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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.

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 [

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 [

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

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

If

We recall that, for a production function

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 [

Let

where

where

where

In the last section of the paper, we generalize the above theorem for an arbitrary number of inputs

Consider the following.

(i) We first suppose that

But with

or

where

From (

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

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 (

From (

where

Therefore, from (

where

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

Let

A production function is said to satisfy the proportional marginal rate of substitution property if and only if

Let

(i) The elasticity of production is a constant

where

The elasticity of production is a constant

and

where

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

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

On the other hand, since

where

If we settle

Replacing now (

and solving the partial differential equations in (

where

The conclusion follows now easily from (

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

(iii) It is easy to show that if

then

On the other hand, since

From (

Finally, from the above system of partial differential equations, we obtain the solution

where

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.