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doi: 10.3934/jimo.2020099

Application of a modified VES production function model

Maolin Cheng 1,, and  Bin Liu 2,

1. 

School of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou, 215009, China
2. 

School of Business, Suzhou University of Science and Technology, Suzhou, 215009, China

* Corresponding author: Maolin Cheng

Received  November 2019 Revised  February 2020 Published  May 2020

Fund Project: This work is supported in part by the National Natural Science Foundation of China(11401418)

In the analyses on economic growth factors, researchers generally use the production function model to calculate the contribution rates of influencing factors to economic growth. The paper proposes a new modified VES production function model. As for the model's parameter estimation, the conventional optimization methods are complicated, generally require information like the gradient of objective function, and have the poor convergence rate and precision. The paper gives a modern intelligent algorithm, i.e., the cuckoo search algorithm, which has the strong robustness, can be realized easily, has the fast convergence rate and can be used flexibly. To enhance the convergence rate and precision, the paper improves the conventional cuckoo search algorithm. Using the new model, the paper gives a method calculating the contribution rates of economic growth influencing factors scientifically. Finally, the paper calculates the contribution rates of influencing factors to economic growth in Shanghai City, China.

Keywords: Economic growth, VES production function, cuckoo search algorithm, contribution rate.
Mathematics Subject Classification: Primary: 65K10, 91B02; Secondary: 93B40.
Citation: Maolin Cheng, Bin Liu. Application of a modified VES production function model. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020099
References:
[1]

A. Assad and K. Deep, A Hybrid Harmony search and Simulated Annealing algorithm for continuous optimization, Information Sci., 450 (2018), 246-266.  doi: 10.1016/j.ins.2018.03.042.  Google Scholar

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J. Cheng, L. Wang, Q. Jiang and Y. Xiong, A novel cuckoo search algorithm with multiple update rules, Appl. Intell., 48 (2018), 4192-4211.  doi: 10.1007/s10489-018-1198-y.  Google Scholar

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M. L. Cheng, A Grey CES production function model and its application in calculating the contribution rate of economic growth factors, Complexity, 2019 (2019), 8pp. doi: 10.1155/2019/5617061.  Google Scholar

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J. Chongela, V. Nandala and S. Korabandi, Estimation of constant elasticity of substitution (CES) production function with capital and labour inputs of agri-food firms in Tanzania, African J. Agricultural Res., 8 (2013), 5082-5089.   Google Scholar

[8]

F. Erdal, A firefly algorithm for optimum design of new-generation beams, Engineering Optim., 49 (2017), 915-931.  doi: 10.1080/0305215X.2016.1218003.  Google Scholar

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H. Guo, J. Hu, S. Yu, H. Sun and Y. Chen, Computing of the contribution rate of scientific and technological progress to economic growth in Chinese regions, Expert Systems Appl., 39 (2012), 8514-8521.  doi: 10.1016/j.eswa.2011.12.032.  Google Scholar

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Y. He, S. W. Gao and N. Liao, An intelligent computing approach to evaluating the contribution rate of talent on economic growth, Comput. Econ., 48 (2016), 399-423.  doi: 10.1007/s10614-015-9536-1.  Google Scholar

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G. V. S. K. Karthik and S. Deb, A methodology for assembly sequence optimization by hybrid cuckoo-search genetic algorithm, J. Adv. Manufacturing Systems, 17 (2018), 47-59.  doi: 10.1142/S021968671850004X.  Google Scholar

[12]

Y. N. Kiselev, S. N. Avvakumov and M. V. Orlov, Optimal control in the resource allocation problem for a two-sector economy with a CES production function, Comput. Math. Model., 28 (2017), 449-477.  doi: 10.1007/s10598-017-9375-0.  Google Scholar

[13]

X. J. Meng, J. X. Chang, X. B. Wang and Y. M. Wang, Multi-objective hydropower station operation using an improved cuckoo search algorithm, Energy, 168 (2019), 425-439.  doi: 10.1016/j.energy.2018.11.096.  Google Scholar

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S. K. Mishra, A brief history of production functions, IUP J. Managerial Econ., 8 (2010), 6-34.  doi: 10.2139/ssrn.1020577.  Google Scholar

[15]

Y. Nakamura, Productivity versus elasticity: A normalized constant elasticity of substitution production function applied to historical Soviet data, Appl. Econ., 47 (2015), 5805-5823.  doi: 10.1080/00036846.2015.1058909.  Google Scholar

[16]

S. Opuni-Basoa, F. T. Oduro and G. A. Okyere, Population dynamics in optimally controlled economic growth models: Case of Cobb-Douglas production function, J. Adv. Math. Comput. Sci., 25 (2017), 1-24.  doi: 10.9734/JAMCS/2017/36753.  Google Scholar

[17]

A. S. Pillai, K. Singh, V. Saravanan, A. Anpalagan, I. Woungang and L. Barolli, A genetic algorithm-based method for optimizing the energy consumption and performance of multiprocessor systems, Soft Comput., 22 (2018), 3271-3285.  doi: 10.1007/s00500-017-2789-y.  Google Scholar

[18]

F. E. Sarac, A general evaluation on estimates of Cobb-Douglas, CES, VES and Translog production functions, Bull. Econ. Theory Anal., 2 (2017), 235-278.   Google Scholar

[19]

C. Takeang and A. Aurasopon, Multiple of hybrid lambda iteration and simulated annealing algorithm to solve economic dispatch problem with ramp rate limit and prohibited operating zones, J. Electrical Engineering Tech., 14 (2019), 111-120.  doi: 10.1007/s42835-018-00001-z.  Google Scholar

[20]

K. Thirugnanasambandam, S. Prakash, V. Subramanian, S. Pothula and V. Thirumal, Reinforced cuckoo search algorithm-based multimodal optimization, Applid Intell., 49 (2019), 2059-2083.  doi: 10.1007/s10489-018-1355-3.  Google Scholar

[21]

H. Wang, X. Zhou, H. Sun, X. Yu, J. Zhao, H. Zhang and L. Cui, Firefly algorithm with adaptive control parameters, Soft Comput., 21 (2017), 5091-5102.  doi: 10.1007/s00500-016-2104-3.  Google Scholar

[22]

D. Zha, A. S. Kavuri and S. Si, Energy-biased technical change in the Chinese industrial sector with CES production functions, Energy, 148 (2018), 896-903.  doi: 10.1016/j.energy.2017.11.087.  Google Scholar

[23]

C. Zhao, Y. Xu and Y. Feng, A study on contribution rate of management elements in economic growth, in International Conference on Information and Management Engineering, Communications in Computer and Information Science, 234, Springer, Berlin, Heidelberg, 2011,151–158. doi: 10.1007/978-3-642-24091-1_21.  Google Scholar

show all references

References:
[1]

A. Assad and K. Deep, A Hybrid Harmony search and Simulated Annealing algorithm for continuous optimization, Information Sci., 450 (2018), 246-266.  doi: 10.1016/j.ins.2018.03.042.  Google Scholar

[2]

V. O. Bohaienko and V. M. Popov, Optimization of operation regimes of irrigation canals using genetic algorithms, in International Conference on Computer Science, Engineering and Education Applications, Advances in Intelligent Systems and Computing, 754, Springer, Cham, 2019,224–233. doi: 10.1007/978-3-319-91008-6_23.  Google Scholar

[3]

M. A. Cardenete, M. C. Lima and F. Sancho, An assessment of the impact of EU funds through productivity boosts using CES functions, Appl. Econ. Lett., 26 (2019), 872-876.  doi: 10.1080/13504851.2018.1508867.  Google Scholar

[4]

J. Cheng, L. Wang, Q. Jiang and Y. Xiong, A novel cuckoo search algorithm with multiple update rules, Appl. Intell., 48 (2018), 4192-4211.  doi: 10.1007/s10489-018-1198-y.  Google Scholar

[5]

M. L. Cheng, A generalized constant elasticity of substitution production function model and its application, J. Systems Sci. Info., 4 (2016), 269-279.  doi: 10.21078/JSSI-2016-269-11.  Google Scholar

[6]

M. L. Cheng, A Grey CES production function model and its application in calculating the contribution rate of economic growth factors, Complexity, 2019 (2019), 8pp. doi: 10.1155/2019/5617061.  Google Scholar

[7]

J. Chongela, V. Nandala and S. Korabandi, Estimation of constant elasticity of substitution (CES) production function with capital and labour inputs of agri-food firms in Tanzania, African J. Agricultural Res., 8 (2013), 5082-5089.   Google Scholar

[8]

F. Erdal, A firefly algorithm for optimum design of new-generation beams, Engineering Optim., 49 (2017), 915-931.  doi: 10.1080/0305215X.2016.1218003.  Google Scholar

[9]

H. Guo, J. Hu, S. Yu, H. Sun and Y. Chen, Computing of the contribution rate of scientific and technological progress to economic growth in Chinese regions, Expert Systems Appl., 39 (2012), 8514-8521.  doi: 10.1016/j.eswa.2011.12.032.  Google Scholar

[10]

Y. He, S. W. Gao and N. Liao, An intelligent computing approach to evaluating the contribution rate of talent on economic growth, Comput. Econ., 48 (2016), 399-423.  doi: 10.1007/s10614-015-9536-1.  Google Scholar

[11]

G. V. S. K. Karthik and S. Deb, A methodology for assembly sequence optimization by hybrid cuckoo-search genetic algorithm, J. Adv. Manufacturing Systems, 17 (2018), 47-59.  doi: 10.1142/S021968671850004X.  Google Scholar

[12]

Y. N. Kiselev, S. N. Avvakumov and M. V. Orlov, Optimal control in the resource allocation problem for a two-sector economy with a CES production function, Comput. Math. Model., 28 (2017), 449-477.  doi: 10.1007/s10598-017-9375-0.  Google Scholar

[13]

X. J. Meng, J. X. Chang, X. B. Wang and Y. M. Wang, Multi-objective hydropower station operation using an improved cuckoo search algorithm, Energy, 168 (2019), 425-439.  doi: 10.1016/j.energy.2018.11.096.  Google Scholar

[14]

S. K. Mishra, A brief history of production functions, IUP J. Managerial Econ., 8 (2010), 6-34.  doi: 10.2139/ssrn.1020577.  Google Scholar

[15]

Y. Nakamura, Productivity versus elasticity: A normalized constant elasticity of substitution production function applied to historical Soviet data, Appl. Econ., 47 (2015), 5805-5823.  doi: 10.1080/00036846.2015.1058909.  Google Scholar

[16]

S. Opuni-Basoa, F. T. Oduro and G. A. Okyere, Population dynamics in optimally controlled economic growth models: Case of Cobb-Douglas production function, J. Adv. Math. Comput. Sci., 25 (2017), 1-24.  doi: 10.9734/JAMCS/2017/36753.  Google Scholar

[17]

A. S. Pillai, K. Singh, V. Saravanan, A. Anpalagan, I. Woungang and L. Barolli, A genetic algorithm-based method for optimizing the energy consumption and performance of multiprocessor systems, Soft Comput., 22 (2018), 3271-3285.  doi: 10.1007/s00500-017-2789-y.  Google Scholar

[18]

F. E. Sarac, A general evaluation on estimates of Cobb-Douglas, CES, VES and Translog production functions, Bull. Econ. Theory Anal., 2 (2017), 235-278.   Google Scholar

[19]

C. Takeang and A. Aurasopon, Multiple of hybrid lambda iteration and simulated annealing algorithm to solve economic dispatch problem with ramp rate limit and prohibited operating zones, J. Electrical Engineering Tech., 14 (2019), 111-120.  doi: 10.1007/s42835-018-00001-z.  Google Scholar

[20]

K. Thirugnanasambandam, S. Prakash, V. Subramanian, S. Pothula and V. Thirumal, Reinforced cuckoo search algorithm-based multimodal optimization, Applid Intell., 49 (2019), 2059-2083.  doi: 10.1007/s10489-018-1355-3.  Google Scholar

[21]

H. Wang, X. Zhou, H. Sun, X. Yu, J. Zhao, H. Zhang and L. Cui, Firefly algorithm with adaptive control parameters, Soft Comput., 21 (2017), 5091-5102.  doi: 10.1007/s00500-016-2104-3.  Google Scholar

[22]

D. Zha, A. S. Kavuri and S. Si, Energy-biased technical change in the Chinese industrial sector with CES production functions, Energy, 148 (2018), 896-903.  doi: 10.1016/j.energy.2017.11.087.  Google Scholar

[23]

C. Zhao, Y. Xu and Y. Feng, A study on contribution rate of management elements in economic growth, in International Conference on Information and Management Engineering, Communications in Computer and Information Science, 234, Springer, Berlin, Heidelberg, 2011,151–158. doi: 10.1007/978-3-642-24091-1_21.  Google Scholar

Figure 1.  Two algorithms' objective function value variation curves with the changes of the number of iteration
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Figure 2.  The distribution diagram of contribution rates of influencing factors to economic growth in Shanghai City, China
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Table 1.  Related data about the economic growth of Shanghai city, China
Year $ Y $ $ K $ $ L $
1999 4222.30 1856.72 733.76
2000 4812.15 1869.67 745.24
2001 5257.66 1994.73 752.26
2002 5795.02 2187.06 792.04
2003 6762.38 2452.11 813.05
2004 8165.38 3084.66 836.87
2005 9365.54 3542.55 863.32
2006 10718.04 3925.09 885.51
2007 12668.12 4458.61 909.08
2008 14275.80 4829.45 1053.24
2009 15285.58 5273.33 1064.42
2010 17433.21 5317.67 1090.76
2011 19533.84 5067.09 1104.33
2012 20553.52 5254.38 1115.50
2013 22257.66 5647.79 1137.35
2014 24060.87 6016.43 1197.31
2015 25643.47 6352.70 1361.51
2016 28178.65 6755.88 1365.24
2017 30632.09 7246.60 1372.65
2018 32679.87 7623.42 1430.82
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Year $ Y $ $ K $ $ L $
1999 4222.30 1856.72 733.76
2000 4812.15 1869.67 745.24
2001 5257.66 1994.73 752.26
2002 5795.02 2187.06 792.04
2003 6762.38 2452.11 813.05
2004 8165.38 3084.66 836.87
2005 9365.54 3542.55 863.32
2006 10718.04 3925.09 885.51
2007 12668.12 4458.61 909.08
2008 14275.80 4829.45 1053.24
2009 15285.58 5273.33 1064.42
2010 17433.21 5317.67 1090.76
2011 19533.84 5067.09 1104.33
2012 20553.52 5254.38 1115.50
2013 22257.66 5647.79 1137.35
2014 24060.87 6016.43 1197.31
2015 25643.47 6352.70 1361.51
2016 28178.65 6755.88 1365.24
2017 30632.09 7246.60 1372.65
2018 32679.87 7623.42 1430.82
Table 2.  The comparison of results of two CS algorithms
Method Conventional CS Improved CS
$ A_{0} $ 4.9963 3.9414
$ \sigma $ 0.0450 0.0433
$ \delta_{1} $ 1.1503 1.4430
$ \delta_{2} $ 4.9788 6.6868
$ a $ 4.9940 5.0012
$ b $ 3.2084 3.1064
$ c $ 0.1988 0.1497
$ \mu $ 0.8231 0.8293
Number of Iteration 252 54
$ G, $ Optimal Value of Objective Function 1.0319e$ + $07 9.9439e$ + $06
$ R^{2}, $ Coefficient of Determination of Model 0.9935 0.9937
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Method Conventional CS Improved CS
$ A_{0} $ 4.9963 3.9414
$ \sigma $ 0.0450 0.0433
$ \delta_{1} $ 1.1503 1.4430
$ \delta_{2} $ 4.9788 6.6868
$ a $ 4.9940 5.0012
$ b $ 3.2084 3.1064
$ c $ 0.1988 0.1497
$ \mu $ 0.8231 0.8293
Number of Iteration 252 54
$ G, $ Optimal Value of Objective Function 1.0319e$ + $07 9.9439e$ + $06
$ R^{2}, $ Coefficient of Determination of Model 0.9935 0.9937
Table 3.  Verification results of conditions of production function
Year $ f_{1} $ $ f_{2} $ $ f_{11} $ $ f_{22} $ $ f_{12} $ $ ff $
1999 1.2536 6.5002 -9.275e-5 -13.7e-4 -5.69e-5 1.2364e-7
2000 1.2387 6.2050 -7.511e-5 -11.8e-4 -9.53e-5 7.9713e-8
2001 1.2487 6.1583 -6.335e-5 -10.9e-4 -1.15e-4 5.5835e-8
2002 1.2665 6.1774 -5.439e-5 -9.92e-4 -1.23e-4 3.8894e-8
2003 1.2922 6.2664 -4.787e-5 -9.33e-4 -1.27e-4 2.8559e-8
2004 1.3090 6.3446 -3.959e-5 -8.48e-4 -1.21e-4 1.8918e-8
2005 1.3397 6.4705 -3.565e-5 -7.93e-4 -1.19e-4 1.4196e-8
2006 1.3780 6.6312 -3.337e-5 -7.56e-4 -1.18e-4 1.1381e-8
2007 1.4150 6.7923 -3.086e-5 -7.15e-4 -1.14e-4 8.9777e-9
2008 1.4472 6.9094 -2.805e-5 -6.34e-4 -1.06e-4 6.5751e-9
2009 1.4950 7.1239 -2.709e-5 -6.20e-4 -1.06e-4 5.6539e-9
2010 1.5549 7.3862 -2.754e-5 -6.24e-4 -1.09e-4 5.2946e-9
2011 1.6274 7.7058 -2.929e-5 -6.53e-4 -1.17e-4 5.4122e-9
2012 1.6907 7.9924 -2.949e-5 -6.59e-4 -1.20e-4 5.0563e-9
2013 1.7496 8.2616 -2.888e-5 -6.48e-4 -1.19e-4 4.5043e-9
2014 1.8078 8.5239 -2.803e-5 -6.28e-4 -1.17e-4 3.9252e-9
2015 1.8582 8.7431 -2.631e-5 -5.82e-4 -1.10e-4 3.1723e-9
2016 1.9281 9.0661 -2.625e-5 -5.83e-4 -1.11e-4 2.9566e-9
2017 1.9985 9.3922 -2.600e-5 -5.81e-4 -1.11e-4 2.7283e-9
2018 2.0695 9.7171 -2.563e-5 -5.71e-4 -1.10e-4 2.4778e-9
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Year $ f_{1} $ $ f_{2} $ $ f_{11} $ $ f_{22} $ $ f_{12} $ $ ff $
1999 1.2536 6.5002 -9.275e-5 -13.7e-4 -5.69e-5 1.2364e-7
2000 1.2387 6.2050 -7.511e-5 -11.8e-4 -9.53e-5 7.9713e-8
2001 1.2487 6.1583 -6.335e-5 -10.9e-4 -1.15e-4 5.5835e-8
2002 1.2665 6.1774 -5.439e-5 -9.92e-4 -1.23e-4 3.8894e-8
2003 1.2922 6.2664 -4.787e-5 -9.33e-4 -1.27e-4 2.8559e-8
2004 1.3090 6.3446 -3.959e-5 -8.48e-4 -1.21e-4 1.8918e-8
2005 1.3397 6.4705 -3.565e-5 -7.93e-4 -1.19e-4 1.4196e-8
2006 1.3780 6.6312 -3.337e-5 -7.56e-4 -1.18e-4 1.1381e-8
2007 1.4150 6.7923 -3.086e-5 -7.15e-4 -1.14e-4 8.9777e-9
2008 1.4472 6.9094 -2.805e-5 -6.34e-4 -1.06e-4 6.5751e-9
2009 1.4950 7.1239 -2.709e-5 -6.20e-4 -1.06e-4 5.6539e-9
2010 1.5549 7.3862 -2.754e-5 -6.24e-4 -1.09e-4 5.2946e-9
2011 1.6274 7.7058 -2.929e-5 -6.53e-4 -1.17e-4 5.4122e-9
2012 1.6907 7.9924 -2.949e-5 -6.59e-4 -1.20e-4 5.0563e-9
2013 1.7496 8.2616 -2.888e-5 -6.48e-4 -1.19e-4 4.5043e-9
2014 1.8078 8.5239 -2.803e-5 -6.28e-4 -1.17e-4 3.9252e-9
2015 1.8582 8.7431 -2.631e-5 -5.82e-4 -1.10e-4 3.1723e-9
2016 1.9281 9.0661 -2.625e-5 -5.83e-4 -1.11e-4 2.9566e-9
2017 1.9985 9.3922 -2.600e-5 -5.81e-4 -1.11e-4 2.7283e-9
2018 2.0695 9.7171 -2.563e-5 -5.71e-4 -1.10e-4 2.4778e-9
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