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

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 :Xtimes 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 :Xrightarrow {mathbb {R}}. We give necessary and sufficient conditions for the existence of homeomorhic solutions of this system.