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# A convexity result for the range of vector measures with applications to large economies

## Author

Listed:
• Urbinati, Niccolò

## Abstract

On a Boolean algebra we consider the topology $u$ induced by a finitely additive measure $\mu$ with values in a locally convex space and formulate a condition on $u$ that is sufficient to guarantee the convexity and weak compactness of the range of $\mu$. This result à la Lyapunov extends those obtained in (Khan, Sagara 2013) to the finitely additive setting through a more direct and less involved proof. We will then give an economical interpretation of the topology $u$ in the framework of coalitional large economies to tackle the problem of measuring the bargaining power of coalitions when the commodity space is infinite dimensional and locally convex. We will show that our condition on $u$ plays the role of the "many more agents than commodities" condition introduced by Rustichini and Yannelis in (1991). As a consequence of the convexity theorem, we will obtain two straight generalizations of Schmeidler's and Vind's Theorems on the veto power of coalitions of arbitrary economic weight.

## Suggested Citation

• Urbinati, Niccolò, 2018. "A convexity result for the range of vector measures with applications to large economies," MPRA Paper 87185, University Library of Munich, Germany.
• Handle: RePEc:pra:mprapa:87185
as

File URL: https://mpra.ub.uni-muenchen.de/87185/1/MPRA_paper_87185.pdf
File Function: original version

## References listed on IDEAS

as
1. Herves-Beloso, Carlos & Moreno-Garcia, Emma & Nunez-Sanz, Carmelo & Rui Pascoa, Mario, 2000. "Blocking Efficacy of Small Coalitions in Myopic Economies," Journal of Economic Theory, Elsevier, vol. 93(1), pages 72-86, July.
2. Schmeidler, David, 1972. "A Remark on the Core of an Atomless Economy," Econometrica, Econometric Society, vol. 40(3), pages 579-580, May.
3. Evren, Özgür & Hüsseinov, Farhad, 2008. "Theorems on the core of an economy with infinitely many commodities and consumers," Journal of Mathematical Economics, Elsevier, vol. 44(11), pages 1180-1196, December.
4. Armstrong, Thomas E. & Richter, Marcel K., 1984. "The core-walras equivalence," Journal of Economic Theory, Elsevier, vol. 33(1), pages 116-151, June.
5. Grodal, Birgit, 1972. "A Second Remark on the Core of an Atomless Economy," Econometrica, Econometric Society, vol. 40(3), pages 581-583, May.
6. Vind, Karl, 1972. "A Third Remark on the Core of an Atomless Economy," Econometrica, Econometric Society, vol. 40(3), pages 585-586, May.
7. Basile, Achille, 1993. "Finitely Additive Nonatomic Coalition Production Economies: Core-Walras Equivalence," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 34(4), pages 983-994, November.
8. Avallone, Anna & Basile, Achille, 1993. "On the Liapunov-Richter theorem in the finitely additive setting," Journal of Mathematical Economics, Elsevier, vol. 22(6), pages 557-561.
9. repec:spr:etbull:v:1:y:2013:i:2:d:10.1007_s40505-013-0018-0 is not listed on IDEAS
Full references (including those not matched with items on IDEAS)

### Keywords

Lyapunov's Theorem; finitely additive measures; correspondences; coalitional economies;

### JEL classification:

• C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
• C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
• D51 - Microeconomics - - General Equilibrium and Disequilibrium - - - Exchange and Production Economies

### NEP fields

This paper has been announced in the following NEP Reports:

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