Complementary Inputs and the Existence of Stable Outcomes in Large Trading Networks

This paper studies a model of large trading networks with bilateral contracts. Contracts capture exchange, production, and prices, as well as frictions such as complex taxes and the absence of transfers. In our setting, under standard continuity and convexity conditions, a stable outcome exists in any acyclic network, as long as all firms regard sales as substitutes and the market is large. Thus, complementarities between inputs do not preclude the existence of stable outcomes in large markets, unlike in discrete markets. Even when sales are not substitutable, tree stable outcomes exist in our setting. The model presented in this paper generalizes and unifies versions of general equilibrium models with divisible and indivisible goods, matching models with continuously divisible contracts, models of large (two-sided) matching with complementarities, and club formation models. Additional results provide intuition for the role of uni-directional substitutability conditions and acyclicity in the main existence results, and explain what kinds of equilibria are guaranteed to exist even when these conditions are relaxed. Unlike in two-sided large-market settings, the sufficient conditions described in this paper pin down maximal domains for the existence of equilibria.

[1]  A. Siow,et al.  Who Marries Whom and Why , 2006, Journal of Political Economy.

[2]  David Gale,et al.  Stable schedule matching under revealed preference , 2003, J. Econ. Theory.

[3]  S. Battiston,et al.  Credit Chains and Bankruptcy Propagation in Production Networks , 2007 .

[4]  L. McKenzie,et al.  On Equilibrium in Graham's Model of World Trade and Other Competitive Systems , 1954 .

[5]  Roy Radner,et al.  COMPETITIVE EQUILIBRIUM UNDER UNCERTAINTY11This research was supported in part by the Office of Naval Research under Contract ONR 222(77) with the University of California. Reproduction in whole or in part is permitted for any purpose of the United States Government. , 1968 .

[6]  Patrick J. Kehoe,et al.  Money, Interest Rates, and Exchange Rates with Endogenously Segmented Markets , 2002, Journal of Political Economy.

[7]  David M. Kreps,et al.  Martingales and arbitrage in multiperiod securities markets , 1979 .

[8]  Paul R. Milgrom,et al.  The Economics of Modern Manufacturing: Technology, Strategy, and Organization , 1990 .

[9]  Nicole Immorlica,et al.  Incentives in Large Random Two-Sided Markets , 2015, TEAC.

[10]  T. Jacobson,et al.  Trade Credit and the Propagation of Corporate Failure: An Empirical Analysis , 2015 .

[11]  L. McKenzie,et al.  ON THE EXISTENCE OF GENERAL EQUILIBRIUM FOR A COMPETITIVE MARKET , 1959 .

[12]  K. Arrow,et al.  EXISTENCE OF AN EQUILIBRIUM FOR A COMPETITIVE ECONOMY , 1954 .

[13]  Robert J. Aumann,et al.  EXISTENCE OF COMPETITIVE EQUILIBRIA IN MARKETS WITH A CONTINUUM OF TRADERS , 2020, Classics in Game Theory.

[14]  W. Hildenbrand.  Existence of Equilibria for Economies with Production and a Measure Space of Consumers , 1970 .

[15]  J. Hatfield,et al.  Matching in Networks with Bilateral Contracts , 2010 .

[16]  Lars-Gunnar Svensson,et al.  Non-Manipulable House Allocation With Rent Control , 2014 .

[17]  F. Echenique,et al.  A Theory of Stability in Many-to-Many Matching Markets , 2004 .

[18]  R. Starr.  Quasi-Equilibria in Markets with Non-Convex Preferences , 1969 .

[19]  M. Ostrovsky,et al.  Stability and Competitive Equilibrium in Trading Networks , 2013, Journal of Political Economy.

[20]  L. Shapley,et al.  College Admissions and the Stability of Marriage , 1962 .

[21]  Tamás Fleiner,et al.  A Fixed-Point Approach to Stable Matchings and Some Applications , 2003, Math. Oper. Res..

[22]  Jeremy T. Fox,et al.  Measuring the Efficiency of an Fcc Spectrum Auction , 2005 .

[23]  Roy Radner,et al.  Existence of Equilibrium of Plans, Prices, and Price Expectations in a Sequence of Markets , 1972 .

[24]  Jeremy T. Fox.  Specifying a Structural Matching Game of Trading Networks with Transferable Utility , 2017 .

[25]  O. Hart.  On the optimality of equilibrium when the market structure is incomplete , 1975 .

[26]  Charles Blair,et al.  The Lattice Structure of the Set of Stable Matchings with Multiple Partners , 1988, Math. Oper. Res..

[27]  Bryan Ellickson,et al.  Clubs and the Market , 1999 .

[28]  N. Agarwal.  An Empirical Model of the Medical Match , 2014, The American economic review.

[29]  M. Pycia.  Stability and Preference Alignment in Matching and Coalition Formation , 2010 .

[30]  Ahmet Alkan,et al.  A class of multipartner matching markets with a strong lattice structure , 2002 .

[31]  Zaifu Yang,et al.  Equilibria and Indivisibilities: Gross Substitutes and Complements , 2006 .

[32]  Susanna Loeb,et al.  Analyzing the Determinants of the Matching of Public School Teachers to Jobs: Disentangling the Preferences of Teachers and Employers , 2013, Journal of Labor Economics.

[33]  Kazuo Murota,et al.  Discrete convexity and equilibria in economies with indivisible goods and money , 2001, Math. Soc. Sci..

[34]  Matteo Maggiori.  Financial Intermediation, International Risk Sharing, and Reserve Currencies , 2017 .

[35]  V. Crawford,et al.  Job Matching with Heterogeneous Firms and Workers , 1981 .

[36]  B. Klaus,et al.  Stable Many-to-Many Matchings with Contracts , 2008 .

[37]  X. Gabaix,et al.  International Liquidity and Exchange Rate Dynamics , 2014 .

[38]  L. Shapley,et al.  QUASI-CORES IN A MONETARY ECONOMY WITH NONCONVEX PREFERENCES , 1966 .

[39]  Charles R. Plott,et al.  Understanding Price Controls and Nonprice Competition with Matching Theory , 2012 .

[40]  L. E. Ward.  Partially ordered topological spaces , 1954 .

[41]  F. Echenique,et al.  Core many-to-one matchings by fixed-point methods , 2004 .

[42]  Zaifu Yang,et al.  A Double‐Track Adjustment Process for Discrete Markets With Substitutes and Complements , 2009 .

[43]  Hiroyuki Adachi,et al.  Stable matchings and fixed points in trading networks: A note , 2017 .

[44]  J. Broome.  Approximate equilibrium in economies with indivisible commodities , 1972 .

[45]  A. Roth.  The Evolution of the Labor Market for Medical Interns and Residents: A Case Study in Game Theory , 1984, Journal of Political Economy.

[46]  Lars Peter Hansen,et al.  THE ROLE OF CONDITIONING INFORMATION IN DEDUCING TESTABLE RESTRICTIONS IMPLIED BY DYNAMIC ASSET PRICING MODELS1 , 1987 .

[47]  Morten Sorensen,et al.  How Smart is Smart Money? A Two-Sided Matching Model of Venture Capital , 2006 .

[48]  S. Kakutani.  A generalization of Brouwer’s fixed point theorem , 1941 .

[49]  E. Glen Weyl,et al.  Walrasian Equilibrium in Large, Quasilinear Markets , 2012 .

[50]  M. Ostrovsky.  Stability in Supply Chain Networks , 2005 .

[51]  A. Roth.  Stability and Polarization of Interests in Job Matching , 1984 .

[52]  Paul Milgrom,et al.  Complementarities and fit strategy, structure, and organizational change in manufacturing , 1995 .

[53]  N. Z. Shapiro,et al.  Values of Large Games, I: A Limit Theorem , 1978, Math. Oper. Res..

[54]  Parag A. Pathak,et al.  Appendix to "Incentives and Stability in Large Two-Sided Matching Markets" , 2009 .

[55]  Oliver Hart,et al.  On the existence of equilibrium in a securities model , 1974 .

[56]  Michael A Newton,et al.  Two-Sided Estimation of Mate Preferences for Similarities in Age, Education, and Religion , 2008 .

[57]  Martin Skutella,et al.  Stable Flows over Time , 2013, Algorithms.

[58]  John William Hatfield,et al.  Substitutes and stability for matching with contracts , 2010, J. Econ. Theory.

[59]  Paul Klemperer,et al.  Understanding Preferences: 'Demand Types', and The Existence of Equilibrium with Indivisibilities , 2018 .

[60]  Scott Duke Kominers,et al.  Contract design and stability in many-to-many matching , 2017, Games Econ. Behav..

[61]  A. Roth.  A natural experiment in the organization of entry-level labor markets: regional markets for new physicians and surgeons in the United Kingdom. , 1991, The American economic review.

[62]  A. Galichon,et al.  Costly Concessions: An Empirical Framework for Matching with Imperfectly Transferable Utility , 2017, Journal of Political Economy.

[63]  Andreu Mas-Colell,et al.  Indivisible commodities and general equilibrium theory , 1977 .

[64]  Alvin E. Roth,et al.  Conflict and Coincidence of Interest in Job Matching: Some New Results and Open Questions , 1985, Math. Oper. Res..

[65]  Ennio Stacchetti,et al.  The English Auction with Differentiated Commodities , 2000, J. Econ. Theory.

[66]  R. Aumann.  Markets with a continuum of traders , 1964 .

[67]  Yeon-Koo Che,et al.  Expanding 'Choice' in School Choice , 2008 .

[68]  Hiroyuki Adachi.  On a characterization of stable matchings , 2000 .

[69]  Paul R. Milgrom,et al.  Matching with Contracts , 2005 .

[70]  Faruk Gul,et al.  WALRASIAN EQUILIBRIUM WITH GROSS SUBSTITUTES , 1999 .

[71]  Parag A. Pathak,et al.  Matching with Couples: Stability and Incentives in Large Markets , 2010 .

[72]  V. Crawford,et al.  Job Matching, Coalition Formation, and Gross Substitutes , 1982 .

[73]  Jacques H. Dreze,et al.  Existence of an Exchange Equilibrium under Price Rigidities , 1975 .

[74]  L. S. Shapley,et al.  Values of Large Games II: Oceanic Games , 1978, Math. Oper. Res..

[75]  J. Geanakoplos,et al.  Existence,regularity, and constrained suboptimality of competitive allocations when the asset market is incomplete , 1985 .

[76]  C. Plott,et al.  Price Controls, Non-Price Quality Competition, and the Nonexistence of Competitive Equilibrium , 2015 .

[77]  Itai Ashlagi,et al.  Stability in Large Matching Markets with Complementarities , 2014, Oper. Res..