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doi: 10.3934/jimo.2020099

## Application of a modified VES production function model

 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.

Citation: Maolin Cheng, Bin Liu. Application of a modified VES production function model. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020099
##### References:

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##### References:
Two algorithms' objective function value variation curves with the changes of the number of iteration
The distribution diagram of contribution rates of influencing factors to economic growth in Shanghai City, China
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
 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
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
 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
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
 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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