**Proof** **of** **Lemma** **1.** Because

f is a QH function of degree

q with weight vector

g, it follows that the generalized Euler identity in Equation (

7) is satisfied. In the following, we use the method of characteristics to solve the quasi-linear PDE in Equation (

7). The characteristic equations in the nonparametric form are

However, since

g has at least one nonzero entry, it is clear that there exists at least one index

$i\in \{1,\dots ,n\}$ such that

${g}_{i}\ne 0$. Next, we set this index

i. Consequently, the above characteristic equations can be written as

for

$j=1,\dots ,i-1,i+1,\dots ,n$, and

Integrating the above equations, we derive

for

$j=1,\dots ,i-1,i+1,\dots ,n$, where

${C}_{1},\dots ,{C}_{i-1},{C}_{i+1},\dots ,{C}_{n}$ are arbitrary real constants, and

where

C is an arbitrary real constant.

Therefore, the general solution of Equation (

7) can be written in the implicit form as

where

$\mathsf{\Phi}$ is an arbitrary function, and the conclusion follows immediately. □

**Proof** **of** **Theorem** **1.** i. The "if" part of the assertion follows by simple direct computation. Next, we prove the "only if" part of the assertion. Suppose that the production has constant elasticity

${k}_{i}$ with respect to an input

${x}_{i}$. Then, we have from Equation (

1):

We can distinguish now two cases.

Case 1. If there exists

$j\ne i$ such that

${g}_{j}\ne 0$, then it follows from Lemma 1 that the function

f can be expressed as

where

Taking into account that

$i\ne j$, from Equation (

17), we get

Using now Equations (

16) and (

19), we deduce

and, solving the partial differential equation (Equation (

20)), we obtain the solution

where

F is a twice differentiable real valued function of

$n-2$ variables and “

$\widehat{}$” over

${u}_{k}$ indicates that

${u}_{k}$ is omitted.

Replacing now Equation (

21) into Equation (

17), and taking account of Equation (

18), we derive that

f takes the form of Equation (

10). Hence, we have Case (b) of the statement.

Case 2. If

${g}_{j}=0$, for all

$j\ne i$, then it follows that

${g}_{i}\ne 0$. Therefore, Equation (

7) reduces to

From Equations (

16) and (

22), it follows immediately that

$q\ne 0$ and

${g}_{i}=\frac{q}{{k}_{i}}$, and the function

f is given by Equation (

9). Hence, we deduce Case (a) of the statement.

ii. If

f is given by Equation (

13) and the weights satisfy the condition in Equation (

12), then we can easily check by direct computation that the production has constant elasticity

${k}_{i}$ with respect to all inputs

${x}_{i}$,

$i\in \{1,2,\dots ,n\}$. Conversely, suppose that the partial output elasticity

${E}_{{x}_{i}}$ with respect to the input

${x}_{i}$ is a constant

${k}_{i}$, for all

$i\in \{1,2,\dots ,n\}$. Then, according to the number of non-zero values of weights, we distinguish the following two situations.

Case 1. Only one weight is non-zero. Then, we suppose that

${g}_{i}\ne 0$ and

${g}_{j}=0$, for all

$j\ne i$. Now, from the condition

${E}_{{x}_{i}}={k}_{i}$, we deduce from i. that

${k}_{i}=\frac{q}{{g}_{i}}$ and

f reduces to Equation (

9). Imposing now the condition

and taking account of Equations (

1) and (

9), we get the following system of PDEs

with the solution

where

C is a positive real constant. Replacing now Equation (

23) into Equation (

9), we obtain that

f reduces to the CD production function given by Equation (

13). Moreover, we remark that Equation (

12) is valid in this case.

Case 2. At least two weights are non-zero. Then, we suppose that

${g}_{i}\ne 0$ and

${g}_{j}\ne 0$, where

$i,j\in \{1,\dots ,n\}$,

$i\ne j$. Next, from the condition

${E}_{{x}_{i}}={k}_{i}$, we deduce from i. that

f reduces to Equation (

10). Imposing now the conditions

and taking account of Equations (

1), (

10) and (

11), we get the following system of PDEs

having the solution

where

C is a positive real constant. Replacing now Equation (

24) into Equation (

10) and taking account of Equation (

11), we obtain that

f takes the form

where

C is a positive real constant.

Finally, we impose the condition

and taking account of Equations (

1) and (

25), we derive that the weights must satisfy the relation

which is equivalent to the condition in Equation (

12). Consequently, replacing Equation (

26) into Equation (

25), we deduce that

f reduces to the CD production function given by Equation (

13).

iii. If either of Situation (a) or (b) occurs, then one can check easily by a direct calculation that the production function

f satisfies the PMRS property. Next, we prove the “only if” part of the assertion. Assume that a quasi-homogeneous production function

f of degree

q with weight vector

$({g}_{1},\dots ,{g}_{n})$ satisfies the PMRS property. Then, we deduce from Equations (

2) and (

3) that

Using the generalized Euler identity in Equation (

7) and the above relation in Equation (

27), we derive

We can distinguish now the following two situations.

Case 1. If the weights satisfy

${\sum}_{i=1}^{n}{g}_{i}\ne 0$, then it follows from Equation (

28) that

$q\ne 0$ and we get the following system of PDEs

having the solution

where

C is a positive real constant. Hence, we have Case (b) of the statement.

Case 2. If the weights satisfy

${\sum}_{i=1}^{n}{g}_{i}=0$, then it follows from Equation (

28) that

$q=0$. Next, we set an index

j such that

${g}_{j}>0$, and we deduce from Lemma 1 that

f can be written as

where

h is a differentiable function of

$n-1$ variables

$({u}_{1},\dots ,{u}_{n-1})$ expressed by Equation (

18).

Replacing now Equation (

29) into Equation (

27), we derive

for

$1\le i\le n-1$, with

$i\ne j$. Taking into account now that

${g}_{j}\ne 0$ and the weights satisfy

${\sum}_{i=1}^{n}{g}_{i}=0$, we find that Equation (

30) reduces to

Solving the above system of PDEs, we obtain the solution

where

F is a twice differentiable function of one variable. Finally, replacing Equations (

32) and (

18) into Equation (

29), we derive that

f is given by

and taking into account that

${\sum}_{i=1}^{n}{g}_{i}=0$, we conclude that

f takes the form of Equation (

14). Hence, we have Case (a) of the statement. □

In the particular case of two inputs, Theorem 1 reduces to the following.