Constant Elasticity Of Substitution Production Function Research Articles

Econometric Modeling of Input-Driven Output Risk through a Versatile CES Production Function

The conventional functional form of the Constant-Elasticity-of-Substitution (CES) production function is a general production function nesting a number of other forms of production functions. Examples of such functions include Leontief, Cobb–Douglas, and linear production functions. Nevertheless, the conventional form of the CES production specification is still restrictive in multiple aspects. One example is the fact that the marginal effect of increasing input use always has to be to increase the variability of output quantity by the conventional construction of this function. This paper proposes a generalized variant of the CES production function that allows for various input effects on the probability distribution of output. Failure to allow for this possible input–output risk structure is indeed one of the limitations of the conventional form of the CES production function. This limitation may result in false inferences about input-driven output risk. In light of this, the present paper proposes a solution to this problem. First, it is shown that the familiar CES formulation suffers from very restrictive structural assumptions regarding risk considerations, and that such restrictions may lead to biased and inefficient estimates of production quantity and production risk. Following the general theme of Just and Pope’s approach, a CES-based production-function specification that overcomes this shortcoming of the original CES production function is introduced, and a three-stage Nonlinear Least-Squares (NLS) estimation procedure for the estimation of the proposed functional form is presented. To illustrate the proposed approaches in this paper, two empirical applications in irrigation and fertilizer response using the famous Hexem–Heady experimental dataset are provided. Finally, implications for modeling input-driven production risks are discussed.

A novel method for calculating the contribution rates of economic growth factors

<abstract><p>The common production functions include the Leontief production function, the Cobb-Douglas (C-D) production function, the constant elasticity of substitution (CES) production function, the variable elasticity of substitution (VES) production function and so on. With different elasticity of substitution of factor, the production functions have different ranges of applications. In the production functions, the C-D production function is used the most widely because of its simple form, while the CES production function and the VES production function have limitations in applications due to their complicated forms. However, the C-D production function has the elasticity of substitution of factors of 1, and the CES production function has the elasticity of substitution of factors which is not 1 but a constant, so the two production functions both have limitations in applications. The VES production function with the variable elasticity of substitution is more practical in some application cases. This paper studies the applications of the VES production function model and gives a method of calculating the contribution rates of economic growth factors scientifically. As for the parameter estimation of the model, this paper gives an improved Sine Cosine Algorithm (SCA) to enhance the convergence rate and precision. Finally, the paper makes an empirical analysis on the contribution rates of economic growth factors of Shanghai City, China, using the method proposed.</p></abstract>

Generalizing the General: generalizing the CES production function to allow for the viability of input thresholds

ABSTRACT The original specification of the Constant-Elasticity-of-Substitution (CES) production function introduced by Arrow, Chenery, Minhas, and Solow is considered to be a general production specification that nests multiple types of production functions, i.e. Leontief, Cobb-Douglas, and linear. However, even this general specification of production functions is restrictive in several ways. This paper proposes a generalized variant of the CES production function that allows for the inclusion of the minimum required levels of inputs. Not allowing for this potential attribute is, in fact, one shortcoming of the original CES production-function specification, which in turn could result in misleading conclusions about essential levels of inputs. Accordingly, a solution is proposed to overcome the mentioned shortcoming. Input thresholds are incorporated in the CES production specification, and empirical applications are provided for irrigation and nitrogen. To illustrate the proposed approach in this paper, two empirical applications in irrigation and fertilizer response using the famous Hexem-Heady experimental dataset as well as several datasets produced using Monte-Carlo experiments with different data-generating processes are provided. Finally, implications for modelling input thresholds are considered and discussed.

Elasticity of substitution and biased technical change in the CES production function for China's metal-intensive industries

To achieve sustainable and green development, it is imperative for China, as the world's major metal consumer, to improve utilization efficiency and conservation of metal resources. Thus, understanding the industrial endogenous growth patterns of China regarding metal utilization and technical change becomes the prerequisite. Therefore, this paper regards metal as one of production factors nested with capital and labor, employs normalized supply-side system approach to estimate the elasticity of substitution in constant elasticity of substitution (CES) function and analyzes technical change bias based on the analytical framework built in this paper. Results show that: the optimal nesting structure for most metal-intensive industries are metal-capital composite nested with labor. And the metal input is less efficient than capital or labor in all metal-intensive industries. The technical change biases for metal-intensive industries are diversified, but only metal mining industries and heavy machinery manufacturing industries tend to save metal when expanding the production effectively. Targeted suggestions are made per industry according to the heterogeneity of production factors allocation and technical change biases.

Macroeconomic Growth in Vietnam Transitioned to Market: An Unrestricted VES Framework

The Vietnamese economy has increased at high speed over the transformation decades; however, most recent studies on the economic growth of this country used the Cobb-Douglas or CES (Constant Elasticity of Substitution) production functions, which are unable to explore the relationship between the elasticity of capital-labour substitution and development process, and hence, are not relevant to accessing a dynamic economic system. For that reason, this study is conducted to specify an unrestricted VES (Variable Elasticity of Substitution) production function in a one-sector growth model of Vietnam, highlighted by two characteristics: successful transition from plan to market and rapid progress. The VES is given preference over the CES and the Cobb-Douglas having the elasticity of substitution between capital and labour varying with economic development. By employing a Bayesian nonlinear regression through MCMC methods, the study reported the following findings: (1) the above-unity variable elasticity of capital-labour substitution in an aggregate unrestricted VES function specified for Vietnam shows that the model generates the possibility of endogenous economic growth; (2) the capital share tends to increase, while the labour share faces a downward trend along with the development of Vietnam; (3) the VES is empirically proven through a Bayes factor test to be superior to the CES and Cobb-Douglas for analysis of the growth process of Vietnam, an emerging transition economy.

Application of a modified CES production function model based on improved PSO algorithm

In the analysis of the economic growth factors, people often use the CES (constant elasticity of substitution) production function model to calculate the contribution rate of influencing factors to economic growth. However, the traditional CES production function model does not consider the period characteristic of technological progress on economic growth, so this paper puts forward a modified CES production function model. On the parameter estimation of the model, this paper presents an improved PSO (particle swarm optimization) algorithm with high precision and fast convergence. Using the modified CES production function model, this paper provides a scientific method to calculate the contribution rates of all factors influencing the economic growth. Finally, this paper performs the empirical analysis of the contribution rates of all factors influencing China’s economic growth.

How to Explain When the ES Is Lower Than One? A Bayesian Nonlinear Mixed-Effects Approach

Most studies in Vietnam use the Cobb-Douglas production function and its modifications for economic analysis. Extremely rigid presumptions are a main weak point of this functional form, particularly if the elasticity of factor substitution (ES) is equal to one, which hides the role of the ES for economic growth. The CES (constant elasticity of substitution) production function with more flexible presumptions, concretely its ES, is not unitary, and has been used more and more widely in economic investigations. So, this study is conducted to estimate the average ES through the specification of an aggregate CES function for the Vietnamese nonfinancial enterprises. By performing Bayesian nonlinear mixed-effects regression via Random-walk Metropolis Hastings (MH) algorithm, based on the data set of the listed nonfinancial enterprises of Vietnam, the author found that the CES function estimated for the researched enterprises has an ES lower than one, i.e., capital and labor are complimentary. This finding shows that Vietnamese nonfinancial enterprises can confront a downward trend of output growth.

A Note on Concavity Conditions of Cobb–Douglas and CES Production Function with at Least Two Inputs

In microeconomics, the strict concavity is a very important property of a production function. In 2010, Avvakumov et al. gave a necessary and sufficient condition for the strict concavity of the Cobb–Douglas and constant elasticity of substitution (CES) production function with at least two inputs. To derive these conditions, the negative definiteness of the Hessian for both production functions was examined using certain recurrences for the principal corner minors. The purpose of this note is to complement the proof of Avvakumov, Kiselev, Orlov & Taras’ev ( 2010 , Computational Mathematics and Modeling, 21(3), 336–378) by showing that the use of recurrences and mathematical induction is not necessary, and that a necessary and sufficient condition for the strict concavity can be obtained by considering a particular square matrix, whose determinant can be calculated directly using the rule for the determinant of a lower or upper triangular matrix. JEL Classifications: C60, C65, D21, D24

A Modified CES Production Function Model and Its Application in Calculating the Contribution Rate of Energy and Other Influencing Factors to Economic Growth

Abstract In the analysis of economic growth factors, the constant elasticity of substitution (CES) production function model is used to calculate the contribution rates of influencing factors to economic growth. However, the traditional CES production function model fails to consider the staged characteristics of economic growth. Therefore, this study provides a modified model of the CES production function. With regard to its application, a new method for calculating the contribution rates of energy and other influencing factors to economic growth is proposed using a modified CES production function model. This work concludes by calculating the contribution rates of Chinese energy and other influencing factors to economic growth.

A Grey CES Production Function Model and Its Application in Calculating the Contribution Rate of Economic Growth Factors

In analyses of economic growth factors, people generally use the CES (Constant Elasticity of Substitution) production function model to calculate the contribution rates of the factors that influence economic growth. However, the traditional CES function model that is built directly from economic data often shows apparent errors in parameter estimation due to data fluctuations. Such a model also may cause a negative calculation of the contribution rates of economic growth factors, or it may create abnormal fluctuations for some periods, and thus it fails to meet economic growth laws. In this paper, we propose a grey CES production function that can eliminate the random fluctuations of data and make the estimated parameters more reasonable, and this model can reflect the relationship between inputs and outputs more accurately. With regard to model application, the paper puts forward a scientific calculation method to avoid the calculation deviations caused by the substitution of difference equation for a differential equation with Solow’s formula. With the grey two‐level nested CES production function model and the calculation method proposed, the paper makes an empirical analysis of the contribution rates of factors that influence China’s economic growth.

Parameter Estimation of a Mixed Production Function Model Based on Improved Firefly Algorithm and Model Application

Abstract In the analysis on economic growth factors, researchers usually use the production function model to calculate and measure influencing factors’ contribution rates to economic growth. Common production functions include the CD (Cobb-Douglas) production function, the CES (Constant Elasticity of Substitution) production function, the VES (Variable Elasticity of Substitution) production function, and so on. In consideration of the diversity and complementarity of models, the paper combines the CD production function with the CES production function and then proposes a mixed production function. With regard to the parameter estimation of model, the paper gives an improved firefly algorithm with the high precision and a fast rate of convergence. With regard to the calculation of factors’ contribution rates, traditional methods generally have big errors and are not applicable to complicated models, so the paper offers a new method which can calculate contribution rates scientifically. Finally, the paper calculates the contribution rates of factors affecting Chinese economic growth and gets a good result.

Capital-labour-energy substitution in a nested CES framework: A replication and update of Kemfert (1998)

The ease with which firms can substitute away from energy to other inputs is an important determining factor in the costs of climate change mitigation policies. Climate policy simulation models usually represent this substitutability by using the Constant Elasticity of Substitution (CES) function with parameter values often taken from econometric studies. Hence, the accuracy of the estimated substitution parameters has a strong influence on the validity of the climate policy simulation. In this article, we attempt to replicate the results presented in a widely cited article in this field: Kemfert (1998) (‘Estimated substitution elasticities of a nested CES production function approach for Germany’, Energy Economics, 20, 249–264). We first use the data and software reported in that article and compare our results with those reported in the original study. We then test the same data and a new, more recent, data set on German industrial data with an improved econometric approach. Despite applying various approaches and modifications, we are not able to replicate the results in Kemfert (1998). We furthermore conclude that the data sets that are typically used to estimate nested CES functions often have too few observations and too little independent variation of the explanatory variables to obtain reliable estimates when using a direct non-linear approach.

Energy-biased technical change in the Chinese industrial sector with CES production functions

We develop a theoretical framework to study energy-biased technical change considering capital, labor and energy as inputs. The framework involves a first order condition estimation of elasticity and technical change parameters for a three factor-nested Constant Elasticity of Substitution (CES) function. Technical change parameters, elasticities and time derivatives of marginal products are combined to compute technical change bias. Conceptually, we introduce total bias in order to estimate the direction without requiring a direct comparison with another factor. For Chinese industries from 1990 to 2012, the optimal structure is capital and energy to be combined at the composite level and then with labor to form total output. Technical change is found to be unambiguously energy biased, it increases in every year, and the bias is predominately away from labor. The results show that Chinese industrialization was fuelled by fossil fuels and energy-intensive technologies. Nonetheless, the growth rate of energy-biased technical change decreased during the 2000s that may result from more energy efficient development.

Optimal Control in the Resource Allocation Problem for a Two-Sector Economy with a CES Production Function

We consider the resource allocation problem in a two-sector economy with a Constant Elasticity of Substitution (CES) production function on a given sufficiently long finite planning horizon. The performance criterion being optimized is one of the phase coordinates at the terminal time. The scalar control u satisfies the geometrical constraint u ∈ [0, 1]. The optimal control contains a singular section. Analyzing the Pontryagin maximum principle boundary-value problem we find the extremal triple and prove its optimality using a special representation of the functional increment (the theorem of sufficient conditions of optimality in terms of maximum-principle constructs). The solution is constructed with the aid of a Lambert special function y = W(x), that solves the equation ye y = x. The optimal control consists of three sections: the initial section that involves motion toward the singular ray L sng = {x 1 = x 2 > 0} , the singular section that involves motion along the singular ray, and terminal section that involves motion under zero control. When the initial state is on the singular ray, the optimal control consists of two sections — singular and terminal. The article is related to a number of the authors’ publications using other production function.

Estimasi Nonlinear Least Trimmed Squares (NLTS) pada Model Regresi Nonlinier yang Dikenai Outlier

Constant Elasticity of Substitution (CES) production function is the intrinsic nonlinear regression models that are often used to estimate the data in an industry. Intrinsic nonlinear regression model is a kindof nonlinear regression that can not be linearized, so as to estimate the beta parameters nonlinear statistical model used was Nonlinear Least Squares (NLS) using a first order taylor series approach used in the GaussNewton iteration. One of the problems often encountered in the analysis of data is an outlier, the presence of outliers in the data analysis greatly influence the results of the analysis so it becomes less valid and the estimationbecome biased. One method that is resistant to outliers regression is a method of Nonlinear Least Trimmed Squares. This research aims to determine the characteristics of parameter CES production function whichcontains outlier. The result shows that parameter of the production function CES which contains outliers are bias, inconsistent. So the CES production function which does not contain outliers better than the are containsoutliers.

Productivity versus elasticity: a normalized constant elasticity of substitution production function applied to historical Soviet data

This study tests the extensive growth hypothesis, which would attribute the Soviet economic slowdown to low elasticity of substitution and over-investment rather than deteriorating productivity growth. To circumvent the low availability of data, widely applicable new methods for estimating productivity and elasticity of substitution were developed based on the normalized constant elasticity of substitution (CES) production function. One is an extended version of the Solow residual calculation, and the other is a direct estimation of the time-varying parameter normalized CES production function. The application of the methods to the Soviet data showed a decreasing trend in productivity growth and a low elasticity of substitution of approximately 0.25. The results neither reject nor support the extensive growth hypothesis because of uncertainty in the interpretation of the low elasticity of substitution. The results suggest that it is reasonable to assert that both deteriorating productivity growth and low elasticity of substitution caused the Soviet economic slowdown. Further empirical studies on productivity and elasticity of substitution in the Soviet and other economies are necessary to fully understand relations between productivity growth, elasticity of substitution and economic development.

Bayesian estimation of the CES production function with capital- and labour-augmenting technical change

In this paper, we propose a Bayesian approach for analysing factor-augmenting technical changes based on a constant elasticity of substitution (CES) production function. To estimate trends in capital- and labour-augmenting technical change, a set of Bayesian linear models is constructed based on a smoothness prior approach. A statistical model constructed for the CES production function can then be expressed in a regression model with time-varying coefficients. However, the multicollinearity between the time-series data for the explanatory variables makes parameter estimation difficult. Therefore, we express the original model using two simplified models. Estimates of the parameters for each simplified model are obtained using Bayesian linear modelling and the maximum likelihood method. We then obtain a set of synthetic estimates for the time-varying coefficients based on Bayesian model averaging. As a practical application, we examine technical changes in Taiwan and South Korea at the macroeconomic level. We find that labour-augmenting technical progress contributed greatly to Taiwan's economic growth during the period under investigation. On the other hand, in South Korea, the contribution of capital-augmenting technical change to economic growth was greater than that of labour-augmenting technical change. The results show that the proposed Bayesian approach can capture movements in technical change more rigorously than conventional approaches.

Impact analysis of government investment on water projects in the arid Gansu Province of China

In this paper, we introduced three-nested Constant Elasticity of Substitution (CES) production function into a static Computable General Equilibrium (CGE) Model. Through four levels of factor productivity, we constructed three nested production function of land use productivity in the conceptual modeling frameworks. The first level of factor productivity is generated by the basic value-added land. On the second level, factor productivity in each sector is generated by human activities that presents human intervention to the first level of factor productivity. On the third level of factor productivity, water allocation reshapes the non-linear structure of transaction among first and second levels. From the perspective of resource utilization, we examined the economic efficiency of water allocation. The scenario-based empirical analysis results show that the three-nested CES production function within CGE model is well-behaved to present the economy system of the case study area. Firstly, water scarcity harmed economic production. Government investment on water projects in Gansu thereby had impacts on economic outcomes. Secondly, huge governmental financing on water projects bring depreciation of present value of social welfare. Moreover, water use for environment adaptation pressures on water supply. The theoretical water price can be sharply increased due to the increasing costs of factor inputs. Thirdly, water use efficiency can be improved by water projects, typically can be benefited from the expansion of water-saving irrigation areas even in those expanding dry area in Gansu. Therefore, increasing governmental financing on water projects can depreciate present value of social welfare but benefit economic efficiency for future generation.

Usage of Production Functions in the Comparative Analysis of Transport Related Fuel Consumption

Abstract This contribution aims to examine the relationship between the transport sector and the macroeconomy, particularly in fossil energy use, capital and labour relations. The authors have investigated the transport related fossil fuel consumption 2003 -2010 in a macroeconomic context in Hungary and Germany. The Cobb-Douglas type of production function could be justified empirically, while originating from the general CES (Constant Elasticity of Substitution) production function. Furthermore, as a policy implication, the results suggest that a solution for the for the reduction of anthropogenic CO2 driven by the combustion of fossil fuels presupposes technological innovation to reach emission reduction targets. Other measures, such as increasing the fossil fuel price by levying taxes, would consequently lead to an undesirable GDP decline.

The elasticity of substitution and the way of nesting CES production function with emphasis on energy input

This article proposes the way to choose the optimal bundle among inputs by combing the translog cost function and Constant-Elasticity-of-Substitution (CES) function. The translog cost function is applied to select the most appropriate nested structure by estimating elasticities of substitution between inputs. And then the CES function is used to get the elasiticities among inputs. It introduces the empirical simulation model for the Chinese industrial sectors focusing on how energy combines with non-energy inputs. Derived from the estimated elasticities of substitution, the results present that (capital-labor)-energy is the most appropriate way. And the two-level elasticity of (capital-labor) and energy is 0.9432.