期刊
AIMS MATHEMATICS
卷 6, 期 9, 页码 9194-9206出版社
AMER INST MATHEMATICAL SCIENCES-AIMS
DOI: 10.3934/math.2021534
关键词
two-scale economics; two-scale fractal derivative; scale-dependent law; fractal variational principle; fractal economics
资金
- Research Council of Shahid Chamran University of Ahvaz [SCU.EM99.98]
This paper argues that economic phenomena should be observed in two different scales, and economic laws are scale-dependent. It proposes the concept of two-scale price dynamics and establishes a fractal variational theory for profit maximization. The paper explores the application of the Lagrange multiplier method to solve complex economic problems.
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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