A Rigidity Theorem for Flat Graph Hypersurfaces of Variable-Returns-to-Scale Cobb--Douglas Production Functions
Abstract
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.
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BibTeX
@misc{chen2026rigidity,
title = {{A Rigidity Theorem for Flat Graph Hypersurfaces of Variable-Returns-to-Scale Cobb--Douglas Production Functions}},
author = {Bang-Yen Chen and Sharief Deshmukh and Nasser Bin Turki and Alina-Daniela Vîlcu and Gabriel-Eduard Vîlcu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27646},
eprint = {2609.27646},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27646},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Rigidity Theorem for Flat Graph Hypersurfaces of Variable-Returns-to-Scale Cobb--Douglas Production Functions
AU - Bang-Yen Chen
AU - Sharief Deshmukh
AU - Nasser Bin Turki
AU - Alina-Daniela Vîlcu
AU - Gabriel-Eduard Vîlcu
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27646
UR - https://arxiv.org/abs/2609.27646
AB - 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.
ER -