Cobb-Douglas Technology Research Articles

Plant capacity utilization with piecewise Cobb-Douglas technology: Definition and interpretation

Convex and nonconvex nonparametric technologies have been applied to estimate plant capacity utilization. However, they face challenges in capturing the production characteristic that involves the simultaneous consideration of increasing, constant, and decreasing marginal production rates along the production surfaces, which may be inconsistent with the standard microeconomic production theory. This inconsistency raises concerns regarding the potential bias of plant capacity utilization estimates. Thus, this paper introduces a new plant capacity utilization measure based on the piecewise Cobb–Douglas technology, aligned with the standard microeconomic production theory. This measure is defined using two multiplicative directional distance functions with optimal endogenous directions. It can be interpreted by the Euclidean distance between two points associated with optimal capacities and the Euclidean norm of one optimal endogenous direction vector. The new measure captures potential production slacks to measure plant capacity in the sense of Pareto–Koopmans efficiency. It also suggests a remedy of non-existence of a maximal plant capacity in the Cobb–Douglas function by utilizing the piecewise Cobb–Douglas technology. The proposed plant capacity utilization measure is validated using a secondary dataset of 19 Chilean hydroelectric power plants.

Identifying the elasticity of substitution with biased technical change: a structural panel GMM estimator

Summary: This paper provides a structural panel GMM (P-GMM) estimator of the elasticity of substitution between capital and labour that does not depend on external instruments, and which can be applied in the presence of biased technical change. We identify the conditions under which P-GMM is a consistent estimator and compare it to a fixed effects estimator. Using a Monte Carlo study, we find that the P-GMM estimator is nearly unbiased provided the number of time periods (T) is not too small. We show analytically how the small-T bias is related to metrics of weak identification. In an application on manufacturing firms in Norway, we estimate the elasticity of substitution to be 1.9 using the P-GMM and 1.0 using the fixed effects estimator. Neglecting simultaneity may thus lead to the conclusion that capital and labour are complements or can be described by Cobb–Douglas technology, when, in fact, they are substitutes.

Technologies for endogenous growth

I microfound endogenous growth through neoclassical technologies with substitutable inputs created by monopolistically competitive innovators. Investment delivers innovations of declining profitability, but increasing labor force generates growth depending on structural technological parameters that determine the elasticities of profits and output with respect to the mass of inputs. For instance, with a Cobb–Douglas technology in labor and a CES aggregator of inputs, the growth rate declines with the substitutability between inputs and increases in the share of inputs in production. I also explore other classes of technologies and novel specifications that can deliver long run growth.

On a Geometric Programming Approach to Profit Maximization: The Case of CES Technology

The profit maximization problem takes a central place in the theory of the firm, especially when conditions for perfect competition hold. In this paper, we solve the profit maximization problem of a perfectly competitive firm when the constant elasticity of substitution (CES) production function with n≥2 inputs describes its technology. Commonly, this problem is solved by using multivariable differential calculus. However, to avoid tedious algebraic manipulations and bypass checking nontrivial necessary and sufficient conditions, we employ geometric programming (GP), and the power mean inequality (PMI) as an elegant complementary tool to multivariable calculus. Since the GP and the PMI are simple optimization techniques without derivatives, they can provide new insights into the given problem to managers, students, and other audiences who may be unfamiliar with multivariable differential calculus. Additionally, by using the properties of limits, we show that the solution to the profit maximization problem with Cobb-Douglas technology is a limiting case of our result.

Human capital and welfare

AbstractWe introduce the Human Development Index (HDI) in a growth model to show that if households cared about human capital, even if infinitesimally, then human capital would increase forever protecting economies indirectly. Here, human capital will have an additional positive effect on social welfare through the quality of individual health and education. In a simple economy with a Cobb–Douglas technology and logarithmic preferences, we provide the explicit trajectories for human capital, consumption, and the HDI, which correspond to the balanced growth path (BGP). Using a two-step maximization strategy, we compute the optimal initial value of the control variable, in this case, the initial optimal labour supply. In other words, we prove the optimality of the BGP. We highlight a HDI crossing property: the propensity to consume has a positive effect on the HDI in the short run, but negative in the long run.

Utility maximization in a multidimensional semimartingale model with nonlinear wealth dynamics

We explore martingale and convex duality techniques to maximize expected risk-averse utility from consumption in a general multi-dimensional (non-Markovian) semimartingale market model with jumps and non-linear wealth dynamics. The model allows to incorporate additional cash flows via non-linear margin payment functions in the drift term that depend on the allocation proportion. These can be used to cast frictions such as the impact of the portfolio choices of a ‘large’ investor on the expected assets’ returns, funding costs arising from differential borrowing and lending rates, and the cash inflow of a firm in a neoclassical economy with constant return-to-scale Cobb–Douglas technology subject to exogenous aggregate shocks. We provide a general verification theorem for random utility fields satisfying the usual Inada conditions, find conditions under which jumps in our model lead to precautionary saving, and present an explicit characterization for CRRA. We report two-fund separation-type results which assert that optimal allocations move along one-dimensional segments, and illustrate our results numerically for various margin payment functions and bounded variation tempered $$\alpha $$ -stable jumps.

Vertical mergers with input substitution: Double marginalization, foreclosure and welfare

We consider differentiated duopolists that face symmetric linear demands and produce using Cobb–Douglas technologies with a monopolized input and a competitively supplied input. A merger between the input monopolist and either firm eliminates double marginalization but – unlike​ with fixed-proportions technologies – can lead to foreclosure and reduce welfare. The same can occur under a CES technology with greater input substitutability than Cobb–Douglas. With identical Cobb–Douglas technologies, the merged firm raises the rival’s cost by more, and the welfare effects are worse, when the input it controls constitutes a low rather than high share of input costs. With different technologies, the welfare effects can be non-monotonic in that input’s share of costs.

Vertical Mergers with Input Substitution: Double Marginalization, Foreclosure and Welfare

We consider differentiated duopolists facing symmetric linear demands and using Cobb-Douglas technologies with two inputs: a monopolized input and a competitively supplied input. Unlike with fixed-proportions technologies, a merger between the input monopolist and either firm can reduce welfare. The merged firm raises the input price to the rival by more—sometimes inducing complete foreclosure—when that input constitutes a lower share of the rival’s input costs. If that share is sufficiently low then welfare declines, while rising elsewhere despite foreclosure. The merger also can reduce welfare under a CES technology with greater input substitutability than Cobb-Douglas, though the foreclosure and welfare effects are no longer monotonic in the monopoly input’s share of downstream costs.

Maternal subjective expectations about the technology of skill formation predict investments in children one year later

A growing literature reports significant socio-economic gaps in investments in the human capital of young children. Because the returns to these investments may be huge, parenting programs attempt to improve children’s environments by increasing parental expectations about the importance of investments for their children’s human capital formation. We contribute to this literature by investigating the relevance of maternal subjective expectations (MSE) about the technology of skill formation in predicting investments in the human capital of children. We develop and implement a framework to elicit and analyze MSE data. We launch a longitudinal study with 822 participants, all of whom were women in the second trimester of their first pregnancy at the date of enrollment. In the first wave of the study, during pregnancy, we elicited the woman’s MSE. In the second wave, approximately one year later, we measured maternal investments using the Home Observation for the Measurement of the Environment (HOME) Inventory. The vast majority of study participants believe that the Cobb–Douglas technology of skill formation describes the process of child development accurately. We observed substantial heterogeneity in MSE about the impact of human capital at birth and investments in child development at age two. Family income explains part of this heterogeneity in MSE. The higher the family income, the higher the MSE about the impact of investment in child development. We find that a one-standard-deviation of MSE measured at pregnancy is associated with 11% of a standard deviation in investments measured when the child is approximately nine months old.

Driving force for China’s photovoltaic industry output growth: Factor-driven or technological innovation-driven?

At present, China’s PV industry is experiencing a policy-driven to market-driven transition. Whether the endogenous driving force for China’s PV industry output growth has been changed is a question of substantial concern? To scientifically answer the question of the endogenous driving force of the output growth of China’s PV industry, we empirically analyse the driving force of China’s PV industry output growth by constructing an expanded Cobb-Douglas technology production function. The results demonstrate that the output elasticity of research and development (R&D) is between 0.13 and 0.18, which is higher than the level of 0.1 in traditional Chinese manufacturing. Compared with traditional industries, PV industries are more capable of translating their R&D inputs into output revenue; The contribution rate of R&D input to output growth exceeds physical capital and labour factors, indicating that the endogenous driving force of China’s PV industry output growth has changed from factor-driven to technological innovation-driven.

Investissement et embauche avec coûts d’ajustement fixes et asymétriques

Une entreprise évoluant dans un monde déterministe en temps continu utilise une technologie Cobb-Douglas à rendements décroissants et fait face à des coûts d’ajustement complexes de ses facteurs. D’une part, un coût d’embauche proportionnel s’applique au facteur travail irréversible. D’autre part, investir nécessite de s’acquitter d’un coût fixe mais proportionnel à la main d’œuvre présente dans l’entreprise. L’entreprise détermine alors sa politique d’embauche et d’investissement qui maximise la valeur actualisée de ses flux de trésorerie. La combinaison de coûts fixe et variables implique un cycle non trivial d’ajustement des facteurs au cours duquel se succèdent un épisode d’embauche continue, suivi par un gel des embauches puis par un pic simultané d’investissement et d’embauche. L’article propose une évaluation numérique des effets de la taille du coût fixe sur le cycle de vie de l’entreprise. Codes JEL : C61, D92, E22, E24

POLICY OPTIONS ON SUSTAINABLE RESOURCE UTILIZATION AND FOOD SECURITY IN HAOR AREAS OF BANGLADESH: A THEORETICAL APPROACH

This study consists of two theoretical approaches on the policy options regarding the link between entrepreneurship, resource utilization and food security in the haor-basin areas. The first one deals with a Cobb-Douglas technology to investigate the relationship between entrepreneurship and food security. In particular, we suggest that, while more entrepreneurship could mean more productivity in the production sectors, productivity growth in turn could affect the individual arbitrage between different haor’s occupations (including entrepreneurship) and expected payoffs. The second question is concerned with the types of haor resource management to which the individuals or local communities direct their talents. We distinguish between community management and government management on the efficient utilization of haor resources. The impact on household food security and resource utilization can be assessed in static and dynamic frameworks.

The effects of globalization and technology on the elasticity of substitution

The elasticity of substitution between capital and labor ( $$\sigma$$ ) is usually considered a “deep parameter”. This paper shows, in contrast, that $$\sigma$$ is affected by both globalization and technology, and that different intensities in these drivers have different consequences for the OECD and the non-OECD economies. In the OECD, we find that the elasticity of substitution between capital and labor is below unity; that it increases along with the degree of globalization; but it decreases with the level of technology. Although results for the non-OECD area are more heterogeneous, we find that technology enhances the substitutability between capital and labor. We also find evidence of a non-significant impact of the capital-output ratio on the labor share irrespective of the degree of globalization (which would be consistent with an average aggregate Cobb–Douglas technology). Given the relevance of $$\sigma$$ for economic growth and the functional distribution of income, the intertwined linkage among globalization, technology and the elasticity of substitution should be taken into account in any policy makers’ objective function.

An alternative approach to solving cost minimization problem with Cobb–Douglas technology

We propose a new method for solving the production cost minimization problem with Cobb–Douglas technology. The method is based on weighted arithmetic–geometric-mean inequality (weighted AM–GM) and it does not use calculus. In comparison to methods which use calculus (like substitution method or the Lagrange multiplier method), our method derives the global minimum costs and unique global minimizer in case of the Cobb–Douglas production function in a new and easier manner. Another benefit of this method is that it derives the exact formula for the optimal level of inputs as well as the formula for minimum costs in general case of $$n>2$$ inputs. Also, it is important to emphasize that our method does not require checking the first and the second order conditions, which is a necessary step in methods using calculus. The result is first derived for the case of two inputs and then generalized for the problem with $$n>2$$ inputs.

Stochastic continuous time growth models that allow for closed form solutions

We find a closed form solution that maximises the expected utility of an agent’s inter-temporal consumption subject to a stochastic technology, which is a linear combination of AK and Cobb–Douglas technologies. Additionally, we consider two cases of agent preferences: (i) Constant Relative Risk Aversion (CRRA) preferences, which treat optimal consumption as a linear function of capital, and (ii) Hyperbolic Absolute Risk Aversion (HARA) preferences, which treat optimal consumption as an affine function of capital. By establishing a minimum (subsistence) level of consumption in the HARA model, we are able to create a framework that more accurately represents real-world circumstances than previous studies have done. Furthermore, for both the CRRA and HARA cases we show the suitable, consistent stochastic differential equation which describes the capital dynamics. Finally, we perform a numerical simulation based on the CRRA case and calibrate US data for the HARA case.

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Comment

The Effects of Globalization and Technology on the Elasticity of Substitution

The elasticity of substitution between capital and labor (?) is usually considered a deep parameter. This paper shows, in contrast, that ? is affected by both globalization and technology, and that different intensities in these drivers have different consequences for the OECD and the non-OECD economies. In the OECD, we find that the elasticity of substitution between capital and labor is below unity; that it increases along with the degree of globalization; but it decreases with the level of technology. Although results for the non-OECD area are more heterogeneous, we find that technology enhances the substitutability between capital and labor. We also find evidence of a non-significant impact of the capital-output ratio on the labor share irrespective of the degree of globalization (which would be consistent with an average aggregate Cobb-Douglas technology). Given the relevance of ? for economic growth and the functional distribution of income, the intertwined linkage among globalization, technology and the elasticity of substitution should be taken into account in any policy makers' objective function.

Who provides the capital for Chinese growth: the public or the private sector?

ABSTRACTWe focus on the role of the government in the provision of investment in China, through the medium of a Dynamic Stochastic General Equilibrium model of the economy in which the form of the production function reflects this governmental role. Using indirect inference, we estimate and test for the elasticity of substitution between government and nongovernment capital in both Constant Elasticity of Substitution (CES) and Cobb–Douglas technologies. The results underscore the strong substitution relationship between government and nongovernment capital from 1949, supporting CES rather than the Cobb–Douglas technology. They also show that the orientation of public investment changed after the start of the ‘Socialist Market Economy’ in 1992: government capital became more complementary to nongovernment capital as it focused more on infrastructure and withdrew from industrial production, intervening only in times of crisis, for stabilization purposes, indirectly via the state banks.

Generational Risk -- Is it a Big Deal? Simulating an 80-Period OLG Model with Aggregate Shocks

The theoretical literature presumes generational risk is large enough to merit study and that such risk can be meaningfully shared via appropriate government policies. This paper assesses these propositions. It develops a computational technique to overcome the curse of dimensionality and measure generational risk in an 80-period OLG model. The model features isoelastic preferences, moderate risk aversion, Cobb-Douglas technology, and shocks to both TFP and capital depreciation. When shocks are calibrated using empirically plausible parameter values (our baseline model), the model reproduces the U.S. variability of output and wages. But it overstates the variability of the economy’s overall return to aggregate wealth. Even so, the model understates the economy’s risk premium (the mean return on aggregate wealth less the mean return on safe assets). But, as shown by Constantinides, Donaldson, and Mehra (2002) and Hasanhodzic (2014), adding hard or increasing borrowing costs can readily reduce equilibrium risk-free rates to essentially any desired level with little or no impact on the economy’s key macro variables. Based on this calibration, we nd that generational risk is small and that Social Security can exacerbate it. When depreciation shocks are accentuated to reproduce the observed variability of equity returns, as in Krueger and Kubler (2006), the model’s risk premium is close to the equity premium observed in the stock market. Yet, with this calibration, the model greatly overstates the variability of output and wages. It also produces substantial generational risk|risk that Social Security can materially reduce. Under both calibrations, we nd that even a one-period bond market is very eective, indeed far more eective than Social Security, in sharing risks among contemporaneous generations. We also nd that policy-induced intergenerational redistribution can produce far larger generational risk than macroeconomic uctuations.

New Approach to the Ramsey Model

In this paper it is suggested a new approach to the Ramsey model with the Constant Relative Risk Aversion (CRRA) utility function and Cobb-Douglas technology. The suggested approach allows finding a steady state which is new and unknown before. It is proved that the found steady state is stable. With help of the method of a small parameter analytical solution of the model has been obtained. Some opportunities for improving the obtained analytical solution are suggested. It is shown that transversely condition, which is a component part of the Rumsey model, holds true for the new approach. DOI: 10.5901/mjss.2015.v6n6p284