Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 761832, 6 pages http://dx.doi.org/10.1155/2013/761832

Research Article

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

Xiaoshu Wang1 and Yu Fu2

1 School of Investment and Construction Management, Dongbei University of Finance and Economics, Dalian 116025, China

2 School of Mathematics and Quantitative Economics, Dongbei University of Finance and Economics, Dalian 116025, China

Correspondence should be addressed to Yu Fu; yufudufe@gmail.com Received 5 November 2013; Accepted 10 December 2013 Academic Editor: Grzegorz Lukaszewicz

Copyright © 2013 X. Wang and Y. 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. Vilcu (Vilcu, 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. Vilcu and G. E. Vllcu, 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 = bLkC1-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

F(xi,...,xn)=Axa11 •••<», (2)

where xt > 0 (i = 1 ,...,n), A is a positive constant, and a1,.. ,,an are nonzero constants.

In 1961, Arrow et al. [2] introduced another two-input production function given by

Q = b(aKr + (1-a)Lr)1/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

n \ylP

F(Xl,...,xn) = A(

where a¡, y, A, and p are nonzero constants and A, at > 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 En+1 through the map f : R+ ^ R++1,

f(xi ,---,xn) = (xi,...,xn,F(xi,...,xn)). (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 En+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(x1t..., xn) canbeidentifiedwith a graph of a nonparametric hypersurface of an Euclidean (n + 1)-space En+1 given by

f(x1 ,---,Xn) = (x1,...,Xn,F(x1,...,Xn)).

We recall some basic concepts in the theory of hypersurfaces in Euclidean spaces.

Let M" be a hypersurface in an n+1-dimension Euclidean space. The Gauss map v is defined by

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 Ç(p) of Mn 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 e 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),

where u,w e TpM and g is the metric tensor on M".

Moreover, the second fundamental form h is related to the shape operator Sp by

g (SpU, w) = g(h (u, w), Ç (p))

for u,w e TpM.

It is well known that the determinant of the shape operator Sp 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 dF/dx{, d2F/dxiXp ..., and so forth by Fj, Fjj,..., and so forth. Put

Recall some well-known results for a graph of hypersurface (6) in En+1 from [7,15,16].

Proposition 1. For a production hypersurface of En+1 defined

f(x1,...,x„) = (x1,...,Xn,F(x1,,x„)) (11) One has the following.

(1) The unit normal Ç is given by

(2) The coefficient g^ = g(d/dxi,d/dxj) of the metric tensor is given by

0ij = sij + FiFj>

where = 1 if i = j, otherwise 0. (3) The volume element is

v :Mn ~^Sn c En+1,

dV = ygîjdx1 A-- - Adxn.

(4) The inverse matrix (g,} ) of (gt: ) is

g,} = stj -

PJPj w2

(5) The matrix of the second fundamental form h is

h> = w

(6) The matrix of the shape operator Sp is

Note that the generalized Cobb-Douglas production function (23) is homogeneous of degree h = 1"=i a;. It follows from (23) that

F, = —F, i=l,...,n,

a (a - ^ . Fu =-2-p> i=1,---,n,

a.¡a:

Fu =-F, i = j, i,j= 1,..., n.

' XjXj

J Vi kj 1 ^pikpkpj

(7) The mean curvature H is given by

(8) The Gauss-Kronecker curvature G is

det htj det F¡j det gtj = Wn+2 .

^ 1W,) (18)

By assumption, the production hypersurface is minimal; that is, H = 0. Thus, by (18) and (10) we have

which reduces to

IF>i )(1 + IFM = IF-FJF.

i=l J i,j

IF , + IiFfFjj -F,FjF,j) = °. (26) •=i i,j

Note that 1 iJ(F^Fjj - FiFjFij) = 0 for i = j. Hence (26)

(19) becomes

(9) The sectional curvature K^ of the plane section spanned by d/dxt and d/dxj is

F F ■ ■- F2.

K = r"rJJ rt)

ij W2 (1 + F2 + F2)'

(10) The Riemann curvature tensor R satisfies

(r(JL JL)_L 9 ) FtlFjk - FikFji 0( (9X, 'dXj )dxk 'dxt ) W4 .

3. 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 En+ .

yF,, + -F,FjFlj) = 0. (27)

i=i i = j

Substituting (24) into (27), we obtain

(at - 1) -yxfxj o

¿—i v-2 ¿—i ,-2,-2

¡=i Ai ¡=jAiAj

Differentiating (28) with respect to %k, one has

2«k {a.k -1) - 2^iF2 y^^l + 2^KF2y «i«k + a2 xk xk i = jxixj xk ¡=1 xixk

which reduces to

Oi-1 + p2 yOlOl r2yaak + a

A i ' t v'V • A - •_-i A • A a

= 0. (30)

k i = j t j i=1 t k Substituting (28) into (30), we have

Proof. Let M be a generalized Cobb-Douglas production hypersurface in E"+1 given by a graph

f(xi,...,xn) = (Xi,...,Xn,F{Xi,...,X„)), (22)

where F is the generalized Cobb-Douglas production function

F(x1,...,xn) = Ax^ ■■■<».

-i "a(a-1) r2^aiak + O

2 /, 2 2 2 ^ I I I I Y \

xk i=1 xi i=1 xixk

= 0. (31)

Moreover, differentiating (31) with respect to Xt (I = k), we get

2ai (ai - 0 2aiak + 2ai r2 2al 2-n aiak + 02

3 2 xi xk

.F2 -tHLF2y:

v, ¿-1

A 7 1 .A- « A ?

l t=l i k

Combining (32) with (31) gives

« -1 ak + al 2 «к - 1 , v « (at - 0

2 2 2 xl xl xk

к i=1 i

= 0. (33)

Differentiating (33) with respect to xi again, one has

(ak + ai) (ai - 1L2 (ai - 1) («1 + 1)

3 2 xl xk

which yields either щ = 1 (I = k) or

(ak + «i)f2 = (ai + l)xk

= 0, (34)

Since F is defined by (23) with nonzero constants at for i = 1,...,n,it follows that (35) is impossible. Hence we have a-i = 1.

Without loss of generality, we may choose k = 1. In this case, ai = 1 for 1=1 reduces to

a2 = ■■■ = an = 1.

Therefore, (33) becomes

a1 + 1 2 al - 1

—1-F2 + —-= 0

2 2 2 у ^ у ^ 'у*

x1xi x1

The above equation implies that either at = -1 or

F2 = (a1 -1)xf. (38)

It is easy to see that the latter case is impossible. At this moment, we obtain the function F as follows:

F{x1,...,xn) = Ax-1x2 ■■■xn. Consider (31) for k = 1, which turns into

1 F2 ( 1 1\

— - — (— + ■■■+— )=°.

X1 Xk \X2 Xn)

Substituting (39) into (40), we have

2 2 2 1 1 2 A x2 •••xn [~2 +-+x? ¡=xk.

x2 xn ,

In view of (41), by comparing the degrees of the left and the right of the equality we have n = 3. Hence (41) becomes

A2 (x2 +xl) = x2k, k = 2,3,

which is impossible since A = 0.

Therefore, there is not a minimal generalized Cobb-Douglas production hypersurface in En+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

F(xi ,...,xn) = Taixi

for some nonzero constants a1,...,an.

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

Theorem 3. An n-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 that M is a generalized CES production hypersurface in En+ given by a graph

f(x1,...,xn) = (x1,...,xn,F(x1,...,xn)), (44) where F is the generalized CES production function

F(x1,...,xn) = A( T

with A,ai > 0 and xi > 0. The generalized CES production function (45) is homogeneous of degree y. By assumption that the production hypersurface is minimal, from (18) the minimality condition is equivalent to

IF i ){1 + TF ) = T F'F?'r

We define another minimal production hypersurface as f (Xx1,... ,Xxn) = (Xx1,... ,Xxn,F (Xx1,... ,Xxn)),

X e R+.

Since the CES production function is homogeneous of degree y, we conclude that the hypersurface given by

f(xu. ..,xn) = (x1,...,xn, Xy-1F (x1,..., xn)) (48)

is also minimal. In this case, the minimality condition (46) becomes

lFt i )(X2-2y + IF*) = I FtFjFtj. (49)

Comparing (49) with (46), we obtain that either y = 1 or y = 1 and

IF t = °> t=1

IWÜ =

••j

Moreover, it follows from (45) that

F, = AyaÇxr1(i

F,, = Aya?^2 ((p-1) Z44 + (Y-1) afx^)

n \v/p-1

1 Iapxp ) , (52)

n \ yip-2

x I Yapxp

F¡j = Ay (y- p) apafPxp 1^pj 1( Iapxp ) > l = ^

We divide it into two cases.

Case A (y=1). Substituting (52) and (54) into (51), we get

n X 3y'p-4

A3y3 (y-p)[Tajxp

: I (a2

2 p 2 p 2 p-2 2 p-2

x I (ai a.rXj Xj i •j

If p = y, then the above equation reduces to

2 p 2 p 2 p-2 2 p-2

2P-2) = 0,

¡•J

which is equivalent to

I2Px2P-2j =o

Ia22px22p-2 = 0.

Since p = 1, this case is impossible.

Therefore we conclude that p = y. Substituting (53) into (50), we have

Ay(y-l)(Xapxp-2

n n yi p-2

*(IaPxP)(I<xP) =o.

Noting y = 1,it is easy to see that (59) is impossible.

Case B (y = 1). We note that the minimality condition reduces to

IF,, + Y(pfpii -FtFjFtj) = 0. (60) ,=1 , = j

Substituting (52)-(54) into (60), we have

A^-i)!^)!!^^,

1/p-2 / n

(.P-1)\I

p p UkXk

p p apxp

=1 3/p-4

x I (a ' a'.x.' x'.

\ i J i J

2p p 2p-2 p-2 \

-A3 (1-p)(Xapx>

x X (aipap^ip-2^p-2) =

Note that p = 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 p = 1 is impossible. In fact, if p= 1, (61) reduces to

KOH^-p ) ll-p-p-2

■A2 H^IK^p^D (62)

\k = i ) i = j

2 2 p 2 p 2 p-2 2 p-2

+ A I \a{ a. x{ Xj

) = o.

Since xt > ° and at > ° 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 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.

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).

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