On the Maximal Domain Theorem

Yang, Yi-You (2015): On the Maximal Domain Theorem.

Abstract

The maximal domain theorem by Gul and Stacchetti (J. Econ. Theory 87 (1999), 95-124) implies that for markets with indivisible objects and sufficiently many agents, the set of gross substitutable preferences is a largest set for which the existence of a competitive equilibrium is guaranteed, and hence no relaxation of the gross substitutability can ensure the existence of a competitive equilibrium. However, we note that there is a flaw in their proof, and give an example to show that a claim used in the proof may fail to be true. We correct the proof and sharpen the result by showing that even there are only two agents in the market, if the preferences of one agent are not gross substitutable, then gross substitutable preferences can be found for another agent such that no competitive equilibrium exists. Moreover, we introduce the new notion of implicit gross substitutability, which is weaker than the gross substitutability condition and is still sufficient for the existence of a competitive equilibrium when the preferences of some agent are monotone.

• Item Type: MPRA Paper
• Language: English
• Keywords: Competitive equilibrium; gross substitutability; indivisibility
• Subjects: D - Microeconomics > D5 - General Equilibrium and Disequilibrium; D - Microeconomics > D5 - General Equilibrium and Disequilibrium > D51 - Exchange and Production Economies; D - Microeconomics > D6 - Welfare Economics; D - Microeconomics > D6 - Welfare Economics > D61 - Allocative Efficiency ; Cost-Benefit Analysis

References

1. C. Bevia, M. Quinzii, and J. A. Silva, Buying several indivisible goods, Math. Soc. Sci. 37 (1999), 1-23.

2. S. Bikhchandani and J. W. Mamer, Competitive equilibrium in an exchange economy with indivisibilities, J. Econ. Theory 74 (1997), 385-413.

3. F. Gul and E. Stacchetti, Walrasian equilibrium with gross substitutes, J. Econ. Theory 87 (1999), 95-124.

4. F. Gul and E. Stacchetti, The English auction with differentiated commodities, J. Econ. Theory 92 (2000), 66-95.

5. C. Hara, Existence of equilibria in economies with bads, Econometrica 73 (2005), 647-658.

6. C. Hara, Bargaining set and anonymous core without the monotonicity assumption, J. Math. Econ. 41 (2005), 545-556.

7. A. S. Kelso and V. P. Crawford, Job matching, coalition formation, and gross substitutes, Econometrica 50 (1982), 1483-1504.

8. J. Ma, Competitive equilibrium with indivisibilities, J. Econ. Theory 82 (1998), 458-468.

9. A. M. Manelli, Monotonic preferences and core equivalence, Econometrica 59 (1991), 123-138. 10. N. Sun and Z. Yang, Equilibria and indivisibilities: Gross substitutes and complements, Econometrica 74 (2006), 1385-1402.

11. Gross substitutes and complements: A simple generalization, Econ. Lett. 123 (2014), 135-138.