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.
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
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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