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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  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 & Management Optimization, doi: 10.3934/jimo.2019018
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##### References:
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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