Remarks on a linearization of Koopmans recursion

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).