On a linearization of the recursion $$\varvec{U(x_0,x_1,x_2,\ldots )}\varvec{=\varphi (x_0, U(x_1,x_2,\ldots ))}$$U(x0,x1,x2,…)=φ(x0,U(x1,x2,…)) and its application in economics

Let I be an interval, X be a metric space and $$\succeq $$ be an order relation on the infinite product $$X^{\infty }$$ . Let $$U:X^{\infty }\rightarrow {\mathbb {R}}$$ be a continuous mapping, representing $$\succeq $$ , that is such that $$(x_0,x_1,x_2,\ldots )\succeq (y_0,y_1,y_2,\ldots )\Leftrightarrow U(x_0,x_1,x_2,\ldots )\ge U(y_0,y_1,y_2,\ldots )$$ . 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 $$U(x_0,x_1,x_2,\ldots )=\varphi (x_0, U(x_1,x_2,\ldots )),$$ where $$\varphi :X\times I \rightarrow I$$ is a continuous function strictly increasing in its second variable such that each function $$\varphi (x,\cdot )$$ 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 $$V(x_0,x_1,x_2,\ldots ) = \alpha (x_0)V(x_1,x_2,\ldots )+ \beta (x_0)$$ . We prove that this property holds if and only if there exists a homeomorphic solution of the system of simultaneous affine functional equations $$ F(\varphi (x,t))=\alpha (x) F(t)+ \beta (x), x \in X, t \in I$$ for some functions $$\alpha , \beta :X\rightarrow {\mathbb {R}}$$ . We give necessary and sufficient conditions for the existence of homeomorhic solutions of this system.