On Homothetic Functions
为位似生产函数（或效用函数）的扩张路径均为从原点出发的射线这一重要命题，提供了一个简短、初等的证明，避免了复杂的偏微分方程或集合论方法。
In economic analysis, the importance of the homotheticity of production functions (or utility functions), which is due to Shephard (1953), has been well recognized. Its important feature lies in the fact that every expansion path is a ray from the origin and the underlying production (or utility) function is homothetic. Although the proof of the part of this statement is easy, the proof of the only if part, at least as it appears in the literature, is not easy, requiring a few pages for the proof. Lau (1969) proved it by way of partial differential equations (cf. pp. 379-81), and Fare and Shephard (1977a, b) used a set-theoretic approach, whereas Sandler and Swimmer (1978) proved it for the two-input case. F0rsund (1975) provides a simple proof for a closely related proposition (cf. his Proposition 2), and yet his proof requires the use of partial differential equations. It would thus be desirable to obtain an elementary, short alternative proof of the above important proposition. The purpose of this note is to offer such a proof. In this paper we prove both the only if and the part simultaneously. Although our proof is carried out in terms of production theory, the same proof obviously applies to the theory of consumption. Let f(x) be the production function of a firm which produces a single output, where x is an n-dimensional input vector. The firm minimizes its