‘Mixing apples and oranges’: Note on a bewildering application of Euler's theorem to the s-shaped total product and related curves

Abstract

In this research note, the Sigmoid-shaped total product (TP) curve and some extended versions (previously known as the symmetry) of the three stages of production are taken as subjects for critical discussion. We make the following arguments: (1) The extended version, which plots all four of APL, MPL, APK and MPK as inverted U-shaped curves on the same coordinate plane utilizing Euler’s theorem to narrate and prove that the four curves can generate perfectly symmetric three stages, constitutes not only illogical drawing but also may involve improper reasoning that transgresses the boundary between disjoint sets. (2) When Euler’s theorem is applied precisely to some of its accredited objects of TP curves, of which the Cobb-Douglas function often serves as the standard representation, the distinction among the alleged simple three stages
of production would be inherently nonexistent in the first instance, thus dissolving the extended three-stage version accordingly. (3) If our potentially unprecedented theoretical elaboration and deduction in the penultimate section are validated and accepted, it appears there is no possibility for another proposed extended version, of which K/L has been directly assigned as an argument, to bear fruit in the end.