变规模报酬Cobb--Douglas生产函数的平坦图超曲面刚性定理

中文摘要 AI 辅助

我们研究了与高维（n>2）变规模报酬Cobb--Douglas生产函数相关的图超曲面。我们证明了一个刚性定理，表明该类中每个平坦模型必然是齐次的。证明依赖于一个仿射降维原理，该原理将高维平坦性条件转化为合适的二维限制。随后，在非一次情形下我们得到了完整分类：每个这样的平坦模型都是某个正线性形式的幂。对于一次情形，我们给出了在秩一集合上用包络描述的局部结构刻画，并在开秩零区域上具有线性性。这些结果将文献[8]中最近发展的两输入分析推广到高维情形，其中一次情形表现出本质上不同的行为。

英文摘要

We study graph hypersurfaces associated with higher-dimensional ($n>2$) Cobb--Douglas production functions with variable returns to scale. We prove a rigidity theorem showing that every flat model in this class is necessarily homogeneous. The proof relies on an affine dimension-reduction principle that transfers the higher-dimensional flatness condition to suitable two-dimensional restrictions. We then obtain a complete classification in the non-degree-one case: every such flat model is a power of a positive linear form. For degree one, we give a local structural description in terms of envelopes on the rank-one set, with linearity on open rank-zero regions. These results extend the two-input analysis recently developed in [8] to the higher-dimensional setting, where the degree-one case exhibits a genuinely different behavior.

发表机构

• Michigan State University（密歇根州立大学）
• King Saud University（沙特国王大学）
• Petroleum-Gas University of Ploieşti（普洛耶什蒂石油天然气大学）
• "Gheorghe Mihoc-Caius Iacob" Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy（罗马尼亚科学院“格奥尔基·米霍克-卡尤斯·雅各布”数学统计与应用数学研究所）
• National University of Science and Technology Politehnica Bucharest（布加勒斯特理工大学国家科技大学）

机构由 AI 辅助整理，请以论文原文为准。