Some Characterizations of the Cobb-Douglas and CES Production Functions in Microeconomics

Xiaoshu Wang,Yu Fu

doi:10.1155/2013/761832

Abstract

It is well known that the study of the shape and the properties of the production possibility frontier is a subject of great interest in economic analysis. Vîlcu (Vîlcu, 2011) proved that the generalized Cobb-Douglas production function has constant return to scale if and only if the corresponding hypersurface is developable. Later on, the authors A. D. Vîlcu and G. E. Vîlcu, 2011 extended this result to the case of CES production function. Both results establish an interesting link between some fundamental notions in the theory of production functions and the differential geometry of hypersurfaces in Euclidean spaces. In this paper, we give some characterizations of minimal generalized Cobb-Douglas and CES production hypersurfaces in Euclidean spaces.

Highlights

In microeconomics and macroeconomics, the production functions are positive nonconstant functions that specify the output of a firm, an industry, or an entire economy for all combinations of inputs
At each point p ∈ Mn, the tangent space TpM is an inner product space, the shape operator Sp can be defined as a linear operator on this space by the relation g (Spu, w) = g (dV (u), w), (8)
It is well known that the determinant of the shape operator Sp is called the Gauss-Kronecker curvature

Summary

Introduction

The production functions are positive nonconstant functions that specify the output of a firm, an industry, or an entire economy for all combinations of inputs. The Cobb-Douglas (CD) production function is especially notable for being the first time an aggregate or economy-wide production function was developed, estimated, and presented to the profession for analysis. It gave a landmark change in how economists approached macroeconomics. The author proved that the generalized Cobb-Douglas production function has constant return to scale if and only if the corresponding hypersurface is developable. We give further study of the generalized Cobb-Douglas and CES production functions as hypersurfaces in Euclidean space En+1. We give some characterizations of the generalized Cobb-Douglas and CES production hypersurfaces under the minimality condition in Euclidean spaces

Some Basic Concepts in Theory of Hypersurfaces

Generalized Cobb-Douglas Production Hypersurfaces

Generalized CES Production Hypersurfaces