On a linearization of the recursion U(x_0,x_1,x_2,… )=φ (x_0, U(x_1,x_2,… )) and its application in economics

Let I be an interval, X be a metric space and ≽ be an order relation on the infinite product X^∞ . Let U:X^∞→ℝ be a continuous mapping, representing ≽ , that is such that (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 ≽ 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,… )=φ (x_0, U(x_1,x_2,… )), where φ :X× I → I is a continuous function strictly increasing in its second variable such that each function φ (x,· ) has a unique fixed point. We consider an open problem in economics, when the relation ≽ can be represented by another continuous function V satisfying the affine recursion 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(φ (x,t))=α (x) F(t)+ β (x), x ∈ X, t ∈ I for some functions α , β :X→ℝ . We give necessary and sufficient conditions for the existence of homeomorhic solutions of this system.