A Geometrical Interpretation of Production Functions in Economics in terms of Second Fundamental Form

Abstract

The two well-known production models in microeconomics are Cobb-Douglas and ACMS production functions. We study such production functions with 2-inputs in terms of the differential-geometrical properties of their graphs. In particular, we investigate the production functions when their graphs have the second fundamental forms of constant length. We obtain that the ACMS production surface has the second fundamental forms of constant length if and only if the ACMS production function is a perfect substitute. Furthermore, in the case of Cobb-Douglas production function, we provide a non-existence result.