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.

令我为一个区间，X为度量空间，\（\ succeq \）为无穷乘积\（X ^ {\ infty} \）的顺序关系。令\（U：X ^ {\ infty} \ rightarrow {\ mathbb {R}} \）是一个连续映射，表示\（\ succeq \），即\（（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）\）。我们将X解释为消费结果的空间，并且关系\（\ succeq \）代表个人如何对所有消费序列进行排名。有人假设U（称为效用函数）满足递归\（U（x_0，x_1，x_2，\ ldots）= \ varphi（x_0，U（x_1，x_2，\ ldots）），\）其中\（\ varphi：X \ times I \ rightarrow I \）是连续的函数严格增加其第二个变量，以使每个函数\（\ varphi（x，\ cdot）\）具有唯一的固定点。我们认为在经济一个开放的问题，当关系\（\ succeq \）可以由另一个连续函数来表示V满足仿射递归\（V（X_0，X_1，X_2，\ ldots）= \α（X_0）V（ x_1，x_2，\ ldots）+ \ beta（x_0）\）。我们证明只有当同时存在仿射功能方程组\（F（\ varphi（x，t））= \ alpha（x）F（t）+ \ beta（x ），x \ in X，t \ in I \）对于某些功能\（\ alpha，\ beta：X \ rightarrow {\ mathbb {R}} \）。我们为该系统的同胚解决方案的存在提供了充分必要的条件。