Abstract
The current non-linear programming method does not derive regional economic surpluses and may derive an imprecise maximized value of the total economic surplus. The main reason is that the integrals for supply functions will automatically take regional non-economic producer surpluses into account if any intercepts of supply functions is negative. Consequently, the derived values are always lower than the real regional and total economic surpluses. The unknown regional economic surpluses and the imprecise total economic surplus will limit the suitable application of the model for broader contexts including game theory analysis, international trade policy analysis, and even GDP calculation. This paper recommends two formulae applied for two types of functions, namely original and inverse supply and demand functions, to calculate the regional and total economic surpluses of commodities. The two methods can be converted to each other conveniently, for example by using an inverse matrix of coefficients of original supply and demand functions to solve coefficients of inverse supply and demand functions. A numerical example is used to illustrate the spatial equilibrium model for 2 products and 3 regions with original linear supply and demand functions.
Highlights
The concept of the spatial equilibrium model was raised in the late 19th century [1]
If an economist does not take into account signs of the intercepts of original linear supply functions, and values of regional imports and exports, the imprecise total economic surplus generated by the General Algebraic Modeling System (GAMS) programming is 6224.6
The optimal solution of the total economic surplus obtained by the current non-linear programming method can be imprecise
Summary
The concept of the spatial equilibrium model was raised in the late 19th century [1]. Non-linear programming is the most common method used to solve spatial equilibrium models with a non-linear objective function for the total economic surplus including linear and non-linear supply and demand functions as presented in journal articles and mathematical programming books including Takayama and Judge (1964) [3] and McCarl and Spreen (2002) [4]. Linear programming is used to solve spatial equilibrium models with linear supply functions, linear demand functions, and linear objective function of the total transportation cost as presented in Phan, Harrison, and Lamb (2011) [5]. Mixed complementary programming is used to derive optimal solutions for spatial equilibrium models without any objective function, as presented in Goletti et al (1996) [6], but requires some strict conditions and in particular equal numbers of constraints and equations [4]. The following sections examine why the two limitations exist and proposes how to solve precisely the economic surpluses of commodities, regional economic surpluses, and the total economic surplus generated by the spatial equilibrium models with linear supply and demand functions
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