({ω} ∪ Φ)), or cannot own both (in which case the marginal utility of having the two goods is −∞). Simple preferences play a similar role to the unit-demand preferences used by Gul and Stacchetti (1999) to show the non-existence of competitive equilibria in their setting. However, in our setting we must allow an individual agent to act as a set of unit-demand consumers, as each contract specifies both the buyer and the seller, and the violation of substitutability may only occur for an agent i when he holds multiple contracts with another agent. Our necessity result also requires sufficient “richness” of the set of trades. Specifically, we require that the set of trades Ω is exhaustive, i.e. that for each i 6= j ∈ I there exist ωi , ωj ∈ Ω such that b (ωi ) = s (ωj ) = i and b (ωj ) = s (ωi ) = j. Theorem 7. Suppose that there exist at least four agents and that the set of trades is exhaustive. Then if the preferences of some agent are not fully substitutable, there exist simple preferences for all other agents such that no stable match exists.20 To understand the result, consider the following example. 20

The proof of this result also shows that for two-sided markets with transferable utility, if any agent’s preferences do not satisy substitutability, then if there exists at least one other agent on the same side of the market, simple preferences can be constructed such that no stable allocation exists.

17

Example 1. Agent i is just a buyer, and has perfectly complementary preferences over the contracts ψ and ω, and is not interested in other contracts, that is, 1 = ui ({ψ, ω}) 0 = ui ({ψ}) = ui ({ω}) = ui (∅) = 0 ˆ Now suppose that s (ψ) and s (ω) also have contracts ψ and ω ˆ (where s ψˆ = s (ψ) and s (ˆ ω ) = s (ω)) with another agent j, and let the other agents have preferences of the form: n o n o us(ψ) ψˆ = us(ψ) ({ψ}) = us(ψ) (∅) = 0 us(ψ) ψ, ψˆ = −∞ us(ω) ω })o = us(ω) n ({ω}) o= us(ω) (∅) = 0 n({ˆ ˆ ω ψˆ = uj ({ˆ ω }) = 3 ψ, ˆ = uj uj 4

us(ω) ({ω, ω ˆ }) = −∞ uj (∅) = 0

ˆ and similarly Then in any stable outcome s (ψ) will sell at most one of ψ and ψ, for s (ω). It can not be that {ψ, ω} is part of a stable outcome, as their total price is at most 1, meaning at least one of them has a price at most 21 ; suppose it o nis ω—we 5 ˆ ψ, pψˆ then have that ω ˆ , 8 is a blocking set. It also can not be the case that 3 or {(ˆ ω , pωˆ )} is stable: in the former case, pψˆ must be less than 4 , in which case 1 ψ, 78 , ω, 16 is a blocking set. A symmetric construction holds for the latter case. The proof of Theorem 7 essentially generalizes Example 1. Since a stable outcome does not necessarily exist when preferences are not fully substitutable, and any competitive equilibrium generates a stable outcome (by Theorem 5), we can use Theorem 7 to show that if the preferences of even one agent are not fully substitutable, then there exist simple preferences for the other agents such that no competitive equilibrium exists. Corollary 1. Suppose that there exist at least four agents and that the set of trades is exhaustive. Then, if the preferences of some agent are not fully substitutable, there exist simple preferences for all other agents such that no competitive equilibrium exists.

5

Relation with Other Solution Concepts

In this section, we describe the relationship between competitive equilibrium, stability, and other solutions concepts that have played important roles in the previous 18

literature.

5.1

The Core and Strong Group Stability

We start by introducing a classical solution concept: the core. Definition 8. An outcome A is in the core if it is core unblocked : there does not exist a set of contracts Z such that, for all i ∈ b (Z) ∪ s (Z), Ui (Z) > Ui (A). The core forces deviating agents to break off all relationships with outsiders. We now consider a stronger notion of stability which allows agents to maintain trading relationships with non-deviating agents. Definition 9. An outcome A is strongly group stable if it is 1. Individually rational; 2. Strongly unblocked: There does not exist a nonempty set Z ⊆ X such that (a) Z ∩ A = ∅, and (b) for all i ∈ b (Z) ∪ s (Z), there exists a Y i ⊆ Z ∪ A such that Z ⊆ Y i and Ui (Y i ) > Ui (A). The key difference to the stability concept used in the previous sections is that strong group stability does not require deviations to be self-enforcing. We call this notion strong group stability as it is stronger than both strong stability (introduced by Hatfield and Kominers (2010a)), which also required that each Y i be individually rational, and group stability (introduced by Roth and Sotomayor (1990) and ¨ extended to the setting of many-to-many matching by Konishi and Unver (2006)), b(y) s(y) which required that if y ∈ Y , then y ∈ Y , i.e. that the deviating agents agreed on which contracts they presently had would be kept. Strong stability and group stability are themselves strengthenings of the notion of setwise stability, introduced by Echenique and Oviedo (2006) and Klaus and Walzl (2009), who imposed both of the above requirements.21 Given these definitions, the following result is immediate. 21

The notion of setwise stability used in these works is slightly stronger than the definition of setwise stability introduced by Sotomayor (1999); see Klaus and Walzl (2009) for a discussion of the slight differences in these two definitions.

19

Theorem 8. Any strongly group stable outcome is stable and in the core. Furthermore, any core outcome is efficient. However, without additional assumptions on preferences, no additional structure is necessarily present. In particular, it may be that both stable and core outcomes exist for a given set of preferences, but that no outcome is both stable and core. Example 2. There are two agents i and j, and two trades ψ and ω, where s (ψ) = s (ω) = i and b (ψ) = b (ω) = j. This trading environment is pictured below, along with a table specifying utilities. Ψ ui (Ψ) uj (Ψ)

i ω

ψ

j

∅ {ψ} {ω} {ψ, ω}

0 −2 −2 −6

0 0 0 7

The set of core outcomes is given by {{(ψ, pψ ) , (ω, pω )} : 6 ≤ pψ + pω ≤ 7}. However, the unique stable outcome is ∅. Even when an outcome is both stable and in the core, it is not necessarily strongly group stable. An example in the appendix demonstrates this. However, even when preferences are fully substitutable, the relationship between strong group stability and stability is unclear. For models without transferable utility (see e.g., Sotomayor (1999), Hatfield and Kominers (2010a), Westkamp (2010)), strong group stability is strictly more restrictive than stability. However, with quasilinear utility and fully substitutable preferences, these equilibrium notions coincide. Theorem 9. If preferences are fully substitutable and A is a stable outcome, then A is strongly group stable and in the core. Moreover, for any core outcome A, there exists a stable outcome Aˆ such that τ (A) = τ Aˆ . However, for some fully substitutable preferences, the core is strictly larger than the set of stable outcomes. Consider again the setting of Example 2 but take preferences to be:

20

Ψ ui (Ψ) uj (Ψ) ∅ {ψ} {ω} {ψ, ω}

0 0 0 −3

0 5 5 9

.

In this case, {(ψ, 2) , (ω, 2)} is a core outcome, but it is not individually rational for agent i; he will choose to drop one of the two contracts. We therefore see that the set of imputed utilities of a core outcome may not correspond to the set of imputed utilities for any stable outcome: in this example, the payoff of agent i in any stable outcome is at least 3, while it is only 1 in this core outcome.

5.2

Chain Stability

The stability concept used in this paper also appears substantively different and noticeably stronger than the chain stability concept used in Ostrovsky (2008): the former (Definition 1) requires robustness to all blocking sets, while the latter requires robustness only to very specific blocking sets—chains of contracts. However, Hatfield and Kominers (2010b) show that under full substitutes, in the setting of Ostrovsky (2008) these conditions are in fact equivalent. In this section, we prove an analogous result for the current setting with continuous prices. First, we need to define the notion of chain stability. It is more general than that in the earlier work, because in the current setting trading cycles are allowed, and the definition needs to accommodate that. If trading cycles are prohibited (i.e., there is an upstream–downstream partial ordering on the set of agents), then the definition is essentially the same as in Ostrovsky (2008), adjusted only for the presence of continuous prices. Definition 10. A set of contracts Z is a chain if its elements can be arranged in some order y 1 , . . . , y |Z| such that s(y `+1 ) = b(y ` ) for all ` < |Z|. Note that under this definition, the buyer in contract y |Z| is allowed to be the seller in contract y 1 (in which case the chain becomes a cycle), and also the same agent can be involved in the chain multiple times (because there is no enforced upstream– downstream partial ordering). Definition 11. An outcome A is chain stable if it is 21

1. Individually rational ; 2. Not blocked by a chain: There does not exist a nonempty blocking chain Z ⊆ X such that (a) Z ∩ A = ∅, and (b) for all agents i involved in contracts in Z, for all of i’s choices Y ∈ Ci (Z ∪ A), we have Zi ⊆ Y . This definition is weaker than the defintion of stability (Definition 1), and so it is immediate that any stable outcome is chain stable, even if preferences are not fully substitutable. The converse, under the assumption of full substitutes, is a direct corollary of the following, more general result, which shows that any blocking set of an arbitrary feasible outcome A can be “decomposed” into blocking chains. Theorem 10. Suppose agents’ preferences are fully substitutable, and consider any feasible outcome A that is blocked by some nonempty set Z. Then for some M ≥ 1 m we can partition the set Z into a collection of M chains W m such that Z = ∪M m=1 W , A is blocked by W 1 , and for any m ≤ M − 1, set A ∪ W 1 ∪ · · · ∪ W m is blocked by chain W m+1 . Corollary 2. If agents’ preferences are fully substitutable, then any chain stable outcome A is stable. Full substitutability is critical for these results. Without it, chain stability is strictly weaker than stability. For instance, consider Example 1 above. There is no stable outcome in the example, yet {(ˆ ω , 0)} is chain stable: any set that blocks {(ˆ ω , 0)} includes two contracts of the form (ψ, pψ ) and (ω, pω ), and so the blocking set does not constitute a blocking chain.

5.3

Competitive Equilibria without Personalized Prices

The notion of competitive equilibrium considered in this paper considers trades as the basic unit of analysis and a price vector specifies one price for each trade. When an agent i has one indivisible object to sell (and is not interested in acquiring any object), a competitive equilibrium price vector would thus set one price for each possible buyer of i’s good. Thus, in particular, competitive equilibria in the sense defined above 22

allow for personalized pricing. This is in contrast to standard notions of competitive equilibrium (e.g., those of Gul and Stacchetti (1999) or Sun and Yang (2006)), which only establish a single price for each good. We now introduce a condition on utilities that allows for non-personalized pricing. Definition 12. Trades in Ψ ⊆ Ωi are mutually incompatible for i, if for all Ξ ⊆ Ωi such that |Ξ ∩ Ψ| ≥ 2, ui (Ξ) = −∞. Definition 13. Trades in Ψ ⊆ Ωi are perfect substitutes for i, if for all Ξ ⊆ Ωi − Ψ and all ω, ω 0 ∈ Ψ, ui (Ξ ∪ {ω}) = ui (Ξ ∪ {ω 0 }). Theorem 11. Let agents’ preferences be fully substitutable. Suppose trades in Ψ ⊆ Ωi are mutually incompatible and perfect substitutes for i and let [Ξ; p] be an arbitrary competitive equilibrium. (a) If Ψ ⊆ Ωi→ , let p = maxψ∈Ψ pψ and define q by qψ = p for all ψ ∈ Ψ and qψ = pψ for all ψ ∈ Ω − Ψ. Then [Ξ; q] is a competitive equilibrium. (b) If Ψ ⊆ Ω→i , let p = minψ∈Ψ pψ and define q by qψ = p for all ψ ∈ Ψ and qψ = pψ for all ψ ∈ Ω − Ψ. Then [Ξ; q] is a competitive equilibrium. The cases listed in the Theorem 11 are exhaustive, since a trade ω ∈ Ωi→ can never be a perfect substitute for a trade in ω 0 ∈ Ω→i when the preferences of i are fully substitutable. An important corollary of this result is existence of competitive equilibria for the setting of Sun and Yang (2006, 2009): an exchange economy with a finite set of agents I, a set of indivisible objects N , and a numeraire. The main assumption of Sun and Yang (2006, 2009) is (GSC)—the set of objects can be partitioned as N = S1 ∪ S2 , where, if the price of one object in S1 (S2 ) is increased, then an agent weakly increases his demand for all objects in S1 (S2 ) that have not become more expensive and weakly decreases his demands for all objects in S2 (S1 ).22 Under these assumptions, Sun and Yang (2006, 2009) show that a competitive equilibrium without personalized prices exists. To embed the Sun and Yang (2006, 2009) setting into our model, let the set of upstream trades of agent i be given by Ω→i = {(i, s) : s ∈ S1 } and the set of 22

This model in turn is a generalization of the model of Gul and Stacchetti (1999), which is the special case one obtains if one of the sets in the partition is empty so that agents have substitutable preferences.

23

downstream trades be given by Ωi→ = {(i, s) : s ∈ S2 }. We treat object s ∈ S1 ∪ S2 as an agent whose utility us over sets of trades is zero for any singleton or empty set of trades and −∞ for any other set of trades (since each object can only be sold once). Our existence result (Theorem 2) implies that a competitive equilibrium exists if we allow the price at which i can acquire good s (the price of the trade (i, s)) to differ from the price at which j 6= i can acquire good s (the price of trade (j, s)). Given that all trades are mutually incompatible and perfect substitutes for all s, Theorem 11 implies the following result. Corollary 3. For any Sun and Yang (2006, 2009) economy with (GSC) preferences, the set of single price competitive equilibria is a non-empty subset of the set of competitive equilibria.

6

Conclusion

We have introduced a general model in which a network of agents trades via bilateral contracts. In this setting, when continuous transfers are allowed and agents’ preferences are quasilinear, full substitutability of preferences is sufficient and necessary for the guaranteed existence of stable outcomes. Furthermore, full substitutability implies that the set of stable outcomes is equivalent to the set of competitive equilibria, and that all stable outcomes are strongly group stable, in the core, and efficient. Previous results have shown that for contractual sets over which agents’ preferences are fully substitutable, supply chain structure (i.e. the acyclicity of the contractual set) is sufficient to guarantee the existence of stable outcomes. Moreover, both full substitutability and supply chain structure are necessary to ensure existence of stable outcomes if the contractual set is otherwise fully general. Our work shows that, for contractual sets that incorporate a numeraire, supply chain structure is not necessary, although full substitutability is. It is an open question why the presence of a numeraire can replace supply chain structure. Our work generalizes a number of results on the existence of competitive equilibria in networks (e.g., those of Kranton and Minehart (2001)). However, unlike the network literature, we have assumed that the network structure is exogenous. We leave the question of endogenous trade network formation to future work.

24

Appendix Proof of Theorem 1 (MFS) ⇒ (DFS) Given any price vector p ∈ R|Ω| , define the set of contracts available to agent i by X(p) ≡ {(ω, pˆ) : ∀ω ∈ Ω→i , pˆ ≥ pω and ∀ω ∈ Ωi→ , pˆ ≤ pω }. Let p, p0 ∈ R|Ω| be such that pω = p0ω for each ω ∈ Ωi→ and pω ≤ p0ω for each ω ∈ Ω→i . Let Y = X(p) and Z = X(p0 ). Clearly, Zi→ = Yi→ and Z→i ⊆ Y→i . Take any Ψ ∈ Di (p), let (Ψ, π(Ψ)) = Y ∗ , and note that Y ∗ ∈ Ci (Y ). By (MFS), ∗ ∗ ∗ there exists Z ∗ ∈ Ci (Z) such that Y→i ∩ Z ⊆ Z ∗ and Zi→ ⊆ Yi→ . Let Ψ0 = τ (Z ∗ ) and note that Ψ0 ∈ Di (p0 ). We show that Ψ0 satisfies the conditions in part 1 of Definition 4. ∗ ∗ ∗ ∩Z ⊆ Z ∗ : pω = p0ω and ω ∈ Ψ} and hence Y→i ∩Z = {(ω, pω ) ∈ Y→i Note that Y→i ∗ ∗ and pω = p0ω for each ⊆ Yi→ implies {ω ∈ Ψ→i : pω = p0ω } ⊆ Ψ0 . Furthermore, Zi→ ω ∈ Ωi→ imply Ψ0i→ ⊆ Ψi→ . The proof for case 2 of the definition is analogous. (DFS) ⇒ (MFS) Let Π be a price higher than any agent’s highest possible utility from any combination of trades, that is, Π > maxi∈I max{Ψ⊆Ωi :ui (Ψ)∈R} |ui (Ψ)|. Given any set of contracts Y ⊆ Xi , the price vector relevant for a utility maximizing agent i is p(Y ) ∈ R|Ωi | , where for each ω ∈ Ω→i , pω = min{p : (ω, p) ∈ Y or Π}, and for each ω ∈ Ωi→ , pω = max{p : (ω, p) ∈ Y or − Π}. Let Y, Z ⊆ Xi be such that Yi→ = Zi→ and Y→i ⊇ Z→i . Let p ≡ p(Y ) and p0 ≡ p(Z). Clearly, pω = p0ω for each ω ∈ Ωi→ , and pω ≤ p0ω for each ω ∈ Ω→i . Take any Y ∗ ∈ Ci (Y ), let τ (Y ∗ ) = Ψ, and note that Ψ ∈ Di (p). By (DFS), there exists Ψ0 ∈ Di (p0 ) such that {ω ∈ Ψ→i : pω = p0ω } ⊆ Ψ0→i and Ψ0i→ ⊆ Ψi→ . Let Z ∗ = (Ψ0 , π(Ψ0 )) and note that Z ∗ ∈ Ci (Z). We show that this set of contracts satisfies the conditions in part 1 of Definition 3. ∗ Note that Y→i ∩ Z = {(ω, p0ω ) : ω ∈ Ψ→i and pω = p0ω } ⊆ {(ω, p0ω ) : ω ∈ ∗ Ψ0→i and pω = p0ω } ⊆ Z ∗ . Furthermore, Zi→ = {(ω, p0ω ) : ω ∈ Ψ0i→ } ⊆ {(ω, p0ω ) : ∗ ω ∈ Ψi→ } = {(ω, pω ) : ω ∈ Ψi→ } = Yi→ . The proof for case 2 of the definitions is analogous. 25

(AFS) ⇒ (DFS) The proof follows directly, upon observing that the second part of the definition of (DFS) can be equivalently expressed as follows: Lemma 1 (Sun and Yang (2006), Lemma 1). For all price vectors p, p0 ∈ R|Ω| such that pω = p0ω for each ω ∈ Ω→i and pω ≤ p0ω for each ω ∈ Ωi→ , and for each Ψ ∈ Di (p), there exists Ψ0 ∈ Di (p0 ) such that {ω ∈ (Ω − Ψ)i→ : pω = p0ω } ⊆ (Ω − Ψ0 )i→ and Ψ→i ⊆ Ψ0→i . (DFS) ⇒ (AFS) Let p, p0 be such that pω ≤ p0ω for each ω ∈ Ωi and fix some Ψ ∈ Di (p). Let p∗ be such that p∗ω = pω for each ω ∈ Ωi→ , and p∗ω = p0ω for each ω ∈ Ω→i . By the first part of (DFS), there exists Ψ∗ ∈ Di (p∗ ) such that {ω ∈ Ψ→i : pω = p∗ω } ⊆ Ψ∗→i and Ψ∗i→ ⊆ Ψi→ . Now let prices of all downstream trades increase to p0 while keeping prices of all upstream trades fixed. By Lemma 1, there exists Ψ0 ∈ Di (p0 ) such that {ω ∈ (Ω − Ψ∗ )i→ : pω = p0ω } ⊆ (Ω − Ψ0 )i→ and Ψ∗→i ⊆ Ψ0→i . Combining the two demand changes, we have that (i) for ω ∈ Ω→i such that pω = p0ω , eω (Ψ) = 1 implies eω (Ψ0 ) = 1, and (ii) for ω ∈ Ωi→ such that pω = p0ω , eω (Ψ) = 0 implies eω (Ψ0 ) = 0 (this follows since Ψ∗i→ ⊆ Ψi→ so that i does not sell any additional objects at p∗ compared to p). This completes the proof. Submodularity The equivalence with submodularity follows directly from the results of Sun and Yang (2009).23

Proof of Theorem 2 Step 1: We show how to transform a fully substitutable but potentially unbounded from below utility function ui into a fully substitutable and bounded utility function uî . For this purpose, we now introduce a very high price Π. Specifically, for each agent i, let ui be the highest possible utility of agent i from a combination of trades, 23

In addition to full substitutability, Sun and Yang (2009) assume monotonicity of ui . The proof of the equivalence of (generalized) submodularity (Theorem 2 of Sun and Yang (2009)), however, does not use this additional assumption and thus carries over to our setting.

26

P that is, ui ≡ max{Ψ∈Ωi :|ui (Ψ)| ui (Ψ \ {ψ}) + q(Ψ \ {ψ}) + = uî (Ψ) + p(Ψ), ˆ i (p). Next, take any Ξ such that ψ ∈ Ξ and uî (Ξ) = ui (Ξ). Then so that Ψ ∈ /D uî (Ψ) + p(Ψ) = ui (Ψ \ {ψ}) + q(Ψ \ {ψ}) + ≥ ui (Ξ) + q(Ξ \ {ψ}) + Π + = uî (Ξ) + p(Ξ), so that in particular ui (Ψ \ {ψ}) + q(Ψ \ {ψ}) ≥ ui (Ξ) + q(Ξ). Now let q 0 be defined by qω0 = p0ω for each ω 6= ψ and qψ0 = min{p0ψ , Π}. Suppose that Ψ ∈ / Di (q), so that Ψ \ {ψ} ∈ Di (q) by Claim 2 above (the argument in case of Ψ ∈ Di (q) is completely analogous). By full substitutability, there exists a set Φ ∈ Di (q 0 ) such that {ω ∈ (Ψ \ {ψ})i→ : p0ψ = pψ } = {ω ∈ Ψ \ {ψ}i→ : qψ0 = qψ } ⊆ Φ ˆ i (q 0 ) and hence Φ ∈ D ˆ i (q 0 ). and Φ→i ⊆ Ψ \ {ψ}. If qψ0 < Π, we must have Di (q 0 ) = D Since q 0 = p0 , this completes the proof. So suppose qψ0 = Π. Since the agent exactly breaks even when buying out of ψ, we can assume without loss of generality that ˆ i (q 0 ). The arguments are similar to the ones employed to prove Claims Φ ∪ {ψ} ∈ D 1 and 2 above and the details are omitted. In particular, under uî and at the price vector q 0 , Φ ∪ {ψ} gives i a weakly higher payoff than any other subset of Ωi that ˆ i (p0 ) must include ψ: Take an includes ψ. Now note that if p0ψ > Π, any set in D arbitrary set Ξ such that ψ ∈ / Ξ and ui (Ξ) > −∞. Then uî (Ξ ∪ {ψ}) + p0 (Ξ ∪ {ψ}) ≥ ui (Ξ) + p0 (Ξ) + p0ψ − Π = uî (Ξ) + p0 (Ξ) + p0ψ − Π > uî (Ξ) + p0 (Ξ), 28

ˆ i (p0 ). Since Φ∪{ψ} gives i a weakly higher payoff than any other subset so that Ξ ∈ /D ˆ i (p0 ). of Ωi that includes ψ at q 0 , the same must be true at p0 and we obtain Φ∪{ψ} ∈ D The argument in case the prices of some upstream trades increase, while prices of downstream trades stay constant, are completely analogous. Finally, the proof is easily adapted to show that uî satisfies full substitutes in case Ω0i = {ψ : ψ ∈ Ω→i }. Finally, note that for any Ω0i ⊆ Ωi , utility function uˆ can be obtained from original utility function u by allowing agent i to buy out all of his trades, ony by one, and since each such transformation preserves substitutability, function uˆ is substitutable as well. Step 2: We transform the modified economy with bounded and fully substitutable utilities uî into an associated (two-sided matching) market in which the set of firms is I and the set of workers is Ω. In this market, firm i’s utility from hiring a set of workers Ψ ⊆ Ωi is given by vi (Ψ) = uî (Ψ→i ∪ (Ω − Ψ)i→ ) . In the associated market firms negotiate wages freely with workers. Hence, a wage vector is a vector in R2|Ω| , where pω,s(ω) is the wage at which s(ω) can hire ω and pω,b(ω) is the wage at which b(ω) can hire ω. Given a wage vector p ∈ R2|Ω| , firm i’s profit from demanding Ψ ⊆ Ωi is given by vi (Ψ) −

X

pω,s(ω) −

X

pω,b(ω) .

ω∈Ψ→i

ω∈Ψi→

The associated demand correspondence is denoted by Div . We now analyze the implications of full substitutability for firms’ preferences in the associated market. First, given p ∈ R2|Ω| define the wage vector faced by firm i by setting pω,i = pω,s(ω) for each ω ∈ Ωi→ and pω,i = pω,b(ω) for each ω ∈ Ω→i (just to get a wage vector in R|Ω| set prices arbitrarily for ω ∈ Ω − Ωi ). ˆ i (pi ). This follows Note that Ψ ∈ Div (p) if and only if Ψ→i ∪ (Ω − Ψ)i→ ∈ D immediately from an inspection of the maximization problems. Lemma 3. The preferences of firm i in the associated market satisfy the gross substitutes (GS) condition.

29

Proof. Let i ∈ I, p ∈ R2|Ω| , and Ψ ∈ Div (p) all be arbitrary. Then, Ψ→i ∪ (Ω − Ψ)i→ ∈ ˆ i (pi ). We distinguish two cases: D Case 1: Let p0 ∈ R2|Ω| be such that p0ω,b(ω) ≥ pω,b(ω) for some ω ∈ Ω→i and p0ξ,i = pξ,i for all ξ ∈ Ω \ {ω} (rest of the price vector can be arbitrary since it doesn’t concern ˆ i (p0 ) such that (Ψ→i ) \ {ω} ⊆ Ψ0 i). By full substitutability, there exists a set Ψ0 ∈ D i 0 0 and Ψi→ ⊆ (Ω − Ψ)i→ (or Ψi→ ⊆ (Ω − Ψ )i→ ). But this completes the proof in this ˆ i (p0 ) implies Ψ0 ∪ (Ω − Ψ0 )i→ ∈ Dv (p0 ). case since Ψ0 ∈ D i →i i Case 2: Let p0 ∈ R2|Ω| be such that p0ω,s(ω) ≥ pω,s(ω) for some ω ∈ Ωi→ and p0ξ,i = pξ,i ˆ i (p0i ) for all ξ ∈ Ω\{ω}. By full substitutability and Lemma 1, there exists a set Ψ0 ∈ D such that Ψ→i ⊆ Ψ0 and (Ψi→ ) \ {ω} ⊆ Ω − Ψ0 . But this completes the proof in this case since it implies Ψ→i ∪ (Ψi→ ) \ {ω} ⊆ Ψ0→i ∪ (Ω − Ψ0 )i→ ∈ Div (p0 ). Step 3: The existence of a competitive equilibrium for the associated market follows essentially by applying Kelso and Crawford (1982)’s proof strategy. We now describe the details of this derivation. A (full employment) matching is a mapping µ : Ω → I. Given some matching µ and i ∈ I, let µ(i) ≡ {ω ∈ Ω : µ(ω) = i}. An outcome in the associated market is a pair (µ, p), where µ is a matching and p ∈ R2|Ω| is a wage vector. An outcome (µ, p) is in the strict core, if there is no deviation (i, Ξ, q) consisting of a firm i ∈ I, a set of workers Ξ ⊆ Ωi , and a wage vector q ∈ R|Ωi | such that (1) vi (Ξ) −

P

ξ∈Ξ qξ

≥ vi (µ(i)) −

P

ω∈µ(i)

pω,i ,

(2) qω ≥ pω,µ(ω) for each ω ∈ Ξ, and (3) at least one of the inequalities in (1) and (2) is strict. An outcome is in the core, if there is no deviation (i, Ξ, q) such that all inequalities in (1) and (2) are strict. We argue first that a core outcome exists if wages are required to be multiples of some δ > 0. Note first that since utility functions are bounded, there clearly is a wage vector p1 ∈ R2|Ω| which satisfies this requirement and, in addition, is such that

30

Ωi ∈ Div (p1 ) for each i ∈ I.24 Proceeding from this wage vector, we can apply the salary-adjustment procedure of Kelso and Crawford (1982) to obtain a core outcome of the discrete market:25 All that is needed for this property of the procedure is that it starts from a wage vector at which each worker receives an offer and that firm preferences are substitutable. Proceeding as in the proof of Theorem 2 of Kelso and Crawford (1982), we can then show that a strict core outcome exists for the associated market when wages are allowed to vary continuously. The details are omitted.

Step 4: Take a strict core outcome (µ, p) for the associated matching market. Let p∗ ∈ R|Ω| be defined by p∗ω ≡ pω,µ(ω) for each ω ∈ Ω, and let Ψ∗ ≡ ∪i∈I [µ(i)]→i . Note that Ψ∗i = [µ(i)]→i ∪ [Ω − µ(i)]i→ since ∪i∈I µ(i) = Ω. We now show that (Ψ∗ , p∗ ) is a competitive equilibrium of the modified economy with utility functions uî : Since (µ, p) is in the strict core of the associated market, there is no deviation (i, Ξ, q) such P P that vi (Ξ) − ξ∈ξ qξ > vi (µ(i)) − ω∈µ(i) p∗ω and qω ≥ p∗ω for each ω ∈ Ξ. This implies in particular, that we must have µ(i) ∈ Div (p∗ ) for each i ∈ I. But then ˆ i (p∗ ), which is precisely what we wanted to show. [µ(i)]→i ∪ [Ω − µ(i)]i→ ∈ D To complete the proof, we now show that (Ψ∗ , p∗ ) is an equilibrium of our original economy (with potentially unbounded from below utilities) as well. Suppose to the contrary that there exists an agent i and an alternative set of trades involving agent i, Ξ, such that Ui ([Ξ; p∗ ]) > Ui ([Ψ∗i ; p∗ ]). Since Uî ([Ξ; p∗ ]) ≤ Uî ([Ψ∗i ; p∗ ]), and by the construction of uî , Uî ([Ξ; p∗ ]) ≥ Ui ([Ξ; p∗ ]), it follows that Uî ([Ψ∗i ; p∗ ]) > Ui ([Ψ∗i ; p∗ ]). This, in turn, implies that uî (Ψ∗i ) 6= ui (Ψ∗i ). Thus, by construction, for some nonempty set Φ, uî (Ψ∗i ) = ui (Ψ∗i − Φ) − Π · |Φ| ≤ ui − Π. This implies that P P P P P ˆj (Ψ∗j ) = uî (Ψ∗i ) + j6=i uˆj (Ψ∗j ) ≤ ui − Π + j6=i uj = j uj − Π = − j ui − 1 < ju P j uj (∅), contradicting the first welfare theorem. Let v i ≡ maxΨ⊆Ωi |vi (Ψ)| and Π0 be the smallest multiple of δ that is greater than 2v i + 1. Set p1ω,i = −Π0 for each ω ∈ Ωi . Now consider an arbitrary Ψ ⊂ Ωi and ω ∈ Ωi \ Ψ. We have P P vi (Ψ) − vi (Ψ ∪ {ω}) ≤ 2v i < −p1ω,i so that vi (Ψ) − ψ∈Ψ p1ψ,i < vi (Ψ ∪ {ω}) − ψ∈Ψ p1ψ,i − p1ω,i . Hence, Ωi = Div (p1 ). 25 In our application, workers care only about their monetary compensation and may break ties however they like. 24

31

Proof of Theorem 3 If [Ψ; p] is a competitive equilibrium, then for any other set of trades Ξ, we have that ui (Ψ) +

X

pψ −

ψ∈Ψi→

X

X

pψ ≥ ui (Ξ) +

ψ∈Ψ→i

pψ −

ψ∈Ξi→

X

pψ

ψ∈Ξ→i

for each i ∈ I. By summing over all agents then, we have that X

ui (Ψ) ≥

i∈I

X

ui (Ξ)

i∈I

and hence Ψ is an efficient set of trades.

Proof of Theorem 4 We prove the latter half of the theorem first. The proof is analogous to the proof of Lemma 6 in (Gul and Stacchetti, 1999): Suppose that [Ξ; p] is a competitive equilibrium and that Ψ is an efficient allocation. Since [Ξ; p] is a competitive equilibrium, we have that ui (Ξ) +

X

pψ −

X

pψ ≥ ui (Ψ) +

pψ −

X

pψ ,

ψ∈Ψ→i

ψ∈Ψi→

ψ∈Ξ→i

ψ∈Ξi→

X

as agents are choosing optimally. However, since Ψ is efficient, we have X

ui (Ψ) ≥

i∈I

i∈I

# X

ui (Ψ) +

X

pψ −

ψ∈Ψi→

ui (Ξ)

i∈I

" X

X

"

pψ ≥

X

# ui (Ξ) +

i∈I

ψ∈Ψ→i

X

pψ −

ψ∈Ξi→

X

pψ

ψ∈Ξ→i

and both inequalities can only hold if ui (Ξ) +

X ψ∈Ξi→

pψ −

X

pψ = ui (Ψ) +

ψ∈Ξ→i

X ψ∈Ψi→

pψ −

X

pψ .

ψ∈Ψ→i

Hence Ψi is an optimal choice for each i ∈ I at the prices p and so [Ψ; p] is a competitive equilibrium. The proof of the first part of the theorem is completely analogous to the proof of

32

Theorem 3 in (Sun and Yang, 2009) except for some differences in notation: Given P some price vector p ∈ R|Ω| , let V (p) ≡ i Vi (p). Let Ψ∗ ⊆ Ω be any efficient allocation P and set U ∗ ≡ i ui (Ψ∗i ). We claim that p∗ is an equilibrium price vector if and only if p∗ minimizes V (·) and V (p∗ ) = U ∗ .26 To prove this let first p∗ be an equilibrium price vector and p be an arbitrary price vector. For agent i, let Ψi ∈ Di (p) be arbitrary. Then we have

 V (p) =

X

 X

ui (Ψi ) +

i

X

pω −

ω∈(Ψi )i→

pω 

ω∈(Ψi )→i

 ≥

X

 X

ui (Ψ∗i ) +

pω −

ω∈(Ψ∗i )i→

i

X

pω 

ω∈(Ψ∗i )→i

= V (p∗ ), where the first inequality follows from utility maximization and the second equality follows since (Ψ∗i )i is an allocation so that prices cancel out. This shows that p∗ must minimize V (p). Next, let p be any price vector that minimizes V (·) and satisfies V (p) = U ∗ . We claim that (p, Ψ∗ ) is a competitive equilibrium. To see this note first that utility maximization implies Vi (p) ≥ ui (Ψ∗i ) +

X

pω −

ω∈(Ψ∗i )i→

X

pω .

ω∈(Ψ∗i )→i

If this inequality was strict for one or more i, we would obtain V (p) > U ∗ , a contradiction. Now suppose p and q are two equilibrium price vectors, let p∧q denote their meet, and p ∨ q denote their join. To see that p ∧ q must be an equilibrium price vector, note that

U ∗ ≤ V (p ∧ q) ≤ V (p) + V (q) − V (p ∨ q) ≤ 2U ∗ − U ∗ = U ∗ , 26

This is the reformulation of Lemma 1 in (Sun and Yang, 2009) that applies in our setting.

33

where the first inequality follows since U ∗ is the minimal value of V (·), the second follows from the submodularity of V (·) (which is guaranteed since the indirect utility of each agent is submodular), and the last inequality follows since p and q are equilibrium price vectors. The same argument shows that p ∨ q is an equilibrium price vector.

Proof of Theorem 5 Let A ≡ λ ([Ψ; p]). Suppose A is not stable; then either it is not individually rational or there exists a blocking set. If A is not individually rational, then Ai ∈ / Ci (A) for some i ∈ I. Hence Ai ∈ / arg max Ui (Z) Z⊆Ai

Ai ∈ / arg max ui (Ψ) + Ψ⊆τ (Ai )

X

pψ −

ψ∈Ψi→

X

Ai ∈ / arg max ui (Ψ) + Ψ⊆Ωi

ψ∈Ψi→

pψ −

X

pψ

ψ∈Ψi→

X

pψ

ψ∈Ψi→

Ai ∈ / Di (p) , where the first statement follows from the definition of the choice function, the second from the definition of Ui (·), the third as τ (Ai ) ⊆ Ωi , and the fourth from the definition of Di (p); hence, [Ψ; p] is not a competitive equilibrium, a contradiction. Suppose now that there exists a blocking set Z = {(ω, p˜ω ) ∈ X : (ω, p˜ω ) ∈ / A}. Let J = {j ∈ I : Zj 6= ∅}. For each j ∈ J, fix Y j ∈ Cj (Z ∪ A); note that due to the definition of a blocking set, Zj ⊆ Y j . Since Z ∩ A = ∅, and for all Y ∈ Cj (Z ∪ A) we have that Zj ⊆ Y , Aj ∈ / Cj (Z ∪ A). Hence, for all j ∈ J,

   

Uj Y j > Uj (Aj )  " # uj (τ (Y j )) +  P P u (τ (A )) + j j > P p˜ψ − ψ∈τ (Zj→ ) p˜ψ + . P  P ψ∈τ (Zj→ ) P ψ∈τ (Aj→ ) pψ − ψ∈τ (Aj→ ) pψ pψ − ψ∈τ (Y j −Z) pψ ψ∈τ (Y j −Z) j→

→j

Summing over individuals, and noting that both the buyer and the seller of every

34

contract in Z choose the contract, we have "

uj (τ (Y j )) + P P ψ∈τ (Y j −Z)j→ pψ − ψ∈τ (Y j −Z)→j pψ

#

uj (τ (Y j )) + X P P   P ψ∈τ (Zj→ ) pψ − Pψ∈τ (Zj→ ) pψ + j∈J ψ∈τ (Y j −Z)j→ pψ − ψ∈τ (Y j −Z)→j pψ " X uj (τ (Y j )) + P P pψ − ψ∈τ (Y j ) pψ ψ∈τ (Y j ) j∈J



X j∈J



j→

>

"

uj (τ (Aj )) + P P ψ∈τ (Aj→ ) pψ − ψ∈τ (Aj→ ) pψ

#

"


#

"


#

X j∈J

 X >  j∈J

# >

X j∈J

j→

where the second statement follows from again noting that the buyer and the seller of every contract in Z chooses the contract. (The third follows from summation.) The above equation states that the sum of the utilities for all agents in J is strictly higher if each j ∈ J chooses Y j instead of Aj . However, for that to be true, it must be the case that for some j ∈ J, we have "

uj (τ (Y j )) + P P pψ − ψ∈τ (Y j ) ψ∈τ (Y j ) j→

j→

# pψ

" >

uj (τ (Aj )) + P P ψ∈Aj→ pψ ψ∈Aj→ pψ −

#

and hence Aj ∈ / Dj (p), as the allocation τ (Yj ) provides strictly higher utility, given the prices p. Hence, [Ψ; p] is not a competitive equilibrium; a contradiction.

Proof of Theorem 6 Consider a feasible set A of contracts. Suppose it is stable. For every agent i, consider a modified utility uî , defined on sets of contracts Ψi that do not have any trades in common with A. Utility uî (Ψi ) is equal to the max utility attainable by agent i by combining contracts in Ψi with various subsets of Ai . It is clear that uî satisfies ˆ i generated by uî : since Di satisfies full substitutability. To see this, consider the D ˆ i satisfies demand-theory demand-theory full substitutability by assumption, then D full substitutability as there are less prices that can be changed. Now, consider a modified economy with the same agents as before, the set of trades Ω − τ (A), and utilities uˆ. By Theorem 2, this economy has a CE, and moreover, this CE is efficient. If there is a CE that involves no trades, we are done: combining the prices in this CE with the prices in A gives us a CE of the original economy as no 35

,

agent wants to add trades not in A; and also, by the individual rationality of A, no agent wants to drop any trades in A. Suppose there is no CE of this modified economy that involves no trades. Pick any P CE, [Ψ, pˆ]. It has to be the case that in this CE, the sum of utilities i Uî ([Ψi ; p]) is P strictly greater than the sum of utilities i Uî ([∅; pˆ]), because for each i, Uî ([Ψi ; pˆ]) ≥ Uî ([∅; pˆ]), and if for all i, Uî ([Ψi ; p]) = Uî ([∅; p]), then [∅, p] is also a competitive equilibrium, contradicting our earlier assumption. P P P P Next note that i uî (Ψi ) = i Uî ([Ψi ; p]) > i Uî ([∅; p]) = i uî (∅), so P δ≡

i

P uî (Ψi ) − i uî (∅) > 0. 2|Ωi |

Consider another modification of the utility function, u˜i (Ψi ) = uî (Ψi ) − δ|Ψi |. It is clear that u˜i satisfies full substitutability: let Vî be the associated indirect utility function for Uî . Then Vî is submodular by Theorem 1, so Vî (p ∧ p0 ) + Vî (p ∨ p0 ) ≤ Vî (p) + Vî (p0 ) and since this is true for every price vector p and p0 , it is also true for p˜ and p˜0 , where p˜ω = pω + δ if b (ω) = i, p˜ω = pω − δ if s (ω) = i, and p˜ω = pω otherwise, with analogous definitions for p˜0 . Hence Vî (˜ p ∧ p˜0 ) + Vî (˜ p ∨ p˜0 ) ≤ Vî (˜ p) + Vî (˜ p0 ) V˜i (p ∧ p0 ) + V˜i (p ∨ p0 ) ≤ V˜i (p) + V˜i (p0 ) and hence U˜i satisfies full substitutability by Theorem 1. P P Note that i u˜i (Ψi ) > i u˜i (∅), and while that does not necessarily imply that Ψi can be supported as a CE set of trades under utilities u˜, it does guarantee that the empty set cannot be supported. Take any competitive equilibrium [Φ, q] of this economy. We know that Φ 6= ∅. Moreover, since for every i, Φi is an optimal choice of i under utility U˜ and prices q (and in particular, no subset of Φi gives i a higher utility), it has to be the case under utility Uˆ and prices q, Φi gives i a strictly higher utility than any proper subset. But this, in turn, implies that for all i, under original utility U in the original economy with trades Ω, the set of trades {(ϕ, qϕ ) |ϕ ∈ Φi } belongs to every set in Ci (A ∪ {(ϕ, qϕ ) |ϕ ∈ Φi }). Thus, {(ϕ, qϕ ) |ϕ ∈ Φi } is a blocking

36

set for A, contradicting the assumption that A is stable.

Proof of Theorem 7 We need the following lemma for the proof. Lemma 4. Suppose the demand correspondence Di satisfies conditions 1 and 2 of Definition 4 for all price vectors p such that Di (p) is single-valued. Then Di satisfies conditions 1 and 2 of Definition 4 for all price vectors p. The main idea of the proof is that whenever demand is not single valued, we can always find “small” price variations to make demand single-valued (since we assume quasi-linearity) and then use the assumed (DFS) conditions on these single valued demand sets. Proof. Take two price vectors p and p0 such that for some ω ∈ Ωi→ , p0ω < pω and p0ξ = pξ for all ξ 6= ω. Fix some Ψ1 ∈ Di (p). We need to show that there exists some Ψ0 ∈ Di (p0 ) such that (Ψ \ {ω})i→ ⊆ Ψ0 and Ψ0→i ⊆ Ψ. As the statement is trivial when Di (p0 ) = {Ψ : Ψ ⊂ Ωi }, we assume the contrary. Furthermore, we assume that Di (p) 6= {Ψ : Ψ ⊂ Ωi } and the arguments below are easily extended to the case where this assumption is not satisfied. As in the proof of Lemma 2, given some price vector P P q ∈ R|Ω| , an agent i, and a set of trades Ξ ⊆ Ω, let qi (Ξ) ≡ ξ∈Ξi→ pξ − ξ∈Ξ→i pξ . Set 1 ≡ minΞ∈Ωi \Di (p) [vi (p) − (ui (Ξ) + pi (Ξ))] and 2 ≡ minΞ∈Ωi \Di (p0 ) [vi (p0 ) − (ui (Ξ) + p0i (Ξ))]. Let ≡ min{1 , 2 , pω − p0ω } and ˜ ≡ 2|Ω i | . Using these definitions, we define a price vector q 1 by    pξ − ˜, ξ ∈ Ωi→ \ Ψ,     p + ˜, ξ ∈ Ω \ Ψ, ξ →i 1 qξ =   pξ − ˜, ξ ∈ Ψ→i ,     p + ˜, ξ ∈ Ψ . ξ

i→

Clearly, we must have Di (q 1 ) = {Ψ}. Now define q 2 by qξ2 = qξ1 for all ξ 6= ω and qω2 = p0ω . We claim that Di (q 2 ) ⊆ Di (p0 ). To see this, fix an arbitrary Ψ ∈ Di (p0 ) and

37

an arbitrary Ξ ∈ {Ψ : Ψ ⊂ Ωi } \ Di (p0 ). Then we must have u(Ψ) + qi2 (Ψ) ≥ u(Ψ) + p0i (Ψ) − |Ψ|˜ > u(Ξ) + p0i (Ξ) + |Ξ|˜ ≥ u(Ξ) + qi2 (Ξ), where the first and third inequalities follow directly from the definitions of q 2 , and the second inequality follows from (|Ξ| − |Ψ|)˜ ≤ |Ωi |˜ < u(Ψ) + p0i (Ψ) − (u(Ξ) + p0i (Ξ). Now fix some Ψ2 ∈ Di (q 2 ). Similar to above, we define δ1 ≡ minΞ∈Ωi \Di (q2 ) [vi (p) − (ui (Ξ)+qi2 (Ξ))] and δ2 ≡ minΞ∈Ωi \Di (q1 ) [vi (q 1 )−(ui (Ξ)+qi1 (Ξ))]. Let δ ≡ min{δ1 , δ2 , pω − p0ω } and δ˜ ≡ 2|Ωδ i | . Using these definitions, we define a price vector q 3 by   ˜  qξ2 − δ,     q 2 + δ, ˜ ξ 3 qξ = q 2 − δ, ˜   ξ    q 2 + δ, ˜ ξ

ξ ∈ Ωi→ \ Ψ2 , ξ ∈ Ω→i \ Ψ2 , ξ ∈ Ψ2→i , ξ ∈ Ψ2i→ .

Clearly, we must have Di (q 3 ) = {Ψ2 }. Now define q 4 by qξ4 = qξ3 for all ξ 6= ω and qω4 = qω1 . Similar to above, we can show that Di (q 4 ) ⊆ Di (q 1 ), so that in particular Di (q 4 ) = {Ψ}. Since qω3 < qω4 and qξ3 = qξ4 for all ξ 6= ω, we can now apply the full substitutes condition for singleton demand sets to conclude that Ψ2 must be a set with the desired properties. Suppose that the preferences of i are not fully substitutable, and in particular fail part (i) of the definition of (DFS) with unique demands (Definition 4). Then there exist price vectors p and p0 and trades ω and ψ where b (ω) = i and p0−ω = p−ω , p0ω > pω and {Ψ} = Di (p) and {Ψ0 } = Di (p0 ) and either Case 1 b (ψ) = i and ψ ∈ Ψ but ψ ∈ / Ψ0 , or Case 2 s (ψ) = i and ψ ∈ / Ψ but ψ ∈ Ψ0 .

38

For every trade ξ ∈ Ωi − {ψ, ω}, if b(ξ) = i we let us(ξ) (Ξ − {ξ}) − us(ξ) (Ξ ∪ {ξ}) = pξ for all Ξ ⊆ Ω and similarly, if s (ξ) = i, then we let ub(ξ) (Ξ ∪ {ξ}) − ub(ξ) (Ξ − {ξ}) = pξ for all Ξ ⊆ Ω. It is clear that these preferences, restricted to Ωi − {ψ, ω}, are simple for every agent. Without further specification of agents’ preferences we can infer that whenever a stable outcome A exists, the outcome A¯ = (A − {(ξ, qξ ) : ξ ∈ [τ (Ai ) − {ψ, ω}] and (ξ, qξ ) ∈ A})∪{(ξ, pξ ) : ξ ∈ [τ (Ai ) − {ψ, ω}]} will also be stable. To see this, note that if (ξ, qξ ) ∈ A for some ξ 6= ψ, ω such that b(ξ) = i then qξ ≥ pξ due to the preferences defined above. If qξ > pξ then A˜ ≡ [A − (ξ, qξ )] ∪ {(ξ, pξ )} is also a stable match, as it is clearly individually rational ˜ it would also be a as A is individually rational, and if Z was a blocking set for A, blocking set for A. Similarly, if (ξ, qξ ) ∈ A, s (ξ) = i, and ξ 6= ψ, ω then qξ ≤ pξ and so [A − (ξ, qξ )] ∪ {(ξ, pξ )} is also a stable match. The above statement now follows by induction and we will use it repeatedly in our proof below. It will be helpful to define the marginal utility agent i obtains from having available trades in some set Φ ⊆ {ψ, ω} in addition to having trades in Ωi − {ψ, ω} at their prices according to the price vector p by ( vi (Φ) ≡

max

Ξ⊆Ωi −{ψ,ω} ˆ Φ⊆Φ

ui

X X ˆ Ξ∪Φ + pξ − pξ ξ∈Ξi→

ξ∈Ξ→i

We now proceed to discuss the two possible cases. Case 1 b (ψ) = i and ψ ∈ Ψ but ψ ∈ / Ψ0 : Note that vi ({ψ, ω}) − vi ({ω}) > vi ({ψ}) − vi (∅) ≥ 0

39

) .

as otherwise we would have that ψ ∈ Ψ0 , as if vi ({ψ, ω}) − vi ({ω}) ≤ vi ({ψ}) − vi (∅) then i must demand ψ at prices (p−ω , p0ω ) as i demanded ψ at prices p. ˆ ω Now let ψ, ˆ be two trades such that s (ψ) = s ψˆ , s (ω) = s (ˆ ω ), and b ψˆ = b (ˆ ω ) = j 6= i (such a trade must exist as there are at least four agents and the set of trades is exhaustive). Let s (ψ) , s (ω) have preferences such that n o us(ψ) (Ξ ∪ {ψ}) − us(ψ) (Ξ) = us(ψ) Ξ ∪ ψˆ − us(ψ) (Ξ) = 0 n o us(ψ) Ξ ∪ ψ, ψˆ = −∞ n o ˆ for all Ξ ⊆ Ω − ψ, ψ and us(ω) (Ξ ∪ {ω}) − us(ω) (Ξ) = us(ω) (Ξ ∪ {ˆ ω }) − us(ω) (Ξ) = 0 us(ω) (Ξ ∪ {ω, ω ˆ }) = −∞ for all Ξ ⊆ Ω − {ω, ω ˆ }. It is possible that s (ψ) = s (ω). Let j’s preferences satisfy n o 2 [vi ({ψ, ω}) − vi ({ω})] + [vi ({ψ}) − vi (∅)] ≡ t(ψ) ψˆ ∪ Ξ − uj (Ξ) = 3 2 [vi ({ψ, ω}) − vi ({ψ})] + [vi ({ω}) − vi (∅)] uj ({ˆ ω } ∪ Ξ) − uj (Ξ) = ≡ t(ω) 3 n o ˆ ω uj ψ, ˆ ∪ Ξ − uj (Ξ) = −∞ uj

n o ˆ for all Ξ ⊆ Ω − ψ, ω ˆ . Note that due to the above inequality we must have 0 < t(ψ) < vi ({ψ, ω}) − vi ({ω}) and 0 < t(ω) < vi ({ψ, ω}) − vi ({ψ}). Clearly, the preferences of all agents but i can be extended to simple preferences

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on Ω.27 There are four subcases to consider to show that A¯ can not be stable: ˆ and ω ¯ , then A¯ is not individually rational 1. τ A¯ ∩ {ψ, ω} = ∅: If both ψ ˆ ∈ τ A n o ˆ ω ˆ for j. If ψ, ˆ∈ / τ A¯ , then ψ, is a block. Hence, exactly one of ψˆ and ω ˆ ˆ q ˆ ∈ A¯ for some q ˆ ∈ R+ . Individual rationality for is in τ A¯ . Suppose ψ, ψ ψ j requires that qψˆ ≤ t(ψ) < vi ({ψ, ω}) − vi ({ω}). But then Z≡

o n ξ, pξ + Ib(ξ)=i − Is(ξ)=i : ξ ∈ Ψ − τ A¯i ∪ ψ, qψˆ + , (ω, )

is a blocking set for some small > 0. Note that (ω, ) strictly increases by the utility of s (ω), no matter what other contracts s (ω) chooses. Similarly, for all ξ ∈ Ψ − τ A¯ , ξ, pξ + Ib(ξ)=i − Is(ξ)=i strictly increases by the utility of the agent other than i associated with what other contracts this contract, no matter that agent chooses. Agent s ψˆ will choose contract ψ, qψˆ + and not choose ˆ q ˆ , regardless of other contracts he chooses. Finally, agent i’s choice A¯ ∪ Z ψ, ψ is single valued and includesZ, as the above inequality implies that if i chooses (ω, ), he must also choose ψ, qψˆ + . We also have that vi ({ω}) ≥ vi (∅), implying that for small enough i will choose both ψ, qψˆ + and (ω, ) from A¯ ∪ Z, and hence i will choose all of the contracts associated with trades in Ψ as such contracts are optimal at prices p. If (ˆ ω , qωˆ ) ∈ A¯ for some qωˆ ∈ R+ , we obtain a similar contradiction since individual rationality for j requires that qωˆ ≤ t(ω) < vi ({ψ, ω}) − vi ({ψ}). 2. (ψ, qψ ) ∈ A¯ for some qψ ∈ R+ and ω ∈ / τ A¯ : In this case we must have (ˆ ω , qωˆ ) ∈ A¯ for some qωˆ ∈ R+ , as otherwise {(ˆ ω , )} for some small > 0 would be a blocking set since j’s incremental utility of signing ω ˆ is t(ω) > 0. Individual rationality for j requires qωˆ ≤ t(ω) < vi ({ψ, ω}) − vi ({ω}). 27

Throughout the proof we only specify the parts of the preferences that are important to show that no stable outcome can exist.

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Furthermore, we must have qψ ≤ vi ({ψ}) − vi (∅) as otherwise either A¯ is not individually rational for i or

ξ, pξ + Ib(ξ)=i − Is(ξ)=i : ξ ∈ Ψ0 − τ A¯

is a blocking set for > 0 sufficiently small. However, these inequalities imply that

ξ, pξ + Ib(ξ)=i − Is(ξ)=i : ξ ∈ Ψ − τ A¯ ∪ {(ω, pωˆ + )}

is a blocking set for some small > 0. 3. (ω, qω ) ∈ A¯ for some qω ∈ R+ and ψ ∈ / τ A¯ : The reasoning is symmetric to the previous subcase. 4. {(ψ, qψ ), (ω, qω )} ⊆ A¯ for some qψ , qω ∈ R≥0 : It must be the case that qψ + qω ≤ vi ({ψ, ω}) − vi (∅) as otherwise A¯ is not individually rational for i or

¯ ξ, pξ + Ib(ξ)=i − Is(ξ)=i : ξ ∈ Ψ0 − τ (A)

is a blocking set for some small > 0. In order to prevent a block by s(ψ) ˆ qψ + ) for some small > 0), we must have qψ ≥ t(ψ). and j (using (ψ, Similarly, to prevent a block by s(ω) and j, we must have qω ≥ t(ω). Simple algebra shows that t(ψ)+t(ω) > vi ({ψ, ω})−vi (∅) is equivalent to the inequality vi ({ψ, ω})+vi (∅) > vi ({ψ})+vi ({ω}), which holds as explained in the beginning of the proof of Case 1. Hence, we must have qψ + qω > vi ({ψ, ω}) − vi (∅), contradicting our earlier statement.

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Case 2 s (ψ) = i and ψ ∈ / Ψ but ψ ∈ Ψ0 : Note that vi ({ω}) − vi (∅) > vi ({ψ, ω}) − vi ({ψ}) as otherwise we would have ψ ∈ Ψ0 , as if vi ({ω}) − vi ({ψ, ω}) ≤ vi (∅) − vi ({ψ}) then i must demand to sell ψ at prices p if i demanded to sell ψ at prices (p−ω , p0ω ). Similar to the previous case, we use the following conventions to simplify notation 2 [vi ({ω}) − vi ({ψ, ω})] + [vi (∅) − vi ({ψ})] ≡ t(ψ) 3 and 2 [vi ({ω}) − vi (∅)] + [vi ({ψ, ω}) − vi ({ψ})] ≡ t(ω). 3 Note that due to the inequality above we must have 0 < t(ψ) < vi ({ω}) − vi ({ψ, ω}), and 0 < t(ω) < vi ({ω}) − vi (∅). We have to consider two cases according to whether s(ω) is equal to b(ψ) or not. 1. s (ω) 6= b (ψ): Consider the trade ω ˆ (which must exist by exhaustivity), where s (ˆ ω ) = s (ω) and b (ˆ ω ) = b (ψ) ≡ j. Let s (ω) have preferences such that us(ω) (Ξ ∪ {ω}) − us(ω) (Ξ) = us(ω) (Ξ ∪ {ˆ ω }) − us(ω) (Ξ) = 0 us(ω) (Ξ ∪ {ω, ω ˆ }) = −∞ for all Ξ ⊆ Ω − {ω, ω ˆ }.

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Now let j’s preferences satisfy uj ({ˆ ω } ∪ Ξ) − uj (Ξ) = t(ω) uj ({ψ} ∪ Ξ) − uj (Ξ) = t(ψ) uj ({ψ, ω ˆ } ∪ Ξ) − uj (Ξ) = −∞ for all Ξ ⊆ Ω − {ψ, ω ˆ }. As in the first case preferences of the above type can be extended to simple preferences over all sets of trades. We now show that no stable match can exist if preferences satisfy the above properties by distinguishing four cases. ¯ = ∅: In this case (a) {ψ, ω, ω ˆ } ∩ τ (A)

ξ, pξ + Ib(ξ)=i − Is(ξ)=i : ξ ∈ Ψ − τ A¯ − {ω} ∪ {(ω, t(ω))}

is a blocking set as it is increases the utility of each agent except i and s(ω) by at least , increases the utility of s(ω) by at least t(ω) > 0, and increases i’s utility for sufficiently small > 0, since t(ω) < vi ({ω})−vi (∅). (b) (ˆ ω , qωˆ ) ∈ A¯ for some qωˆ ∈ R≥0 : Given our assumptions about preferences, ¯ and qωˆ ≤ individual rationality (for s(ω) and j) requires that ψ, ω ∈ / τ (A) t(ω). Since t(ω) < vi ({ω}) − vi (∅), this implies that

ξ, pξ + Ib(ξ)=i − Is(ξ)=i : ξ ∈ Ψ − τ A¯ − {ω} ∪ {(ω, qωˆ + )}

¯ is a blocking set and this shows that we cannot have ω ˆ ∈ τ (A). ¯ In this case j obtains a utility (c) (ω, qω ) ∈ A¯ for some qω ∈ R+ and ψ ∈ / τ (A): of zero under A¯ and in order to prevent a block by s(ω) and j, we must have qω ≥ t(ω). But then

ξ, pξ + Ib(ξ)=i − Is(ξ)=i : ξ ∈ Ψ0 − τ A¯ − {ψ} ∪ {(ψ, t(ψ) − )}

is a blocking set for sufficiently small > 0. To see this note first that j will clearly accept all of his contracts in the blocking set, since each of these contracts increases his utility by > 0. Note that i’s utility after the

44

block is vi ({ψ}) + t(ψ) − M , where M is the order of the blocking set, whereas his utility before the block is at most vi ({ω}) − t(ω). Subtracting from the former the latter expression we obtain after some algebra [vi ({ω}) − vi (∅)] − [vi ({ω, ψ}) − vi ({ψ})] − M , 3 which is positive for > 0 sufficiently small. (d) {(ψ, qψ ), (ω, qω )} ⊆ A¯ for some qψ , qω ∈ R≥0 : We must have that qψ ≥ vi ({ω}) − vi ({ψ, ω}) , since otherwise either A¯ would not be individually rational for i, or

ξ, pξ + Ib(ξ)=i − Is(ξ)=i : ξ ∈ Ψ − τ A¯

would be a blocking set. Similarly, we must have qω ≤ vi ({ω, ψ}) − vi ({ψ}) . We claim that {(ˆ ω , qω + )} is a blocking set for > 0 sufficiently small. It will clearly be chosen by s (ω), and b (ψ) obtains a utility increase of at least [t(ω) − (qω + )] − [t(ψ) − qψ ] . Substituting and using the price inequalities we just derived, we find that this expression is greater than or equal to [vi ({ω}) − vi (∅)] − [vi ({ψ, ω}) − vi ({ψ})] − , which is positive for sufficiently small.

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¯ Then we must have (e) (ψ, qψ ) ∈ A¯ for some qψ ∈ R and ω ∈ / τ (A): qψ ≤ t(ψ) − t(ω) ≤ vi (∅) − vi ({ψ, ω}) ¯ But then for {(ˆ ω , )} to not be a blocking set for A. Z≡

ξ, pξ + Ib(ξ)=i − Is(ξ)=i : ξ ∈ Ψ − τ A¯ − {ω} ∪ {(ω, )}

is a blocking set for > 0 sufficiently small, as s (ω) will clearly accept this contract, and i’s utility before is at most vi ({ψ}) + vi (∅) − vi ({ψ, ω}) and choosing from A¯ ∪ Z, i obtains vi ({ω}) − M where M is the order of the blocking set. Subtracting the former expression from the latter, we obtain [vi ({ω}) − vi (∅)] − [vi ({ψ, ω}) − vi ({ψ})] − M which is greater than 0 for > 0 sufficiently small. 2. s (ω) = b (ψ) ≡ j: Let j’s preferences satisfy, for all Ξ ⊆ Ω − {ω, ψ}, uj ({ω} ∪ Ξ) − uj (Ξ) = −t(ω) uj ({ψ, ω} ∪ Ξ) − uj (Ξ) = t(ψ) − t(ω) uj ({ψ} ∪ Ξ) − uj (Ξ) = −∞ There are four subcases to consider to show that A¯ can not be stable: (a) τ A¯ ∩ {ψ, ω} = ∅: The argument from Case 2.(a) can be used to show that i and j = s(ω) have an incentive to deviate. ¯ (b) (ω, qω ) ∈ A¯ for some qω ∈ R≥0 : Suppose that ψ ∈ / τ (A). Individual 46

rationality for j requires that qω ≥ t(ω). The argument from Case 2.(c) can then be used to establish that

ξ, pξ + Ib(ξ)=i − Is(ξ)=i : ξ ∈ Ψ0 − τ A¯ − {ψ} ∪ {(ψ, t(ψ) − )}

is a blocking set for sufficiently small > 0. (c) {(ψ, qψ ), (ω, qω )} ⊆ A¯ for some qψ , qω ∈ R≥0 : We must have qψ ≥ vi ({ω}) − vi ({ψ, ω}) , as otherwise either A¯ would not be individually rational for i, or

ξ, pξ + Ib(ξ)=i − Is(ξ)=i : ξ ∈ Ψ − τ A¯

would be a blocking set. Similarly, we must have qω ≤ vi ({ω, ψ}) − vi ({ψ}) . The first inequality implies that A¯ cannot be individually rational for j since the incremental utility of signing ψ on top of ω is t(ψ) − qψ ≤ t(ψ) − [vi ({ω}) − vi ({ψ, ω})] < 0. (d) ψ ∈ τ A¯ and ω ∈ / τ A¯ : This can clearly not be individually rational for j, given that he obtains −∞ utility if he signs ψ but not ω no matter which other trades he signs. The case where the preferences of i do not satisfy part (ii) of Definition 4 of substitutability is analogous.

Proof of Corollary 1 Consider the preferences given in the proof of the theorem above. Suppose [Ψ; p] is a competitive equilibrium. Then, by Theorem 5, λ ([Ψ; p]) is a stable outcome, and yet we know that for these preferences no stable outcome exists.

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Counterexample for Section 5 Let I = {i, j}, Ω = {χ, ψ, ω}, and s (χ) = s (ψ) = s (ω) = i and b (χ) = b (ψ) = b (ω) = j. Furthermore, let the utility functions be given by Ψ ui (Ψ) uj (Ψ) ∅ 0 0 0 2 {χ} {ψ} −2 1 {ω} −2 1 −2 3 {χ, ψ} {χ, ω} −2 3 −9 2 {ψ, ω} {χ, ψ, ω} −20 15 In this case, any outcome of the form {(χ, pχ )} such that 0 ≤ pχ ≤ 2 is both stable and in the core. However, it is not strongly group stable, as {(ψ, 6) , (ω, 6)} constitutes a block.

Proof of Theorem 9 It is clear that every strongly stable outcome is stable. Consider an arbitrary stable outcome A and suppose to the contrary that A is not strongly stable. Take a set Z that strongly blocks A. Note that the second part of the proof of Theorem 5 actually shows that in this case there cannot be a vector of prices p such that [τ (A) ; p] is a competitive equilibrium. This contradicts Theorem 6, completing the proof.

Proof of Theorem 10 We first define a version of the Laws of Aggregate Demand and Supply for our setting.28 Definition 14. Preferences of i satisfy the Law of Aggregate Demand if for all ˆ ∈ Ci (Yˆ ) Y ⊆ X and Yˆ ⊆ Y such that Yî→ = Yî→ , for any W ∈ Ci (Y ) there exists W 28

Hatfield and Kominers (2010b) introduced this notion for the trading networks setting where nodes can be both buyers and sellers, generalizing the notion of the law of aggregate demand introduced by Hatfield and Milgrom (2005).

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such that ˆ →i | ≥ |Wi→ | − |W ˆ i→ |. |W→i | − |W Preferences of i satisfy the Law of Aggregate Supply if for all Y ⊆ X and Yˆ ⊆ Y ˆ ∈ Ci (Yˆ ) such that such that Yˆ→i = Yˆ→i , for any W ∈ Ci (Y ) there exists W ˆ i→ | ≥ |W→i | − |W ˆ →i |. |Wi→ | − |W Theorem 7 of Hatfield and Milgrom (2005) shows that any substitutable preferences in the quasilinear two-sided matching setting satisfy the Law of Aggregate Demand. The extension of that theorem, that fully substitutable preferences in our setting satisfy the law of aggregate supply and demand is a straightforward generalization. Lemma 5. If the preferences of agent i are fully substitutable, then they satisfy the Laws of Aggregate Demand and Supply.29 Consider the modified economy obtained in the proof of Theorem 2. The proof then follows as in that of Theorem 7 of Hatfield and Milgrom (2005). We now prove the following result, which implies Theorem 10 by a natural inductive argument: Lemma 6. For any feasible outcome A blocked by a nonempty set Z, if Z is not itself a chain, then there exists a nonempty chain W ⊆ Z such that set A is blocked by Z − W and set A ∪ (Z − W ) is blocked by W . Note that the second part of the statement, that A ∪ (Z − W ) is blocked by W , is true for any subset W of Z: Set Z blocks A and thus for every agent i involved in W (and thus in Z), every choice of i from A ∪ Z = (A ∪ (Z − W )) ∪ W contains every contract in Zi (and thus in Wi ). So we only need to prove the first part, that we can “remove” some chain W from Z so that the remaining set Z − W still blocks A. Consider any contract y ∈ Z. We will algorithmically “grow” a chain W `s ,`b ≡ {y `s , . . . , y `b } by applying (generally in both directions from y 0 ) the following analysis, starting with W 0,0 = {y 0 } where y 0 = y. 29

Note that the quasilinearity of the agent’s preferences is essential for this result.

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Our chain-growing algorithm has two general steps: Buyer Step: Let j = b(y `b ). If every Y ∈ Cj A ∪ Z − W `s ,`b contains (Z − W `s ,`b )j as a subset, stop. Otherwise, note that by full substitutability, each such Y must contain (Z − W `s ,`b )→j as a subset. Also, by the Laws of Aggregate Supply and Demand, each such Y excludes at most one contract in (Z − W `s ,`b )j→ . Pick any such optimal choice that excludes exactly one contract, and denote that excluded contract by y `b +1 . Note that by construction, every optimal choice of j from A ∪ (Z − W `s ,`b +1 ) contains (Z − W `s ,`b +1 )j . Seller Step: Let k = s y `s . Consider its optimal choices from A ∪ Z − W `s ,`b . If all of them contain all contracts in (Z − W `s ,`b )k , stop. Otherwise, by analogy with the previous case, pick contract y `s −1 . So long as s y `s 6= b(y `b ), we sequentially iterate these two steps on the input W `s ,`b , decrementing `s and incrementing `b with each iteration. Over the course of this process, it may happen that s y `s = b(y `b ) ≡ h. If ev ery Y ∈ Ch A ∪ Z − y `s , y `b contains Z − y `s , y `b j , we are done—the chain W `s ,`b suffices for the result. Otherwise, note that as before, by the Laws of Aggre excludes at most one gate Supply and Demand, each Y ∈ Ch A ∪ Z − y `s , y `b upstream contract and at most one downstream contract from (Z − y `s , y `b )h . In this case, we take any such excluded contract y, and suppose it is a downstream contract for agent h (the argument if it is an upstream contract for h is analogous). Let y `b +1 = y. Each Y ∈ Ch A ∪ Z − y `s , y `b , y `b +1 is, by construction, also in Ch A ∪ Z − y `s , y `b . Each such Y contains (Z − y `s , y `b , y `b +1 )h→ and excludes at most one contract in (Z − y `s , y `b , y `b +1 )→h . If it so happens that each Y con tains (Z − y `s , y `b , y `b +1 )→h , then one end of the chain is complete; we continue with the Buyer Step on input W `s ,`b +1 . Otherwise, we select any excluded contract in (Z − y `s , y `b , y `b +1 ), take it to be y `s −1 , and continue as above. This algorithm must terminate as |Z| is finite. At every iteration, we ensured, by construction, that for each agent i all contracts in (Z − W `s ,`b )i were contained ˆ ˆ in every Y ∈ Ci A ∪ (Z − W `s ,`b ) . It follows that A is blocked by the Z − W `s ,`b . ˆ ˆ Since W `s ,`b is a chain, we are done.

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Proof of Theorem 11 (a) We show first that Ξi ∈ Di (q). In the following let ξ ∈ Ψ \ {ω} be any trade such that pξ = p. Note that |Ξi ∩ Ψ| ∈ {0, 1} due to mutual incompatibility, and if ω ∈ Ξi ∩ Ψ then pω = p: Otherwise, perfect substitutability would imply Ui ([Ξ − {ω} ∪ {ξ}; p]) > Ui ([Ξ; p]), contradicting the assumption that [Ξ; p] is a competitive equilibrium. This implies that prices for trades in Ξi have not been changed in going from p to q and in particular implies that Ui ([Ξi ; p]) = Ui ([Ξi ; q]). Since prices for trades in Ω − Ψ have also not been changed, we must have Ui ([Ξi ; q]) = Ui ([Ξi ; p]) ≥ Ui ([Φ; p]) = Ui ([Φ; q]) for all Φ ⊆ Ω − Ψ. Now take any set Φ ⊆ Ω − Ψ and any ω ∈ Ψ. By perfect substitutability, Ui ([Φ ∪ {ω}; q]) = Ui ([Φ ∪ {ξ}; q]) = Ui ([Φ ∪ {ξ}; p]) ≤ Ui ([Ξi ; p]) = Ui ([Ξi ; q]). Hence, Ξi ∈ Di (q). Now consider an arbitrary agent j 6= i. If Ξj ∩ Ψ = {ω}, we must have pω = p, implying Ui ([Ξj ; q]) = Uj ([Ξj ; p]). If Ξj ∩ Ψ = ∅ the last statement is evidently true as well. Let Φ ⊆ Ωj be arbitrary and note that Uj ([Φ; q]) ≤ Ui ([Φ; p]) since trades in Φ ∩ Ψ→j have become weakly more expensive. Since Uj ([Ξj ; q]) = Uj ([Ξj ; p]) ≥ Uj ([Φ; p]), we obtain Ξj ∈ Dj (q). This completes the proof. (b) Analogous to part (a).

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