January  2018, 5(1): 1-7. doi: 10.3934/jdg.2018001

Hyperopic topologies on $l^{∞}$

1. 

FGV EPGE, Escola Brasileira de Economia e Finanças, Rio de Janeiro RJ 22250-900, Brazil

2. 

Graduate School of Economics, Catholic University of Brasilia 70790-160, Brazil

3. 

Departamento de Ciencias -Sección Matemática, Pontifical Catholic University of Peru, San Miguel, Lima, Peru

* Corresponding author

Received  January 2017 Revised  September 2017 Published  January 2018

Fund Project: P. K. Monteiro acknowledges the financial support of CNPq-Brazil
J. Orrillo acknowledges CNPq-Brazil for financial support project 309525/2013-6.

Myopic economic agents are well studied in economics. They are impatient. A myopic topology is a topology such that every continuous preference relation is myopic. If the space is $l^{∞}$, the Mackey topology $τ _{M}(l^{∞},l^{1})$, is the largest locally convex such topology. However there is a growing interest in patient consumers. In this paper we analyze the extreme case of consumers who only value the long run. We call such a consumer hyperopic. We define hyperopic preferences and hyperopic topologies. We show the existence of the largest locally convex hyperopic topology, characterize its dual and determine its relationship with the norm dual of $l^{∞}$.

Citation: Paulo Klinger Monteiro, Jaime Orrillo, Rudy José Rosas Bazán. Hyperopic topologies on $l^{∞}$. Journal of Dynamics & Games, 2018, 5 (1) : 1-7. doi: 10.3934/jdg.2018001
References:
[1] C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis: a Hitchhiker's Guide, Springer-Verlag, New York, 1999.  doi: 10.1007/3-540-29587-9.  Google Scholar
[2]

A. AraujoR. Novinski and M. R. Pascoa, General equilibrium wariness and efficient bubbles, Journal of Economic Theory, 146 (2011), 785-811.  doi: 10.1016/j.jet.2011.01.005.  Google Scholar

[3]

T. Bewley, Existence of equilibria in economies with infinitely many commodities, Journal of Economic Theory, 4 (1972), 514-540.  doi: 10.1016/0022-0531(72)90136-6.  Google Scholar

[4]

J. Brown and L. M. Lewis, Myopic economic agents, Econometrica, 49 (1981), 359-368.  doi: 10.2307/1913315.  Google Scholar

[5]

Ch. Gilles, Charges as equilibrium prices and asset bubbles, Journal of Mathematical Economics, 18 (1989), 155-167.  doi: 10.1016/0304-4068(89)90019-0.  Google Scholar

[6]

Ch. Gilles and S. F. LeRoy, Bubbles and charges, International Economic Review, 33 (1992), 323-339.  doi: 10.2307/2526897.  Google Scholar

[7]

J. Martinez-Legaz, On Weierstrass extreme value theorem, Optimization Letters, 8 (2014), 391-393.  doi: 10.1007/s11590-012-0587-0.  Google Scholar

[8]

L. K. Raut, Myopic topologies on general commodity spaces, Journal of Economic Theory, 39 (1986), 358-367.  doi: 10.1016/0022-0531(86)90050-5.  Google Scholar

[9]

P. Reny, On the existence of pure and mixed strategy Nash equilibria in discontinuous games, Econometrica, 67 (1999), 1029-1056.  doi: 10.1111/1468-0262.00069.  Google Scholar

[10]

H. H. Schaefer, Topological Vector Spaces, second editon, GTM 3, Springer-Verlag, 1999. doi: 10.1007/978-1-4612-1468-7.  Google Scholar

[11]

M. Sion, On general minimax theorems, Pacific Journal of Mathematics, 8 (1958), 171-176.  doi: 10.2140/pjm.1958.8.171.  Google Scholar

[12]

J. Werner, Arbitrage, bubbles, and valuation, International Economic Review, 38 (1997), 453-464.  doi: 10.2307/2527383.  Google Scholar

show all references

References:
[1] C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis: a Hitchhiker's Guide, Springer-Verlag, New York, 1999.  doi: 10.1007/3-540-29587-9.  Google Scholar
[2]

A. AraujoR. Novinski and M. R. Pascoa, General equilibrium wariness and efficient bubbles, Journal of Economic Theory, 146 (2011), 785-811.  doi: 10.1016/j.jet.2011.01.005.  Google Scholar

[3]

T. Bewley, Existence of equilibria in economies with infinitely many commodities, Journal of Economic Theory, 4 (1972), 514-540.  doi: 10.1016/0022-0531(72)90136-6.  Google Scholar

[4]

J. Brown and L. M. Lewis, Myopic economic agents, Econometrica, 49 (1981), 359-368.  doi: 10.2307/1913315.  Google Scholar

[5]

Ch. Gilles, Charges as equilibrium prices and asset bubbles, Journal of Mathematical Economics, 18 (1989), 155-167.  doi: 10.1016/0304-4068(89)90019-0.  Google Scholar

[6]

Ch. Gilles and S. F. LeRoy, Bubbles and charges, International Economic Review, 33 (1992), 323-339.  doi: 10.2307/2526897.  Google Scholar

[7]

J. Martinez-Legaz, On Weierstrass extreme value theorem, Optimization Letters, 8 (2014), 391-393.  doi: 10.1007/s11590-012-0587-0.  Google Scholar

[8]

L. K. Raut, Myopic topologies on general commodity spaces, Journal of Economic Theory, 39 (1986), 358-367.  doi: 10.1016/0022-0531(86)90050-5.  Google Scholar

[9]

P. Reny, On the existence of pure and mixed strategy Nash equilibria in discontinuous games, Econometrica, 67 (1999), 1029-1056.  doi: 10.1111/1468-0262.00069.  Google Scholar

[10]

H. H. Schaefer, Topological Vector Spaces, second editon, GTM 3, Springer-Verlag, 1999. doi: 10.1007/978-1-4612-1468-7.  Google Scholar

[11]

M. Sion, On general minimax theorems, Pacific Journal of Mathematics, 8 (1958), 171-176.  doi: 10.2140/pjm.1958.8.171.  Google Scholar

[12]

J. Werner, Arbitrage, bubbles, and valuation, International Economic Review, 38 (1997), 453-464.  doi: 10.2307/2527383.  Google Scholar

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