In this paper we provide a statement of dynamic spatial price equilibrium (DSPE) in continuous time as a basis for modeling freight flows in a network economy. The model presented describes a spatial price equilibrium due to its reliance on the notion that freight movements occur in response to differences between the local and distant prices of goods for which there is excess demand; moreover, local and distant delivered prices are equated at equilibrium. We propose and analyze a differential variational inequality (DVI) associated with dynamic spatial price equilibrium to study the Nash-like aggregate game at the heart of DSPE using the calculus of variations and optimal control theory. Our formulation explicitly considers inventory and the time lag between shipping and demand fulfillment. We stress that such a time lag cannot be readily accommodated in a discrete-time formulation. We provide an in-depth analysis of the DVI's necessary conditions that reveals the dynamic user equilibrium nature of freight flows obtained from the DVI, alongside the role played by freight transport in maintaining equilibrium commodity prices and the delivered-price-equals-local-price property of spatial price equilibrium. By intent, our contribution is wholly theoretical in nature, focusing on a mathematical statement of the defining equations and inequalities for dynamic spatial price equilibrium (DSPE), while also showing there is an associated differential variational inequality (DVI), any solution of which is a DSPE. The model of spatial price equilibrium we present integrates the theory of spatial price equilibrium in a dynamic setting with the path delay operator notion used in the theory of dynamic user equilibrium. It should be noted that the path delay operator used herein is based on LWR theory and fully vetted in the published dynamic user equilibrium literature. This integration is new and constitutes a significant addition to the spatial price equilibrium and freight network equilibrium modeling literatures. Among other things, it points the way for researchers interested in dynamic traffic assignment to become involved in dynamic freight modeling using the technical knowledge they already possess. In particular, it suggests that algorithms developed for dynamic user equilibrium may be adapted to the study of urban freight modelled as a dynamic spatial price equilibrium. As such, our work provides direction for future DSPE algorithmic research and application. However, no computational experiments are reported herein; instead, the computing of dynamic spatial price equilibria is the subject of a separate manuscript.

中文翻译：

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动态空间价格均衡、动态用户均衡与连续时间中的货运：基于微分变分不等式视角


在本文中，我们提供了连续时间的动态空间价格均衡 （DSPE） 声明，作为对网络经济中的货运流进行建模的基础。所提出的模型描述了一种空间价格均衡，因为它依赖于以下概念：货运移动是响应本地和远距离商品价格之间的差异而发生的，这些商品存在过剩需求;此外，本地和远距离交货价格等同于均衡。我们提出并分析了与动态空间价格均衡相关的差分变分不等式 （DVI），以使用变分微积分和最优控制理论研究 DSPE 核心的类似纳什的聚合博弈。我们的公式明确考虑了库存以及运输和需求履行之间的时间滞后。我们强调，这种时间滞后不能轻易地用离散时间公式来容纳。我们对 DVI 的必要条件进行了深入分析，揭示了从 DVI 获得的货物流的动态用户均衡性质，以及货运在维持均衡商品价格方面的作用以及空间价格均衡的交付价格等于本地价格属性。从本质上讲，我们的贡献完全是理论性的，专注于动态空间价格均衡 （DSPE） 的定义方程和不等式的数学陈述，同时还表明存在相关的微分变分不等式 （DVI），其任何解决方案都是 DSPE。我们提出的空间价格均衡模型将动态设置中的空间价格均衡理论与动态用户均衡理论中使用的路径延迟算子概念相结合。 应该注意的是，本文使用的路径延迟算子基于 LWR 理论，并在已发表的动态用户均衡文献中进行了充分审查。这种集成是新的，构成了对空间价格均衡和货运网络均衡建模文献的重要补充。除其他外，它还为对动态交通分配感兴趣的研究人员利用他们已经拥有的技术知识参与动态货运建模指明了方向。特别是，它表明为动态用户均衡开发的算法可以适用于建模为动态空间价格均衡的城市货运研究。因此，我们的工作为未来的 DSPE 算法研究和应用提供了方向。但是，此处未报告任何计算实验;相反，动态空间价格均衡的计算是单独手稿的主题。