On Homogeneous Production Functions with Proportional Marginal Rate of Substitution

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 𝑓 of class 𝐶^∞, 𝑓∶ℝ^𝑛₊→ℝ₊, 𝑓=𝑓(𝑥₁,𝑥₂,…,𝑥_𝑛), where 𝑓 is the quantity of output, 𝑛 is the number of the inputs, and 𝑥₁,𝑥₂,…,𝑥_𝑛 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 𝑝 such that 𝑄(𝜆⋅𝑥)=𝜆^𝑝𝑄(𝑥), for all 𝑥∈ℝ^𝑛₊ and 𝜆∈ℝ₊, that means if the inputs are multiplied by same factor, then the output is multiplied by some power of this factor. If 𝜆=1, then the function is said to have a constant return to scale, if 𝜆>1, then we have an increased return to scale, and if 𝜆<1, 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 𝑓=𝐶𝐾^𝛼𝐿^𝛽, (1) where 𝑓 is the total production, 𝐾 is the capital input, 𝐿 is the labor input, and 𝐶 is a positive constant which signifies the total factor productivity. We note that, in the original definition of Cobb and Douglas, we have 𝛼+𝛽=1, 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 𝑓(𝐾,𝐿)=^𝑛_𝑖=1𝛼_𝑖𝑐_𝑖1𝐾^{𝑝𝑖1+𝑝𝑖2}+𝑐_𝑖2𝐾^𝑝𝑖1𝐿^𝑝𝑖2+𝑐_𝑖3𝐿^{𝑝𝑖1+𝑝𝑖2𝑝𝑖3}, (2) where 𝑛≥1, 𝛼_𝑖≥0, 𝑝_𝑖3∈(−∞,0)∪[1,∞), 𝑝_𝑖1𝑝_𝑖2>0, 𝑝_𝑖3(𝑝_𝑖1+𝑝_𝑖2)=1, ∑^𝑛_𝑖=1(𝑐_𝑖2+𝑐_𝑖1𝑐_𝑖3)>0, and 𝑐_𝑖𝑗≥0, forall𝑖∈{1,2,…,𝑛}, forall𝑗∈{1,2,3}.

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 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 MRS=𝜕𝑓/𝜕𝐿,𝜕𝑓/𝜕𝐾 (3) where the elasticities of 𝐿 and 𝐾 are defined as 𝐸_𝐿=𝜕𝑓/𝜕𝐿𝑓/𝐿,𝐸_𝐾=𝜕𝑓/𝜕𝐾,𝑓/𝐾 (4) while the elasticity of substitution is given by 𝐾+𝐿×−𝜕𝜎=1/𝜕𝑓/𝜕𝐾1/𝜕𝑓/𝜕𝐿₂𝑓/𝜕𝐾²/𝜕𝑓/𝜕𝐾²+2𝜕²/−𝜕𝑓/𝜕𝐾𝜕𝐿𝜕𝑓/𝜕𝐾𝜕𝑓/𝜕𝐿₂𝑓/𝜕𝐿²/𝜕𝑓/𝜕𝐿²⁻¹. (5)

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: 𝜎=𝜕𝑓/𝜕𝐿𝜕𝑓/𝜕𝐾𝑓𝜕².𝑓/𝜕𝐾𝜕𝐿 (6)

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 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.(i)𝑓 has a constant elasticity of labor 𝑘 if and only if it is a Cobb-Douglas production function given by 𝑓(𝐾,𝐿)=𝐶𝐾^𝑟−𝑘𝐿^𝑘, (7) where 𝐶 is a positive constant. (ii)𝑓 has a constant elasticity of capital 𝑘 if and only if it is a Cobb-Douglas production function given by 𝑓(𝐾,𝐿)=𝐶𝐾^𝑘𝐿^𝑟−𝑘, (8) where 𝐶 is a positive constant. (iii)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 𝑓(𝐾,𝐿)=𝐶𝐾^{𝑟/(𝑘+1)}𝐿^{𝑟𝑘/(𝑘+1)}, (9) where 𝐶 is a positive constant.

In the last section of the paper, we generalize the above theorem for an arbitrary number of inputs 𝑛≥3. 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 𝑓 has a constant elasticity of labor 𝑘. Then, we have 𝜕𝑓𝑓𝜕𝐿=𝑘𝐿. (10) But with𝑓 being homogeneous of degree 𝑟, it follows that it can be written in the form 𝑓(𝐾,𝐿)=𝐾^𝑟ℎ(𝑢) (11) or 𝑓(𝐾,𝐿)=𝐿^𝑟ℎ(𝑢), (12) where 𝑢=𝐿/𝐾 (with 𝐾≠0), respectively, 𝑢=𝐾/𝐿 (with 𝐿≠0), 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)=𝐾^𝑟ℎ(𝑢), with 𝑢=𝐿/𝐾. Then, we have 𝜕𝑓𝜕𝐿=𝐾^𝑟−1ℎ^′(𝑢). (13) From (10) and (13), we obtain 𝐾^𝑟−1ℎ^′(𝑢)𝐾=𝑘^𝑟ℎ(𝑢)𝐿, (14) and therefore we deduce that the constant elasticity of labor property implies the following differential equation: ℎ^′(𝑢)ℎ(𝑢)=𝑘𝑢. (15) Solving the above separable differential equation, we obtain ℎ(𝑢)=𝐶𝑢^𝑘, (16) where 𝐶 is a positive constant. Finally, from (11) and (16), we derive that 𝑓 is a Cobb-Douglas production function given by 𝑓(𝐾,𝐿)=𝐶⋅𝐾^𝑟−𝑘𝐿^𝑘. (17) 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 𝜕𝑓𝐾𝜕𝐿=𝑘𝐿𝜕𝑓.𝜕𝐾 (18) On the other hand, from Euler’s homogeneous function theorem, we have 𝐾𝜕𝑓𝜕𝐾+𝐿𝜕𝑓().𝜕𝐿=𝑟𝑓𝐾,𝐿 (19) Combining now (18) and (19), we obtain 𝜕𝑓=𝑟𝜕𝐾𝑓𝑘+1𝐾. (20) From (20), we deduce that 𝑓(𝐾,𝐿)=𝐶𝐾^{𝑟/(𝑘+1)}𝑢(𝐿), (21) where 𝐶 is a real constant. But with𝑓 being a homogeneous function of degree 𝑟, it follows from (21) that 𝑢(𝐿)=𝐿^{𝑟𝑘/(𝑘+1)}. (22) Therefore, from (21) and (22), we derive that 𝑓(𝐾,𝐿)=𝐶𝐾^{𝑟/(𝑘+1)}𝐿^{𝑟𝑘/(𝑘+1)}, (23) 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,…,𝑥_𝑛, 𝑛>2. Then, the elasticity of production with respect to a certain factor of production 𝑥_𝑖 is defined as 𝐸_𝑥𝑖=𝜕𝑓/𝜕𝑥_𝑖𝑓/𝑥_𝑖, (24) while the marginal rate of technical substitution of input 𝑗 for input 𝑖 is given by MRS_𝑖𝑗=𝜕𝑓/𝜕𝑥_𝑗𝜕𝑓/𝜕𝑥_𝑖. (25)

A production function is said to satisfy the proportional marginal rate of substitution property if and only if MRS_𝑖𝑗=𝑥_𝑖/𝑥_𝑗, for all 1≤𝑖≠𝑗≤𝑛. 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,…,𝑥_𝑛) defined on 𝐷=ℝ^𝑛₊, where 𝑛>2. Then, one has the following. (i) The elasticity of production is a constant 𝑘_𝑖 with respect to a certain factor of production 𝑥_𝑖 if and only if 𝑓𝑥₁,𝑥₂,…,𝑥_𝑛=𝑥^𝑘𝑖_𝑖𝑥^{𝑟−𝑘𝑖}_𝑗𝐹𝑢₁,…,𝑢_𝑛−2, (26) 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 𝑢₁,…,𝑢_𝑛−2=𝑥_𝑘𝑥_𝑗∣𝑘∈{1,…,𝑛}⧵𝑖,𝑗. (27)(ii) The elasticity of production is a constant 𝑘_𝑖 with respect to all factors of production 𝑥_𝑖, 𝑖∈{1,2,…,𝑛}, if and only if 𝑘₁+𝑘₂+⋯+𝑘_𝑛=𝑟, (28) and 𝑓 reduces to the Cobb-Douglas production function given by 𝑓𝑥₁,𝑥₂,…,𝑥_𝑛=𝐶𝑥^𝑘1₁𝑥^𝑘2₂⋯𝑥^𝑘𝑛_𝑛, (29) where 𝐶 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 𝑓𝑥₁,𝑥₂,…,𝑥_𝑛=𝐶𝑥₁^𝑟/𝑛𝑥₂^𝑟/𝑛⋯𝑥_𝑛^𝑟/𝑛, (30) 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 𝑥_𝑖 is a constant 𝑘_𝑖, we have 𝜕𝑓𝜕𝑥_𝑖=𝑘_𝑖𝑓𝑥_𝑖. (31) On the other hand, since 𝑓 is a homogeneous of degree 𝑟, it follows that it can be expressed in the form 𝑓𝑥₁,…,𝑥_𝑛=𝑥^𝑟_𝑗ℎ𝑢₁,…,𝑢_𝑛−1, (32) where 𝑗 can be settled in the set {1,…,𝑛} and 𝑢_𝑘=𝑥_𝑘𝑥_𝑗𝑥,1≤𝑘≤𝑗−1,_𝑘+1𝑥_𝑗,𝑗≤𝑘≤𝑛−1. (33) If we settle 𝑗 such that 𝑗≠𝑖, then we derive from (32) 𝜕𝑓𝜕𝑥_𝑖=𝑥_𝑗^𝑟−1𝜕ℎ𝜕𝑢_𝑖𝑥,if𝑖<𝑗,_𝑗𝑟−1𝜕ℎ𝜕𝑢_𝑖−1,if𝑖>𝑗. (34) Replacing now (34) in (31), we obtain 𝑘_𝑖𝑢ℎ=^𝑖𝜕ℎ𝜕𝑢_𝑖𝑢,if𝑖<𝑗,_𝑖−1𝜕ℎ𝜕𝑢_𝑖−1,if𝑖>𝑗, (35) and solving the partial differential equations in (35), we derive ℎ𝑢₁,…,𝑢_𝑛−1=𝐶𝑢^𝑘𝑖_𝑖𝐹𝑢₁,…,𝑢_𝑖,…,𝑢_𝑛−1,if𝑖<𝑗,𝐶𝑢^𝑘𝑖_𝑖−1𝐹𝑢₁,…,𝑢_𝑖−1,…,𝑢_𝑛−1,if𝑖>𝑗, (36) 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 𝑓𝑥₁,𝑥₂,…,𝑥_𝑛=𝐶𝑥₁^𝑟/𝑛𝑥₂^𝑟/𝑛⋯𝑥_𝑛^𝑟/𝑛, (37) 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 𝑥₁𝜕𝑓𝜕𝑥₁=𝑥₂𝜕𝑓𝜕𝑥₂=⋯=𝑥_𝑛𝜕𝑓𝜕𝑥_𝑛. (38) On the other hand, since 𝑓 is a homogeneous of degree 𝑟, the Euler homogeneous function theorem implies that 𝑥₁𝜕𝑓𝜕𝑥₁+𝑥₂𝜕𝑓𝜕𝑥₂+⋯+𝑥_𝑛𝜕𝑓𝜕𝑥_𝑛=𝑟𝑓. (39) From (38) and (39), we obtain 𝑥_𝑖𝜕𝑓𝜕𝑥_𝑖=𝑟𝑛𝑓,𝑖∈{1,2,…,𝑛}. (40) Finally, from the above system of partial differential equations, we obtain the solution 𝑓𝑥₁,𝑥₂,…,𝑥_𝑛=𝐶𝑥₁^𝑟/𝑛𝑥₂^𝑟/𝑛⋯𝑥_𝑛^𝑟/𝑛, (41) 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.