Future-blindness and the product topology

Technical Summary: Future-blindness and the Product Topology

Problem Statement
The evaluation of infinite consumption streams is a central issue in social choice theory and intergenerational welfare. A well-known impossibility result (Diamond [1965]) establishes that continuity, Pareto efficiency, and intergenerational equity (anonymity) cannot be simultaneously satisfied in infinite-horizon models. While much literature focuses on reconciling equity with patience, this paper investigates the opposite behavioral extreme: future-blindness. This concept describes agents who exhibit extreme impatience, effectively ignoring sufficiently distant generations even when offered large compensations. The paper seeks to formally characterize the topological structures on the space of bounded sequences ($l^\infty$) that correspond to such behaviors, specifically addressing how continuity assumptions translate into time discounting.

Methodology
The authors employ a topological approach within the framework of locally convex topological vector spaces. They analyze the space $l^\infty$ of bounded real-valued sequences, representing infinite consumption streams. The methodology involves:

1. Defining Behavioral Concepts: Introducing two distinct notions of future-blindness:
   • $N$-blindness: An agent ignores all periods beyond a fixed, exogenous date $N$.
   • Eventual blindness: An agent ignores all but finitely many dates, where the cutoff is endogenous and may depend on the specific streams being compared, even if future streams are compensated at growing rates.
2. Topological Characterization: Identifying the finest locally convex topologies under which all continuous preference relations satisfy these behavioral definitions. This involves analyzing the properties of generating seminorms and the convergence of "tail" operators ($T_n$).
3. Dual Space Analysis: Investigating the topological duals (spaces of continuous linear functionals) associated with these topologies to determine the structure of generalized price vectors in these settings.

Key Contributions and Results

1. Characterization of $N$-Blindness:

   • The paper defines an $N$-blind topology ($\tau^b_N$) as the finest locally convex topology where every continuous preference is $N$-blind.
   • Result: $\tau^b_N$ is generated by the family of seminorms $\{p_k : k = 0, \dots, N\}$, where $p_k(x) = |x_k|$.
   • Properties: This topology is not Hausdorff. The closure of the zero set is the subspace of sequences that are zero up to period $N$ ($T_N$). The dual space $(l^\infty, \tau^b_N)'$ is isomorphic to $c^N_{00}$, the space of sequences that are zero for all $n > N$.

2. Characterization of Eventual Blindness and the Product Topology:

   • The paper defines an eventually blind topology ($\tau^b$) as the finest locally convex topology where every continuous preference is eventually blind.
   • Main Theorem (Proposition 4.3): The finest eventually blind locally convex topology coincides exactly with the product topology ($\tau_p$).
   • Behavioral Interpretation: Continuity with respect to the product topology is behaviorally equivalent to eventual blindness. This means that for any pair of streams where one is strictly preferred, there exists a finite horizon beyond which adding any sequence of improvements to the less preferred stream will not alter the strict preference.
   • Dual Space: The dual space $(l^\infty, \tau^b)'$ is isomorphic to $c_{00}$, the space of sequences with only finitely many non-zero elements.

3. Comparison with Myopia:

   • The paper situates these results within the existing literature on myopia (Brown and Lewis [1981]), where the finest myopic topology coincides with the Mackey topology ($\tau_{Mc}$).
   • The authors establish a strict hierarchy: $\tau^b_N \subsetneq \tau^b = \tau_p \subsetneq \tau_{Mc} = \tau_m$.
   • This implies that agents with preferences continuous in the product topology are "more myopic" (or at least as myopic) than those continuous in the Mackey topology.

Significance and Claims
The paper claims to fill a significant gap in the literature regarding the behavioral foundations of the product topology. While the product topology has been widely used in general equilibrium theory (e.g., Peleg and Yaari [1970]) to ensure the existence of equilibria (via the compactness of budget sets by Tychonoff's theorem), it lacked a direct behavioral interpretation.

• Behavioral Foundation: The authors provide the first explicit characterization showing that product-topology continuity corresponds to eventual blindness. This offers a behavioral justification for using the product topology in infinite-dimensional economic models: it models agents who eventually disregard remote generations regardless of compensation.
• Dual Space Characterization: The paper clarifies the nature of prices in these settings, showing that under eventual blindness, prices must be sequences with finite support ($c_{00}$), contrasting with the $l^1$ prices associated with the Mackey topology (myopia).

The authors note that their results parallel the seminal work of Brown and Lewis [1981] but extend the analysis to the specific case of the product topology, which is distinct from the Mackey topology used in Bewley's [1972] equilibrium existence proofs. The paper concludes that these topological structures provide a rigorous framework for modeling extreme forms of impatience and tail insensitivity in infinite-horizon economic models.