The structure of equilibria in trading networks with frictions

Schlegel, Jan Christoph Article The structure of equilibria in trading networks with frictions Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Schlegel, Jan Christoph (2022) : The structure of equilibria in trading networks with frictions, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 17, Iss. 2, pp. 801-839, https://doi.org/10.3982/TE4405 This Version is available at: https://hdl.handle.net/10419/296371 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Theoretical Economics 17 (2022), 801–839 1555-7561/20220801 The structure of equilibria in trading networks with frictions Jan Christoph Schlegel Department of Economics, City, University of London Several structural results for the set of competitive equilibria in trading networks with frictions are established: The lattice theorem, the rural hospitals theorem, the existence of side-optimal equilibria, and a group-incentive-compatibility result hold with imperfectly transferable utility and in the presence of frictions. While our results are developed in a trading network model, they also imply analogous (and new) results for exchange economies with combinatorial demand and for two-sided matching markets with transfers. Keywords. Trading networks, full substitutability, imperfectly transferable utility, competitive equilibrium, indivisible goods, frictions, lattice, rural hospitals. JEL classification. C78, D47, D52, L14. 1. Introduction The assumption of transferable utility is pervasive in models of matching markets, exchange economies with indivisible goods, trading networks, and in mechanism design. The transferable utility assumption can simplify the analysis considerably since it allows us to exploit the duality between optimal allocations and supporting equilibrium prices. While the assumption of transferable utility simplifies the analysis, it is often empirically problematic. Wealth effects are present in marriage and labor markets so that matching models with transferable utility are unrealistic for these applications. Even if wealth effects are absent, transaction frictions such as those induced by taxation, subsidies, or transaction costs, make a transferable utility model inapplicable. This has motivated researchers to explore how results for matching markets with transfers (Demange and Gale (1985), Legros and Newman (2007), Nöldeke and Samuelson (2018), Galichon et al. (2019)) and for trading networks (Fleiner et al. (2019), Hatfield et al. (2021)) can be generalized beyond transferable utility. Jan Christoph Schlegel: [email protected] An extended abstract under the previous title “Trading networks with general preferences” appeared in the Proceedings of the 20th ACM Conference on Economics and Computation (EC’19). The paper extends and supersedes Schlegel (2018), which proves similar results in the more restrictive model of job matching with salaries. I gratefully acknowledge financial support by the Swiss National Science Foundation (SNSF) under project 100018-150086. I thank Ravi Jagadeesan, Bettina Klaus, Alex Nichifor, Alex Teytelboym, Ning Yu and Klaus Zauner, three anonymous referees, seminar participants in Bristol, Lausanne and Oxford, participants of the 2018 Lisbon Game Theory Meetings, the Matching in Practice workshop in Mannheim, the 5th Match-Up workshop, the 20th ACM Conference on Economics and Computation (EC19), the 2019 North American Summer Meeting of the Econometric Society and the International Conference on Game Theory and the 6th Microeconomics Workshop in Nanjing for valuable comments. ©2022 The Author. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE4405 802 Jan Christoph Schlegel Theoretical Economics 17 (2022) In this paper, we contribute to this discussion and study wealth effects and frictions1 in the context of trading networks (Hatfield et al. (2013)). Trading networks with bilateral contracts model complex supply chains in an industry where firms are engaged in upstream as well as downstream contracts. They generalize two-sided matching markets, in the sense that they replace a bipartite graph of potential relations, by an arbitrary graph where each edge represents a potential trade. We show that important structural results for trading networks do not depend on the assumption of transferable utility, and establish several results about the set of competitive equilibria under minimal assumptions on utility functions. Our results apply even in the case of wealth effects, in the presence of frictions, and if constraints make the execution of certain combinations of trades infeasible. Our results can be summarized as follows: For a model of trading networks with frictions (Fleiner et al. (2019)) and under the assumptions of full substitutability (Sun and Yang (2006), Ostrovsky (2008), Hatfield et al. (2013)) and the laws of aggregate demand and supply (Hatfield et al. (2012,2021)), we show that – the set of competitive equilibria is a sublattice of the price space (first part of Theorem 1), – a generalized “rural hospitals theorem” holds: the difference between the number of signed downstream and the number of signed upstream contracts is the same for each firm in each equilibrium (second part of Theorem 1), – assuming additionally “bounded willingness to pay” (Fleiner et al.,2019),2there is an equilibrium that is most preferred by terminal sellers and an equilibrium that is most preferred by terminal buyers (Theorem 2), – a mechanism that selects buyer-optimal equilibria is group-strategy-proof for terminal buyers on the domain of unit-demand utility functions and similarly a mechanism that selects seller-optimal equilibria is group-strategy-proof for terminal sellers on the domain of unit-supply utility functions (Theorem 3). While our results are established for trading networks, the results are already new for many-to-one matching markets and for exchange economies with combinatorial demand, which are special cases of our model. For matching markets, similar results were so far only known (a) under transferable utility, (b) for models without transfers and with strict preferences, or (c) for one-to-one matching markets with imperfectly transferable utility. For exchange economies with indivisible goods, analogous results were so far only known for (a) quasilinear utility, or for the case of (b) unit demand. 1We use the term “frictions” for any situation where utility is not necessarily a function of the sum of transfers received, but a function of the entire vector of transfers. If frictions are present, it thus not only matters how much a firm receives in total transfers, but also through which trades it receives the transfers. This can, for example, be the case if different trades involve different transaction costs. The results in our paper apply to both wealth effects and frictions. 2Alternatively, the result also holds with “bounding compensating variations” instead of bounded willingness to pay, as we show in the full working paper version (Schlegel (2020)). Theoretical Economics 17 (2022) The structure of equilibria in trading networks 803 Working with imperfectly-transferable utility requires us to develop fundamentally new techniques: Similar results without the transferable utility assumption were so far only known for one-to-one matching markets (Demange and Gale (1985)) and the proof techniques developed in this context (in particular the “decomposition lemma”) do not adapt to more general settings. On the other hand, techniques from transferable utility models do not generalize to our model: Hatfield et al. (2013) use the efficiency of competitive equilibria and the submodularity of the indirect utility function to establish the lattice property. For our model with frictions, competitive equilibria can fail to be efficient. Moreover, full substitutability implies only the weaker notion of quasisubmodularity (Hatfield et al. (2020)). More subtly, as we will discuss below, with wealth effects or frictions there are several nonequivalent definitions of full substitutability and these definitions are not distinguishable through conditions on the indirect utility function alone. Thus, an approach as in Hatfield et al. (2013) that characterizes competitive equilibria through the indirect utility function and uses properties of that function under full substitutability does not generalize. Likewise, the network flow approach to trading networks (Candogan et al. (2021)) obtains structural results on the set of equilibria through the duality between optimal allocations and supporting prices. Since efficiency fails in our setting this duality approach does not generalize. Finally, techniques from trading networks without transfers (Ostrovsky (2008)) that rely on Tarski’s fixedpoint theorem do not apply to the our model. With transfers, the issue of tie-breaking arises that is not present in models without transfers and with strict preferences. While versions of Tarski’s fixed-point theorem for correspondences are known (Zhou (1994)), none of them work under sufficiently weak assumptions to be useful in our environment. Since existing techniques do not work for our setting, we introduce a new approach to establish structural results for the set of competitive equilibria. The approach can be characterized as a tie-breaking approach: we show that for each finite set of (equilibrium) price vectors and each firm a single-valued selection from the demand correspondence can be made such that the properties of full substitutability and the laws of aggregate demand and supply are satisfied by the selection, and moreover, relevant trades that are demanded in the supporting equilibrium allocations are demanded in the selection. The assumption of the laws of aggregate demand and supply is necessary for our tie-breaking argument, and our result does not hold under full substitutability alone (see Example 3). We also make a more technical contributions to the literature on trading networks with imperfectly transferable utility and clarify issues related to the definition of full substitutability: For the transferable utility model, there are various equivalent definitions of full substitutability (Hatfield et al. (2019)). The equivalence, however, breaks down if we go beyond transferable utility, and, for our results, it matters which of the full substitutability notions is used. More specifically, it matters how full substitutability restricts the demand at price vectors at which the demand is multivalued. We consider weak notions of full substitutability and the laws of aggregate demand and supply that only restrict the demand at price vectors where the demand is single-valued and stronger versions that also restrict it at prices where the demand is multivalued. The notions are 804 Jan Christoph Schlegel Theoretical Economics 17 (2022) equivalent for transferable utility, but not in general. The set of competitive equilibria is a lattice only under the strong versions of full substitutability (see Example 1)andthe rural hospitals theorem requires the strong versions of the law of aggregate demand and supply (see Example 2). Our group-strategy-proofness result, however, holds also under the weaker notions (Corollary 1). Thus, the exact definition of full substitutability matters in the model with frictions.3 We proceed as follows: In Section 2, we introduce the model and discuss different versions of the full substitutability conditions and their relation to each other. In Section 3, we prove our main results: the lattice structure of the set of competitive equilibria, the generalized rural hospitals theorem, the existence of extremal equilibria, and groupincentive compatibility for terminal buyers. In Section 4, we apply our main results to two-sided matching markets, and to exchange economies with indivisible goods. 1.1 Related literature The literature on trading networks has its origins in the literature on matching markets with transfers. In a seminal paper, Kelso and Crawford (1982) show that, under the assumption of gross substitutability, competitive equilibria with personalized prices exist and are equivalent to core allocations in a many-to-one labor market matching model. The construction is by an approximation argument where the existence in the continuum is obtained from the existence of an equilibrium in a discrete markets with smaller and smaller price increments. Different versions of a strategy-proofness result for a many-to-one matching model with continuous transfers were established by Hatfield et al. (2014), Schlegel (2018), Jagadeesan et al. (2018). Subsequent to Kelso and Crawford (1982), the question of existence of equilibria has been studied in the context of exchange economies with indivisibilities. See, for example, Gul and Stacchetti (1999) and the recent contribution of Baldwin and Klemperer (2019). Trading networks with bilateral contracts and continuous transfers were introduced by Hatfield et al. (2013). Under the assumption of transferable utility and full substitutability, they establish many results that we generalize to the case of general utility functions. The notion of full substitutability has been studied in detail by Hatfield et al. (2019) who show the equivalence of various different definitions of full substitutability. The existence result of Hatfield et al. (2013) is proved via a reduction to the existence result of Kelso and Crawford (1982). An alternative approach is via a submodular version of a network flow problem (Candogan et al. (2021)). The work of Hatfield et al. (2013) builds on the work of Ostrovsky (2008)ontrading networks without transfers that generalizes matching models with contracts (Hatfield et al. (2005), Fleiner (2003), Roth (1984)) beyond two-sided markets. The matching model with contracts in turn originates in the discrete version of the model of Kelso and Crawford (1982). Hatfield et al. (2012)andFleiner et al. (2016) provide additional results for the discrete trading networks model, which in many ways are parallel to the 3Related issues occur in Hatfield et al. (2021) where the stronger monotone substitutability property is needed that restricts the choice in circumstances where the choice correspondence is multivalued. Theoretical Economics 17 (2022) The structure of equilibria in trading networks 805 results we obtain in the continuous model. Importantly, results for the model without continuous transfers rely on the assumption of strict preferences. All of the above mentioned work for the continuous models make the assumption of transferable utility.4There are few papers that deal with wealth effects, frictions, or constraints and that are particularly close to our work: In a classical paper, Demange and Gale (1985) establish several structural results about the core (or equivalently the set of competitive equilibria) for a one-to-one matching model with continuous transfers. In particular, they show that the core has a lattice structure and an agent that is unmatched in one core allocation receives his reservation utility in each core allocation (the result is often called the rural hospitals theorem in the literature on discrete matching markets). Moreover, they show that the mechanism that selects an extreme point of the bounded lattice is strategy-proof for one side of the market. Importantly, these results are established without assuming transferable utility. They only require that utility is increasing, continuous in transfers and satisfies a full range assumption. We generalize this work to trading networks and to situations in which utility does not satisfy the full range assumption. In recent work, Fleiner et al. (2019) study trading networks with frictions. Their work is in many regards complementary to our work. In particular, Fleiner et al. (2019)establish the existence of a competitive equilibrium under the assumption of full substitutability and mild regularity conditions. Moreover, they study the efficiency of competitive equilibria and provide conditions under which equilibria correspond to allocations satisfying different related cooperative solution concepts. We derive our results for competitive equilibria. However, by the equivalence result of Fleiner et al. (2019) analogous results also would hold for “trail-stable” allocations. All results of Hatfield et al. (2013), except for the maximal domain result (Theorem 7) are generalized to the model with frictions, either in our work or by Fleiner et al. (2019). Table 1summarizes results for trading networks with frictions. Kojima et al. (2020b) introduce constraints in the job matching model of Kelso and Crawford (1982) and characterize constraints that leave the gross substitutes condition invariant. The model with constraints is a special case of the model in the current paper so that we obtain as a corollary of our results a version of a lattice and of the rural hospital theorems for their model of job matching under constraints. In a spin-off paper, Kojima et al. (2020a) study comparative statics for their model and also prove versions of the lattice result and the rural hospitals theorem. These results have been obtained independently and contemporaneously with the results in the current paper.5 2. Model The model follows Hatfield et al. (2013), and the extensions of Fleiner et al. (2019) and Hatfield et al. (2021). We consider a finite set of firms Fand a finite set of trades .Eachtradeω∈is associated with a buyer b(ω)∈Fand a seller s(ω)∈Fwith 4Note however that the existence proof of Kelso and Crawford (1982) is actually more general and applies as long as preferences are continuous, monotonic, and unbounded in transfers for each bundle. 5Weaker versions of these results were obtained prior to that in Schlegel (2018). 806 Jan Christoph Schlegel Theoretical Economics 17 (2022) Table 1. Sufficient conditions for results for trading networks with frictions. Result (Theorem*) Source FS LADS BCV BWP NF Existence of Equil. (1) Fleiner et al. (2019)x x 1st Welfare Theorem (2) Fleiner et al. (2019)x Rural Hospitals (3) Theorem 1(ii) x x Lattice (4) Theorem 1(i) x x Side Optimality (4) Theorem 2xx x Equil. ⇒Stable (5) Fleiner et al. (2019)x Stable ⇒Equil. (6) Fleiner et al. (2019)x x Stable ⇔Group-Stable (8,9) Fleiner et al. (2019)x x x Trail-Stable ⇔Equil. Fleiner et al. (2019)x x Chain-Stable ⇔Stable Hatfield et al. (2021)x x Group-Strategy-Proofness Theorem 3xx x Note:Theorem* Corresponding theorem in Hatfield et al. (2013) under transferable utility. The existence of a side-optimal equilibrium additionally assumes finite valuations. FS stands for Full Substitutability,LADS stands for the Laws of Aggregate Demand and Supply,BCV stands for Bounded Compensating Variations,BWP stands for Bounded Willingness to Pay,andNF stands for No Frictions. b(ω)= s(ω). For a set of trades ⊆and firm f∈F, we define the set of downstream trades for fby f→:={ω∈:s(ω)=f}and the set of upstream trades by →f:={ω∈:b(x)=f}.Moreover,weletf:=f→∪→f.Afirmf∈Fsuch that f→=∅is called a terminal buyer andafirmsuchthat→f=∅is called a terminal buyer. Note that terminal buyers and/or terminal buyers do not need to exist. A contract is a pair (ω,pω)∈×R,wherepωis the price attached to the trade ω. An allocation is a pair (,p)consisting of a set of trades ⊆and a price vector p∈ R. We denote the set of allocations by Aand we let Af:={(f,(pω)ω∈f):(,p)∈A}. An arrangement is a pair [,p]∈2×R. In contrast to an allocation, the price vector also contains prices for unrealized trades. Each firm has a utility function uf:Af→R∪{−∞}. For notational convenience, we extend ufto 2×Rby defining for ⊆and p∈R, the utility uf(,p):= uf(f,(pω)ω∈f). We allow the utility function to take on a value of −∞ in which case the combination of trades is infeasible for the firm.6We require that •if a bundle is infeasible under some prices, then it is infeasible under all prices: if uf(,p)=−∞for p∈Rfthen uf(,p)=−∞for each p∈Rf, •at least one bundle of trades is feasible: there is a ⊆fsuch that uf(,·)>−∞. Moreover, we make the following assumptions on utility functions: •Continuity:For⊆fwith uf(,·)>−∞, the function uf(,·)is continuous on R. 6Infeasibilities can, for example, arise through technological constraints, if producing and selling an output good requires the firm to buy certain input goods. In that case, executing a downstream alone without executing related upstream trades is infeasible. Alternatively, infeasibilities can also arise through institutional constraints that restrict, for example, such as in Kojima et al. (2020b), the number of trades that a firm is allowed to execute. Theoretical Economics 17 (2022) The structure of equilibria in trading networks 807 •Monotonicity:For⊆fwith uf(,·)>−∞ and p,p∈Rwith p= p: (i) If p ω=pωfor ω∈f→and pω≤p ωfor ω∈→f, then uf(,p)>u f(,p). (ii) If p ω=pωfor ω∈→fand pω≥p ωfor ω∈f→, then uf(,p)>u f(,p). Thus, utility is continuous in prices and firms strictly prefer higher sell prices to lower sell prices and lower buy prices to higher buy prices. We allow utility for a set of trades to be different for prices p,p∈R,evenif the transfers received are the same for both price vectors, that is, even if ω∈f→pω− ω∈→fpω=ω∈f→p ω−ω∈→fp ω. This can, for example, be the case if different trades involve different transaction costs. If utility only depends on the set of trades and the transfers received, we have a special case of our model: We say that ufsatisfies no frictions (Fleiner et al. (2019)) if there is a function ˜ uf:2 f×R→R∪{−∞}such that uf(,p)=˜ uf(, ω∈f→ pω− ω∈→f pω). A utility function without frictions has full range if for each ⊆f,= ∅,˜ uf(,·)is a surjective function onto R.Itisquasilinear if there is a valuation function vf:2 f→ R∪{−∞}such that ˜ uf(,t)=vf()+t. A utility function ufinduces an indirect utility function vf:R→Rby vf(p):=max ⊆f uf(,p), and a demand correspondence Df:R⇒2by Df(p):=argmax ⊆f uf(,p). Continuity of the utility function implies (e.g., by Berge’s maximum theorem) that the demand correspondence is upper hemicontinuous. Monotonicity of the utility function implies that price vectors where the demand is single-valued are dense in price space. We will repeatedly use these facts to generate a single-valued selection from the demand-correspondence that inherits its good properties (such as full substitutability or the laws of aggregate demand and supply) by perturbing the price vector such that it becomes single-valued; see, in particular, Lemma 3. The proof is straightforward, and hence omitted. Lemma 1. For a continuous and monotonic utility function uf: (i) the induced demand Dfis upper hemicontinuous, that is, for each p∈Rfthere is an >0such for any q∈Rfwith p−q<(where ·denotes the Euclidean norm) we have Df(q)⊆Df(p), 808 Jan Christoph Schlegel Theoretical Economics 17 (2022) (ii) the set of price vectors such that the induced demand is single-valued is dense in Rf, that is, for each >0and p∈Rfthere is a q∈Rfwith p−q<such that |Df(q)|=1. 2.1 Full substitutability Our results rely on a full substitutability assumption on utility functions. Informally, the condition requires that a firm sees upstream (downstream) trades as substitutes to each other, and upstream and downstream trades as complements to each other. Hatfield et al. (2019) show that for transferable utility various ways of defining full substitutability are equivalent, and hence one can work with either of the definitions discussed in their paper. Going beyond transferable utility makes issues more subtle: Not all equivalence results of Hatfield et al. (2019) generalize and it matters which of the full substitutability conditions are used. More specifically, it matters how the full substitutes condition is defined in instances where indifferences matter, that is, when the demand is multivalued. We will proceed as follows: First, we introduce our main definition of full substitutability which restrict the demand both at price vectors where the demand is singlevalued and where it is multivalued. Second, we introduce a weaker version of full substitutability that only restricts the demand at price vectors where the demand is singlevalued. We provide an example that shows that the single-valued version of full substitutability is strictly weaker than the multivalued version. We later show, using this example, that the single-valued full substitutability condition is not sufficient for establishing the lattice and the rural hospitals theorem. Importantly, the difference between the single-valued and multivalued version of full substitutability only matters for the “cross-side conditions” on firms’ demand functions. In particular, the notions are equivalent for a two-sided market and the results for two-sided markets (see Section 4.1)hold under the single-valued notion of full substitutability. Third, we show that the multivalued and the single-valued versions are, however, closely related in the sense that for each demand correspondence satisfying single-valued full substitutability, a selection from the demand correspondence exists that satisfies multivalued full substitutability and can be rationalized by a utility function inducing the same indirect utility. In particular, this will allow us, later on (Corollary 1), to obtain a group-strategy-proofness result using the single-valued full substitutability notion. 2.1.1 Multivalued full substitutability The following notion of full substitutability is due to Hatfield et al. (2019).7Precursors of the full substitutability notion were introduced for exchange economies (Sun and Yang (2006)) and for trading networks without 7Hatfield et al.,2019 call this version of full substitutability the “demand-language expansion” version of full substitutability (cf. Definition A.3 in Hatfield et al.,2019). Throughout the paper, we use “demand language” definitions of full substitutability that restrict the demand correspondence induced by the utility function. The demand language definitions are generally weaker than the corresponding “choice language” definitions that restrict the choice correspondence induced by the utility function. Consequently, all of our result would also hold under the corresponding “choice language” notions of full substitutability. Alternative multivalued definitions of full substitutability are discussed in the full working paper version (Schlegel (2020)). Theoretical Economics 17 (2022) The structure of equilibria in trading networks 815 demanded at the perturbed prices support an equilibrium,8since not every collection of demanded trades at an equilibrium price vector support these prices as an equilibrium. While a naive perturbation argument fails to work, we can use a more intricate perturbation argument. We perturb prices for each firm individually. Importantly, we can rely on the observation (Lemma 2) that for each firm fthere are prices q(in general different for different firms) close to pwhere the equilibrium set of trades fis the unique demanded bundle of trades. This allows to show that for each firm fthere is a ¯ f∈Df(¯ p)that satisfies (1),(2)and (3)simultaneously (and a f∈Df(p)that satisfies dual properties for the pairwise minimum p). This is the content of the following lemma that is a main ingredient in the proof of the theorem. Lemma 4. Let ufbe a utility function inducing a demand correspondence Dfsatisfying FS,LAD,andLAS.Letp,p∈Rf, and define ¯ p,p∈Rfby pω:=maxpω,p ω,pω:=minpω,p ω. Let ∈Df(p)and ∈Df(p). (i) There is a ¯ ∈Df(¯ p)with ω∈→f:pω≥p ω∪ω∈ →f:p ω>p ω⊆¯ →f, ¯ f→⊆ω∈f→:pω≥p ω∪ω∈ f→:p ω>p ω. (ii) There is a ∈Df(p)with →f⊆ω∈→f:p ω≥pω∪ω∈ →f:pω>p  ω, ω∈f→:p ω≥pω∪ω∈ f→:pω>p  ω⊆f→. (iii) ¯ and can be chosen such that |→f|−|f→|≥|→f|−|f→|≥|¯ →f|−|¯ f→|. With the lemma, the proof of the theorem can be carried out as described before. The lemma and the first part of the theorem fail to hold if we replace FS by weak FS, as the following example shows. Example 1 (cont.). Consider the set of trades ={α1,α2,β1,β2}and firm fwith the utility function ufas defined in Example 1. The induced demand Dfsatisfies weak FS as previously shown. Moreover, for each p∈Rfand ∈Df(p)we have |f→|=|→f|. Thus, Dfsatisfies LAD and LAS. Consider four additional firms s1,s2,b1,b2with s1= 8This is related to the observation that the set of competitive equilibrium price vectors for our model can fail to be connected. See the example in Roth and Sotomayor (1988) for the case of one-to-one matching with transfers which is a special case of our model. In contrast to this, for the transferable utility case it is easy to show that the set of competitive equilibrium price vectors is convex, and thus, in particular, connected. 816 Jan Christoph Schlegel Theoretical Economics 17 (2022) s(α1),s2=s(α2),b1=b(β1),andb2=b(β2). Define utility functions for the additional firms as follows: For i=1, 2, define usiαi,pαi=pαi, ubiβi,pβi=2−pβi, usi(∅)=ubi(∅)=0. Observe that the equilibria for uare [,(1, 1, 1, 1)] and [{αi,βj},(0, 0, 2, 2)] for i,j= 1, 2. In particular, the vector (1, 1, 2, 2)is not an equilibrium price vector, since Ds1(1, 1, 2, 2)={{α1}} and Ds2(1, 1, 2, 2)={{α2}} but Df(1, 1, 2, 2)={{α1,β1},{α1,β2}, {α2,β1},{α2,β2}}.♦ Similarly, the second part of the theorem fails if LAD (LAS) is replaced by weak LAD (weak LAS) as the following example shows. Example 2 (cont.). Consider the set of trades ={ω1,ω2}and firm fwith the utility function ufas defined in Example 2. As observed before, Dfsatisfies FS, weak LAD, (and, trivially, LAS), but not LAD. Consider a second firm fwith f=s(ω1)=s(ω2)with utility function ufdefined by uf{ωi},pωi=pωi,fori=1, 2, uf{ω1,ω2},p=pω1+pω2−1.5, uf(∅)=0. The induced demand satisfies FS and LAS (and, trivially, LAD). The set of equilibrium vectors is E(u)={p:1≤pω1=pω2≤1.5}∪{(2, 2)}.Eachp∈E(u)\{(2, 2)}is supported by {ω1}and by {ω2}. The equilibrium prices (2, 2)are supported by {ω1,ω2}.Ananalogous example can be constructed to show that LAS and not just weak LAS is necessary for the rural hospitals theorem. ♦ It is well known that the theorem fails to hold without FS, even for transferable utility. The following example shows that the first part of the theorem fails without LAD. More generally, the example shows that without LAD the set of equilibria can even fail to be a lattice with respect to the (weaker) partial ordering induced by terminal sellers’ preferences. The example relies on the previous logic highlighted in the discussion of Theorem 1: The FS condition can be used to show that there is no excess supply of trades at the pairwise maximum of two equilibrium price vectors. However, without the LAD there can still be strict excess demand of trades at the pairwise maximum (or at price vectors dominating it). Example 3. Let ={ω1,ω2,ω3}.Letb(ωi)=ffor i=1, 2, 3, and s(ωi)= s(ωj)for i= j.Weletus(ωi)(ωi,pωi)=pωi,us(ωi)(∅)=0fori=1, 2 and us(ω3)(ω3,pω3)=−∞, Theoretical Economics 17 (2022) The structure of equilibria in trading networks 817 us(ω3)(∅)=0. We define ufby uf(∅)=0, uf{ω1,ω2},(pω1,pω2)=2−pω1−pω2, uf{ω1,ω2,ω3},p=1−1 1+exp−(pω1+pω2+pω3), and uf(,·)=−∞else. Consider the price vectors p=(0, 1, 0)and p=(1, 0, 0). Note that {ω1,ω2}∈Df(p) and {ω1,ω2}∈Df(p). Moreover, we have Ds(ω1)(p)={{ω1}} =Ds(ω1)(p),Ds(ω2)(p)= {{ω2}} =Ds(ω2)(p)and Ds(ω3)(p)={∅}=Ds(ω3)(p).Thus,pand pare equilibrium price vectors. Suppose there is a ¯ pthat each terminal seller weakly prefers to pand p, that is, vs(ω1)(¯ p)≥max{vs(ω1)(p),vs(ω1)(p)}=1, vs(ω2)(¯ p)≥max{vs(ω2)(p),vs(ω2)(p)}= 1, and vs(ω3)(¯ p)≥vs(ω3)(p)=vs(ω3)(p)=us(ω3)(∅)=0. Thus, ¯ pω1≥1and ¯ pω2≥1. But then Df(¯ p)={{ω1,ω2,ω3}}.Moreover,Ds(ω3)(¯ p)={∅}. Thus, there is no such equilibrium price vector ¯ p. To check that ufsatisfies FS, first note that for each p∈Rf,wehaveuf({ω1,ω2,ω3}, p)>0=uf(∅). Thus, at each p∈Rf,wehaveDf(p)⊆{{ω1,ω2},{ω1,ω2,ω3}} and the only possible FS violation could occur for p≤pwith p ω3=pω3and {ω1,ω2,ω3}∈ Df(p). However, if uf({ω1,ω2,ω3},p)≥uf({ω1,ω2},p),thenasuf({ω1,ω2,ω3},p)− uf({ω1,ω2},p)is increasing in pω1and in pω2for each pω3,wehaveuf({ω1,ω2,ω3}, p)≥uf({ω1,ω2},p). Thus, FS holds.9♦ 3.2 Extremal equilibria So far, we have not considered whether competitive equilibria exist in our model and, in principle, the lattice in Theorem 1could be empty. Next, we show that under the additional assumption of bounded willingness to pay (we follow the terminology of Fleiner et al.,2019), side-optimal equilibria exist, that is, there exist an equilibrium that is a most preferred equilibrium for all terminal buyers and an equilibrium that is a most preferred equilibrium for all terminal sellers. Bounded willingness to pay (BWP) The utility function ufsatisfies bounded willingness to pay if there exists a K≥0suchthatforallp∈Rfand ∈Df(p)if ω∈→f, then pω<Kand if ω∈f→, then pω>−K. The condition rules out, for example, the case that for a trade the seller would never sell under any price and the buyer would buy under any price. BWP guarantees that equilibrium prices of trades realized in equilibrium are bounded. It follows straightforwardly from the continuity of utility functions that equilibrium prices of trades realized in equilibrium form a closed set. Thus, the set of equilibrium prices of trades realized in equilibrium is compact. The existence of side-optimal equilibria follows straightforwardly from this. 9The example violates the BWP and the BCV conditions that we consider in Section 3.2. 818 Jan Christoph Schlegel Theoretical Economics 17 (2022) Theorem 2 (Existence of extremal equilibria). Under the assumption of BWP, FS, LAD, LAS, there exists a seller-optimal equilibrium, that is, a ¯ p∈E(u)such that for each terminal seller f∈F: vf(¯ p)≥vf(p)for each p∈E(u), and a buyer-optimal equilibrium, that is, a p∈E(u)such that for each terminal buyer f∈F: vf(p)≥vf(p)for each p∈E(u). Remark 3. Under the assumptions of weak FS and BWP, Fleiner et al. (2019) establish that equilibrium allocations are equivalent to trail-stable allocations.10 Thus, under BWP, FS, LAD, LAS there is a seller-optimal trail-stable allocation and a buyer-optimal trail-stable allocation. In the case of no frictions, BWP is implied by requiring, that uf(∅)>−∞ and utility functions have full range. Fleiner et al. (2019) also introduce an alternative regularity condition, called bounded compensating variations (BCV), which guarantees that utility of individually rational allocations is bounded for all agents. Bounded compensating variations The utility function of firm fsatisfies bounded compensating variations if for each ⊆we have inf p∈R:uf(,p)>uf(∅) ω∈f→ pω− ω∈→f pω>−∞. Remark 4. The previous result also holds if BWP is replaced by BCV. See the full working paper version (Schlegel (2020)) for the proof. Theorem (Existence of extremal equilibria with BCV, Schlegel (2020)). Under the assumption of BCV, FS, LAD, LAS, there exists a seller-optimal equilibrium and a buyeroptimal equilibrium. 3.3 Strategic considerations The existence of buyer-optimal equilibria established in Theorem 2allows us to obtain a group-incentive compatibility result.11 In the following, a domain of utility profiles is a set U=×f∈FUfwhere Ufis a set of (continuous and monotonic) utility functions 10Fleiner et al. (2019) use the choice-language definition of weak FS. They show that the choice-language definition is equivalent to the demand-language definition when the price space is amended by infinite prices. Under BWP, it is easy to see that the equivalence between the choice-language and the demandlanguage versions of weak FS and of FS also holds on the standard price space R. 11In the following, we talk about incentives for terminal buyers. A completely analogous result also holds for terminal sellers. Theoretical Economics 17 (2022) The structure of equilibria in trading networks 819 for firm f.Amechanism is a function M:U→A. A mechanism is (weakly) groupstrategy-proof for a set of workers F⊆Fon the domain U⊆Uif for each u,˜ u∈Uwith ˜ u−F=u−F,thereexistaf∈Fwith ufM(u)≥ufM(˜ u). Theorem 2allows us to define a class of focal mechanisms on the domain of utility profiles satisfying BWP, FS, LAD, and LAS: A buyer-optimal mechanism maps to each utility profile a buyer-optimal equilibrium allocation. To obtain a group-strategy-proofness results for terminal buyers for buyer-optimal mechanisms, we have to restrict the domain. In the following, a unit demand utility function is a ufsuch that for the induced demand Dfat each p∈Rfand ∈Df(p) we have |→f|≤1. For the case without transfers and with strict preferences, analogous results are proved by Hatfield et al. (2012) (for the case of acyclic networks and the solution concept of stability) and Fleiner et al. (2016) (for arbitrary networks and the solution concept of trail-stability). Theorem 3 (Group-strategy-proofness). Each buyer-optimal mechanism is groupstrategy-proof for terminal buyers on the domain of utility profiles such that terminal buyers’ utility functions satisfy unit demand and BWP and all other firms’ utility functionssatisfyBWP,FS,LAD,andLAS. In view of Proposition 1, we can extend the construction to profiles satisfying BWP, weak (!) FS, weak LAD, and weak LAS. For each such profile u, there exists a corresponding profile ˜ usatisfying BWP, FS, LAD, and LAS such that the indirect utility functions are the same for both profiles. The mechanism that assigns to each profile uabuyeroptimal equilibrium allocation under a corresponding profile ˜ uis group-strategy-proof for terminal buyers (since for terminal buyers (and terminal sellers) the weak FS and the FS condition coincide), and the assigned allocations are equilibrium allocations under uas well. Corollary 1 (Group-strategy-proofness under weak FS). On the domain of utility profiles such that terminal buyers’ utility functions satisfy unit demand and BWP and all other firms’ utility functions satisfy BWP, weak FS, weak LAD, and weak LAS, there exists a group-strategy-proof mechanisms for terminal buyers that implements a competitive equilibrium. Remark 5. As noted in Remark 4, the existence of extremal equilibria can alternatively be proved with BWP replaced by BCV. Analogously, we can obtain a group-strategyproofness result with BWP replaced by BCV. The proof remains unchanged in that case. 820 Jan Christoph Schlegel Theoretical Economics 17 (2022) 4. Applications 4.1 Two-sided matching markets The results in the previous sections immediately apply to two-sided matching markets. In this case, the results generalize previously known results for two-sided matching markets in two directions: we provide a lattice result, a rural hospitals theorem, and a groupstrategy-proofness result for markets with a) wealth effects and frictions for both sides of the market b) the possibility that it is infeasible for a hospital to hire certain groups of doctors. As remarked in Section 2.1, the weak version of full substitutability is sufficient to obtain the results for two-sided markets. Instead of a set of firms, the economy now consists of a finite set of hospitals Hand a finite set of doctors D. Each hospital hhas a utility function uh:{(D,p):D⊆D,p∈ RD}→R∪{−∞}that assigns to each D⊆Dand price vector p∈RDa utility level. We extend uhto 2D×RDby letting uh(D,p):=uh(D,(pd)d∈D). We allow the utility function to take on a value of −∞ to indicate that it is infeasible for the hospital to hire a particular group of doctors. This allows us, for example, to incorporate institutional constraints such as the “generalized interval constraints” characterized by Kojima et al. (2020b), which specify a lower and an upper bound on the number of doctors a hospital can hire. We assume that uh(D,p)=−∞implies uh(D,p)=−∞for each p∈RD. We assume that there is at least one group of doctors D⊆Dthat is feasible to hire, that is, such that uh(D,·)>−∞. Moreover, we require that for uh(D,·)= −∞, the utility function uh(D,·)is continuous and strictly decreasing in prices. The utility function induces a demand correspondence Dh:RD⇒2Dby Dh(p):=argmaxD⊆Duh(D,p).We assume that doctors are gross substitutes for hospitals. We only need to require the condition for price vectors where the demand is single-valued. Weak gross substitutability For p,p∈RDwith p≤p,Dh(p)={D}and Dh(p)={D}, we have {d∈D:p d=px}⊆D. Moreover, we require the law of aggregate demand. Law of aggregate demand For p,p∈RDwith p≤pand each D∈Df(p),thereisa ˜ D∈Df(p)with |˜ D|≥|D|. Each doctor dhas a utility function ud:H×R∪{∅}→Rthat is strictly increasing and continuous in its second argument. We extend udto H∪{∅}×RHby letting ud(h,p):= ud(h,phd )and ud(∅,p):=ud(∅). Amatching is a function μ:H×D→2D∪Hwith μ(h)⊆Dfor each h∈Hand μ(d)∈H∪{∅}for each d∈Dsuch that d∈μ(h)if and only if h=μ(d).Acompetitive equilibrium (μ,p)is a pair consisting of a matching μ, and a price vector p∈RH×D such that for each h∈Hand ph:=(phd )d∈Dwe have μ(h)∈Dh(ph), and for each d∈D and pd=(phd )h∈Hwe have ud(μ(d),pd)=maxh∈H∪{∅}ud(h,pd). The following is an immediate consequence of Theorems 1,2,and3and generalizes results of Hatfield et al. (2013,2014) for the transferable utility model.12 12It is important that we use the “multivalued” version of the law of aggregate demand. Otherwise, Example 3demonstrates that the lattice result can fail. An example similar to Example 2but with two instead of one seller demonstrates that the rural hospitals theorem can fail. Theoretical Economics 17 (2022) The structure of equilibria in trading networks 821 Corollary 2. For each matching market such that doctors are weak gross substitutes for hospitals and the law of aggregate demand holds, the following is true: (i) Let p,p∈RH×Dbe equilibrium prices. Then ¯ p,p∈RH×Ddefined by ¯ phd =maxphd,p hd,phd =minphd,p hd are equilibrium prices. (ii) Let p,p∈RH×Dbe equilibrium prices. For each matching μsupporting pas an equilibrium (μ,p), there is a matching μsupporting pas an equilibrium (μ,p) such that (a) a doctor is unemployed in μif and only if he is unemployed in μ,thatis, μ(d)=∅⇔μ(d)=∅, for each d∈D, (b) each hospital hires the same number of doctors in μand μ, that is, |μ(h)|= |μ(h)| for each h∈H. (iii) If utility functions satisfy, moreover, BWP, then there exists a worker-optimal equilibrium allocation and a hospital-optimal equilibrium allocation. (iv) The worker-optimal mechanism is group-strategy-proof for workers on the domain of utility profiles such that workers’ utility functions satisfy unit supply and BWP and hospitals’ utility functions satisfy BWP, weak GS, and LAD. Proof. We can construct a corresponding trading network with =H∪Dand ˜ uh(,p)=uh({d:(h,d)∈},p)for ⊆hfor each h∈H,and˜ ud({(h,d)},phd )= ud(h,phd ),˜ ud(∅)=ud(∅)and ˜ ud(,·)=−∞if ⊆dwith ||>1 for each d∈D.The weak gross substitutes condition then corresponds to the weak SSS condition. Weak SSS is equivalent to SSS as shown in Appendix D of Fleiner et al. (2019) (SSS corresponds to the conjunction of the two properties that Fleiner et al. (2019) call “increasing price full substitutability for sales” and “decreasing price full substitutability for purchases”). Since the market is two-sided, SSS and FS are equivalent. The corollary follows from Theorems 1,2,and3. 4.2 Exchange economies with uniform pricing Next, we apply the model to the exchange of indivisible objects. The result extends results of Gul and Stacchetti (1999)andHatfield et al. (2013) (see the discussion in their Section IV.B) to imperfectly transferable utility. As in Gul and Stacchetti (1999), we maintain the assumption that the market is cleared through transfers of a perfectly divisible good and there is no constraint on the amount of the divisible good an agent can consume. Moreover, negative quantities of the divisible good can be consumed. However, we do not assume that utility in the divisible good is quasilinear. Similar assumptions are standard in the object allocation literature with general preferences; see, for example, Morimoto and Serizawa (2015). 822 Jan Christoph Schlegel Theoretical Economics 17 (2022) Inthefollowing,weletXbe a finite set of heterogeneous indivisible objects.From now on, we use the term agents in lieu of firms. Agents have utility functions over bundles of objects and transfers, ˜ uf:2 X×R→Rsuch that for each Y⊆X,˜ uf(Y,·)is continuous, strictly increasing, and has full range,13 and for each t∈Rand Y⊆Y⊆X, we have ˜ uf(Y,t)≤˜ uf(Y,t). Each agent fis endowed with a bundle of objects Xf⊆X such that Xf∩Xf=∅for f= fand f∈FXf=X.Anexchange economy is a pair (˜ u,(Xf)f∈F)of utility functions and endowments for each agent. We define for each f∈Fademand correspondence ˜ Df:RX +×2X⇒2Xby ˜ Df(p,Xf):=argmax Y⊆X ˜ ufY, x∈Xf\Y px− x∈Y\Xf px. Remark 6. In contrast to quasilinear utility, demand can depend on the endowment, that is, in general ˜ Df(p,Xf)= ˜ Df(p,˜ Xf)for Xf= ˜ Xf. We assume that objects are gross substitutes for agents.14 Gross Substitutability (GS) For p,p∈RX +with p≤p,ifp x=pxfor x∈Xf,thenfor each Y∈˜ Df(p,Xf)there exists a Y∈˜ Df(p,Xf)such that {x∈Y:p x=px}⊆Y,and if p x=pxfor x∈X\Xf, then for each Y∈˜ Df(p,Xf)there exists a Y∈˜ Df(p,Xf), such that {x∈Y:p x=px}⊆Y. Moreover, we assume the law of aggregate demand. Law of Aggregate Demand (LAD) For p,p∈RX +with p≤p,ifp x=pxfor x∈Xf,then for each Y∈˜ Df(p,Xf)there exists a Y∈˜ Df(p,Xf), and if p x=pxfor x∈X\Xf, then for each Y∈˜ Df(p,Xf)there exists a Y∈˜ Df(p,Xf),suchthat|Y|≥|Y|. Remark 7. We assume that there is only one copy of each object. More generally, we can extend the model to multiple units of the same object by creating identical copies of objects. In this case, we can use the strong substitutes condition (Baldwin and Klemperer (2019)) that requires that objects are gross substitutes for agents if each of the identical copies of an object is treated as a separate object. The law of aggregate demand can be generalized in an analogous way. One can show that under the assumption of strong substitutes and the generalized law of aggregate demand, an equilibrium with uniform prices (identical copies of the same good have the same price) exists whenever an equilibrium with nonuniform prices (identical copies of the same good can have different prices) exists. All subsequent results generalize to this setting. An allocation of objects is a partition Y=(Yf)f∈Fwith Yf⊆Xand Yf∩Yf=∅ for f= f.Acompetitive equilibrium oftheexchangeeconomy(˜ u,(Xf)f∈F)is a pair 13This assumption is only necessary for the existence of side-optimal allocations and otherwise redundant. 14As in Section 3.1 and in contrast to Section 4.1, we need a multivalued version of gross substitutability to obtain corresponding results for exchange economies. This is because gross substitutability between a good that an agent owns and one that he does not own corresponds to cross-side complementarity in a trading network. Theoretical Economics 17 (2022) The structure of equilibria in trading networks 823 [Y,p]where Yis an allocation of objects and p∈RX +such that for each f∈Fwe have Yf∈˜ Df(p,Xf). For each exchange economy (˜ u,(Xf)f∈F), a corresponding trading network can be defined as follows: The set of trades is :=(x,f1,f2)∈X×F×F:x∈Xf1,f2= f1 where for ω=(x,f1,f2)∈we have s(ω)=f1= f2=b(ω).Wewritex(ω)for the object involved in trade ω.For⊆fand p∈Rf, define Xf():=x(ω):ω∈→f∪Xf\x(ω):ω∈f→, pf():= ω∈f→ pω− ω∈→f pω. Utility functions are induced by utility functions over bundles of objects and transfers; for ⊆fand p∈Rf +we let uf(,p)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˜ ufXf(),pf(),if x(ω):ω∈f→⊆Xfand x(ω)= xω for ω,ω∈with ω= ω, −∞,else. To apply the results from the previous sections, we also extend the utility functions to negative prices; for ⊆fand p∈R\R +we let uf(,p):=uf,max{pω,0 }ω∈+ ω∈f→ min{pω,0 }− ω∈→f min{pω,0 }. Remark 8. Extending utility for negative prices in this way implies (see the proof of Lemma 5) that the induced demand Dfsatisfies FS on Rfwhenever it satisfies FS on Rf +. Moreover, is easy to see that for each ⊆fwith uf(,·)>−∞,ufis continuous (as ufis continuous on R +,andmin and max are continuous) and monotonic on R. This will allow us to apply the results from previous sections. Equilibrium prices in the trading network are nonnegative by our assumption that ˜ uf(Y,t)≤˜ uf(Y,t)for t∈Rand Y⊆Y⊆X:Letp∈Rand define −:={ω∈: pω<0}. First, note that for ∈Df(p)we have ∩− f→=∅: Define p+∈Rby p+ ω:= max{pω,0 }for ω∈f→and p+ ω:=pωelse. Note that Xf()⊆Xf(\− f→).Thus,by monotonicity uf(,p)≤uf,p+≤uf\− f→,p+=uf\− f→,p. As ∈Df(p), all inequalities hold with equality, in particular, uf(,p)=uf(,p+) and, therefore, by monotonicity, ∩− f→=∅. By a similar argument, if ∈Df(p)and − →f= ∅, then ∩− →f= ∅. Thus, for p∈R\R +there is excess demand, and for each p∈E(u),wehavepω≥0 for each ω∈. 824 Jan Christoph Schlegel Theoretical Economics 17 (2022) The gross substitutes condition for ˜ ufcorresponds to the full substitutability condition for ufand the law of aggregate demand for ˜ ufimplies the laws of aggregate demand and supply for uf. Lemma 5. If ˜ ufsatisfies GS, then ufsatisfies FS. If ˜ ufsatisfies LAD, then ufsatisfies LAD and LAS. In general, different trades involving the same object can be priced differently. In the following, we call p∈E(u)acompetitive equilibrium of the trading network with uniform pricing,ifforω,ω∈,withx(ω)=x(ω)we have pω=pω.Tradesinthesame object are perfect substitutes to each other for the seller of the object, and he will sell the object to a buyer who is offering the highest price. Thus, we can always construct an equilibrium with uniform pricing from an equilibrium with nonuniform pricing by setting the price of the nonrealized trades to the highest price for the involved object over all trades in the trading network. Similarly, a competitive equilibrium in the exchange economy, induces a competitive equilibrium with uniform pricing in the trading network. The following theorem can be interpreted as a generalization of Theorem 10 of Hatfield et al. (2013). Proposition 2. (i) If p∈R +are equilibrium prices in the trading network induced by an exchange economy, then (maxω∈,x=x(ω)pω)x∈X∈RX +are equilibrium prices in the exchange economy. (ii) If p∈RX +are equilibrium prices in an exchange economy, then (px(ω))ω∈∈R +are equilibrium prices in the trading network induced by the exchange economy. Proof.Let[,p]be an equilibrium in the induced trading network. Let q:= (maxω∈,x=x(ω)pω)x∈Xand consider the allocation [(Xf())f∈F,q]in the exchange economy. By construction, we have pω≤qx(ω)for each ω/∈and pω=qx(ω)for ω∈. Thus, f∈Df(p)⇒Xf()∈˜ Df(q,Xf) and [(Xf())f∈F,q]is an equilibrium of the exchange economy. For the second part, let [Y,p]be an equilibrium of the exchange economy. Define q:=(px(ω))ω∈and consider the set of trades ⊆defined by :=ω∈:x(ω)∈Yb(ω)∩Xs(ω). By construction, we have Yf∈˜ Df(p,Xf)⇒f∈Df(q). Therefore, [,q]is an equilibrium of the induced trading network. Proposition 2and the previous results for trading networks imply the following. Theoretical Economics 17 (2022) The structure of equilibria in trading networks 831 Df(p),wehave¯ q−¯ p≤+1<,andthusDf(¯ q)⊆Df(¯ p),andwehaveq−p≤ +1<,andthusDf(q)⊆Df(p).LetP:={˜ q∈Rf:˜ qω∈{qω,q ω}for all ω∈f}. By Lemma 3, there is a single-valued selection ˜ Df:P→2ffrom Dfsatisfying FS, LAD, and LAS. Let ¯ :=˜ Df(¯ q),:=˜ Df(q)and  :=˜ Df(q).AsDf(q)={},wehave ˜ Df(q)=. First, we show that ω∈→f:pω≥p ω∪ω∈ →f:p ω>p ω⊆¯ →f. For ˜ q∈P,suchthat ˜ qω=¯ qωfor ω∈→fand ˜ qω=qωfor ω∈f→,let ˜ :=˜ Df(˜ q).By CSC for ˜ Df,wehave ˜ →f⊆¯ →f. By SSS for ˜ Df,wehave{ω∈→f:qω=¯ qω≥q ω}⊆ ˜ →fand, therefore, {ω∈→f:qω=¯ qω≥q ω}⊆¯ →f. Similarly, for ˜ q∈Psuch that ˜ qω=¯ qωfor ω∈→fand ˜ qω=q ωfor ω∈f→let ˜ :=˜ Df(˜ q).ByCSCfor ˜ Df,wehave ˜ →f⊆¯ →f. By SSS for ˜ Df,wehave{ω∈ →f:q ω=¯ qω≥qω}⊆˜ →fand, therefore, {ω∈ →f:q ω>q ω}⊆{ω∈ →f:q ω=¯ qω≥qω}⊆¯ →f. Moreover, by Claim 2and as  ∈Df(q)∈Df(p+),wehave{ω∈:p ω= pω}⊆ and {ω/∈:p ω= pω}∩  =∅. Therefore, ω∈→f:pω≥p ω∪ω∈ →f:p ω>p ω ⊆ω∈→f:qω≥q ω∪ω∈ →f:q ω>q ω⊆¯ →f. Next, we show that ¯ f→⊆ω∈f→:pω≥p ω∪ω∈ f→:p ω>p ω. Let ¯ω∈¯ f→. We consider two cases. Either ¯ p¯ω=p¯ωor ¯ p¯ω=p¯ω>p¯ω. In the first case, consider ˜ q∈Pwith ˜ qω=¯ qωfor ω∈→fand ˜ qω=qωfor ω∈f→.Let ˜ :=˜ Df(˜ q).By SSS of ˜ Df,wehave ¯ω∈˜ f→.ByCSCof ˜ Df,wehave ˜ f→⊆f→, and hence ¯ω∈f→. Similarly, if ¯ p¯ω=p¯ω>p¯ω,consider˜ q∈Pwith ˜ qω=¯ qωfor ω∈→fand ˜ qω=q ωfor ω∈f→.Let˜ :=˜ Df(˜ q). By SSS of ˜ Df,wehave ¯ω∈˜ f→.ByCSCof˜ Df,wehave ˜ f→⊆ f→, and hence ¯ω∈ f→. Since {ω/∈:p ω= pω}∩ =∅and p¯ω= p¯ω,this implies ¯ω∈ f→. Finally, let ˜ q∈Psuch that ˜ qω=¯ qωfor ω∈→fand ˜ qω=qωfor ω∈f→and ˜ := ˜ Df(˜ q).ByLADfor ˜ Dfat qand ˜ qand LAS for ˜ Dfat ˜ qand ¯ q,wehave |→f|−|f→|≥|˜ →f|−|˜ f→|≥|¯ →f|−|¯ f→|. A completely dual proof shows that has the desired properties. Proof of Theorem 1 Proof.Let∈E(u,p)and ∈E(u,p). First, we show that for the pairwise minimum p∈E(u). For each firm f∈F, there is by Lemma 4applied to f∈Df(p)and  f∈ Df(p)af∈Df(p)with →f⊆ω∈→f:p ω≥pω∪ω∈ →f:pω>p  ω,(7) 832 Jan Christoph Schlegel Theoretical Economics 17 (2022) ω∈f→:p ω≥pω∪ω∈ f→:pω>p  ω⊆f→(8) and |→f|−|f→|≥|→f|−|f→|.(9) Taking the union over all firms of (7) and (8), we have  f∈F →f⊆ω∈:p ω≥pω∪ω∈:pω>p  ω⊆ f∈F f→, (10) and summing inequality (9) over all firms  f∈F|→f|−|f→|≥ f∈F|→f|−|f→|=0. (11) This implies f∈F|→f|≥f∈F|f→|, which together with (10)implies f∈Ff→= f∈F→f=:and [,p]is an equilibrium. Moreover, since f∈Ff→=f∈F→f, the left-hand side of inequality (11) is also equal to 0 and the inequality holds with equality. This implies that for each f∈F, inequality (9) holds with equality as well and we have |→f|−|f→|=|→f|−|f→|. A completely dual argument shows that there is a ¯ ∈E(u,¯ p)with |→f|−|f→|=|¯ →f|−|¯ f→|. The same argument as before with ¯ intheroleof,and ¯ pintheroleofpestablishes (note that the pairwise minimum of ¯ pand pis again p) that there is a ⊆such that [,p]is an equilibrium, and for each f∈Fwe have |→f|−|f→|=|→f|−|f→|. Since |→f|−|f→|=|→f|−|f→|, this concludes the proof. Proof of Theorem 2 Proof. Following an idea of Kelso and Crawford (1982), we can characterize competitive equilibria by a zero-surplus condition. Define a surplus function Z:R→Rby Z(p):=min ⊆max f∈Fmax ⊆f uf,p−uf(,p). By definition, for each f∈F,wehaveDf(p)=argmax⊆fuf(,p). Thus, for each arrangement [,p],wehavemaxf∈Fmax⊆fuf(,p)−uf(,p)≥0 with equality if Theoretical Economics 17 (2022) The structure of equilibria in trading networks 833 and only if ∈E(u,p).Thus,p∈E(u)if and only if Z(p)=0. The surplus function is continuous, as uf(,p)−uf(,p)is continuous in pand the maximum, respectively, minimum of finitely many continuous functions is continuous. Thus, E(u)is a closed set, as it is the preimage of the closed set {0}under the continuous function Z. By BWP, there is a K>0suchthatforallf∈F,p∈Rfand ∈Df(p)if ω∈→f, then pω<Kand if ω∈f→,thenpω>−K.LetE(u):=E(u)∩[−K,K].ByBWP, for each p∈E(u), the vector p∈Rdefined by p ω=pωfor −K<p ω<K,p ω=K for pω>K,andp ω=−Kfor pω<−Kis an equilibrium price vector p∈E(u)with vf(p)=vf(p)for each f∈F. By Corollary 2 in Fleiner et al. (2019) (as indicated in Footnote 10, under BWP the choice-language version of weak FS used by Fleiner et al. (2019) is equivalent to the demand-language version), E(u)is nonempty, and hence E(u)is nonempty. As E(u)is closed, E(u)is compact. From Theorem 1, and observing that the pairwise maximum (minimum) of two vectors in [−K,K]is an element of [−K,K], we conclude that E(u)is a nonempty, compact sublattice of R. This implies that E(u) has a maximal element ¯ pand a minimal element p. By monotonicity and the previous observation that for each p∈E(u),thereisap∈E(u)with vf(p)=vf(p)for each f∈F, for each terminal seller fand p∈E(u)we have vf(¯ p)≥vf(p).Thus,¯ pis a terminal seller optimal equilibrium. Similarly, pis a terminal buyer optimal equilibrium p under u. Proof of Theorem 3 Proof.LetF⊆Fbe the set of terminal buyers. Let U=×f∈FUfwhere for f∈Fthe set Ufis the set of unit demand and BWP utility functions and for each f∈F\Fthe set Ufis the set of BWP, FS, LAD, and LAD utility functions. In the following for ˜ uf,ˆ uf∈Uf, etc. we denote the induced demand by ˜ Df,ˆ Df,etc. Let M:U→Abe a buyer-optimal mechanism. First, we establish that Mis immune to truncation strategies. Claim 3. Let f∈F.Letu,˜ u∈Uwith ˜ u−f=u−f, and let [,p]be a buyer-optimal equilibrium under u.Iff= ∅,˜ uf(ω,·)=uf(ω,·)for each ω∈→fand ˜ uf(∅)>˜ uf(,p), then for each equilibrium [˜ ,˜ p]under ˜ u, we have ˜ f=∅. Proof. Suppose not. Then ˜ f= ∅.Let˜ f={˜ω}. Note that also {˜ω}∈Df(˜ p).Thus, [˜ ,˜ p]is an equilibrium under u. But since uf(˜ω,˜ p˜ω)=˜ uf(˜ω,˜ p˜ω)≥˜ uf(∅)>˜ uf(,p)=uf(,p) this contradicts the buyer optimality of [,p]. Second, we establish that Mis immune to certain strategies where a single terminal buyer changes the utility function for one trade so that it becomes more attractive relative to the other trades. The claim can be interpreted as an adaption of Lemma 1 of Hatfield et al. (2009) to the setting with transfers. 834 Jan Christoph Schlegel Theoretical Economics 17 (2022) Claim 4. Let f∈F.Letu,ˆ u∈Uwith ˆ u−f=u−fsuch that there is a ˆω∈→fwith ˆ uf(ω,·)=uf(ω,·)for ω= ˆωand ˆ uf(∅)=uf(∅).Let [¯ ,¯ p]be a buyer-optimal equilibrium under u. If for all pˆω∈R, we have uf(ˆω,pˆω)≤uf(¯ ,¯ p)⇒ˆ uf(ˆω,pˆω)=uf(ˆω,pˆω), uf(ˆω,pˆω)≥uf(¯ ,¯ p)⇒ˆ uf(ˆω,pˆω)≥uf(ˆω,pˆω), then [¯ ,¯ p]is a buyer-optimal equilibrium under ˆ u. Proof.Let [ˆ ,ˆ p]be a buyer-optimal equilibrium under ˆ u.Ifuf(ˆω,ˆ pˆω)≤uf(¯ ,¯ p), then we have Df(ˆ p)=ˆ Df(ˆ p)and [ˆ ,ˆ p]is an equilibrium under u.Moreover,ˆ uf(ˆω, ¯ pˆω)=uf(ˆω,¯ pˆω)and, therefore, [¯ ,¯ p]is an equilibrium under ˆ u. By buyer-optimality of [¯ ,¯ p]under u,wehave ˆ uf(ˆ ,ˆ p)=uf(ˆ ,ˆ p)≤uf(¯ ,¯ p)=ˆ uf(¯ ,¯ p)for each f∈F. Thus, [¯ ,¯ p]is a buyer-optimal equilibrium under ˆ u. It remains to consider the case that uf(ˆω,ˆ pˆω)>u f(¯ ,¯ p). In this case, consider the two subcases that ˆ f={ˆω}or ˆ f= {ˆω}. If ˆ f= {ˆω}, we can show that [ˆ ,ˆ p]is an equilibrium under u. Suppose not. Then, as ˆ f/∈Df(ˆ p)and uf(ω,ˆ pω)=ˆ uf(ω,ˆ pω)for ω= ˆω,wehaveuf(ˆω,ˆ pˆω)>u f(ˆ ,ˆ p). Thus, ˆ uf(ˆω,ˆ pˆω)≥uf(ˆω,ˆ pˆω)>u f(ˆ ,ˆ p)=ˆ uf(ˆ ,ˆ p)and, therefore, ˆ f/∈ˆ Df(ˆ p).This contradicts the assumption that [ˆ ,ˆ p]is an equilibrium under ˆ u.Thus, [ˆ ,ˆ p]is an equilibrium under uand by the same reasoning as above, [¯ ,¯ p]is a buyer-optimal equilibrium under ˆ u. If ˆ f={ˆω}, consider the utility function ˜ ufobtained from ufby truncating as follows: ˜ uf(ω,·)=uf(ω,·)for all ω∈→fand uf(¯ ,¯ p)<˜ uf(∅)<u f(ˆω,ˆ pˆω).ByClaim3, for each equilibrium [,p]under ˜ u:=(˜ uf,u−f)we have f=∅. Define the utility function ˜ uf ∗by ˜ uf ∗(ˆω,·)=˜ uf(ˆω,·)=uf(ˆω,·),by ˜ uf ∗(ω,·)=−∞for each ω= ˆω,and ˜ uf ∗(∅)= ˜ uf(∅). As for each equilibrium [,p]under ˜ u,wehavef=∅,wehaveE(˜ u)⊆E(˜ u∗)for ˜ u∗:=(˜ uf ∗,u−f), and in particular, there is an equilibrium [˜ ,˜ p]under ˜ u∗with ˜ f=∅. Observe however that ˜ uf ∗(ˆω,ˆ pˆω)=˜ uf(ˆω,ˆ pˆω)=u(ˆω,ˆ pˆω)>˜ uf ∗(∅).Thus, ˜ Df ∗(ˆ p)={{ ˆω}} and [ˆ ,ˆ p]is an equilibrium under ˜ u∗with ˜ uf ∗(ˆ ,ˆ p)>˜ uf ∗(∅). This contradicts the rural hospitals theorem (the second part of Theorem 1). With the claim, we can prove the result. Suppose there are profiles u,˜ u∈Usuch that ˜ u−F=u−Fand for each f∈F,wehaveuf(M(˜ u)) >u f(M(u)).LetM(u)=(¯ ,¯ p)and M(˜ u)=(˜ ,˜ p). We define for each f∈F,aˆ uf∈Ufas follows: Note that ˜ f= ∅ as uf(˜ ,˜ p)> uf(¯ ,¯ p)≥uf(∅).Let˜ω∈˜ be the unique trade in ˜ such that b(˜ω)=f.Welet ˆ uf(ω,·)=uf(ω,·)for ω= ˜ωand we let ˆ uf(∅)=uf(∅).Toconstructˆ uf(˜ω,·), we proceed as follows: Define ˆ uf(˜ω,p˜ω):=uf(˜ω,p˜ω)for each p˜ω∈Rwith uf(˜ω,p˜ω)≤uf(¯ ,¯ p). Define ˆ uf(˜ω,˜ p˜ω):=max ω∈→f uf(ω,˜ pω). Note that ˆ uf(˜ω,˜ p˜ω)≥uf(˜ω,˜ p˜ω)>u f(¯ ,¯ p)=ˆ uf(¯ ,¯ p). Theoretical Economics 17 (2022) The structure of equilibria in trading networks 835 For prices p˜ω= ˜ p˜ωwith uf(˜ω,p˜ω)≥uf(¯ ,¯ p), we can choose any continuous and monotonic extension such that ˆ uf(˜ω,p˜ω)≥uf(˜ω,p˜ω).ByClaim4,[¯ ,¯ p]is a buyeroptimal equilibrium for (ˆ uf,u−f). Iterating for all f∈F,[¯ ,¯ p]is a buyer-optimal equilibrium under ˆ u:=(ˆ uF,u−F). Note however that by construction of ˆ u, for each f∈F, we have ˜ f∈ˆ Df(˜ p).Thus, [˜ ,˜ p]is an equilibrium under ˆ uwith ˆ uf(˜ ,˜ p)>ˆ uf(¯ ,¯ p) for each f∈F. This contradicts the buyer-optimality of [¯ ,¯ p]under (ˆ uF,u−F). Appendix C: Proofs for Section 4 Proof of Lemma 5 Proof. First, we show the result for nonnegative prices. Let p,p∈Rf +. Define q,q∈ RX +by qx:=⎧ ⎪ ⎨ ⎪ ⎩ min ω∈→f:x(ω)=xpω,forx/∈Xf, max ω∈f→:x(ω)=xpω,forx∈Xf,q x:=⎧ ⎪ ⎨ ⎪ ⎩ min ω∈→f:x(ω)=xp ω,forx/∈Xf, max ω∈f→:x(ω)=xp ω,forx∈Xf. By construction, we have Df(p)=⊆f:Xf()∈˜ Df(q),pω=qx(ω)for ω∈, Dfp=⊆f:Xf∈˜ Dfq,p ω=q x(ω)for ω∈. If pω=p ωfor ω∈f→and pω≤p ωfor ω∈→f,thenfor∈Df(p)there is, by gross substitutability a Y∈˜ Df(q)with {x∈Y:q x=qx}⊆Xf().Thus,ifx∈Y\Xfand q x=qx, then x∈Xf(), and if x∈Xf\Xf(),thenasq x=qx,wehavex∈Xf\Y. Therefore, there is a ∈Df(p)with ω∈→f:p ω=q x(ω)=qx(ω)=pω⊆ →f, f→⊆f→. Similarly, by the law of aggregate demand, there is a Y∈˜ Df(q)such that |Y|≥|Xf()|. Then there is a ∈Df(p)with Y=Xf().Butthen |→f|−|f→|=|Y\Xf|−|Xf\Y|=|Y|−|Xf| ≥Xf−|Xf|=Xf\Xf−Xf\Xf= →f−|f→|. An analogous argument shows that Dfsatisfies the second part of the SSS condition, the second part of the CSC condition, and LAS. Next, we establish FS and LAD/LAS on Rf.Letp,p∈Rf, and define q,q∈RX as previously. Moreover, define p0:=(max{pω,0 }ω∈)∈Rfand (p)0:=(max{p ω, 0}ω∈)∈Rf.Byconstructionofufand the assumption that ˜ uf(Y,t)≤˜ uf(Y,t)for Y⊆Y,wehave Df(p)=∈Dfp0:{x∈X:qx<0}⊆Xf(),pω=qx(ω)for ω∈, Dfp=∈Dfp0:x∈X:q x<0⊆Xf,p ω=q x(ω)for ω∈. 836 Jan Christoph Schlegel Theoretical Economics 17 (2022) In particular, for ∈Df(p)we have ∈Df((p)0)and by FS for nonnegative prices, there is a ˜ ∈Df(p0)with ω∈˜ →f:p0 ω=p0 ω⊆ →f, f→⊆˜ f→. We can find a ∈Df(p),suchthat{ω∈˜ :qx(ω)=q x(ω)}⊆.Nowletω∈→fand p ω=pω.Thenq x(ω)=p ω=pω=qx(ω)and ω∈˜ .Moreover,(p)0 ω=p0 ωand, therefore, ω∈ →f. Similarly, for all ω∈f→we have p ω=pω.Ifpω=p ω<q  x(ω)=qx(ω), then ω/∈and ω/∈.Ifpω=p ω=q x(ω)=qx(ω), then ω/∈→fimplies ω/∈˜ →f.Moreover, (p)0 ω=p0 ωand, therefore, ω/∈ →f. To establish LAD, let ∈Df(p). Since ∈Df((p)0)and by LAD for nonnegative price vectors, there is a ˜ ∈Df(p0), and hence a ∈Df(p)with Xf()=Xf(˜ )∪{x∈ X:qx<0}such that |→f|−|f→|=Xf()\Xf−Xf\Xf()≥|˜ →f|−|˜ f→|≥ →f− f→. An analogous argument shows that Dfsatisfies the second part of the SSS condition, the second part of the CSC condition, and LAS. Proof of Corollary 3 Proof. For the first part, consider price vectors in the induces trading network q,q∈ R +defined by qω:=px(ω)and q ω:=p x(ω)for each ω∈. By Proposition 2,qand q are equilibrium prices in the induced trading network. By Lemma 5, utility functions in the induced trading network satisfy FS, LAD, and LAS. Thus, by Theorem 1, price vectors ¯ q,q∈R +with ¯ qω=maxqω,q ω,qω=minqω,q ω, are equilibrium prices in the trading network. By construction of qand q, for each ω,ω∈with x(ω)=x(ω)we have qω=px(ω)=qωand q ω=p x(ω)=q ω. Therefore, ¯ px=max ω∈,x=x(ω) ¯ qωand px=max ω∈,x=x(ω)qω, and, by Proposition 2,¯ pand pare equilibrium price vectors. For the second part, define qand qas before and let :=ω∈:x(ω)∈Yb(ω)∩Xs(ω). As shown in the proof of Proposition 2,[,q]is an equilibrium of the trading network. By the second part of Theorem 1,thereisa⊆such that [,q]is an equilibrium of the trading network with |→f|−|f→|= →f− f→. Theoretical Economics 17 (2022) The structure of equilibria in trading networks 837 Let Y=(Y f)f∈Fwith Y f:=Xf(). As shown in the proof of Proposition 2,[Y,p] is an equilibrium of the exchange economy. Moreover, |Yf|=|Yf\Xf|−|Xf\Yf|+|Xf|=|→f|−|f→|+|Xf| = →f− f→+|Xf|=Y f\Xf−Xf\Y f+|Xf|=Y f. For the third part, we first show that the set of equilibrium price vectors in the induced trading network, E(u)is compact. The same argument as in the proof of Theorem 2establishes that the surplus function Z:R +→Ris continuous, and hence E(u)⊆ R +is closed. To show that E(u)is bounded, note that by the full range assumption there exists a K>0 such that for each f∈Fand Y⊆Xwe have ˜ uf(Y,−K)<˜ uf(Xf,0 ).For each equilibrium [,p]in the trading network and each f∈F,wehave uf(,p)=˜ ufXf(),pf()≥˜ uf(Xf,0 )=uf(∅), and, therefore, by monotonicity of utility in transfers pf()>−K.Moreover, f∈Fpf()=0. Thus, pf()<|F|·Kfor each f∈F. By the full range assumption, there is a ˜ K>0 such that for each f∈Fand Y⊆X,wehave ˜ uf(∅,˜ K)>˜ uf(Y,|F|·K). Note that for each equilibrium [,p]of the trading network, each f∈Fand each ⊆fwith Xf()=∅,wehave ˜ uf∅, ω∈ pω=uf,p≤uf(,p)=˜ ufXf(),pf() <˜ ufXf(),|F|·K<˜ uf(∅,˜ K). Thus, ω∈pω<˜ Kand, as pω≥0 for each ω∈,wehave0≤pω<˜ Kfor each ω∈ . Now note that for each ω∈,thereexistsa⊆s(ω)with X()=∅and ω∈. Thus, for each ω∈we have 0 ≤pω<˜ K.Thus,E(u)is compact and by Proposition 2 nonempty. Moreover, by Theorem 1,E(u)is a sublattice of R. Since E(u)is a nonempty, compact sublattice of R,thereexist ¯ p,p∈E(u)such that for each p∈E(u)we have pω≤pω≤¯ pωfor each ω∈. By the first part of Proposition 2,thevectorsq,¯ q∈RX + defined by qx:=max ω∈,x=x(ω)pω,¯ qx:=max ω∈,x=x(ω) ¯ pω are equilibrium price vectors in the exchange economy. Now let q∈RX +be an equilibrium price vector in the exchange economy. By the second part of Proposition 2,the price vector p∈R +defined by pω:=px(ω)for each ω∈,isinE(u).Letx∈X.Letω∈ with x=x(ω)and qx=pω.Thenqx=pω≤pω=qx. Similarly, let ω∈with x=x(ω) and ¯ qx=¯ pω.Then¯ qx=¯ pω≥pω=px.Thus,¯ q,qare the desired price vectors. References Baldwin, Elizabeth and Paul Klemperer (2019), “Understanding preferences: “demand types”, and the existence of equilibrium with indivisibilities.” Econometrica, 87, 867–932. [804,822] 838 Jan Christoph Schlegel Theoretical Economics 17 (2022) Candogan, Ozan, Markos Epitropou, and Rakesh V. Vohra (2021), “Competitive equilibrium and trading networks: A network flow approach.” Operations Research, 69, 114– 147. [803,804] Demange, Gabrielle and David Gale (1985), “The strategy structure of two-sided matching markets.” Econometrica, 53, 873–888. [801,803,805] Fleiner, Tamás (2003), “A fixed-point approach to stable matchings and some applications.” Mathematics of Operations Research, 28, 103–126. [804] Fleiner, Tamás, Ravi Jagadeesan, Zsuzsanna Jankó, and Alexander Teytelboym (2019), “Trading networks with frictions.” Econometrica, 87, 1633–1661. [801,802,805,806,807, 817,818,821,833] Fleiner, Tamás, Zsuzsanna Jankó, Akihisa Tamura, and Alexander Teytelboym (2016), “Trading networks with bilateral contracts.” Available at SSRN 2457902. [804,813,819] Galichon, Alfred, Scott Duke Kominers, and Simon Weber (2019), “Costly concessions: An empirical framework for matching with imperfectly transferable utility.” Journal of Political Economy, 127, 2875–2925. [801] Gul, Faruk and Ennio Stacchetti (1999), “Walrasian equilibrium with Gross substitutes.” Journal of Economic Theory, 87, 95–124. [804,821] Hatfield, John William, and Scott Duke Kominers (2012), “Matching in networks with bilateral contracts.” American Economic Journal: Microeconomics, 4, 176–208. [802,804, 813,819] Hatfield, John William, Scott Duke Kominers, Alexandru Nichifor, Michael Ostrovsky, and Alexander Westkamp (2013), “Stability and competitive equilibrium in trading networks.” Journal of Political Economy, 121, 966–1005. [802,803,804,805,806,813,820, 821,824] Hatfield, John William, Scott Duke Kominers, Alexandru Nichifor, Michael Ostrovsky, and Alexander Westkamp (2019), “Full substitutability.” Theoretical Economics, 14, 1535– 1590. [803,804,808] Hatfield, John William, Scott Duke Kominers, Alexandru Nichifor, Michael Ostrovsky, and Alexander Westkamp (2021), “Chain stability in trading networks.” Theoretical Economics, 16, 197–234. [801,802,804,805,806,809] Hatfield, John William, Ravi Jagadeesan, and Scott Duke Kominers (2020), “Matching in networks with bilateral contracts: Corrigendum.” American Economic Journal: Microeconomics, 12, 277–285. [803] Hatfield, John William, and Fuhito Kojima (2009), “Group incentive compatibility for matching with contracts.” Games and Economic Behavior, 67, 745–749. [833] Hatfield, John William, Fuhito Kojima, and Scott Duke Kominers (2014), “Strategyproofness, investment efficiency, and marginal returns: An equivalence.” In Proceedings of the Fifteenth ACM Conference on Economics and Computation, 801–801. [804,820] Theoretical Economics 17 (2022) The structure of equilibria in trading networks 839 Hatfield, John William, and Paul R. Milgrom (2005), “Matching with contracts.” American Economic Review, 95, 913–935. [804] Jagadeesan, Ravi, Scott Duke Kominers, and Ross Rheingans-Yoo (2018), “Strategyproofness of worker-optimal matching with continuously transferable utility.” Games and Economic Behavior, 108, 287–294. [804] Kelso, Alexander S. Jr. and Vincent P. Crawford (1982), “Job matching, coalition formation, and Gross substitutes.” Econometrica, 50, 1483–1504. [804,805,832] Kojima, Fuhito, Ning Sun, and Ning Neil Yu (2020a), “Job market interventions.” [805] Kojima, Fuhito, Ning Sun, and Ning Neil Yu (2020b), “Job matching under constraints.” American Economic Review, 110, 2935–2947. [805,806,820] Legros, Patrick and Andrew F. Newman (2007), “Beauty is a beast, frog is a prince: Assortative matching with nontransferabilities.” Econometrica, 75, 1073–1102. [801] Morimoto, Shuhei and Shigehiro Serizawa (2015), “Strategy-proofness and efficiency with non-quasi-linear preferences: A characterization of minimum price Walrasian rule.” Theoretical Economics, 10, 445–487. [821] Nöldeke, Georg and Larry Samuelson (2018), “The implementation duality.” Econometrica, 86, 1283–1324. [801] Ostrovsky, Michael (2008), “Stability in supply chain networks.” American Economic Review, 98, 897–923. [802,803,804,809,813] Roth, Alvin E. (1984), “Stability and polarization of interests in job matching.” Econometrica, 52, 47–57. [804] Roth, Alvin E. and Marilda Sotomayor (1988), “Interior points in the core of two-sided matching markets.” Journal of Economic Theory, 45, 85–101. [815] Schlegel, Jan Christoph (2018), “Group-strategy-proof and stable mechanisms for matching markets with continuous transfers.” Available at SSRN 3157803. [801,804,805] Schlegel, Jan Christoph (2020), “The structure of equilibria in trading networks with frictions.” Available at SSRN 3544605. [802,808,809,810,818] Sun, Ning and Zaifu Yang (2006), “Equilibria and indivisibilities: Gross substitutes and complements.” Econometrica, 74, 1385–1402. [802,808] Zhou, Lin (1994), “The set of Nash equilibria of a supermodular game is a complete lattice.” Games and Economic Behavior, 7, 295–300. [803] Co-editor Federico Echenique handled this manuscript. Manuscript received 7 July, 2020; final version accepted 9 July, 2021; available online 12 July, 2021.