Remarks on a linearization of Koopmans recursion

Abstract

AbstractLet X be a metric space and $$U:X^{\infty }\rightarrow \mathbb {R}$$ U : X ∞ → R be a continuous function satisfying the Koopmans recursion $$U(x_0,x_1,x_2,\ldots )=\varphi (x_0, U(x_1,x_2,\ldots )),$$ U ( x 0 , x 1 , x 2 , … ) = φ ( x 0 , U ( x 1 , x 2 , … ) ) , where $$\varphi :X\times I \rightarrow I$$ φ : X × I → I is a continuous function and I is an interval. Denote by $$\succeq $$ ⪰ a preference relation defined on the product $$X^{\infty }$$ X ∞ represented by a function $$U:X^{\infty }\rightarrow \mathbb {R}$$ U : X ∞ → R , called a utility function, that means $$(x_0,x_1,\ldots )\succeq (y_0,y_1,\ldots )\Leftrightarrow U(x_0,x_1,\ldots )\ge U(y_0,y_1,\ldots )$$ ( 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 $$\succeq $$ ⪰ can be represented by another utility function V satisfying the affine recursion $$V(x_0,x_1,x_2,\ldots ) = \alpha (x_0)V(x_1,x_2,\ldots )+ \beta (x_0)$$ V ( x 0 , x 1 , x 2 , … ) = α ( x 0 ) V ( x 1 , x 2 , … ) + β ( x 0 ) . Under suitable assumptions on relation $$\succeq $$ ⪰ we determine the form of the functions $$\varphi $$ 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).