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

and Applied Analysis 3 (4) The inverse matrix (g) of (gij) is g ij = δij − FiFj W . (15) (5) The matrix of the second fundamental form h is


Introduction
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.Almost all economic theories presuppose a production function, either on the firm level or the aggregate level.Hence, the production function is one of the most key concepts of mainstream neoclassical theories.By assuming that the maximum output technologically possible from a given set of inputs is achieved, economists always using a production function in analysis are abstracting from the engineering and managerial problems inherently associated with a particular production process; see [1].
In 1928, Cobb and Douglas [1] introduced a famous twofactor production function, nowadays called Cobb-Douglas production function, in order to describe the distribution of the national income by help of production functions.Twofactor Cobb-Douglas production function is given by where  denotes the labor input,  is the capital input,  is the total factor productivity, and  is the total production.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 then presented to the profession for analysis.It gave a landmark change in how economists approached macroeconomics.The Cobb-Douglas function has also been applied to many other issues besides production.
In 1961, Arrow et al. [2] introduced another two-input production function given by where  is the output,  the factor productivity,  the share parameter,  and  the primary production factors,  = ( − 1)/, and  = 1/(1−) the elasticity of substitution.Hence this is called constant elasticity of substitution (CES) production function [3,4].
The generalized form of CES production function is given by where   , , , and  are nonzero constants and ,   > 0. The CES production functions are of great interest in economy because of their invariant characteristic, namely, that the elasticity of substitution between the parameters is constant on their domains.
It is easy to see that the CES production function is a generalization of Cobb-Douglas production function, and both of them are homogeneous production functions.
Note that the production function  can be identified with a production hypersurface in the ( + 1)-dimensional Euclidean space E +1 through the map  : R  + → R +1 + ,  ( 1 , . . .,   ) = ( 1 , . . .,   ,  ( 1 , . . .,   )) . (5) In the case of  inputs, we have a hypersurface and an analysis of the generalized CD and CES production functions from the point of view of differential geometry.
In [5], Vîlcu established an interesting link between some fundamental notions in the theory of production functions and the differential geometry of hypersurfaces.The author proved that the generalized Cobb-Douglas production function has constant return to scale if and only if the corresponding hypersurface is developable.The author jointly with A. D. Vîlcu further [6] generalized this result to the case of the generalized CES production function with -inputs.For further study of production hypersurfaces, we refer the reader to Chen's series of interesting papers on homogeneous production functions, quasi-sum production models, and homothetic production functions [7][8][9][10][11][12][13] and G. E. Vîlcu and A. D. Vîlcu's paper [14].
The theory of minimal surfaces (hypersurfaces) is very important in the field of differential geometry.In this paper, we give further study of the generalized Cobb-Douglas and CES production functions as hypersurfaces in Euclidean space E +1 .In particular, 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
Each production function ( 1 , . . .,   ) can be identified with a graph of a nonparametric hypersurface of an Euclidean ( + 1)-space E +1 given by We recall some basic concepts in the theory of hypersurfaces in Euclidean spaces.Let   be a hypersurface in an +1-dimension Euclidean space.The Gauss map V is defined by which maps   to the unit hypersphere   of E +1 .The Gauss map is always defined locally, that is, on a small piece of the hypersurface.It can be defined globally if the hypersurface is orientable.The Gauss map is a continuous map such that V() is a unit normal vector () of   at point .
The differential V of the Gauss map V can be used to define an extrinsic curvature, known as the shape operator or Weingarten map.Since, at each point  ∈   , the tangent space    is an inner product space, the shape operator   can be defined as a linear operator on this space by the relation where ,  ∈    and  is the metric tensor on   .Moreover, the second fundamental form ℎ is related to the shape operator   by for ,  ∈   .
It is well known that the determinant of the shape operator   is called the Gauss-Kronecker curvature.Denote it by ().When  = 2, the Gauss-Kronecker curvature is simply called the Gauss curvature, which is intrinsic due to famous Gauss's Theorema Egregium.The trace of the shape operator   is called the mean curvature of the hypersurfaces.In contrast to the Gauss curvature, the mean curvature is extrinsic, which depends on the immersion of the hypersurface.A hypersurface is called minimal if its mean curvature vanishes identically.
Proposition 1.For a production hypersurface of E +1 defined by One has the following.
(3) The volume element is (4) The inverse matrix (  ) of (  ) is (5) The matrix of the second fundamental form ℎ is (6) The matrix of the shape operator   is The mean curvature  is given by

Generalized Cobb-Douglas Production Hypersurfaces
In this section, we give a nonexistence result of minimal generalized Cobb-Douglas production hypersurfaces in E +1 .
Theorem 2. There does not exist a minimal generalized Cobb-Douglas production hypersurface in E +1 .
Without loss of generality, we may choose  = 1.In this case,   = 1 for  ̸ = 1 reduces to Therefore, (33) becomes The above equation implies that either  1 = −1 or It is easy to see that the latter case is impossible.At this moment, we obtain the function  as follows: Consider (31) for  ̸ = 1, which turns into Substituting (39) into (40), we have In view of (41), by comparing the degrees of the left and the right of the equality we have  = 3. Hence (41) becomes which is impossible since  ̸ = 0. Therefore, there is not a minimal generalized Cobb-Douglas production hypersurface in E +1 .This completes the proof of Theorem 2.

Generalized CES Production Hypersurfaces
In economics, goods that are completely substitutable with each other are called perfect substitutes.They may be characterized as goods having a constant marginal rate of substitution.Mathematically, a production function is called a perfect substitute if it is of the form for some nonzero constants  1 , . . .,   .
In the following, we deal with the case of minimal generalized CES production hypersurfaces in E +1 .Theorem 3.An -factor generalized CES production hypersurface in E +1 is minimal if and only if the generalized CES production function is a perfect substitute.
Proof.Assume that  is a generalized CES production hypersurface in E +1 given by a graph where  is the generalized CES production function with ,   > 0 and   > 0. The generalized CES production function (45) is homogeneous of degree .By assumption that the production hypersurface is minimal, from (18) the minimality condition is equivalent to We define another minimal production hypersurface as f ( 1 , . . .,   ) = ( 1 , . . .,   ,  ( 1 , . . .,   )) ,  ∈ R + . (47) Since the CES production function is homogeneous of degree , we conclude that the hypersurface given by is also minimal.In this case, the minimality condition ( 46) becomes Comparing (49) with (46), we obtain that either  = 1 or  ̸ = 1 and We divide it into two cases.
Case B ( = 1).We note that the minimality condition reduces to (61) Note that  = 1 fulfills (61).Hence, in this case the generalized CES production function  is a perfect substitute.
In the following, we will show that the case  ̸ = 1 is impossible.In fact, if  ̸ = 1, (61) reduces to Since   > 0 and   > 0 for  = 1, . . ., , it follows that the left of equality (62) is positive.Therefore, it is a contradiction.Conversely, it is easy to verify that if the generalized CES production function is a perfect substitute then the -factor generalized CES production hypersurface in E +1 is minimal.This completes the proof of Theorem 3.
Remark 4. Remark that Chen proved in [7] a more general result for 2-factor: ℎ-homogeneous production function is a perfect substitute if and only if the production surface is a minimal surface.

( 9 )
The sectional curvature   of the plane section spanned by /  and /  is Note that the generalized Cobb-Douglas production function (23) is homogeneous of degree ℎ = ∑  =1   .It follows from (23) that