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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 761832, 6 pages
Some Characterizations of the Cobb-Douglas and CES Production Functions in Microeconomics
1School of Investment and Construction Management, Dongbei University of Finance and Economics, Dalian 116025, China
2School of Mathematics and Quantitative Economics, Dongbei University of Finance and Economics, Dalian 116025, China
Received 5 November 2013; Accepted 10 December 2013
Academic Editor: Grzegorz Lukaszewicz
Copyright © 2013 Xiaoshu Wang and Yu Fu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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 .
In 1928, Cobb and Douglas  introduced a famous two-factor production function, nowadays called Cobb-Douglas production function, in order to describe the distribution of the national income by help of production functions. Two-factor 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.
The generalized form of the Cobb-Douglas production function is written as where , is a positive constant, and are nonzero constants.
In 1961, Arrow et al.  introduced another two-input production function given by where is the output, the factor productivity, the share parameter, and the primary production factors, , and 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 . 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 -dimensional Euclidean space through the map , 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 , 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  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 [7–13] and G. E. Vîlcu and A. D. Vîlcu's paper .
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 . In particular, we give some characterizations of the generalized Cobb-Douglas and CES production hypersurfaces under the minimality condition in Euclidean spaces.
2. Some Basic Concepts in Theory of Hypersurfaces
Each production function can be identified with a graph of a nonparametric hypersurface of an Euclidean -space given by We recall some basic concepts in the theory of hypersurfaces in Euclidean spaces.
Let be a hypersurface in an -dimension Euclidean space. The Gauss map is defined by which maps to the unit hypersphere of . 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 is a unit normal vector of at point .
The differential of the Gauss map 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 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 , 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 defined by One has the following.(1)The unit normal is given by (2)The coefficient of the metric tensor is given bywhere if , otherwise 0. (3)The volume element is (4)The inverse matrix of is(5)The matrix of the second fundamental form is(6)The matrix of the shape operator is(7)The mean curvature is given by(8)The Gauss-Kronecker curvature is(9)The sectional curvature of the plane section spanned by and is (10)The Riemann curvature tensor satisfies
3. Generalized Cobb-Douglas Production Hypersurfaces
In this section, we give a nonexistence result of minimal generalized Cobb-Douglas production hypersurfaces in .
Theorem 2. There does not exist a minimal generalized Cobb-Douglas production hypersurface in .
Proof. Let be a generalized Cobb-Douglas production hypersurface in given by a graph
where is the generalized Cobb-Douglas production function
Note that the generalized Cobb-Douglas production function (23) is homogeneous of degree . It follows from (23) that
By assumption, the production hypersurface is minimal; that is, . Thus, by (18) and (10) we have
which reduces to
Note that for . Hence (26) becomes
Substituting (24) into (27), we obtain
Differentiating (28) with respect to , one has
which reduces to
Substituting (28) into (30), we have
Moreover, differentiating (31) with respect to (), we get
Combining (32) with (31) gives
Differentiating (33) with respect to again, one has
which yields either () or
Since is defined by (23) with nonzero constants for , it follows that (35) is impossible. Hence we have .
Without loss of generality, we may choose . In this case, for reduces to Therefore, (33) becomes The above equation implies that either or It is easy to see that the latter case is impossible. At this moment, we obtain the function as follows: Consider (31) for , 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 . Hence (41) becomes which is impossible since .
Therefore, there is not a minimal generalized Cobb-Douglas production hypersurface in . This completes the proof of Theorem 2.
4. 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 .
In the following, we deal with the case of minimal generalized CES production hypersurfaces in .
Theorem 3. An -factor generalized CES production hypersurface in 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 given by a graph
where is the generalized CES production function
with and . 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
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 or and
Moreover, it follows from (45) that
We divide it into two cases.
Case A (). Substituting (52) and (54) into (51), we get If , then the above equation reduces to which is equivalent to Hence Since , this case is impossible.
Therefore we conclude that . Substituting (53) into (50), we have Noting , it is easy to see that (59) is impossible.
Case B (). We note that the minimality condition reduces to Substituting (52)–(54) into (60), we have Note that fulfills (61). Hence, in this case the generalized CES production function is a perfect substitute.
In the following, we will show that the case is impossible. In fact, if , (61) reduces to Since and for , 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 is minimal. This completes the proof of Theorem 3.
Remark 4. Remark that Chen proved in  a more general result for -factor: -homogeneous production function is a perfect substitute if and only if the production surface is a minimal surface.
The authors would like to thank the referees for their very valuable comments and suggestions to improve this paper. This work is supported by the Mathematical Tianyuan Youth Fund of China (no. 11326068), the general Financial Grant from the China Postdoctoral Science Foundation (no. 2012M510818), and the General Project for Scientific Research of Liaoning Educational Committee (no. W2012186).
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