Evans model for dynamic economics revised

Ji-Huan He; Chun-Hui He; Hamid M. Sedighi; Ji-Huan He; Chun-Hui He; Hamid M. Sedighi

doi:10.3934/math.2021534

AIMS Mathematics

2021, Volume 6, Issue 9: 9194-9206. doi: 10.3934/math.2021534

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Research article

Evans model for dynamic economics revised

1.
School of Science, Xi'an University of Architecture and Technology, Xi'an, China
2.
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, China
3.
National Engineering Laboratory for Modern Silk, College of Textile and Clothing Engineering, Soochow University, 199 Ren-Ai Road, Suzhou, China
4.
School of Civil Engineering, Xi'an University of Architecture & Technology, Xi'an 710055, China
5.
Mechanical Engineering Department, Faculty of Engineering, Shahid Chamran University of Ahvaz, Ahvaz, Iran

Received: 06 April 2021 Accepted: 14 June 2021 Published: 21 June 2021
MSC : 35Q91, 28A80, 31E05

This paper argues that any economic phenomena should be observed by two different scales, and any economic laws are scale-dependent. A one-scale law arising in either macroeconomics or microeconomics might be mathematically correct and economically relevant, however, sparking debates might arise for a different scale. This paper re-analyzes the basic assumptions of the Evans model for dynamic economics, and it concludes that they are quite reasonable on a large time-scale, but the assumptions become totally invalid on a smaller scale, and a fractal modification has to be adopted. A two-scale price dynamics is suggested and a fractal variational theory is established to maximize the profit at a given period. Furthermore Evans 1924 variational principle for the maximal profit is easy to be solved for a quadratic cost function using the Lagrange multiplier method. Here a quadratic-cubic cost function and a nonlinear demand function are used, and the stationary condition of the variational formulation is derived step by step, and a more complex dynamic system is obtained. The present derivation process can be extended to a more complex cost function and a more complex demand function, and the paper sheds a promising light on mathematics treatment of complex economic problems.
- two-scale economics,
- two-scale fractal derivative,
- scale-dependent law,
- fractal variational principle,
- fractal economics
Citation: Ji-Huan He, Chun-Hui He, Hamid M. Sedighi. Evans model for dynamic economics revised[J]. AIMS Mathematics, 2021, 6(9): 9194-9206. doi: 10.3934/math.2021534

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Abstract

This paper argues that any economic phenomena should be observed by two different scales, and any economic laws are scale-dependent. A one-scale law arising in either macroeconomics or microeconomics might be mathematically correct and economically relevant, however, sparking debates might arise for a different scale. This paper re-analyzes the basic assumptions of the Evans model for dynamic economics, and it concludes that they are quite reasonable on a large time-scale, but the assumptions become totally invalid on a smaller scale, and a fractal modification has to be adopted. A two-scale price dynamics is suggested and a fractal variational theory is established to maximize the profit at a given period. Furthermore Evans 1924 variational principle for the maximal profit is easy to be solved for a quadratic cost function using the Lagrange multiplier method. Here a quadratic-cubic cost function and a nonlinear demand function are used, and the stationary condition of the variational formulation is derived step by step, and a more complex dynamic system is obtained. The present derivation process can be extended to a more complex cost function and a more complex demand function, and the paper sheds a promising light on mathematics treatment of complex economic problems.

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