A Rigidity Theorem for Flat Graph Hypersurfaces of Variable-Returns-to-Scale Cobb--Douglas Production Functions
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.