MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 732643 10.1155/2013/732643 732643 Research Article On Homogeneous Production Functions with Proportional Marginal Rate of Substitution Vîlcu Alina Daniela 1 0000-0001-6922-756X Vîlcu Gabriel Eduard 2, 3 Milovanovic Gradimir 1 Department of Information Technology, Mathematics and Physics Petroleum-Gas University of Ploieşti Bulevardul Bucureşti No. 39 100680 Ploieşti Romania upg-ploiesti.ro 2 Faculty of Mathematics and Computer Science Research Center in Geometry, Topology and Algebra University of Bucharest Street Academiei No. 14, Sector 1 70109 Bucharest Romania unibuc.ro 3 Department of Mathematical Modelling, Economic Analysis and Statistics Petroleum-Gas University of Ploieşti Bulevardul Bucureşti No. 39 100680 Ploieşti Romania upg-ploiesti.ro 2013 14 3 2013 2013 11 12 2012 10 02 2013 2013 Copyright © 2013 Alina Daniela Vîlcu and Gabriel Eduard Vîlcu. 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.

We completely classify homogeneous production functions with proportional marginal rate of substitution and with constant elasticity of labor and capital, respectively. These classifications generalize some recent results of C. A. Ioan and G. Ioan (2011) concerning the sum production function.

1. Introduction

It is easy to see that this production function is homogeneous of degree 1 and integrates in an unitary expression various production functions, including CD, CES, and VES. In , C. A. Ioan and G. Ioan compute the principal indicators of the sum production function and prove three theorems of characterization for the functions with a proportional marginal rate of substitution, with constant elasticity of labor and for those with constant elasticity of substitution, as follows.

Theorem 1 (see [<xref ref-type="bibr" rid="B14">13</xref>]).

The sum production function has a proportional marginal rate of substitution if and only if it reduces to the Cobb-Douglas function.

Theorem 2 (see [<xref ref-type="bibr" rid="B14">13</xref>]).

The sum production function has a constant elasticity of labor if and only if it reduces to the Cobb-Douglas function.

Theorem 3 (see [<xref ref-type="bibr" rid="B14">13</xref>]).

If n=1, then the sum production function has constant elasticity of substitution if and only if it reduces to the Cobb-Douglas or CES function.

We recall that, for a production function f with two factors (K-capital and L-labor), the marginal rate of substitution (between capital and labor) is given by (3)MRS=f/Lf/K, where the elasticities of L and K are defined as (4)EL=f/Lf/L,EK=f/Kf/K, while the elasticity of substitution is given by (5)σ=((1/(K(f/K)))+(1/(L(f/L))))×(-((2f/K2)/(f/K)2)+((2(2f/KL))/((f/K)(f/L)))-((2f/L2)/(f/L)2)(f/K)2)-1.

It is easy to verify that, in the case of constant return to scale, Euler’s theorem implies the following more simple expression for the elasticity of substitution: (6)σ=(f/L)(f/K)f(2f/KL).

We note that it was proved by Losonczi  that twice differentiable two-input homogeneous production functions with constant elasticity of substitution (CES) property are Cobb-Douglas and ACMS production functions, which is obviously a more general result than Theorem 3. This result was recently generalized by Chen for an arbitrary number of inputs . In the next section, we prove the following result which is a generalization of Theorems 1 and 2.

Theorem 4.

Let f be a twice differentiable, homogeneous of degree r, nonconstant, real valued production function with two inputs (K-capital and L-labor). Then, one has the following.

f has a constant elasticity of labor k if and only if it is a Cobb-Douglas production function given by (7)f(K,L)=CKr-kLk,

where C is a positive constant.

f has a constant elasticity of capital k if and only if it is a Cobb-Douglas production function given by (8)f(K,L)=CKkLr-k,

where C is a positive constant.

f satisfies the proportional rate of substitution property between capital and labor (i.e., MRS=k(K/L), where k is a positive constant) if and only if it is a Cobb-Douglas production function given by (9)f(K,L)=CKr/(k+1)Lrk/(k+1),

where C is a positive constant.

In the last section of the paper, we generalize the above theorem for an arbitrary number of inputs n3. We note that other classification results concerning production functions were proved recently in .

2. Proof of Theorem <xref ref-type="statement" rid="thm1.4">4</xref> Proof.

Consider the following.

(i) We first suppose that f has a constant elasticity of labor k. Then, we have (10)fL=kfL.

But withf being homogeneous of degree r, it follows that it can be written in the form (11)f(K,L)=Krh(u)

or (12)f(K,L)=Lrh(u),

where u=L/K (with K0), respectively, u=K/L (with L0), and h is a real valued function of u, of class C2 on its domain of definition. We can suppose, without loss of generality, that the first situation occurs, so f(K,L)=Krh(u), with u=L/K. Then, we have (13)fL=Kr-1h(u).

From (10) and (13), we obtain (14)Kr-1h(u)=kKrh(u)L,

and therefore we deduce that the constant elasticity of labor property implies the following differential equation: (15)h(u)=kh(u)u.

Solving the above separable differential equation, we obtain (16)h(u)=Cuk,

where C is a positive constant. Finally, from (11) and (16), we derive that f is a Cobb-Douglas production function given by (17)f(K,L)=C·Kr-kLk.

The converse is easy to verify.

(ii) The proof follows similarly as in (i).

(iii) Since the production function satisfies the proportional rate of substitution property, it follows that (18)fL=kKLfK.

On the other hand, from Euler’s homogeneous function theorem, we have (19)KfK+LfL=rf(K,L).

Combining now (18) and (19), we obtain (20)fK=rk+1fK.

From (20), we deduce that (21)f(K,L)=CKr/(k+1)u(L),

where C is a real constant. But withf being a homogeneous function of degree r, it follows from (21) that (22)u(L)=Lrk/(k+1).

Therefore, from (21) and (22), we derive that (23)f(K,L)=CKr/(k+1)Lrk/(k+1),

where C is a real constant. Finally, since f is a nonconstant production function, it follows that f>0, and therefore we deduce that C is in fact a positive constant. So, f is a Cobb-Douglas production function.

The converse is easy to check, and the proof is now complete.

3. Generalization to an Arbitrary Number of Inputs

Let f be a homogeneous production function with n inputs x1,x2,,xn, n>2. Then, the elasticity of production with respect to a certain factor of production xi is defined as (24)Exi=f/xif/xi, while the marginal rate of technical substitution of input j for input i is given by (25)MRSij=f/xjf/xi.

A production function is said to satisfy the proportional marginal rate of substitution property if and only if MRSij=xi/xj, for all 1ijn. Now, we are able to prove the following result, which generalizes Theorem 4 for an arbitrary number of inputs.

Theorem 5.

Let f be a twice differentiable, homogeneous of degree r, nonconstant, real valued function of n variables (x1,x2,,xn) defined on D=+n, where n>2. Then, one has the following.

(i) The elasticity of production is a constant ki with respect to a certain factor of production xi if and only if (26)f(x1,x2,,xn)=xikixjr-kiF(u1,,un-2),

where j is any element settled from the set {1,,n}{i} and F is a twice differentiable real valued function of n-2 variables (27){u1,,un-2}={xkxjk{1,,n}{i,j}}.

The elasticity of production is a constant ki with respect to all factors of production xi, i{1,2,,n}, if and only if (28)k1+k2++kn=r,

and f reduces to the Cobb-Douglas production function given by (29)f(x1,x2,,xn)=Cx1k1x2k2xnkn,

where C is a positive constant.

The production function satisfies the proportional marginal rate of substitution property if and only if it reduces to the Cobb-Douglas production function given by (30)f(x1,x2,,xn)=Cx1r/nx2r/nxnr/n,

where C is a positive constant.

Proof.

Consider the following.

(i) The if part of the statement is easy to verify. Next, we prove the only if part. Since the elasticity of production with respect to a certain factor of production xi is a constant ki, we have (31)fxi=kifxi.

On the other hand, since f is a homogeneous of degree r, it follows that it can be expressed in the form (32)f(x1,,xn)=xjrh(u1,,un-1),

where j can be settled in the set {1,,n} and (33)uk={xkxj,1k  j-1,xk+1xj,jkn-1.

If we settle j such that ji, then we derive from (32) (34)fxi={xjr-1hui,if  i<j,xjr-1hui-1,if  i>j.

Replacing now (34) in (31), we obtain (35)kih={uihui,if  i<j,ui-1hui-1,if  i>j,

and solving the partial differential equations in (35), we derive (36)h(u1,,un-1)={CuikiF(u1,,ui^,,un-1),if  i<j,Cui-1kiF(u1,,ui-1^,,un-1),if  i>j,

where C is a positive constant, F is a twice differentiable real valued function of n-2 variables and the symbol “^” means that the corresponding term is omitted.

The conclusion follows now easily from (32) and (36), taking into account (33).

(ii) This assertion follows immediately from (i).

(iii) It is easy to show that if f is a Cobb-Douglas production function given by (37)f(x1,x2,,xn)=Cx1r/nx2r/nxnr/n,

then f satisfies the proportional marginal rate of substitution property. We prove now the converse. Since f satisfies the proportional marginal rate of substitution property, it follows that (38)x1fx1=x2fx2==xnfxn.

On the other hand, since f is a homogeneous of degree r, the Euler homogeneous function theorem implies that (39)x1fx1+x2fx2++xnfxn=rf.

From (38) and (39), we obtain (40)xifxi=rnf,  i{1,2,,n}.

Finally, from the above system of partial differential equations, we obtain the solution (41)f(x1,x2,,xn)=Cx1r/nx2r/nxnr/n,

where C is a positive constant and the conclusion follows.

Acknowledgments

The authors would like to thank the referees for carefully reading the paper and making valuable comments and suggestions. The second author was supported by CNCS-UEFISCDI, Project no. PN-II-ID-PCE-2011-3-0118.

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