A New Class of Production Function

In this article, we propose a new class of production functions in the new classical theory of economics and study its qualities based on Euler’s relation of quasi-homogeneous functions. In a market economy environment, it is crucial to establish a firm’s profitability and draw conclusions about the op-eration and to make a variety of assumptions using a function that is more consistent with reality in the future. We showed that the quasi-homogeneous function is a general form of the well known production functions such as Cobb-Douglas and Constant Elasticity of Substitution (CES). We have made some qualitative and quantitative analysis and compare our results with the classical models using statistical data of Japan.


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 is satisfy the generalized Euler identity: is chosen from the set M and the point ( ) Proof of Theorem 1 Let us set down composite function where 1 λ = , then on the other hand ( ) where the 1 λ = then 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 γ . 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 ( ) f is concave when 0 1 λ < < and convex when 1 λ > .

Proof of Proposition
From the definition of f function the following condition is satisfied if is a function defined on (*) as a sin- Differentiating the ( ) ϕ λ function twice, we can obtain the following equa- if we recall that f is positive then concave when 0 1 γ < < , and convex when 1 γ > .
Remark 1 We normalize that γ always 1, if so function's concave and con-

Quasi-Concave Functions and Properties
Definition 2 (Arrow, 1961) f is said to be quasi-convex on f is called quasi-linear if it is both quasi-concave and quasi-convex.
If the inequalities are strict, and, 1 2 x x ≠ , then the functions are called a strict convex and concave functions.
Proof Definition 2  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 1 2 , Proof of Theorem 4 Suppose first that f is quasi-concave on M, then As 0 λ + → , that is, taking the limit as λ approaches through positive numbers, we obtain ( )( ) establishing the quasi-concavity of f.

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): , where X ++ is said to be "economic domain" (when the resource's price increase, there is no change on production function on this domain) , this means that the production function is a quasi-concave on the domain.

4)
where A X 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 γ , 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 γ , then by substitution 1 2 1 2 , α α α α γ γ = = then we can obtain the following equation when 1 λ = . and it is traceability that 1 2 , 1 α α ≥ from the properties of concave function.
Again by substituting , z x z x α α = = , where 1 2 , z z 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.
( ) If we choose ( ) n n α σ = then (24) function have the following form: We can show that it is a general form of the production function. For instance 1) In the Equation (25), expressing by 1 2 , , x x y , we have 2 1 1 1 1 (27) is a Cobb-Douglas function (Romer, 1996), 2) Express ( ) if we express, express by 1 2 , , x x y then 6) The concavity and convexity conditions of the function (31)  The function is concave when and quasi-concave when

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. Firstly, we consider the Cobb-Duglas function that is

Conclusion
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.