On a linearization of the recursion $$\varvec{U(x_0,x_1,x_2,\ldots )}\varvec{=\varphi (x_0, U(x_1,x_2,\ldots ))}$$ and its application in economics

Abstract

AbstractLet I be an interval, X be a metric space and $$\succeq $$ ⪰ be an order relation on the infinite product $$X^{\infty }$$ X ∞ . Let $$U:X^{\infty }\rightarrow {\mathbb {R}}$$ U : X ∞ → R be a continuous mapping, representing $$\succeq $$ ⪰ , that is such that $$(x_0,x_1,x_2,\ldots )\succeq (y_0,y_1,y_2,\ldots )\Leftrightarrow U(x_0,x_1,x_2,\ldots )\ge U(y_0,y_1,y_2,\ldots )$$ ( x 0 , x 1 , x 2 , … ) ⪰ ( y 0 , y 1 , y 2 , … ) ⇔ U ( x 0 , x 1 , x 2 , … ) ≥ U ( y 0 , y 1 , y 2 , … ) . We interpret X as a space of consumption outcomes and the relation $$\succeq $$ ⪰ represents how an individual would rank all consumption sequences. One assumes that U, called the utility function, satisfies the 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 strictly increasing in its second variable such that each function $$\varphi (x,\cdot )$$ φ ( x , · ) has a unique fixed point. We consider an open problem in economics, when the relation $$\succeq $$ ⪰ can be represented by another continuous 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 ) . We prove that this property holds if and only if there exists a homeomorphic solution of the system of simultaneous affine functional equations $$ F(\varphi (x,t))=\alpha (x) F(t)+ \beta (x), x \in X, t \in I$$ F ( φ ( x , t ) ) = α ( x ) F ( t ) + β ( x ) , x ∈ X , t ∈ I for some functions $$\alpha , \beta :X\rightarrow {\mathbb {R}}$$ α , β : X → R . We give necessary and sufficient conditions for the existence of homeomorhic solutions of this system.