1. Quasi-Homogeneous Production Functions and Its Properties
The production function is one of the key concepts of mainstream neoclassical theories. By assuming that the maximum output technologically possible from a given set of inputs is achieved, economists use a production function in analysis, representing the mathematical formalization of the relationship between production and the factors that actually contribute to that production. Such a function is a mapping
, given by
where n denotes the number of factors of production (inputs), are the inputs and f is the level of output (production).
Definition 1 (Alina-Daniela & Gabriel-Eduard, 2019) Let f be a differentiable function of n variables defined on the open set then is said to be a quasi-homogeneous function if the following condition holds
of degree with weight vector and all . For instance, the production function defined by is a quasi-homogeneous production function of degree 1 with weight vector .
Theorem 1 (Alina-Daniela & Gabriel-Eduard, 2019) Let f be a differentiable function of n variables defined on the open set of degree with weight vector and all . then is satisfy the generalized Euler identity:
Let point is chosen from the set M and the point , .
Proof of Theorem 1 Let us set down composite function , , .
Functions and f are differentiable where the then the f can be differentiable where the point . We recall
and solve the where , then
on the other hand , whereas
where the then
is the any point selected from M.
Theorem 2 If f be a differentiable function of n variables defined on the open set of degree with weight vector and all then by substituting , it can be a quasi-homogeneous function with variables .
Proof of Theorem 2 According to the given condition of the f function, the following equation must be true
substitution of f function, the equation can be defined as follows
to use the composite-function differential rule then
by substitution, we can obtain the following equation
We recall Euler’s theorem, we can prove that f is quasi-homogeneous function of degree . Now let’s construct the general form of the quasi-homogeneous function. The equation that was mentioned theorem 1, for a f function. If we integrate the Equation (1) and estimate the independent original integrals then we can obtain the solution as follows
if we substitute y for main variables then
(11) are quasi-homogeneous function of n variables , defined on the open set , of exponent degree will be exhausted.
If f is quasi-homogeneous function of degree . In accordance with the following curve,
f is concave when and convex when .
Proof of Proposition
From the definition of f function the following condition is satisfied if
therefore we can imagine is a function defined on (*) as a single variable as follows
Differentiating the function twice, we can obtain the following equation
if we recall that f is positive then concave when , and convex when .
Remark 1 We normalize that always 1, if so function’s concave and convex conditions are defined by . A quasi-homogeneous production function of n variables defined on the open set of degree 1 with weight vector is concave when .
2. Quasi-Concave Functions and Properties
Definition 2 (Arrow, 1961) f is said to be quasi-convex on if
f is said to be quasi-concave on if
f is called quasi-linear if it is both quasi-concave and quasi-convex.
If the inequalities are strict, and, , then the functions are called a strict convex and concave functions.
Proof Definition 2
f is quasi-convex if and only if is quasi-concave. Note that
First, suppose that f is quasi-concave. If the upper contour set of f
and the lower-contour set of f
are convex sets for each . Let and . Assume, without loss of generality, that, , letting, , we have . By convexity of , we have .
Conversely, suppose is convex and empty or contains only one point, it is clear that it is convex, but suppose it contains at least two points and then and , so
by hypothesis, and so , hence is a convex set and f is quasi concave. Quasi-concave function is a generalization of concave functions since we can show that the set of all quasi-concave functions contains the set of all concave functions.
Theorem 3 (Mayer, 2007) If f is concave on M, it is also quasi concave on M. If f is convex on M, it is also quasi-convex on M.
We prove the first part of the theorem and the second part will be proved analogously.
Proof of Theorem 3 Suppose f is concave, then, for all and we have
Theorem 4 (Intriligator, 2002) let f be a differentiable function on M, where is convex and open. Then f is quasi-concave on M if and only if and .
Proof of Theorem 4 Suppose first that f is quasi-concave on M, then
since f is quasi-concave, we have
As , that is, taking the limit as approaches through positive numbers, we obtain , proving one part of the result.
Assume that for all such that , we have . Pick any , and suppose without loss of generality that , we also have
establishing the quasi-concavity of f.
3. Production Function and Its Properties
An economy’s output of goods and services depends on its quantity of inputs, called the factors of production, and its ability to turn inputs into outputs. The two most important factors of production are capital and labor, also the technology change alters the production function (Solow, 1956). We consider the functions that satisfy the following conditions (Intriligator, 2002):
1) , where is the domain of function.
2) , where is said to be “economic domain” (when the resource’s price increase, there is no change on production function on this domain)
3) , this means that the production function is a quasi-concave on the domain.
4) . where is said to be the set of profit.
5) The production function is twice differentiable.
6) The production function is a quasi-homogeneous of degree , with weight vector .
Let’s consider production function with two variables. According to the condition (6) that was mentioned above, then the production function has the following general form:
according to the Euler’s theorem
if we divide both sides , then by substitution then we can obtain the following equation
when . and it is traceability that from the properties of concave function.
Again by substituting , , , where are said to be effective labors, then we get the following equation.
Then the production function has the following general form:
where S is per labor capita and U is per capita production. By the economic low, U and S’s growth velocity rate is defined by the following equation.
we can solve the equation then
If we express the equation by S then we have
If we choose then (24) function have the following form:
We can show that it is a general form of the production function.
1) In the Equation (25), If then
expressing by , we have
(27) is a Cobb-Douglas function (Romer, 1996), .
2) Express then .
3) In the Equation (25), and then
expressing by then
is a Solow’s function.
4) When then , it is a function with degree and constant elasticity.
5) In Equation (25), then
if we express, then
express by then
6) The concavity and convexity conditions of the function (31) are
The function is concave when
and quasi-concave when
4. Numerical Experiments
In this part, we evaluate and compare the parameters of the quasi homogeneous function and classical production functions using specific economic indicators of Japan which are shown in Table 1.
Table 1. The main indicators of Japan.
Statistics Bureau of Japan https://www.stat.go.jp/english/ ( Statistics Bureau of JAPAN).
Firstly, we consider the Cobb-Duglas function that is
and the parameters are evaluated as
and determination coefficient is
Second, we evaluate the parameters of our new function given by
then we have following results
In this paper, we proposed a quasi-homogeneous production function and showed how to construct the production function based on Euler’s theorem and the hypothesis that the production function is a quasi-homogeneous and quasi-concave. We also proved that the classical production functions are the special cases of the quasi-homogeneous production function. The numerical results show that the quasi-homogeneous function is practically more useful.