A note on scale effects

JN recent years, economists have presented empirical evidence that improvements in efficiency have accounted for most of the growth in aggregate output in the United States and many other countries during the past half century. Robert M. Solow has presented a Cobb-Douglas production function model in which he explicitly assumes that this increased efficiency (technical progress) is due to improvements which are embodied in new physical equipment.' The model is easily extended to the case in which improvements may be embodied in new physical equipment, or may be due to other causes (i.e., disembodied improvements).2 Furthermore, Solow has demonstrated that the explicit assumption that technical progress must be embodied in new physical equipment can be incorporated in a constant elasticity of substitution (CES) production function model.3 Such a model is more general than the embodied Cobb-Douglas model in that the latter implies that the elasticity of substitution is not only a constant, but also equal to one. Solow did not consider the case in which technical progress may also be disembodied, but again it is easy to incorporate this into the CES model. This will be done below. The purpose of this paper is to point out one striking difference between Solow's Cobb-Douglas, and his CES model, when it is assumed that technical progress may be either embodied or disembodied. Embodied technical progress in the Cobb-Douglas model has no influence on the elasticities of output with respect to labor and with respect to the net capital stock. In that model, it is not possible to isolate the influence of embodied technical progress from that of disembodied progress. In the case of the CES model, however, the elasticity of output with respect to capital relative to the elasticity with respect to labor is influenced by the rate of embodied progress. Another feature of the CES model is that it is possible to state output per labor unit as a function of the marginal product of labor, and of a time shift factor which is accounted for only by disembodied technical progress. Given these facts, given data on output, labor, and the net capital stock, and finally, given the assumption that labor and capital tend to be paid their marginal products, it is possible to obtain upper and lower bound estimates of the influence of embodied and disembodied progress on output. The procedure in this paper will be: first, to outline the derivation of the Cobb-Douglas model in which progress is assumed to be either embodied or disembodied; second, to describe the extension to the CES model; and third, to demonstrate how the CES model allows us to distinguish between the effects of embodied and disembodied progress on output, and to discuss some of the implications of the CES model.

Interpreting the Hours-Technology Time-Varying Relationship

We contribute to an emerging literature that brings the constant elasticity of substitution (CES) specification of the production function into the analysis of business cycle fluctuations. Using US data, we estimate by Bayesian-Maximum-Likelihood methods a standard medium-sized DSGE model with a CES rather than Cobb–Douglas (CD) technology. We estimate a elasticity of substitution between capital and labour well below unity at 0.15–0.18. In a marginal likelihood race CES decisively beats the CD production and this is matched by its ability to fit the data better in terms of second moments. We show that this result is mainly driven by the implied fluctuations of factor shares under the CES specification. The CES model performance is further improved when the estimation is carried out under an imperfect information assumption. Hence the main message for DSGE models is that we should dismiss once and for all the use of CD for business cycle analysis.

In this article we shall give a generalization of Cobb–Douglas, Constant Elasticity of Substitution (CES), Lu–Fletcher, Liu–Hildebrand, Variable Elasticity of Substitution (VES) and Kadiyala production functions. We compute the principal indicators such as the marginal products, the marginal rate of substitution, the elasticities of factors and the elasticity of substitution. Finally, we formulate three theorems of characterization for the functions with a proportional marginal rate of substitution, with constant elasticity of labour and for those with constant elasticity of substitution (for n = 1).

In the past, growth models have been constructed and analyzed with the help of a production function subject to certain restrictive features. For quite some time, the Cobb-Douglas production function (CD) with its input exponents adding upto unity and a unitary elasticity of substitution were mostly used by the economists for analysis. In recent times, the constant elasticity of substitution production function (CES) which includes CD, as well as Leontif production function as its special case has been widely used in various studies. One major limitation of this production function is that the elasticity of substitution parameter is not variable along an isoquant, though it can take different values for different values for different industries. We have made an attempt to test whether the estimated values of elasticity of substitution between labour and capital for a sector tend to change from one year to another and also whether it is CD or CES production function that characterizes a particular sector.

The Translog cost function extimation approach has been used in this paper, to estimate important industrial policy parameters. The translog approach differs from the more frequently used techniques viz, the generalized Cobb-Dauglas production function (C0D) approach and the Constant Elasticity of Substitution (CES or ACMS) approach, in that restrictive assumptions about the elasticity of substitution between factors do not have to be made. The translog approach also has the additional advantage of providing cost shares for each factor, pair-wise elasticity of substitution, and own and cross price elasticities for the inputs. Thus, a whole range of policy parameters can be simultaneously estimated. While estimates of elasticity of substitution between capital and labour, and capital and materials are not significantly different from one, the parameter estimate is significantly below unity for substitution between labour and materials. The cross price elasticities show that the three inputs capital, labour and materials are complements in production. The own price elasticities have the appropriate signs and are all significantly below one. As expected the cost share of raw materials is the largest (78.9%) followed by labour’s share (14.6%) and capital’s share (6.5%). Alternative hypotheses regarding scale economies have also been tested. Data from the manufacturing sector of Bangladesh for the period 1969/70 to 1978/79 have been used.

This article analyzes the constant elasticity of substitution (CES) production function when there are increasing returns to scale and the elasticity of substitution exceeds 1, which I refer to as the explosive case of the CES. For this explosive case of the CES, the article demonstrates a new and surprising result: marginal and average products of labor and capital approach infinity as either labor or capital approach infinity. Obviously, in this explosive case of the CES, the law of diminishing marginal returns is eventually violated in a dramatic way. Some implications of this result for growth theory are discussed. The article concludes by deriving, for this explosive case of the CES, lower and upper bounds for the capital labor ratio which are consistent with the law of diminishing marginal returns.

Purpose: To specify a Variable Elasticity of Substitution function (VES), in which the estimated Elasticity of Substitution (ES) can give some implications for the tendency of economic growth in the Vietnames manufacturing sector. Design/Methodology/Approach: The contribution and the relevant methodology is based on the Bayesian approach having some advantages over the frequentist method: (i) the simulation and prediction results are more reliable in Bayesian analysis due to combining prior knowledge about parameters with obverved data to compose a posterior model, whereas the frequentist approach is based only on available data; (ii) in probability sense, Bayesian credible intervals have a straightforward interpretation compared to frequentist confidence intervals. The Bayesian nonlinear regresion performed is suitable for fitting production functions and depicting economic growth. Findings: The specified VES function has the ES greater than one and this finding contradicts many previous empirical studies in the growth theory. This result points to the possibility of unbounded endogenous growth in the Vietnamese manufacturing sector. Practical implications: Based on the empirical results, in order to realize the possibility of endogenous growth for the studied Vietnamese manufacturing sector, policies of enforcing investment are needed. To raise the level of science and technique, as well as human capital of the Vietnamese enterprises, at the same time, there is great necessity to encourage R&D activities in both the private and public sectors. Originality/Value: Although this study organically builds upon recent studies about the link between the VES, the elasticity of factor subsitution and economic growth, its results proved that the VES is more appropriate than the Cobb-Douglas and the Constant Elasticity of Substitution (CES) to explain economic growth in the view of capital-labor relationship.

This paper clarifies the concepts of factor-augmenting, factor-biased and factor-saving technical change by tracing back to the original definitions and using CES production function. Make a comparison between Hicks factor-biased technical change and Harrod factor-biased technical change. Use time-series data of China from 1978 to 2008 to estimate elasticity of substitution between capital and labor, then compute the value of factor-augmenting and factor-biased technical change. The results show that in the sample period elasticity of substitution between capital and labor is less than 1, and in most years the direction of technical change is factor-biased.

Gravity Redux: Estimation of gravity-equation coefficients, elasticities of substitution, and general equilibrium comparative statics under asymmetric bilateral trade costs

In this paper we examine the relationship between labor supply and industry-level output in the context of the specific factors model. Jones (Trade, balance of payment and growth: essays in honor of Charles P. Kindleberger, Amsterdam, pp 3–21, 1971) shows that a rise in the amount of labor in the economy will increase the output in all industries. We empirically show which industry output is predicted to expand more when the size of labor force grows. Unlike the commonly used Rybczynski Theorem (Economica 22:336–341, 1955) of the Heckscher-Ohlin model, the specific factors model shows that a comparison both of labor intensities and labor demand elasticities plays an important role in determining which output expands relatively more when the size of labor force grows. For this purpose, we illustrate the importance of the parameters of the model in determining how changes in the labor supply affect the output change, with special reference to elasticities of substitution in production. We estimate the elasticity of substitution by using CES production function and show how these estimates describe the general equilibrium of production with one mobile factor (labor) and 25 industries of the US economy using data for 1979–2001. We show that the increase in the supply of labor raise output in all industries, but the magnitudes of the increases in some industries are more than others depending on the value of the elasticity of substitution along with factor intensities between industries. The largest output effect occurs for educational, health care and social service, where a 1 % supply of labor increase would raise output 10.5 %. However, the growth in the labor supply has a small impact on output growth in the range of 0.1–0.6 % in agriculture, petroleum, coal product and finance and insurance industries.

Estimated substitution elasticities of a nested CES production function approach for Germany

Trade and factor returns: Empirical evidence from U.S. economy

Production function estimates are provided for Soviet industrial production and gross national product for the period 1950–86. A variety of alternative specifications is tested, including Cobb-Douglas, constant elasticity of substitution and variable elasticity of substitution production functions, and an error correction mechanism is used to investigate the long-run properties of the estimated equation. The structural stability of the estimates is also examined. Constant-returns-to-scale Cobb-Douglas production functions suggest that the rate of total factor productivity growth in the Soviet economy has declined steadily over time, becoming negative sometime in the period between 1970 and 1980. However the extensive statistical tests can doubt on the validity of any production function estimated on Soviet data.

This paper has a very specific purpose of explaining the differences among Allen elasticity of substitution, Morishima elasticity of substitution and McFadden elasticity of substitution. It is roughly divided into three sections. The first section traces the evolutionary trajectory and provides a summarized description(including formula) of these three measurements. The next section discusses a hypothetical scenario to bring out the intuition behind elasticity of substitution more vividly. In the last section the ASI all India-industry wise (unbalanced) panel data for 29 two digit industry for 2002 to 2013 is taken for a case study to empirically compare the three measures of elasticity of substitution. The square root production function under random effect model is used as it was the only functional form that satisfies the properties of production function which are-monotonicity, quasi-concavity, twice continuous-differentiability, positive marginal physical product and diminishing marginal productivity of factors of production. Each measure of elasticity of substitution has its own merits and demerits.The direct elasticity of substitution between two input factors doesn't account for the other input factors, hence it is the least advisable measure of elasticity of substitution. Allen elasticity of substitution is a ratio of cross price elasticity and share of the input. So basically it just scaling up or down the cross price elasticity. The measure of elasticity of substitution that does overcome the shortcomings of above two is Morishima Elasticity of Substitution. Itmeasures the change in any two input demands (say capital and labor) in a move to new cost minimisation where the cost minimising land derived demand also changes. The decision of which measure to use is the discretion of the researcher who should take the objective of the study as the primary criterion of judgement. Keywords: Elasticity substitution (ES), Allen elasticity, labor ratio