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- Title
Remarks on a linearization of Koopmans recursion.
- Authors
Zdun, Marek Cezary
- Abstract
Let X be a metric space and U : X ∞ → R be a continuous function satisfying the Koopmans recursion U (x 0 , x 1 , x 2 , ...) = φ (x 0 , U (x 1 , x 2 , ...)) , where φ : X × I → I is a continuous function and I is an interval. Denote by ⪰ a preference relation defined on the product X ∞ represented by a function U : X ∞ → R , called a utility function, that means (x 0 , x 1 , ...) ⪰ (y 0 , y 1 , ...) ⇔ U (x 0 , x 1 , ...) ≥ U (y 0 , y 1 , ...) . We consider a problem when the preference relation ⪰ can be represented by another utility function V satisfying the affine recursion V (x 0 , x 1 , x 2 , ...) = α (x 0) V (x 1 , x 2 , ...) + β (x 0) . Under suitable assumptions on relation ⪰ we determine the form of the functions φ defining the utility functions possessing the above property. The problem is reduced to solving a system of simultaneous functional equations. The subject is strictly connected to a problem of preference in economics. In this note we extend the results obtained in Zdun (Aequ Math 94, 2020).
- Subjects
UTILITY functions; FUNCTIONAL equations; SIMULTANEOUS equations; METRIC spaces; CONTINUOUS functions; COMMERCIAL space ventures
- Publication
Aequationes Mathematicae, 2023, Vol 97, Issue 5/6, p1033
- ISSN
0001-9054
- Publication type
Article
- DOI
10.1007/s00010-023-01009-1