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Original Article | Open Access | Asian J. Soc. Sci. Leg. Stud., 2024; 6(2), 40-46 | doi: 10.34104/ajssls.024.040046

A Study of the Economic Productivity of Several Countries Using a Modified Cobb-Douglas Function

Anindo Mahmud* Mail Img

Abstract

The study analyzes the dynamics of economic productivity in 44 countries using a modified Cobb-Douglas production function. Aside from continuous variables such as labor, capital, and energy, this model also accommodates non-continuous variables in order to provide a more accurate representation of the economic relationships. The findings reveal some rather intriguing results regarding the returns to scale and the influence of government and natural disasters on economic output. Negative returns to scale in certain countries can be attributed to factors such as declining working hours and unemployment. The Government variable, despite revealing positive coefficients for most of the countries analyzed, is still nuanced in nature and requires more contextual understanding. The Natural Disaster variable, meanwhile, reveals both negative and positive coefficients, thus challenging conventional wisdom and posing questions about the resilience and adaptability of economic systems in the face of adversity. The study recommends tailored policy interventions, calling for greater specialization and resource allocation in countries with lower productivity levels and sustained innovation in countries with higher productivity levels.

INTRODUCTION

Economic productivity is an important aspect of a countrys well-being and progress. In fact, it is popu-larly believed that a country cannot considerably improve its standard of living without increasing its economic productivity (Krugman, 1997; Bonna and Akter, 2023).

As a result, the topic of economic productivity has been a subject of timeless fascination for economists and policymakers alike. Understanding the relation-ships between various economic factors and their impact on GDP can provide valuable information regarding the economic dynamics of a country. This study examines such relationships using the famous Cobb-Douglas production function (Cobb and Doug-las, 1928). The Cobb-Douglas production function provides a mathematical framework for studying how different factors of production contribute to an economys total output. Over time, the model has adapted to varying contexts, allowing researchers to conduct both microeconomic and macroeconomic analyses of economic productivity. A modified ver-sion of the Cobb-Douglas production function was adopted in this study to address the limitations of previous models by including non-continuous vari-ables (Cheng and Han, 2014). Non-continuous variables included external factors such as the gov-erning party and the severity of natural disasters to make this analysis more consistent with reality. This analysis covers 44 countries and provides a cross-national perspective on economic productivity.

As proxies were being decided for each of the variables, one challenge involved coming up with adequate mathematical adjustments that could in-crease comparability and reduce distortions in the estimates of factor productivity. A prior study using the Cobb-Douglas production function failed to take similar adjustments into account and so ended up producing much distorted results with extreme fluctu-ations (Khatun and Afroze, 2016).

By utilizing the modified Cobb-Douglas function, this study aimed to reveal insights into factor pro-ductivity, uncovering patterns, differences and pot-ential explanatory factors across the 44 countries. The findings not only contributed to a deeper com-prehension of the economic dynamics of the coun-tries under the study but also offered valuable methodological insights into the application of the Cobb - Douglas production function in contemporary economic analysis.

Rationale of the study

The significance of this study goes beyond merely searching for patterns and differences in factor productivity. It provides methodological insights by improving the application of the Cobb-Douglas production function in contemporary economic ana-lysis. As countries deal with economic challenges influenced by global dynamics, the technological progress and environmental concerns, comprehend-sive research on economic productivity is essential. By revealing the complex relationships within the production function, this research aims to provide valuable insights to policymakers, economists and researchers, ultimately helping them make informed decisions and promote sustainable economic deve-lopment.

Limitations of the study

Some limitations that were faced during this study were as follows:

Lack of representativeness: Only 44 countries were selected for this study due to the avail-ability of their data. In such cases, these countries are usually not representative of the entire world as the countries that tend to be omitted consist largely of third-world countries whose data is difficult to collect due to political instability and lack of transparency.

Temporal constraints: The study covers a 26-year time period from 1991 to 2017. The lack of available data prevented a longer time period for analysis. As a result, the effects of some radical game-changing events that took place after 2017 like the Covid-19 pandemic and the Russia-Ukraine war could not be con-sidered in this study.

Constraints in model development: The modi-fied Cobb-Douglas model used in this study is more multifaceted in nature than the conven-tional Cobb-Douglas model as it encompasses considerations of economic, technological, political and environmental factors. Neverthe-less, it excludes the incorporation of certain categories of factors such as social factors which are also considered influential for bus-iness activities (Aguilar, 1967). This omission is primarily attributable to the scarcity of annual secondary data related to social factors, such as public perceptions and opinions, over the temporal scope of this study.

Model Specification

The Cobb-Douglas production function is one of the most widely used production functions in economics because of its versatility. It can be used in both microeconomic and macroeconomic studies, allow-ing researchers and decision-makers to better com-prehend and model economic production. Its scope for mathematical analysis allows for the quantitative estimation of economic parameters and provides valuable insights into the determinants of factor demand and economic growth. As a result, the Cobb-Douglas production function, despite its limitations, continues to remain a cornerstone of economic theory and empirical research, contributing signifi-cantly to our understanding of production processes and the allocation of resources in modern economies.

The form of the general Cobb - Douglas production is expressed as:

"Y = A" "X" _"1" ^("β" _"1"  ) "X" _"2" ^("β" _"2"  ) "… " "X" _"n" ^("β" _"n"  )…………………………….(1)

In the equation above, Xi (i = 1, 2, ..., n) denotes the input of the ith factor and Y denotes the output; βi (i = 1, 2, ..., n) is the output elasticity of the factor Xi and A denotes the level of technical progress or total factor productivity (TFP). However, one problem with the conventional Cobb - Douglas approach is that it does not take into account the influence of some quality non-continuous variables. The conven-tional Cobb - Douglas function also fails to capture the multifaceted effects on production, especially when faced with external non-economic factors. Therefore, a modified Cobb - Douglas production function model will be used in this paper (Cheng and Han, 2014). Its form is expressed as:

"Y = A" "X" _"1" ^("β" _"1"  ) "X" _"2" ^("β" _"2"  ) "… " "X" _"n" ^("β" _"n"  ) "e" ^("p" _"1"  "D" _"1"  "+" "p" _"2"  "D" _"2"  "+⋯+" "p" _"N"  "D" _"N"  )..…….. (2)

Where,


"D=" {█("1" @"0" )",Dummy Variable" ┤

Both sides of equation (2) undergo log transfor-mations in order to be converted into a linear form:


"ln(Y)=ln(A)+" ∑_"i=1" ^"n" ▒〖"β" _"i"  "ln(" "X" _"i"  ")" 〗 "+" ∑_"i=1" ^"n" ▒〖"p" _"i"  "D" _"i"  〗 "+" "ε" _"i"  "…" … "(3)"  

For the sake of this paper, the modified Cobb-Douglas function is expressed as follows:

"ln(" "Y" _"jk"  ")=ln(" "A" _"k"  ")+" "β" _"1"  "ln(" "K" _"jk"  ")+" "β" _"2"  "ln(" "L" _"jk"  ")+"  

"β" _"3"  "ln(" "E" _"jk"  ")+" "p" _"1"  "G" _"jk"  "+" "p" _"2"  "N" _"jk"  "+" "ε" _"jk"  "…………………...(4)"  

Where,

Yjk = Gross Domestic Product,

Ak = Long-term Total Factor Productivity,

Kjk = Adjusted Gross Capital Formation,

Ljk = Adjusted Total Labor Hours,

Ejk = Adjusted Primary Energy Consumption,

Gjk = Governing Party,

Njk = Natural Disaster Incidence

In the equation above, j = 1, 2, …, 27, representing the number of years in the 27-year period from 1991 to 2017 and k = 1, 2, …, 44, representing the number of countries analyzed.  εi is the error term which is assumed to be normally distributed. Total labor hours, gross capital formation and primary energy consumption were adjusted by scaling so that all three of them have minimal effects on the total factor productivity. It was done to isolate the effects of total factor productivity and the elasticities and also to prevent the total factor productivity figures from deviating significantly among the countries studied.

There are two non-continuous variables in the modi-fied Cobb-Douglas function: ‘Government (repre-sented by G) and ‘Natural disaster (represented by N). The ‘government variable was included as the conventional Cobb-Douglas production function contains an economic component (reflected by the economic inputs of labor, capital and energy) and a technological component (reflected by total factor productivity) but no political component (Aguilar, 1967). The ‘natural disaster variable was added as an attempt to contribute further to the study of the relevance of natural disasters to economic growth (Cavallo and Noy, 2009). Both the non-continuous variables are dummy variables. Gik = 1 for country k in the ith year such that the political party that was in power for most of 2017 in country k was also in power for most of the ith year and Gik = 0 for every other year for country k. Meanwhile, Nik = 1 for country k in the ith year such that the deaths from natural disasters as a share of total deaths in the ith year in country k was at least 0.01%. The coeffi-cients p1 and p2 are such that when Gik = 1, the exp-ected annual percentage GDP growth will increase by p1% and when Nik = 1, the expected annual percentage GDP growth will increase by p2%. If the sum of β1, β2 and β3 is negative, that will imply negative returns to scale. In other words, this will mean that the countrys outputs are actually rising even as inputs are falling. 

If the sum of β1, β2 and β3 is non-negative (either positive or 0) but less than 1, it will imply that output growth is less than proportional to input growth. If the sum of β1, β2 and β3 is 1, it will mean that output growth is exactly proportional to input growth. Finally, if the sum of β1, β2 and β3 is greater than 1, this will imply that output growth is more than pro-portional to input growth.

METHODOLOGY

The methodology for this study involved a compre-hensive data collection process from various reput-able sources. 44 countries were chosen for the study as necessary data from 1991 to 2017 was mostly available for those countries. Total Labor Hours was calculated using the following formula:

Total Labor Hours =

Population aged 15-64 x

(1-Unemployment Rate) x

Working hours per year……(5)

Data for the population aged 15-64 and working hours per year for each of the countries was obtained from the ‘Our World in Data website. Data for the unemployment rate was obtained from the ‘Macro-trends website. Data for Gross Capital Formation was sourced from The World Banks Data Bank. Data for Primary Energy Consumption data was also collected from the ‘Our World in Data website.

The Natural Disaster Incidence variable was derived by assigning a value of 1 for each year in which the number of deaths from natural disasters as a share of total deaths exceeded 0.01% in a particular country. Meanwhile, the data for the number of deaths from natural disasters as a share of total deaths itself was sourced from the ‘Our World in Data website. For output, the proxy used was gross domestic product (GDP) at purchasing power parity expressed in international dollars at 2017 prices. The data was also obtained from the ‘Our World in Data website.

One issue that arose during the data collection pro-cess was the occurrence of missing values. Only 44 countries were selected for this study as relevant data was not available for other countries. Even among the 44 countries, data was occasionally missing. For instance, GDP data for Canada was not available from 1991-1996 and GDP data for Iceland was not available from 1991-1994. As a result, analysis was done using only data from 1997-2017 for Canada and only 1995-2017 for Iceland.

RESULTS AND DISCUSSION

The estimated coefficients and exponents of the production functions of each of the 44 selected coun-tries along with their respective returns to scale and coefficients of determination are shown in Table 1. 

In the table above, the adjusted coefficients of deter-mination (R2) for each and every one of the 44 countries was very high (above 0.85). Even the low-est one, that of Spain, was 0.883. That suggested that for all of the countries concerned, the modified Cobb-Douglas model in this paper provided a very good fit.

Interpretations of the total factor productivity figures

Out of all the countries analyzed, it was discovered that the five countries with the lowest total factor productivity were Denmark, Hungary, India, Paki-stan and the Philippines. The figures of none of the five countries had crossed the 50-billion figures. For developing countries like India, Pakistan and the Philippines, low total factor productivity figures could be due to their lack of specializations in high-value industries with global demand. Other deve-loping countries like Bangladesh and Vietnam that specialized, for instance, in the textile industry had higher total factor productivity figures (The Daily Star, 2023). 

In the case of developed countries like Denmark and Hungary, the low total factor productivity figures are especially surprising considering that both are high-income countries whose exports largely consist of high-value products like cars and other electronic goods (World Population Review, 2024; Obser-vatory of Economic Complexity, 2023). There may be some factors at play such as the presence of regu-lations or the allocation of resources but the exact economic impacts of those factors on Denmark and Hungary will require further study in the future. Meanwhile, on the other side, the five countries with the highest total factor productivity were Germany, Japan, Luxembourg, Singapore and the United States. Each of those countries had a figure of over 1 trillion. It is expected of Singapore and Luxembourg to have high factor productivity given the fact that both countries have very limited resources due to their extremely small respective land areas and need 

high productivity to compete economically with the rest of the world. Germany, Japan and the United States, meanwhile, are known for their robust manu-facturing sectors (United Nations Statistics Division, 2024).

Interpretations of the factor exponents and returns to scale

Out of all the countries analyzed, it was discovered that nine of the countries (Austria, Denmark, Ger-many, Hungary, Italy, Japan, South Korea, Spain and the United States) had negative returns to scale. The negative returns to scale were caused exclusively by negative output elasticity of labor for Austria, Hungary, Japan, South Korea and Spain. As per data trends from the ‘Our World in Data and the ‘Macro-trends websites, the negative returns to scale could be explained by severe drops in working hours in the case of Austria, Hungary and South Korea, declining working population in the case of Japan and rela-tively high rate of unemployment in the case of Spain. In the United States, meanwhile, the negative returns to scale were caused exclusively by negative output elasticity of energy. This could be explained by the fact that the primary energy consumption in the United States had been following a downward trend for the latter half of the 27-year period accor-ding to data trends from the ‘Our World in Data website. Finally, for Denmark, Germany and Italy, the negative returns to scale were caused by both negative output elasticities of labor and energy. They were caused respectively by falling working hours and falling primary energy consumption in all three countries as per data trends from the ‘Our World in Data website.

Meanwhile, nine of the countries (Bangladesh, China, Finland, France, Greece, Norway, Portugal, Sri Lanka and Uruguay) had positive but decreasing returns to scale. This means that the output growth in each of those countries was less than proportional to their input growth. A similar prior study on the economic dynamics of Bangladesh and China using the Cobb-Douglas productivity function revealed increasing returns to scale for Bangladesh and China, a sharp contrast to this study (Khatun and Afroze, 2016). That might have been because the study assu-med that output elasticities of labor and capital cannot be negative; something which is not necess-arily true given the fact that countries with declining working hours might have negative output elastic-ities of labor as their output continues to grow des-pite decreasing labor input. In the case of this study, both Bangladesh and China were found to have negative output elasticities of labor as well as nega-tive output elasticities of energy. Four of the coun-tries analyzed (Belgium, Ireland, Turkey and Viet-nam) had near-constant returns to scale. The output growths of each of those countries were almost proportional to their respective input growths. Even though Ireland had a negative output elasticity of labor, its output elasticities of capital and energy were more than enough for the country to achieve near-constant returns to scale. The rest of the coun-tries all had increasing returns to scale. Switzerland had the largest returns to scale at 3.02.

Interpretations of the ‘Government variable

The coefficient of the ‘Government variable was positive for the majority of the 44 countries. Costa Rica had the highest coefficient at 0.15. This sugg-ests that annual percentage growth in output for the majority of the countries including Costa Rica was around 15% to over 0% higher in the years during which the governing party for the majority of 2017 was in power most of the time. It was, however, zero for eight countries (Austria, Canada, China, Japan, Malaysia, Mexico, Singapore, Vietnam) and nega-tive for another nine countries (Argentina, France, Iceland, Italy, Luxembourg, Spain, Sweden, United States, United Kingdom). While due credit is undou-btedly attributable to the governing parties of coun-tries exhibiting positive coefficients, it is crucial to recognize the more subtle nature of these results. Given the temporal scope of this study, it is essential to consider that the observed relationship might have been significantly influenced by the broader context of the historic global economic growth.

Interpretations of the ‘Natural Disaster variable

The coefficient of the natural disaster variable was either negative or zero for most countries. This means that the economic output of most countries was either lower or remained unchanged in the years during which the share of deaths from natural disas-ters was 0.01% or more. In countries with negative coefficients, severe natural disasters were expected to cause large-scale destruction to human capital, thus impeding quick short-term recovery. Countries with a coefficient of zero were more able to exhibit remarkable resilience even in the face of severe natural disasters. They were possibly able to recover more quickly in the short-run from severe natural disasters, mostly due to the presence of proper institutions and effective policymaking that allowed for quick and efficient allocation of resources at times of crises (Kern, 2010). However, what was really surprising was the finding that 11 countries (Argentina, Brazil, Denmark, Iceland, India, Mexico, New Zealand, Pakistan, Sweden, Switzerland and Uruguay) had positive coefficients. In other words, they saw higher economic output in years with relatively severe natural disasters. That could have been due to temporary boosts in output and employ-ment caused by increased recovery spending and investment in those countries in the aftermath of disasters (Baily, 2011).

CONCLUSION

This study used a modified Cobb-Douglas produc-tion function to analyze the relationships among various factor inputs and outputs in 44 countries. The purpose of this modified model was to make the analysis more consistent with reality by accommo-dating non-continuous variables. The results revea-led rather interesting information on factors affecting factor productivity and the impacts of government and natural disasters on economic output. Returns to scale in some countries were negative, caused by factors such as reduced working hours, unemploy-ment and diminishing energy consumption. The "government" variable, despite revealing mostly positive coefficients, continued to remain nuanced due to the historic nature of economic growth. The "natural disaster" variable showed both negative and positive coefficients, thus challenging conventional wisdom. Differences in total factor productivity data among countries meant that targeted policy inter-ventions are required for the countries analyzed. Specialization and resource allocation are required for low-productivity countries while continuous innovation and resource optimization are required for high-productivity countries. This study not only provided valuable insights into the economic dyna-mics of the analyzed countries, but it also offered methodological improvements for the use of the Cobb-Douglas production function in modern econo-mic analysis. In todays continuously evolving global economic landscape, this study aims to serve as a foundation for policymakers, economists, and rese-archers to make well-informed choices in their pursuit of sustainable economic development.

ACKNOWLEDGEMENT

The author of this paper has acknowledged Dr. Javed Mahmud, Adjunct Faculty, Faculty of Business Administration, Sonargaon University, Dhaka, Bangla- desh and Rabib Kamal Preom, Assistant Professor and Head, Department of Business Administration, Gono Bishwabidyalay, Dhaka, Bangladesh for their support in making this research successful.

CONFLICTS OF INTEREST

The author has declared no potential conflicts of interest with respect to his research work.

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Article Info:

Academic Editor

Dr. Antonio Russo, Professor, Dept. of  Moral Philosophy, Faculty of Humanities, University of Trieste, Friuli-Venezia Giulia, Italy.

Received

April 8, 2024

Accepted

March 13, 2024

Published

March 22, 2024

Article DOI: 10.34104/ajssls.024.040046

Corresponding author

Anindo Mahmud*

Lecturer, Department of Business Administration, Gono Bishwa-bidyalay, Savar, Dhaka 1344, Bangladesh.

Cite this article

Mahmud A. (2024). A study of the economic productivity of several countries using a modified Cobb-Douglas function, Asian J. Soc. Sci. Leg. Stud., 6(2), 40-46. https://doi.org/10.34104/ajssls.024.040046 

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