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We consider general trading networks with bilateral contracts. We show that a suitably adapted chain stability concept is equivalent to stability if all agents' preferences are jointly fully substitutable and satisfy the Laws of Aggregate Supply and Demand (a condition we call monotone - substitutability). We also present three examples to show that are results are sharp, demonstrating that: If preferences of some agents do not satisfy the Laws of Aggregate Supply and Demand, then chain stable outcomes may not be stable. " If preferences of some agents are not fully substitutable, then chain stable outcomes may likewise not be stable. " If blocking sets are restricted to chains that do not "cross" themselves (i.e., chains that involve each agent in at most two contracts), then an outcome that is robust to such blocks may not be robust to richer blocks. Our results imply that in trading networks with transferable utility, an outcome is consistent with competitive equilibrium if and only if it is not blocked by any chain of contracts. We show moreover that, from a computational perspective, checking whether an outcome is chain stable is substantially easier than checking whether that outcome is stable directly. Indeed, we show that as the size of the economy grows, the number of chains of trades (corresponding to possible blocking chains) becomes infinitely smaller than the number of general sets of trades (corresponding to possible blocking sets). [ABSTRACT FROM AUTHOR]

Various forms of substitutability are essential for establishing the existence of equilibria and other useful properties in diverse settings such as matching, auctions, and exchange economies with indivisible goods. In this paper, we extend earlier models' canonical definitions of substitutability to a setting in which an agent can be a buyer in some transactions and a seller in others, and show that all the different substitutability concepts are equivalent. Next, we introduce a new class of fully substitutable preferences that models the preferences of intermediaries with production capacity. We then prove that substitutability is preserved under economically important transformations such as trade endowments, mergers, and limited liability. We show that full substitutability can be recast in terms of submodularity of the indirect utility function, the single improvement property, a "no complementarities" condition, and a condition from discrete convex analysis calledM\-concavity. Finally, we show that substitutability implies two key monotonicity conditions known as the Laws of Aggregate Supply and Demand. All of our results explicitly incorporate economically important features such as indifferences, non-monotonicities, and unbounded utility functions that were not fully addressed in prior work. [ABSTRACT FROM AUTHOR]

We introduce a new matching model in which firms trade goods via bilateral contracts which specify a buyer, a seller, and the terms of the exchange. This framework subsumes all classical matching models, including that of Ostrovsky [5]. The generality our model affords allows us to make two substantial contributions. First, we show that two theoretical restrictions are necessary for classical matching theory: · acyclicity — no agent may both buy from and sell to another agent, even through intermediaries, and · full substitutability upon being endowed with an additional item, an agent's demand for other items is lower, both in the sense of a reduced desire to buy additional items and an increased desire to sell items he currently owns. If either condition is violated, then existence of stable allocations cannot be guaranteed. Intuitively, if a contracting relationship contains a cycle, and if a firm in the cycle has an outside option which is preferred to one contract in the cycle, then both the outside option and the complete trading cycle are unstable; the necessity of acyclicity follows. The necessity of full substitutability is more technical, but follows closely upon prior results of Hatfield and Kominers [2]. Second, in the presence of acyclicity and fully substitutable preferences, we fully generalize the key results of classical matching theory. We prove that under these conditions stable allocations correspond bijectively to fixed points of an isotone operator; Tarski's fixed point theorem then guarantees the existence of a lattice of stable allocations. We also prove a generalization of the classical rural hospitals theorem of Roth [6] and the strategy-proofness results of Hatfield and Milgrom [3] and Hatfield and Kojima [1]. These latter results display a surprising structure which can only be elicited within a framework as general as ours: in particular, we show that the difference between the numbers of buyand sell-contracts held, rather than the absolute number of contracts held, is invariant across stable allocations for each agent. In light of our necessity results, our work establishes a frontier of matching theory. Without acyclicity and fully substitutable preferences, stable allocations are not guaranteed to exist in general, and hence the results of classical matching theory fail. Up to the failure of these conditions, however, all of the results of classical matching theory hold. [ABSTRACT FROM AUTHOR]