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

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.

1. 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 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 𝑌=𝑏𝐿^𝑘𝐶^1−𝑘, (1) where 𝐿 denotes the labor input, 𝐶 is the capital input, 𝑏 is the total factor productivity, and 𝑌 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 𝐹𝑥₁,…,𝑥_𝑛=𝐴𝑥^𝛼1₁⋯𝑥^𝛼𝑛_𝑛, (2) where 𝑥_𝑖>0 (𝑖=1,…,𝑛), 𝐴 is a positive constant, and 𝛼₁,…,𝛼_𝑛 are nonzero constants.

In 1961, Arrow et al. [2] introduced another two-input production function given by 𝑄=𝑏𝑎𝐾^𝑟+(1−𝑎)𝐿^𝑟1/𝑟, (3) where 𝑄 is the output, 𝑏 the factor productivity, 𝑎 the share parameter, 𝐾 and 𝐿 the primary production factors, 𝑟=(𝑠−1)/𝑠, and 𝑠=1/(1−𝑟) 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 𝐹𝑥₁,…,𝑥_𝑛=𝐴^𝑛_𝑖=1𝑎^𝜌_𝑖𝑥^𝜌_𝑖^𝛾/𝜌, (4) where 𝑎_𝑖, 𝛾, 𝐴, and 𝜌 are nonzero constants and 𝐴,𝑎_𝑖>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 𝐹 can be identified with a production hypersurface in the (𝑛+1)-dimensional Euclidean space 𝔼^𝑛+1 through the map 𝑓∶ℝ^𝑛₊→ℝ₊^𝑛+1, 𝑓𝑥₁,…,𝑥_𝑛=𝑥₁,…,𝑥_𝑛,𝐹𝑥₁,…,𝑥_𝑛. (5) In the case of 𝑛 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 𝑛-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 [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^𝑛+1. 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 𝐹(𝑥₁,…,𝑥_𝑛) can be identified with a graph of a nonparametric hypersurface of an Euclidean (𝑛+1)-space 𝔼^𝑛+1 given by 𝑓𝑥₁,…,𝑥_𝑛=𝑥₁,…,𝑥_𝑛,𝐹𝑥₁,…,𝑥_𝑛. (6) We recall some basic concepts in the theory of hypersurfaces in Euclidean spaces.

Let 𝑀^𝑛 be a hypersurface in an 𝑛+1-dimension Euclidean space. The Gauss map 𝑣 is defined by 𝑣∶𝑀^𝑛⟶𝑆^𝑛⊂𝔼^𝑛+1, (7) which maps 𝑀^𝑛 to the unit hypersphere 𝑆^𝑛 of 𝔼^𝑛+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(𝑝) is a unit normal vector 𝜉(𝑝) of 𝑀^𝑛 at point 𝑝.

The differential 𝑑𝑣 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 𝑝∈𝑀^𝑛, the tangent space 𝑇_𝑝𝑀 is an inner product space, the shape operator 𝑆_𝑝 can be defined as a linear operator on this space by the relation 𝑔𝑆_𝑝𝑢,𝑤=𝑔(𝑑𝑣(𝑢),𝑤), (8) where 𝑢,𝑤∈𝑇_𝑝𝑀 and 𝑔 is the metric tensor on 𝑀^𝑛.

Moreover, the second fundamental form ℎ is related to the shape operator 𝑆_𝑝 by 𝑔𝑆_𝑝𝑢,𝑤=𝑔ℎ(𝑢,𝑤),𝜉𝑝 (9) for 𝑢,𝑤∈𝑇_𝑝𝑀.

It is well known that the determinant of the shape operator 𝑆_𝑝 is called the Gauss-Kronecker curvature. Denote it by 𝐺(𝑝). When 𝑛=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 𝑆_𝑝 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.

Denote the partial derivatives 𝜕𝐹/𝜕𝑥_𝑖, 𝜕²𝐹/𝜕𝑥_𝑖𝑥_𝑗,…, and so forth by 𝐹_𝑖,𝐹_𝑖𝑗,…, and so forth. Put 𝑊=1+^𝑛_𝑖=1𝐹²_𝑖. (10) Recall some well-known results for a graph of hypersurface (6) in 𝔼^𝑛+1 from [7, 15, 16].

Proposition 1. For a production hypersurface of 𝔼^𝑛+1 defined by 𝑓𝑥₁,…,𝑥_𝑛=𝑥₁,…,𝑥_𝑛,𝐹𝑥₁,,𝑥_𝑛 (11) One has the following.(1)The unit normal 𝜉 is given by 1𝜉=𝑊−𝐹₁,…,−𝐹_𝑛,1. (12)(2)The coefficient 𝑔_𝑖𝑗=𝑔(𝜕/𝜕𝑥_𝑖,𝜕/𝜕𝑥_𝑗) of the metric tensor is given by𝑔_𝑖𝑗=𝛿_𝑖𝑗+𝐹_𝑖𝐹_𝑗, (13)where 𝛿_𝑖𝑗=1 if 𝑖=𝑗, otherwise 0. (3)The volume element is 𝑑𝑉=𝑔_𝑖𝑗𝑑𝑥₁∧⋯∧𝑑𝑥_𝑛. (14)(4)The inverse matrix (𝑔^𝑖𝑗) of (𝑔_𝑖𝑗) is𝑔^𝑖𝑗=𝛿_𝑖𝑗−𝐹_𝑖𝐹_𝑗𝑊². (15)(5)The matrix of the second fundamental form ℎ isℎ_𝑖𝑗=𝐹_𝑖𝑗𝑊. (16)(6)The matrix of the shape operator 𝑆_𝑝 is𝑎^𝑗_𝑖=_𝑘ℎ_𝑖𝑘𝑔^𝑘𝑗=1𝑊𝐹_𝑖𝑗−_𝑘𝐹_𝑖𝑘𝐹_𝑘𝐹_𝑗𝑊². (17)(7)The mean curvature 𝐻 is given by1𝐻=𝑛𝑊_𝑖𝐹_𝑖𝑖−1𝑊²_𝑖,𝑗𝐹_𝑖𝐹_𝑗𝐹_𝑖𝑗. (18)(8)The Gauss-Kronecker curvature 𝐺 is𝐺=detℎ_𝑖𝑗det𝑔_𝑖𝑗=det𝐹_𝑖𝑗𝑊^𝑛+2. (19)(9)The sectional curvature 𝐾_𝑖𝑗 of the plane section spanned by 𝜕/𝜕𝑥_𝑖 and 𝜕/𝜕𝑥_𝑗 is 𝐾_𝑖𝑗=𝐹_𝑖𝑖𝐹_𝑗𝑗−𝐹²_𝑖𝑗𝑊²1+𝐹²_𝑖+𝐹²_𝑗. (20)(10)The Riemann curvature tensor 𝑅 satisfies𝑔𝑅𝜕𝜕𝑥_𝑖,𝜕𝜕𝑥_𝑗𝜕𝜕𝑥_𝑘,𝜕𝜕𝑥_𝑙=𝐹_𝑖𝑙𝐹_𝑗𝑘−𝐹_𝑖𝑘𝐹_𝑗𝑙𝑊⁴. (21)

3. Generalized Cobb-Douglas Production Hypersurfaces

In this section, we give a nonexistence result of minimal generalized Cobb-Douglas production hypersurfaces in 𝔼^𝑛+1.

Theorem 2. There does not exist a minimal generalized Cobb-Douglas production hypersurface in 𝔼^𝑛+1.

Proof. Let 𝑀 be a generalized Cobb-Douglas production hypersurface in 𝔼^𝑛+1 given by a graph 𝑓𝑥₁,…,𝑥_𝑛=𝑥₁,…,𝑥_𝑛,𝐹𝑥₁,…,𝑥_𝑛, (22) where 𝐹 is the generalized Cobb-Douglas production function 𝐹𝑥₁,…,𝑥_𝑛=𝐴𝑥^𝛼1₁⋯𝑥^𝛼𝑛_𝑛. (23) Note that the generalized Cobb-Douglas production function (23) is homogeneous of degree ℎ=∑^𝑛_𝑖=1𝛼_𝑖. It follows from (23) that 𝐹_𝑖=𝛼_𝑖𝑥_𝑖𝐹𝐹,𝑖=1,…,𝑛,_𝑖𝑖=𝛼_𝑖𝛼_𝑖−1𝑥²_𝑖𝐹𝐹,𝑖=1,…,𝑛,_𝑖𝑗=𝛼_𝑖𝛼_𝑗𝑥_𝑖𝑥_𝑗𝐹,𝑖≠𝑗,𝑖,𝑗=1,…,𝑛. (24) By assumption, the production hypersurface is minimal; that is, 𝐻=0. Thus, by (18) and (10) we have ^𝑛_𝑖=1𝐹_𝑖𝑖1+^𝑛_𝑖=1𝐹²_𝑖=_𝑖,𝑗𝐹_𝑖𝐹_𝑗𝐹_𝑖𝑗, (25) which reduces to ^𝑛_𝑖=1𝐹_𝑖𝑖+_𝑖,𝑗𝐹²_𝑖𝐹_𝑗𝑗−𝐹_𝑖𝐹_𝑗𝐹_𝑖𝑗=0. (26) Note that ∑_𝑖,𝑗(𝐹²_𝑖𝐹_𝑗𝑗−𝐹_𝑖𝐹_𝑗𝐹_𝑖𝑗)=0 for 𝑖=𝑗. Hence (26) becomes ^𝑛_𝑖=1𝐹_𝑖𝑖+_𝑖≠𝑗𝐹²_𝑖𝐹_𝑗𝑗−𝐹_𝑖𝐹_𝑗𝐹_𝑖𝑗=0. (27) Substituting (24) into (27), we obtain ^𝑛_𝑖=1𝛼_𝑖𝛼_𝑖−1𝑥²_𝑖−𝐹²_𝑖≠𝑗𝛼²_𝑖𝛼_𝑗𝑥²_𝑖𝑥²_𝑗=0. (28) Differentiating (28) with respect to 𝑥_𝑘, one has −2𝛼_𝑘𝛼_𝑘−1𝑥³_𝑘−2𝛼_𝑘𝑥_𝑘𝐹²_𝑖≠𝑗𝛼²_𝑖𝛼_𝑗𝑥²_𝑖𝑥²_𝑗+2𝛼_𝑘𝑥_𝑘𝐹^2𝑛_𝑖=1𝛼_𝑖𝛼_𝑘+𝛼²_𝑖𝑥²_𝑖𝑥²_𝑘=0, (29) which reduces to 𝛼_𝑘−1𝑥²_𝑘+𝐹²_𝑖≠𝑗𝛼²_𝑖𝛼_𝑗𝑥²_𝑖𝑥²_𝑗−𝐹^2𝑛_𝑖=1𝛼_𝑖𝛼_𝑘+𝛼²_𝑖𝑥²_𝑖𝑥²_𝑘=0. (30) Substituting (28) into (30), we have 𝛼_𝑘−1𝑥²_𝑘+^𝑛_𝑖=1𝛼_𝑖𝛼_𝑖−1𝑥²_𝑖−𝐹^2𝑛_𝑖=1𝛼_𝑖𝛼_𝑘+𝛼²_𝑖𝑥²_𝑖𝑥²_𝑘=0. (31) Moreover, differentiating (31) with respect to 𝑥_𝑙 (𝑙≠𝑘), we get −2𝛼_𝑙𝛼_𝑙−1𝑥³_𝑙+2𝛼_𝑙𝛼_𝑘+2𝛼²_𝑙𝑥³_𝑙𝑥²_𝑘𝐹²−2𝛼_𝑙𝑥_𝑙𝐹^2𝑛_𝑖=1𝛼_𝑖𝛼_𝑘+𝛼²_𝑖𝑥²_𝑖𝑥²_𝑘=0. (32) Combining (32) with (31) gives 𝛼_𝑙−1𝑥²_𝑙−𝛼_𝑘+𝛼_𝑙𝑥²_𝑙𝑥²_𝑘𝐹²+𝛼_𝑘−1𝑥²_𝑘+^𝑛_𝑖=1𝛼_𝑖𝛼_𝑖−1𝑥²_𝑖=0. (33) Differentiating (33) with respect to 𝑥_𝑙 again, one has 𝛼_𝑘+𝛼_𝑙𝛼_𝑙−1𝑥³_𝑙𝑥²_𝑘𝐹²+𝛼_𝑙−1𝛼₁+1𝑥³_𝑙=0, (34) which yields either 𝛼_𝑙=1 (𝑙≠𝑘) or 𝛼_𝑘+𝛼_𝑙𝐹²=𝛼_𝑙+1𝑥²_𝑘. (35) Since 𝐹 is defined by (23) with nonzero constants 𝛼_𝑖 for 𝑖=1,…,𝑛, it follows that (35) is impossible. Hence we have 𝛼_𝑙=1.
Without loss of generality, we may choose 𝑘=1. In this case, 𝛼_𝑙=1 for 𝑙≠1 reduces to 𝛼₂=⋯=𝛼_𝑛=1. (36) Therefore, (33) becomes −𝛼₁+1𝑥²₁𝑥²_𝑙𝐹²+𝛼²₁−1𝑥²₁=0. (37) The above equation implies that either 𝛼₁=−1 or 𝐹²=𝛼₁−1𝑥²_𝑙. (38) It is easy to see that the latter case is impossible. At this moment, we obtain the function 𝐹 as follows: 𝐹𝑥₁,…,𝑥_𝑛=𝐴𝑥₁⁻¹𝑥₂⋯𝑥_𝑛. (39) Consider (31) for 𝑘≠1, which turns into 1𝑥²₁−𝐹²𝑥²_𝑘1𝑥²₂1+⋯+𝑥²_𝑛=0. (40) Substituting (39) into (40), we have 𝐴²𝑥²₂⋯𝑥²_𝑛1𝑥²₂1+⋯+𝑥²_𝑛=𝑥²_𝑘. (41) In view of (41), by comparing the degrees of the left and the right of the equality we have 𝑛=3. Hence (41) becomes 𝐴²𝑥²₂+𝑥²₃=𝑥²_𝑘,𝑘=2,3, (42) which is impossible since 𝐴≠0.
Therefore, there is not a minimal generalized Cobb-Douglas production hypersurface in 𝔼^𝑛+1. 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 𝐹𝑥₁,…,𝑥_𝑛=^𝑛_𝑖=1𝑎_𝑖𝑥_𝑖, (43) for some nonzero constants 𝑎₁,…,𝑎_𝑛.

In the following, we deal with the case of minimal generalized CES production hypersurfaces in 𝔼^𝑛+1.

Theorem 3. An 𝑛-factor generalized CES production hypersurface in 𝔼^𝑛+1 is minimal if and only if the generalized CES production function is a perfect substitute.

Proof. Assume that 𝑀 is a generalized CES production hypersurface in 𝔼^𝑛+1 given by a graph 𝑓𝑥₁,…,𝑥_𝑛=𝑥₁,…,𝑥_𝑛,𝐹𝑥₁,…,𝑥_𝑛, (44) where 𝐹 is the generalized CES production function 𝐹𝑥₁,…,𝑥_𝑛=𝐴^𝑛_𝑖=1𝑎^𝜌_𝑖𝑥^𝜌_𝑖^𝛾/𝜌, (45) with 𝐴,𝑎_𝑖>0 and 𝑥_𝑖>0. The generalized CES production function (45) is homogeneous of degree 𝛾. By assumption that the production hypersurface is minimal, from (18) the minimality condition is equivalent to ^𝑛_𝑖=1𝐹_𝑖𝑖1+^𝑛_𝑖=1𝐹²_𝑖=_𝑖,𝑗𝐹_𝑖𝐹_𝑗𝐹_𝑖𝑗. (46) We define another minimal production hypersurface as 𝑓𝜆𝑥₁,…,𝜆𝑥_𝑛=𝜆𝑥₁,…,𝜆𝑥_𝑛,𝐹𝜆𝑥₁,…,𝜆𝑥_𝑛,𝜆∈ℝ⁺. (47) Since the CES production function is homogeneous of degree 𝛾, we conclude that the hypersurface given by 𝑓𝑥₁,…,𝑥_𝑛=𝑥₁,…,𝑥_𝑛,𝜆^𝛾−1𝐹𝑥₁,…,𝑥_𝑛 (48) is also minimal. In this case, the minimality condition (46) becomes ^𝑛_𝑖=1𝐹_𝑖𝑖𝜆^2−2𝛾+^𝑛_𝑖=1𝐹²_𝑖=_𝑖,𝑗𝐹_𝑖𝐹_𝑗𝐹_𝑖𝑗. (49) Comparing (49) with (46), we obtain that either 𝛾=1 or 𝛾≠1 and ^𝑛_𝑖=1𝐹_𝑖𝑖=0,_𝑖,𝑗𝐹_𝑖𝐹_𝑗𝐹_𝑖𝑗=0. (50)(51) Moreover, it follows from (45) that 𝐹_𝑖=𝐴𝛾𝑎^𝜌_𝑖𝑥_𝑖^𝜌−1𝑛_𝑖=1𝑎^𝜌_𝑖𝑥^𝜌_𝑖^{𝛾/𝜌−1},𝐹_𝑖𝑖=𝐴𝛾𝑎^𝜌_𝑖𝑥_𝑖^𝜌−2𝜌−1_𝑘≠𝑖𝑎^𝜌_𝑘𝑥^𝜌_𝑘+𝛾−1𝑎^𝜌_𝑖𝑥^𝜌_𝑖×^𝑛_𝑖=1𝑎^𝜌_𝑖𝑥^𝜌_𝑖^{𝛾/𝜌−2},𝐹_𝑖𝑗=𝐴𝛾𝛾−𝜌𝑎^𝜌_𝑖𝑎^𝜌_𝑗𝑥_𝑖^𝜌−1𝑥_𝑗^𝜌−1𝑛_𝑖=1𝑎^𝜌_𝑖𝑥^𝜌_𝑖^{𝛾/𝜌−2},𝑖≠𝑗. (52)(53)(54) We divide it into two cases.
Case A (𝛾≠1). Substituting (52) and (54) into (51), we get 𝐴³𝛾³𝛾−𝜌^𝑛_𝑖=1𝑎^𝜌_𝑖𝑥^𝜌_𝑖^{3𝛾/𝜌−4}×_𝑖,𝑗𝑎_𝑖^2𝜌𝑎_𝑗^2𝜌𝑥_𝑖^2𝜌−2𝑥_𝑗^2𝜌−2=0. (55) If 𝜌≠𝛾, then the above equation reduces to _𝑖,𝑗𝑎_𝑖^2𝜌𝑎_𝑗^2𝜌𝑥_𝑖^2𝜌−2𝑥_𝑗^2𝜌−2=0, (56) which is equivalent to ^𝑛_𝑖=1𝑎_𝑖^2𝜌𝑥_𝑖^2𝜌−22=0. (57) Hence ^𝑛_𝑖=1𝑎_𝑖^2𝜌𝑥_𝑖^2𝜌−2=0. (58) Since 𝜌≠1, this case is impossible.
Therefore we conclude that 𝜌=𝛾. Substituting (53) into (50), we have 𝐴𝛾𝛾−1^𝑛_𝑖=1𝑎^𝜌_𝑖𝑥_𝑖^𝜌−2×^𝑛_𝑖=1𝑎^𝜌_𝑖𝑥^𝜌_𝑖^𝑛_𝑖=1𝑎^𝜌_𝑖𝑥^𝜌_𝑖^{𝛾/𝜌−2}=0. (59) Noting 𝛾≠1, it is easy to see that (59) is impossible.
Case B (𝛾=1). We note that the minimality condition reduces to ^𝑛_𝑖=1𝐹_𝑖𝑖+_𝑖≠𝑗𝐹²_𝑖𝐹_𝑗𝑗−𝐹_𝑖𝐹_𝑗𝐹_𝑖𝑗=0. (60) Substituting (52)–(54) into (60), we have 𝐴𝜌−1_𝑘≠𝑖𝑎^𝜌_𝑘𝑥^𝜌_𝑘^𝑛_𝑖=1𝑎^𝜌_𝑖𝑥^𝜌_𝑖^{1/𝜌−2𝑛}_𝑖=1𝑎^𝜌_𝑖𝑥_𝑖^𝜌−2+𝐴³𝜌−1_𝑘≠𝑖𝑎^𝜌_𝑘𝑥^𝜌_𝑘^𝑛_𝑖=1𝑎^𝜌_𝑖𝑥^𝜌_𝑖^3/𝜌−4×_𝑖≠𝑗𝑎_𝑖^2𝜌𝑎^𝜌_𝑗𝑥_𝑖^2𝜌−2𝑥_𝑗^𝜌−2−𝐴³1−𝜌^𝑛_𝑖=1𝑎^𝜌_𝑖𝑥^𝜌_𝑖^3/𝜌−4×_𝑖≠𝑗𝑎_𝑖^2𝜌𝑎_𝑗^2𝜌𝑥_𝑖^2𝜌−2𝑥_𝑗^2𝜌−2=0. (61) Note that 𝜌=1 fulfills (61). Hence, in this case the generalized CES production function 𝐹 is a perfect substitute.
In the following, we will show that the case 𝜌≠1 is impossible. In fact, if 𝜌≠1, (61) reduces to _𝑘≠𝑖𝑎^𝜌_𝑘𝑥^𝜌_𝑘^𝑛_𝑖=1𝑎^𝜌_𝑖𝑥^𝜌_𝑖^{2−2/𝜌𝑛}_𝑖=1𝑎^𝜌_𝑖𝑥_𝑖^𝜌−2+𝐴²_𝑘≠𝑖𝑎^𝜌_𝑘𝑥^𝜌_{𝑘𝑖≠𝑗}𝑎_𝑖^2𝜌𝑎^𝜌_𝑗𝑥_𝑖^2𝜌−2𝑥_𝑗^𝜌−2+𝐴²_𝑖≠𝑗𝑎_𝑖^2𝜌𝑎_𝑗^2𝜌𝑥_𝑖^2𝜌−2𝑥_𝑗^2𝜌−2=0. (62) Since 𝑥_𝑖>0 and 𝑎_𝑖>0 for 𝑖=1,…,𝑛, 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 𝑛-factor generalized CES production hypersurface in 𝔼^𝑛+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.

Acknowledgments

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).