#### Abstract

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 well known that the production function is one of the key concepts of mainstream neoclassical theories, with a lot of applications not only in microeconomics and macroeconomics but also in various fields, like biology [1, 2], educational management [3, 4], and engineering [5–8]. Roughly speaking, the production functions are the mathematical formalization of the relationship between the output of a firm/industry/economy and the inputs that have been used in obtaining it. In fact, a production function is a map of class , , , where is the quantity of output, is the number of the inputs, and are the factor inputs (such as labor, capital, land, and raw materials). In order for these functions to model as well the economic reality, they are required to be homogeneous; that is, there exists a real number such that , for all and , that means if the inputs are multiplied by same factor, then the output is multiplied by some power of this factor. If , then the function is said to have a constant return to scale, if , then we have an increased return to scale, and if , then we say that the function has a decreased return to scale. Among the family of production functions, the most famous is the Cobb-Douglas (CD) production function, introduced in 1928 by Cobb and Douglas [9], in order to describe the distribution of the national income of the USA. In its most standard form for production of a single good with two factors, the Cobb-Douglas production function is given by where is the total production, is the capital input, is the labor input, and is a positive constant which signifies the total factor productivity. We note that, in the original definition of Cobb and Douglas, we have , so the production function had a constant return to scale, but this condition has been later relaxed, and CD production functions were generalized (see [10, 11]). Some very interesting information about other production functions of great interest in economic analysis, like Leontief, Lu-Fletcher, Liu-Hildebrand, Kadiyala, Arrow-Chenery-Minhas-Solow (ACMS), constant elasticity of substitution (CES), and variable elasticity of substitution (VES) production functions, can be found in [12]. Recently, C. A. Ioan and G. Ioan [13] introduced a new class of production functions, called sum production function, as a two-factor production function defined by where , , , , , , and , , .

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 [13], 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 [13]). *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 [13]). *The sum production function has a constant elasticity of labor if and only if it reduces to the Cobb-Douglas function. *

Theorem 3 (see [13]). *If , 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 with two factors (-capital and -labor), the marginal rate of substitution (between capital and labor) is given by where the elasticities of and are defined as while the elasticity of substitution is given by

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:

We note that it was proved by Losonczi [14] 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 [15]. In the next section, we prove the following result which is a generalization of Theorems 1 and 2.

Theorem 4. *Let be a twice differentiable, homogeneous of degree , nonconstant, real valued production function with two inputs (-capital and -labor). Then, one has the following.*(i)* has a constant elasticity of labor if and only if it is a Cobb-Douglas production function given by
* *where is a positive constant. *(ii)* has a constant elasticity of capital if and only if it is a Cobb-Douglas production function given by
* *where is a positive constant. *(iii)* satisfies the proportional rate of substitution property between capital and labor (i.e., , where is a positive constant) if and only if it is a Cobb-Douglas production function given by
* *where is a positive constant. *

In the last section of the paper, we generalize the above theorem for an arbitrary number of inputs . We note that other classification results concerning production functions were proved recently in [16–20].

#### 2. Proof of Theorem 4

*Proof. *
Consider the following. (i) We first suppose that has a constant elasticity of labor . Then, we have
But with being homogeneous of degree , it follows that it can be written in the form
or
where (with ), respectively, (with ), and is a real valued function of , of class on its domain of definition. We can suppose, without loss of generality, that the first situation occurs, so , with . Then, we have
From (10) and (13), we obtain
and therefore we deduce that the constant elasticity of labor property implies the following differential equation:
Solving the above separable differential equation, we obtain
where is a positive constant. Finally, from (11) and (16), we derive that is a Cobb-Douglas production function given by
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
On the other hand, from Euler’s homogeneous function theorem, we have
Combining now (18) and (19), we obtain
From (20), we deduce that
where is a real constant. But with being a homogeneous function of degree , it follows from (21) that
Therefore, from (21) and (22), we derive that
where is a real constant. Finally, since is a nonconstant production function, it follows that , and therefore we deduce that is in fact a positive constant. So, 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 be a homogeneous production function with inputs , . Then, the elasticity of production with respect to a certain factor of production is defined as while the marginal rate of technical substitution of input for input is given by

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

Theorem 5. *Let be a twice differentiable, homogeneous of degree , nonconstant, real valued function of variables defined on , where . Then, one has the following.* *(i) The elasticity of production is a constant with respect to a certain factor of production if and only if
* *where is any element settled from the set and is a twice differentiable real valued function of variables
*(ii)* The elasticity of production is a constant with respect to all factors of production , , if and only if
* *and reduces to the Cobb-Douglas production function given by
* *where is a positive constant. *(iii)* 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
* *where 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 is a constant , we have
On the other hand, since is a homogeneous of degree , it follows that it can be expressed in the form
where can be settled in the set and
If we settle such that , then we derive from (32)
Replacing now (34) in (31), we obtain
and solving the partial differential equations in (35), we derive
where is a positive constant, is a twice differentiable real valued function of 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 is a Cobb-Douglas production function given by
then satisfies the proportional marginal rate of substitution property. We prove now the converse. Since satisfies the proportional marginal rate of substitution property, it follows that
On the other hand, since is a homogeneous of degree , the Euler homogeneous function theorem implies that
From (38) and (39), we obtain
Finally, from the above system of partial differential equations, we obtain the solution
where 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.