A New Class of Production Function Model and Its Application

1 Introduction

Since the mathematician Cobb and the economist Douglas from the America proposed the concept of production function, and later the production function had been widely used in practices. The production function represents the relationship between input factors and the output. Through production function model, it is allowable that the influence rate of capital, labor, scientific and technological progress and other factors to economic development to be analyzed. The production functions have always been an active field in economy studies and applications[1–8], most researchers used C-D (Cobb-Dauglas) production function to analyze the economic issues, but it has limitations, the most notably is the elasticity of substitution σ is constant at 1, while in CES (constant elasticity of substitution) production function, the elasticity of substitution σ is not constant at 1, but this constant variable will keep invariable for one specific production function, such as the production function for certain industry. In fact, the elasticity of substitution will vary with different sample points, for example, it may change according to the ratio of capital to labor K/L, and it is based on such thought that variable eleasticity of substitution (VES) is proposed. For VES, there are many theoretical and methodical achievements from researches[9–11], the most well-known is the model proposed by Revankar in 1971 and the model proposed by Sato and Hoffman in 1968, the former assumes the elasticity of substitution σ is a linear function of K/L, while the latter assumes σ is a function of the time t. As a matter of fact, in some cases, the elasticity of substitution σ perhaps is a non-linear function of K/L, therefore, this paper present a circumstance where σ is a quadratic function of K/L and we call this new function as MVES function model. Finally, based on this new model, this paper derives a calculation formula to accurately measure the influence rate of economic growth factors[12–14], and finally analyzes the influence rate of China’s economic growth factors from empirical perspectives.

2 General Form of Common Production Function Models

The production function is the mathematical expression of relationship between changes in input factors, such as the capital K, the labor L, the technology A and the output Y during production process, that is, Y = F(K,L). In case that Y = F(λK, λL) = λ^mF(K,L), namely, the production function is m homogeneous expression, it means the output Y will grow at λ^m times if input of the capital K and the labor L increase at m times simultaneously. m = 1 denotes constant returns to scale, 0 < m < 1 denotes decreasing returns to scale, and m > 1 denotes increasing returns to scale.

Under the circumstance that two production factors can substitute each other, it can produce the same products with different factor combinations, so, the elasticity of substitution between two factors would be the ratio of the increase percentage rate of proportion two factors account for to the increase percentage rate of marginal rate of substitution of these two factors, that is

It assumes the production function Y = F(K,L) is m homogeneous, that is,

Set k=KL,y=f(k), so the above expression is transformed into

Set the elasticity of substitution of two factors σ = g(k), then

Given that Y = L^my, we have

So

that is,

The above expression so is transformed into

Set z=y′y, then y′ = zy, take its derivative, we have y″ = z′y + zy′. As a result,

that is

Set w=−1z, we have

The general solution of the above equation is

Then

By the integration, we have

We have the general expression of production function in this way,

where k=KL.

3 A New Class of Production Function Model and Parameters Estimation

4 The Method to Measure the Influence Rates of Economic Growth

Express MVES production function as

were (A,K,L) are inputs, Y is the output, (A,b,c,m) are parameters to be estimated.

Take its differential, we have

Set the elasticity coefficient of the capital α=∂Y∂K⋅KY=∂F∂K⋅KF, and the elasticity coefficient of the labor β=∂Y∂L⋅LY=∂F∂L⋅LF, then

Provided that the economic vector (A,K,L,Y) changes into (A^{(t + 1)},K^{(t + 1)},L^{(t + 1)},Y^{(t + 1)}) of the year t + 1 from (A^(t),K^(t),L^(t),Y^(t)) of the year t, then

Record as

then

Given

So

It is obvious that ΔYA(t) is the absolute influence rate value of technological progress to economic growth at t period; ΔYK(t) is the absolute influence rate value of capital to economic growth at t period; ΔYL(t) is the absolute influence rate value of labor to economic growth at t period.

Then, the influence rate of capital growth to economic growth at t period is KCI(t)=ΔYE(t)Y(t), the influence rate of labor growth to economic growth at t period is LCI(t)=ΔYL(t)Y(t).

Use the method of residues to calculate the influence rate of technological progress, from

We have the influence rate of technological progress to economic growth at t period is

where

where

In above equation, k=KL,#1=4c2m2−6,#2=cm,#3=9bk24+94,#4=−6cm2+3m,#5=#14−#13+1.5,#6=#14+#13+1.5,#7=3cm2+cm2#17,#8=3m+m#17,#9=1.5bc2km2#14,#10=3b#17#153b,#11=k−#163b,#12=k+#163b,#13=3b#17k2,#14=cm1.5bk2+1.5,#15=1.5b,#16=3b#17,#17=2c2m2−3.

The curve L^(t) has many forms, this paper selects 3rd order polynomial,

We can get the calculation of ΔYK(t)andΔYK(t) by numerical integration.

5 Empirical Analysis on Influence Rates of China’s Economic Growth Factors

In order to intensively study China’s economic growth and explore its growth pattern, as well as measure the influence rate of input factors to economic growth, we take the gross domestic product (Y, in a hundred million Yuan) as the comprehensive representative indicator of economic growth, and take the number of employees (L, in ten thousand ), fixed asset input(K, in a hundred million Yuan) as the influence factors to the economy, and the information is shown in Table 1.

Here, it establishes the MVES production function model, based on non-linear least square method, and take advantages of MATLAB 2014a, we have

The coefficient of determination is

It can be seen that the model has high fitting accuracy.

We have ΔYA(t)=ΔY(t)−ΔYK(t)−ΔYL(t) by taking numerical integration of ΔYK(t),ΔYL(t), and further have the influence rate of capital growth to economic growth, KCI^(t) at each year; the influence rate of labor growth to economic growth, LCI^(t) at each year; the influence rate of technological progress to economic growth, ACI^(t) at each year; as shown in Table 1.

The annual average influence rate of capital growth to economic growth during 1994–2013 is

The annual average influence rate of labor growth to economic growth during 1994–2013 is

The annual average influence rate of technological progress to economic growth during 1994–2013 is

Namely, from 1994 to 2013, the average annual growth rate of China’s economy was 13.97%, in which, 10.06% was contributed by the capital growth, and 0.37% was contributed by the labor growth, and 3.44% was contributed by technological progress. It is observed that China’s economic growth is mainly dependent on capital input, and then on technological progress, and the influence of labor is less, therefore, the results are consistent with the law of China’s economic growth.

6 Conclusion

This paper derived the general form of common production function model by use of differential equation, and under the circumstance that the elasticity of substitution σ is a quadratic function of K/L, a new class of production function model (MVES) was provided. And then the new model was used to derive the calculation equation that is able to precisely measure the influence rate of economic growth factors, which is very significant for the general application of MVES production function and scientific measurement and calculation. In the end of this paper, the influence ratio of various factors to China’s economic growth was analyzed from empirical perspective, and good results were attained. This paper will provide valuable advices for reasonable arrangement of limited resources, and acceleration of economic growth, as well as powerful theoretical foundation for the government leaders to make decisions.