# American Institute of Mathematical Sciences

July  2020, 16(4): 1571-1584. doi: 10.3934/jimo.2019018

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

 1 School of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou, 215009, China 2 School of Mathematics and Statistics, Huangshan University, Huangshan, 245041, China

* Corresponding author: Maolin Cheng

Received  June 2018 Revised  September 2018 Published  July 2020 Early access  March 2019

Fund Project: The first author is supported by NSF grant 11401418

The conventional CES production function model fails to consider the influences of policy factors on economic growth in different stages. This paper proposes a modified model of the CES production function. Regarding model parameter estimation, the paper proposes a modern intelligent algorithm, the firefly algorithm (FA). The paper improves conventional FA to enhance the convergence rate and precision. To overcome the shortcomings of the conventional method in model application, the paper presents a new method of calculating the contribution rates of factors influencing economic growth and provides examples.

Citation: Maolin Cheng, Mingyin Xiang. Application of a modified CES production function model based on improved firefly algorithm. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1571-1584. doi: 10.3934/jimo.2019018
##### References:
 [1] J. Antony, A dual elasticity of substitution production function with an application to cross-country inequality, Economics Letters, 102 (2009), 10-12.  doi: 10.1016/j.econlet.2008.09.007. [2] P. Balachennaiah, M. Suryakalavathi and P. Nagendra, Optimizing real power loss and voltage stability limit of a large transmission network using firefly algorithm, Engineering Science and Technology, 19 (2016), 800-810.  doi: 10.1016/j.jestch.2015.10.008. [3] A. Baykasoğlu and F. B. Ozsoydan, An improved firefly algorithm for solving dynamic multidimensional knapsack problems, Expert Systems with Applications, 41 (2014), 3712-3725. [4] J. Y. Cao, The aggregate production function and contribution rate of technical change to economic growth in China, Journal of Quantitative & Technical Economics, 11 (2007), 37-46. [5] F. Erdal, A firefly algorithm for optimum design of new-generation beams, Engineering Optimization, 49 (2017), 915-931.  doi: 10.1080/0305215X.2016.1218003. [6] M. Farhoodnea, A. Mohamed, H. Shareef and H. Zayandehroodi, Optimum placement of active power conditioner in distribution systems using improved discrete firefly algorithm for power quality enhancement, Applied Soft Computing, 23 (2014), 249-258.  doi: 10.1016/j.asoc.2014.06.038. [7] Q. Fu, R. Q. Jiang, Z. L. Wang and T. X. Li, Optimization of soil water characteristic curves parameters by modified firefly algorithm, Transactions of the Chinese Society of Agricultural Engineering, 31 (2015), 117-122. [8] H. Ge and Y. J. Feng, Total factor productivity and contribution rate of technological progress based on equivalent benefit surface production function, Quantitative & Technica Economics, 11 (2004), 94-101. [9] S. Gholizadeh, Performance-based optimum seismic design of steel structures by a modified firefly algorithm and a new neural network, Advances in Engineering Software, 81 (2015), 50-65.  doi: 10.1016/j.advengsoft.2014.11.003. [10] A. Gupta and P. K. Padhy, Modified Firefly Algorithm based controller design for integrating and unstable delay processes, Engineering Science and Technology, 19 (2016), 548-558.  doi: 10.1016/j.jestch.2015.09.015. [11] A. Henningsen and G. Henningsen, On estimation of the CES production function-Revisited, Economics Letters, 115 (2012), 67-69.  doi: 10.1016/j.econlet.2011.12.007. [12] R. Klump and M. Saam, Calibration of normalised CES production functions in dynamic models, Economics Letters, 99 (2008), 256-259.  doi: 10.1016/j.econlet.2007.07.001. [13] K. N. Krishnanand and D. Ghose, Detection of multiple source locations using a glowworm metaphor with applications to collective robotics, Proc of IEEE Swarm Intelligence Symposium, IEEE Press, Piscataway, 2005, 84-91. [14] L. Lei, Z. H. Zhang and L. L. Wang, Measurement and analysis on contribution rate of agricultural science and technology progress in Shaanxi province: based on C-D production function, Technology Economics, 30 (2011), 59-63. [15] H. Li, X. Guo and W. Li, PID controller parameter optimization based on improved glowworm swarm optimization, Computer Applications and Software, 34 (2017), 227-230. [16] Z. Li, D. D. Wang, L. Liang and Y. Q. Zhou, Artificial fish school algorithm for estimating parametersin production function, Journal of Chongqing Normal University(Natural Science), 26 (2009), 84–86, 93. [17] L. B. Liu, C. P. Liu, Y. Zhang and Y. Wang, Parameter estimation method for predator prey model based on particle swarm optimization algorithm, Pure and Applied Mathematics, 32 (2016), 19-24. [18] W. Long, J. J. Jiao and S. J. Xu, Parameter estimation for reaction kinetics model based on composite genetic algorithm, Journal of Computer Applications, 32 (2012), 1704-1706.  doi: 10.3724/SP.J.1087.2012.01704. [19] Y. Q. Lv, Z. Lu and X. Deng, Application of improved particle swarm optimization algorithm in estimating parameters of production function, Statistics & Decision, 4 (2014), 21-24. [20] A. Mishra, C. Agarwal, A. Sharma and P. Bedi, Optimized gray-scale image watermarking using DWT–SVD and Firefly Algorithm, Expert Systems with Applications, 41 (2014), 7858-7867.  doi: 10.1016/j.eswa.2014.06.011. [21] F. M. Pavelescu, Impact of collinearity on estimated parameters of CES production function, Procedia Economics and Finance, 22 (2015), 762-769.  doi: 10.1016/S2212-5671(15)00304-4. [22] Y. W. Peng, L. S. Guo and C. Mao, Parameter estimation for fuzzy regression based on improved simulated annealing algorithm, Statistics & Decision, 1 (2014), 17-20. [23] A. Rastgou and J. Moshtagh, Application of firefly algorithm for multi-stage transmission expansion planning with adequacy-security considerations in deregulated environments, Applied Soft Computing, 41 (2016), 373-389.  doi: 10.1016/j.asoc.2016.01.018. [24] M. G. Sundari, M. Rajaram and S. Balaraman, Application of improved firefly algorithm for programmed PWM in multilevel inverter with adjustable DC sources, Applied Soft Computing, 41 (2016), 169-179.  doi: 10.1016/j.asoc.2015.12.036. [25] W. Tang and L. Wu, Parameter estimation for piecewise linear Chen system based on genetic algorithm, Computer Science, 42 (2015), 83–85, 99. [26] S. Verma and V. Mukherjee, Firefly algorithm for congestion management in deregulated environment, Engineering Science and Technology, 19 (2016), 1254-1265.  doi: 10.1016/j.jestch.2016.02.001. [27] A. D. Vîlcu and G. E. Vîlcu, On some geometric properties of the generalized CES production functions, Applied Mathematics and Computation, 218 (2011), 124-129.  doi: 10.1016/j.amc.2011.05.061. [28] H. Wang, X. Y. Zhou, H. Sun, X. Yu, J. Zhao, H. Zhang and L. Z. Cui, Firefly algorithm with adaptive control parameters, Soft Computing, 21 (2017), 5091-5102.  doi: 10.1007/s00500-016-2104-3. [29] F. Xia and Y. L. Wang, Measurement of the rural worker's contribution rate to the economic growth based on production function model, Chinese Journal of Management Science, 16 (2008), 587-591. [30] Z. Yan and H. F. Jiang, CES production function and its application, Quantitative & Technica Economics, 9 (2002), 95-98. [31] Z. G. Yan, H. N. Hu and G. Li, Parameter estimation of Richards model and algorithm effectiveness based on particle swarm optimization algorithm, Journal of Computer Applications, 34 (2014), 2827-2830. [32] X. S. Yang, Nature-inspired metaheuristic algorithms, Luniver press, Beckington, 2008, 83–96. [33] Q. L. Zhan and X. H. Ceng, Factor substitution elasticity estimation of CES production function based on Chinese industrial micro data, Statistics & Decision, 24 (2015), 31-35. [34] G. Q. Zheng and G. Q. Zhang, Parameter estmiations of semi-parametric linear regression models using smiulated annealing algorithm, Journal of South China Agricultural University, 27 (2006), 115-117.

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##### References:
 [1] J. Antony, A dual elasticity of substitution production function with an application to cross-country inequality, Economics Letters, 102 (2009), 10-12.  doi: 10.1016/j.econlet.2008.09.007. [2] P. Balachennaiah, M. Suryakalavathi and P. Nagendra, Optimizing real power loss and voltage stability limit of a large transmission network using firefly algorithm, Engineering Science and Technology, 19 (2016), 800-810.  doi: 10.1016/j.jestch.2015.10.008. [3] A. Baykasoğlu and F. B. Ozsoydan, An improved firefly algorithm for solving dynamic multidimensional knapsack problems, Expert Systems with Applications, 41 (2014), 3712-3725. [4] J. Y. Cao, The aggregate production function and contribution rate of technical change to economic growth in China, Journal of Quantitative & Technical Economics, 11 (2007), 37-46. [5] F. Erdal, A firefly algorithm for optimum design of new-generation beams, Engineering Optimization, 49 (2017), 915-931.  doi: 10.1080/0305215X.2016.1218003. [6] M. Farhoodnea, A. Mohamed, H. Shareef and H. Zayandehroodi, Optimum placement of active power conditioner in distribution systems using improved discrete firefly algorithm for power quality enhancement, Applied Soft Computing, 23 (2014), 249-258.  doi: 10.1016/j.asoc.2014.06.038. [7] Q. Fu, R. Q. Jiang, Z. L. Wang and T. X. Li, Optimization of soil water characteristic curves parameters by modified firefly algorithm, Transactions of the Chinese Society of Agricultural Engineering, 31 (2015), 117-122. [8] H. Ge and Y. J. Feng, Total factor productivity and contribution rate of technological progress based on equivalent benefit surface production function, Quantitative & Technica Economics, 11 (2004), 94-101. [9] S. Gholizadeh, Performance-based optimum seismic design of steel structures by a modified firefly algorithm and a new neural network, Advances in Engineering Software, 81 (2015), 50-65.  doi: 10.1016/j.advengsoft.2014.11.003. [10] A. Gupta and P. K. Padhy, Modified Firefly Algorithm based controller design for integrating and unstable delay processes, Engineering Science and Technology, 19 (2016), 548-558.  doi: 10.1016/j.jestch.2015.09.015. [11] A. Henningsen and G. Henningsen, On estimation of the CES production function-Revisited, Economics Letters, 115 (2012), 67-69.  doi: 10.1016/j.econlet.2011.12.007. [12] R. Klump and M. Saam, Calibration of normalised CES production functions in dynamic models, Economics Letters, 99 (2008), 256-259.  doi: 10.1016/j.econlet.2007.07.001. [13] K. N. Krishnanand and D. Ghose, Detection of multiple source locations using a glowworm metaphor with applications to collective robotics, Proc of IEEE Swarm Intelligence Symposium, IEEE Press, Piscataway, 2005, 84-91. [14] L. Lei, Z. H. Zhang and L. L. Wang, Measurement and analysis on contribution rate of agricultural science and technology progress in Shaanxi province: based on C-D production function, Technology Economics, 30 (2011), 59-63. [15] H. Li, X. Guo and W. Li, PID controller parameter optimization based on improved glowworm swarm optimization, Computer Applications and Software, 34 (2017), 227-230. [16] Z. Li, D. D. Wang, L. Liang and Y. Q. Zhou, Artificial fish school algorithm for estimating parametersin production function, Journal of Chongqing Normal University(Natural Science), 26 (2009), 84–86, 93. [17] L. B. Liu, C. P. Liu, Y. Zhang and Y. Wang, Parameter estimation method for predator prey model based on particle swarm optimization algorithm, Pure and Applied Mathematics, 32 (2016), 19-24. [18] W. Long, J. J. Jiao and S. J. Xu, Parameter estimation for reaction kinetics model based on composite genetic algorithm, Journal of Computer Applications, 32 (2012), 1704-1706.  doi: 10.3724/SP.J.1087.2012.01704. [19] Y. Q. Lv, Z. Lu and X. Deng, Application of improved particle swarm optimization algorithm in estimating parameters of production function, Statistics & Decision, 4 (2014), 21-24. [20] A. Mishra, C. Agarwal, A. Sharma and P. Bedi, Optimized gray-scale image watermarking using DWT–SVD and Firefly Algorithm, Expert Systems with Applications, 41 (2014), 7858-7867.  doi: 10.1016/j.eswa.2014.06.011. [21] F. M. Pavelescu, Impact of collinearity on estimated parameters of CES production function, Procedia Economics and Finance, 22 (2015), 762-769.  doi: 10.1016/S2212-5671(15)00304-4. [22] Y. W. Peng, L. S. Guo and C. Mao, Parameter estimation for fuzzy regression based on improved simulated annealing algorithm, Statistics & Decision, 1 (2014), 17-20. [23] A. Rastgou and J. Moshtagh, Application of firefly algorithm for multi-stage transmission expansion planning with adequacy-security considerations in deregulated environments, Applied Soft Computing, 41 (2016), 373-389.  doi: 10.1016/j.asoc.2016.01.018. [24] M. G. Sundari, M. Rajaram and S. Balaraman, Application of improved firefly algorithm for programmed PWM in multilevel inverter with adjustable DC sources, Applied Soft Computing, 41 (2016), 169-179.  doi: 10.1016/j.asoc.2015.12.036. [25] W. Tang and L. Wu, Parameter estimation for piecewise linear Chen system based on genetic algorithm, Computer Science, 42 (2015), 83–85, 99. [26] S. Verma and V. Mukherjee, Firefly algorithm for congestion management in deregulated environment, Engineering Science and Technology, 19 (2016), 1254-1265.  doi: 10.1016/j.jestch.2016.02.001. [27] A. D. Vîlcu and G. E. Vîlcu, On some geometric properties of the generalized CES production functions, Applied Mathematics and Computation, 218 (2011), 124-129.  doi: 10.1016/j.amc.2011.05.061. [28] H. Wang, X. Y. Zhou, H. Sun, X. Yu, J. Zhao, H. Zhang and L. Z. Cui, Firefly algorithm with adaptive control parameters, Soft Computing, 21 (2017), 5091-5102.  doi: 10.1007/s00500-016-2104-3. [29] F. Xia and Y. L. Wang, Measurement of the rural worker's contribution rate to the economic growth based on production function model, Chinese Journal of Management Science, 16 (2008), 587-591. [30] Z. Yan and H. F. Jiang, CES production function and its application, Quantitative & Technica Economics, 9 (2002), 95-98. [31] Z. G. Yan, H. N. Hu and G. Li, Parameter estimation of Richards model and algorithm effectiveness based on particle swarm optimization algorithm, Journal of Computer Applications, 34 (2014), 2827-2830. [32] X. S. Yang, Nature-inspired metaheuristic algorithms, Luniver press, Beckington, 2008, 83–96. [33] Q. L. Zhan and X. H. Ceng, Factor substitution elasticity estimation of CES production function based on Chinese industrial micro data, Statistics & Decision, 24 (2015), 31-35. [34] G. Q. Zheng and G. Q. Zhang, Parameter estmiations of semi-parametric linear regression models using smiulated annealing algorithm, Journal of South China Agricultural University, 27 (2006), 115-117.
Data on Chinese economic growth
 Period Year $Y$ $L$ $K$ $E$ Period of the 9$^{th}$ five-year plan 1996 71176.6 68950 22913.5 135192 1997 78973.0 69820 24941.1 135909 1998 84402.3 70637 28406.2 136184 1999 89677.1 71394 29854.7 140569 2000 99214.6 72085 32917.7 145531 Period of the 10$^{th}$ five-year plan 2001 109655.2 72797 37213.5 150406 2002 120332.7 73280 43499.9 159431 2003 135822.8 73736 55566.6 183792 2004 159878.3 74264 70477.4 213456 2005 183217.5 74647 88773.6 235997 Period of the 11$^{th}$ five-year plan 2006 211923.5 74978 109998.2 258676 2007 249529.9 75321 137239.0 280508 2008 316228.8 75564 172828.4 291448 2009 343464.7 75828 224598.8 306647 2010 401512.8 76105 251683.8 324939 Period of the 12$^{th}$ five-year plan 2011 473104.0 76420 311485.1 348002 2012 519470.1 76704 374694.7 361732 2013 568845.0 76977 447074.0 375252 2014 636462.7 77253 512760.7 426000 2015 676780.0 77451 562000.0 430000 Period of the 13$^{th}$ five-year plan 2016 744127.0 77603 606466.0 436000 2017 827122.0 77640 641238.0 449000
 Period Year $Y$ $L$ $K$ $E$ Period of the 9$^{th}$ five-year plan 1996 71176.6 68950 22913.5 135192 1997 78973.0 69820 24941.1 135909 1998 84402.3 70637 28406.2 136184 1999 89677.1 71394 29854.7 140569 2000 99214.6 72085 32917.7 145531 Period of the 10$^{th}$ five-year plan 2001 109655.2 72797 37213.5 150406 2002 120332.7 73280 43499.9 159431 2003 135822.8 73736 55566.6 183792 2004 159878.3 74264 70477.4 213456 2005 183217.5 74647 88773.6 235997 Period of the 11$^{th}$ five-year plan 2006 211923.5 74978 109998.2 258676 2007 249529.9 75321 137239.0 280508 2008 316228.8 75564 172828.4 291448 2009 343464.7 75828 224598.8 306647 2010 401512.8 76105 251683.8 324939 Period of the 12$^{th}$ five-year plan 2011 473104.0 76420 311485.1 348002 2012 519470.1 76704 374694.7 361732 2013 568845.0 76977 447074.0 375252 2014 636462.7 77253 512760.7 426000 2015 676780.0 77451 562000.0 430000 Period of the 13$^{th}$ five-year plan 2016 744127.0 77603 606466.0 436000 2017 827122.0 77640 641238.0 449000
Comparison of results of three algorithms
 Algorithm Improved PSO Conventional firefly algorithm Improved firefly algorithm $A$ 0.7638 0.8268 0.7500 $\sigma$ 0.0518 0.0700 0.0572 $\delta_{1}$ 0.6612 0.6460 0.6502 $\delta_{2}$ 0.3229 0.3827 0.3297 $\delta_{3}$ 0.1852 0.1712 0.1834 $\alpha_{1}$ 0.5352 0.5080 0.5184 $\alpha_{2}$ 0.7295 0.6825 0.7164 $\alpha_{3}$ 0.1683 0.2149 0.1986 $\alpha_{4}$ 0.3224 0.3464 0.3223 $\alpha_{5}$ 0.1511 0.1970 0.1538 $\beta_{1}$ 0.1431 0.1878 0.1565 $\beta_{2}$ 0.1072 0.1228 0.1170 $\beta_{3}$ 0.1971 0.1916 0.2027 $\beta_{4}$ 0.2472 0.2645 0.2498 $\beta_{5}$ 0.2183 0.1981 0.2239 $\gamma_{1}$ 0.1102 0.0988 0.1137 $\gamma_{2}$ 0.0710 0.1352 0.0901 $\gamma_{3}$ 0.1006 0.1478 0.1002 $\gamma_{4}$ 0.0478 0.0375 0.0482 $\gamma_{5}$ 0.1205 0.1945 0.1306 $\rho$ 1.5457 1.4934 1.5310 $\mu$ 1.2312 1.1995 1.2203 Iteration number 142 774 22 Objective function $RSS$ 2.4602e$+$09 2.8225e$+$09 1.2426e$+$09 Model's coefficient of determination, $R^{2}$ 0.9980 0.9977 0.9990
 Algorithm Improved PSO Conventional firefly algorithm Improved firefly algorithm $A$ 0.7638 0.8268 0.7500 $\sigma$ 0.0518 0.0700 0.0572 $\delta_{1}$ 0.6612 0.6460 0.6502 $\delta_{2}$ 0.3229 0.3827 0.3297 $\delta_{3}$ 0.1852 0.1712 0.1834 $\alpha_{1}$ 0.5352 0.5080 0.5184 $\alpha_{2}$ 0.7295 0.6825 0.7164 $\alpha_{3}$ 0.1683 0.2149 0.1986 $\alpha_{4}$ 0.3224 0.3464 0.3223 $\alpha_{5}$ 0.1511 0.1970 0.1538 $\beta_{1}$ 0.1431 0.1878 0.1565 $\beta_{2}$ 0.1072 0.1228 0.1170 $\beta_{3}$ 0.1971 0.1916 0.2027 $\beta_{4}$ 0.2472 0.2645 0.2498 $\beta_{5}$ 0.2183 0.1981 0.2239 $\gamma_{1}$ 0.1102 0.0988 0.1137 $\gamma_{2}$ 0.0710 0.1352 0.0901 $\gamma_{3}$ 0.1006 0.1478 0.1002 $\gamma_{4}$ 0.0478 0.0375 0.0482 $\gamma_{5}$ 0.1205 0.1945 0.1306 $\rho$ 1.5457 1.4934 1.5310 $\mu$ 1.2312 1.1995 1.2203 Iteration number 142 774 22 Objective function $RSS$ 2.4602e$+$09 2.8225e$+$09 1.2426e$+$09 Model's coefficient of determination, $R^{2}$ 0.9980 0.9977 0.9990
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