A formulation of spatial price equilibrium model and its computation procedure

自从萨缪尔森（1952）指出存在一个目标函数，其中最大化满足空间分离市场之间完全竞争均衡的条件以来，利用数学规划方法研究空间上的市场行为在这一领域取得了重大进展。迄今为止开发的许多作品一般都是完全竞争模型或垄断模型，然而，很自然地感觉，大多数初级商品和制成品市场的真实世界在某种程度上介于这两个极端之间，采取某种形式的寡头垄断。本研究的目的首先是。对Greenhut模型进行扩展，使其能够在某些区域对之间可能不发生交易的商品市场中进行空间定价；第二，为寻找空间分离的寡头垄断市场中区域间均衡商品流动提供一种高效且易于实现的计算程序；该方法更适用于具有非线性边际生产成本的空间定价和变分等式的子优化方案。空间价格均衡模型的建立。本研究得出了三个主要结论。首先，基本定价方程本质上与交通均衡中使用的复合性能函数相同。然而，在给定需求函数和单个性能函数的情况下，复合性能函数只是解释均衡过程的方便工具，尚未用于计算目的。可以说，G-G模型给出了一个关于如何在给定单个成本（或性能）函数的情况下使复合函数用于计算目的的想法。另一方面，在交通均衡中培养的方法可以从概念上和数值上揭示一种特殊类型的空间寡头均衡模型。其次，在空间寡头垄断市场中，与无空间经济中一样，如果市场上存在过剩数量，市场价格就会提高；如果市场处于均衡状态，过剩数量就会消失。这些结果表明，只要市场是试探性的，就会存在试探性的流动，一些企业会因为亏损而退出市场。因此，构建了一个计算程序来反映这种现象。就第3节所展示的数值例子而言，本文提出的过程似乎不仅以有效的方式收敛到平衡状态，而且易于实现。第三，通过引入双阶段算法，本文方法在具有非线性边际生产成本的空间定价中也可能是有效的。这将大大节省空间寡占模型中寻找平衡流的计算时间。但是，需要对所提出的方法在可变边际成本问题上的应用进行进一步的仔细审查和更详细的分析，以确认该程序的收敛性。少

Since Samuelson (1952) pointed out that there exists an objective function where maximization satisfies the conditions of perfectly competitive equilibria among spatially separated markets, the use of mathematical programming approach to study market behavior over space has been significantly progressed in this field. Many works developed so far generally treat a perfectly competitive model or a monopoly model, however, it is natural feeling that a real world for markets of most primary commodities and manufactured goods lie somewhat between these two extremes,taking some forms of oligopoly. The purpose of this research is first. to extend the Greenhut model so as to permit spatial pricing in commodity market where it could happen that no trading takes place between some pairs of regions and second, to provide an efficient and easily implementable computational procedure for finding the inter-regional equilibrium commodity flow in spatially separated oligopolistic market, which is appl … More icable to spatial pricing with nonlinear marginal- production cost and the sub-optimization program of the variational equality. formulation of the spatial price equilibrium model.Three major conclusions were drawn from this research. First, the fundamental pricing equation is essentially the same as the composite performance function used in traffic equilibrium. However, the composite performance function is no more than convenient device of interpreting equilibrium processes given demand function and individual performance functions and has not been used for computational purposes. It can be said that G-G model gives an idea about how to make the composite function for computational purposes given individual cost (or performance) functions. On the other hand,methodology cultivate in traffic equilibrium may shed light on a special type of spatial oligopolistic equilibrium model, conceptually and numerically. Second, it was shown that in a spatial oligopolistic market, as like in spaceless economy, if there is excess quantity in market it makes to raise market price and that excess quantity vanish if the market is in equilibrium. These results imply that as long as market is tentative, there exists tentative flows and that some firms exit from the market because of suffering deficit. Thus, a computational procedure was structured so as to reflect such phenomena. As far as a numerical example demonstrated in section 3 is concerned, the procedure proposed in this paper seems not only to converge to the equilibrium state in an efficient ways, but also to be easily implementable. Third, the method developed in this paper is likely to be efficient in spatial pricing with nonlinear marginal-production cost as well by introducing double- stage algorithm. This may bring a drastic computational time saving in finding equilibrium flows in spatial oligopolistic model. But further careful examination and more detail analysis on application of the proposed method to variable marginal-cost problem will be required to confirm the convergency of the procedure. Less