Future-blindness and the product topology

TL;DR

This paper explores future-blind preferences in infinite consumption streams, establishing a topological foundation for continuity and characterizing dual spaces, with implications for equilibrium analysis.

Contribution

It introduces two notions of future-blindness and proves the product topology is the finest topology ensuring eventual blindness, linking behavioral preferences to topological properties.

Findings

Product topology is the finest ensuring eventual blindness.

Continuity in the product topology underpins equilibrium existence.

Dual spaces are characterized under these topologies.

Abstract

We study future-blind preferences, which are preferences that heavily discount the future, within the space of infinite consumption streams. We give two definitions: $N$-blindness, where agents ignore periods beyond a fixed date $N$, and eventual blindness, where all but finitely many dates are neglected. Using a topological approach, we show that the finest topology ensuring eventual blindness coincides with the product topology. This provides a behavioral foundation for continuity in the product topology, which was considered for studying equilibrium existence in infinite-dimensional spaces. Finally, we characterize the dual spaces under these topologies.