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

Authors: Xiaoshu Wang; Yu Fu
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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
(1)Y=bLkC1-k,
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(2)F(x1,…,xn)=Ax1α1⋯xnαn,
where xi>0 (i=1,…,n), A is a positive constant, and α1,…,αn are nonzero constants.In 1961, Arrow et al. [2] introduced another two-input production function given by
(3)Q=b(aKr+(1-a)Lr)1/r,
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(4)F(x1,…,xn)=A(∑i=1naiρxiρ)γ/ρ,
where ai, γ, A, and ρ are nonzero constants and A,ai>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 functionF can be identified with a production hypersurface in the (n+1)-dimensional Euclidean space 𝔼n+1 through the map f:ℝ+n→ℝ+n+1,
(5)f(x1,…,xn)=(x1,…,xn,F(x1,…,xn)).
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 space𝔼n+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 functionF(x1,…,xn) can be identified with a graph of a nonparametric hypersurface of an Euclidean (n+1)-space 𝔼n+1 given by
(6)f(x1,…,xn)=(x1,…,xn,F(x1,…,xn)).
We recall some basic concepts in the theory of hypersurfaces in Euclidean spaces.LetMn be a hypersurface in an n+1-dimension Euclidean space. The Gauss map v is defined by
(7)v:Mn⟶Sn⊂𝔼n+1,
which maps Mn to the unit hypersphere Sn of 𝔼n+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 Mn at point p.The differentialdv 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∈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
(8)g(Spu,w)=g(dv(u),w),
where u,w∈TpM and g is the metric tensor on Mn.Moreover, the second fundamental formh is related to the shape operator Sp by
(9)g(Spu,w)=g(h(u,w),ξ(p))
for u,w∈TpM.It is well known that the determinant of the shape operatorSp 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 Sp 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/∂xi, ∂2F/∂xixj,…, and so forth by Fi,Fij,…, and so forth. Put
(10)W=1+∑i=1n‍Fi2.
Recall some well-known results for a graph of hypersurface (6) in 𝔼n+1 from [7, 15, 16].Proposition 1.
For a production hypersurface of𝔼n+1 defined by
(11)f(x1,…,xn)=(x1,…,xn,F(x1,,xn))
One has the following.(1)
The unit normalξis given by(12)ξ=1W(-F1,…,-Fn,1).(2)
The coefficientgij=g(∂/∂xi,∂/∂xj)of the metric tensor is given by(13)gij=δij+FiFj,whereδij=1ifi=j, otherwise
0.(3)
The volume element is(14)dV=gijdx1∧⋯∧dxn.(4)
The inverse matrix(gij)of(gij)is(15)gij=δij-FiFjW2.(5)
The matrix of the second fundamental formhis(16)hij=FijW.(6)
The matrix of the shape operatorSpis(17)aij=∑k‍hikgkj=1W(Fij-∑k‍FikFkFjW2).(7)
The mean curvatureHis given by(18)H=1nW(∑i‍Fii-1W2∑i,j‍FiFjFij).(8)
The Gauss-Kronecker curvatureGis(19)G=dethijdetgij=detFijWn+2.(9)
The sectional curvatureKijof the plane section spanned by∂/∂xiand∂/∂xjis(20)Kij=FiiFjj-Fij2W2(1+Fi2+Fj2).(10)
The Riemann curvature tensorRsatisfies(21)g(R(∂∂xi,∂∂xj)∂∂xk,∂∂xl)=FilFjk-FikFjlW4.


## 3. Generalized Cobb-Douglas Production Hypersurfaces

In this section, we give a nonexistence result of minimal generalized Cobb-Douglas production hypersurfaces in𝔼n+1.Theorem 2.
There does not exist a minimal generalized Cobb-Douglas production hypersurface in𝔼n+1.Proof.
LetM be a generalized Cobb-Douglas production hypersurface in 𝔼n+1 given by a graph
(22)f(x1,…,xn)=(x1,…,xn,F(x1,…,xn)),
where F is the generalized Cobb-Douglas production function
(23)F(x1,…,xn)=Ax1α1⋯xnαn.
Note that the generalized Cobb-Douglas production function (23) is homogeneous of degree h=∑i=1n‍αi. It follows from (23) that
(24)Fi=αixiF,i=1,…,n,Fii=αi(αi-1)xi2F,i=1,…,n,Fij=αiαjxixjF,i≠j,i,j=1,…,n.
By assumption, the production hypersurface is minimal; that is, H=0. Thus, by (18) and (10) we have
(25)(∑i=1n‍Fii)(1+∑i=1n‍Fi2)=∑i,j‍FiFjFij,
which reduces to
(26)∑i=1n‍Fii+∑i,j‍(Fi2Fjj-FiFjFij)=0.
Note that ∑i,j‍(Fi2Fjj-FiFjFij)=0 for i=j. Hence (26) becomes
(27)∑i=1n‍Fii+∑i≠j‍(Fi2Fjj-FiFjFij)=0.
Substituting (24) into (27), we obtain
(28)∑i=1n‍αi(αi-1)xi2-F2∑i≠j‍αi2αjxi2xj2=0.
Differentiating (28) with respect to xk, one has
(29)-2αk(αk-1)xk3-2αkxkF2∑i≠j‍αi2αjxi2xj2+2αkxkF2∑i=1n‍αiαk+αi2xi2xk2=0,
which reduces to
(30)αk-1xk2+F2∑i≠j‍αi2αjxi2xj2-F2∑i=1n‍αiαk+αi2xi2xk2=0.
Substituting (28) into (30), we have
(31)αk-1xk2+∑i=1n‍αi(αi-1)xi2-F2∑i=1n‍αiαk+αi2xi2xk2=0.
Moreover, differentiating (31) with respect to xl (l≠k), we get
(32)-2αl(αl-1)xl3+2αlαk+2αl2xl3xk2F2-2αlxlF2∑i=1n‍αiαk+αi2xi2xk2=0.
Combining (32) with (31) gives
(33)αl-1xl2-αk+αlxl2xk2F2+αk-1xk2+∑i=1n‍αi(αi-1)xi2=0.
Differentiating (33) with respect to xl again, one has
(34)(αk+αl)(αl-1)xl3xk2F2+(αl-1)(α1+1)xl3=0,
which yields either αl=1 (l≠k) or
(35)(αk+αl)F2=(αl+1)xk2.
Since F is defined by (23) with nonzero constants αi for i=1,…,n, it follows that (35) is impossible. Hence we have αl=1.
Without loss of generality, we may choosek=1. In this case, αl=1 for l≠1 reduces to
(36)α2=⋯=αn=1.
Therefore, (33) becomes
(37)-α1+1x12xl2F2+α12-1x12=0.
The above equation implies that either α1=-1 or
(38)F2=(α1-1)xl2.
It is easy to see that the latter case is impossible. At this moment, we obtain the function F as follows:
(39)F(x1,…,xn)=Ax1-1x2⋯xn.
Consider (31) for k≠1, which turns into
(40)1x12-F2xk2(1x22+⋯+1xn2)=0.
Substituting (39) into (40), we have
(41)A2x22⋯xn2(1x22+⋯+1xn2)=xk2.
In view of (41), by comparing the degrees of the left and the right of the equality we have n=3. Hence (41) becomes
(42)A2(x22+x32)=xk2,k=2,3,
which is impossible since A≠0.
Therefore, there is not a minimal generalized Cobb-Douglas production hypersurface in𝔼n+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(43)F(x1,…,xn)=∑i=1n‍aixi,
for some nonzero constants a1,…,an.In the following, we deal with the case of minimal generalized CES production hypersurfaces in𝔼n+1.Theorem 3.
Ann-factor generalized CES production hypersurface in 𝔼n+1 is minimal if and only if the generalized CES production function is a perfect substitute.Proof.
Assume thatM is a generalized CES production hypersurface in 𝔼n+1 given by a graph
(44)f(x1,…,xn)=(x1,…,xn,F(x1,…,xn)),
where F is the generalized CES production function
(45)F(x1,…,xn)=A(∑i=1naiρxiρ‍)γ/ρ,
with A,ai>0 and xi>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
(46)(∑i=1n‍Fii)(1+∑i=1n‍Fi2)=∑i,j‍FiFjFij.
We define another minimal production hypersurface as
(47)f~(λx1,…,λxn)=(λx1,…,λxn,F(λx1,…,λxn)),=(λx1,…,λxn,111111F(λx1,…,λxn)),λ∈ℝ+.
Since the CES production function is homogeneous of degree γ, we conclude that the hypersurface given by
(48)f~(x1,…,xn)=(x1,…,xn,λγ-1F(x1,…,xn))
is also minimal. In this case, the minimality condition (46) becomes
(49)(∑i=1n‍Fii)(λ2-2γ+∑i=1n‍Fi2)=∑i,j‍FiFjFij.
Comparing (49) with (46), we obtain that either γ=1 or γ≠1 and
(50)∑i=1n‍Fii=0,(51)∑i,j‍FiFjFij=0.
Moreover, it follows from (45) that
(52)Fi=Aγaiρxiρ-1(∑i=1naiρxiρ)γ/ρ-1,(53)Fii=Aγaiρxiρ-2((ρ-1)∑k≠i‍akρxkρ+(γ-1)aiρxiρ)×(∑i=1naiρxiρ)γ/ρ-2,(54)Fij=Aγ(γ-ρ)aiρajρxiρ-1xjρ-1(∑i=1naiρxiρ)γ/ρ-2,i≠j.
We divide it into two cases.
Case A (γ≠1). Substituting (52) and (54) into (51), we get
(55)A3γ3(γ-ρ)(∑i=1naiρxiρ)3γ/ρ-4×∑i,j‍(ai2ρaj2ρxi2ρ-2xj2ρ-2)=0.
If ρ≠γ, then the above equation reduces to
(56)∑i,j‍(ai2ρaj2ρxi2ρ-2xj2ρ-2)=0,
which is equivalent to
(57)(∑i=1nai2ρxi2ρ-2)2=0.
Hence
(58)∑i=1n‍ai2ρxi2ρ-2=0.
Since ρ≠1, this case is impossible.
Therefore we conclude thatρ=γ. Substituting (53) into (50), we have
(59)Aγ(γ-1)(∑i=1n‍aiρxiρ-2)×(∑i=1n‍aiρxiρ)(∑i=1naiρxiρ)γ/ρ-2=0.
Noting γ≠1, it is easy to see that (59) is impossible.
Case B (γ=1). We note that the minimality condition reduces to
(60)∑i=1n‍Fii+∑i≠j‍(Fi2Fjj-FiFjFij)=0.
Substituting (52)–(54) into (60), we have
(61)A(ρ-1)(∑k≠i‍akρxkρ)(∑i=1naiρxiρ)1/ρ-2(∑i=1n‍aiρxiρ-2)+A3(ρ-1)(∑k≠i‍akρxkρ)(∑i=1naiρxiρ)3/ρ-4×∑i≠j‍(ai2ρajρxi2ρ-2xjρ-2)-A3(1-ρ)(∑i=1naiρxiρ)3/ρ-4×∑i≠j‍(ai2ρaj2ρxi2ρ-2xj2ρ-2)=0.
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 caseρ≠1 is impossible. In fact, if ρ≠1, (61) reduces to
(62)(∑k≠i‍akρxkρ)(∑i=1naiρxiρ)2-2/ρ(∑i=1n‍aiρxiρ-2)+A2(∑k≠i‍akρxkρ)∑i≠j‍(ai2ρajρxi2ρ-2xjρ-2)+A2∑i≠j‍(ai2ρaj2ρxi2ρ-2xj2ρ-2)=0.
Since xi>0 and ai>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 then-factor generalized CES production hypersurface in 𝔼n+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.



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