skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


This content will become publicly available on March 12, 2026

Title: Dynamic Spatial Price Equilibrium, Nonlinear Freight Pricing, and Alternative Mathematical Formulations
This paper is meant as a guide for researchers interested in dynamic modeling of commodity flows from the perspective of spatial price equilibrium. In particular, we present a type of dynamic spatial price equilibrium (DSPE) in continuous time as a basis for modeling freight flows in a network economy. We consider the circumstance of a known matrix of travel times between all pairs of markets (origindestination pairs) within a network for which paths (routes) are articulated. We also consider the unit cost of transport to be the sum of the price for freight services and a surcharge for backorders. Prices for freight services follow a nonlinear operator explained herein. That operator allows consideration of break-point pricing, as well as other forms of nonlinear pricing. The DSPE model considered is expressed four different ways. The first formulation is a nonlinear complementarity problem with explicit embedded dynamics describing the rate of change of inventories at each node as the net of production, consumption, import, and export, with explicit time shifts that account for shipping latencies. We also provide three alternative formulations: a differential complementarity system, a differential variation inequality, and a variational inequality based on a state operator. We discuss algorithms appropriate to each formulation and close with a discussion of future research needed to make DSPE models applicable to freight systems planning and the pricing of freight services.  more » « less
Award ID(s):
2110880
PAR ID:
10586132
Author(s) / Creator(s):
;
Publisher / Repository:
Springer
Date Published:
Journal Name:
Networks and Spatial Economics
Volume:
24
Issue:
1
ISSN:
1566-113X
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. (, Transportation research Part B Methodological)
    none (Ed.)
    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. 
    more » « less
  2. ; ; ; ; (, Journal of Cloud Computing)
    Abstract In current infrastructure-as-a service (IaaS) cloud services, customers are charged for the usage of computing/storage resources only, but not the network resource. The difficulty lies in the fact that it is nontrivial to allocate network resource to individual customers effectively, especially for short-lived flows, in terms of both performance and cost, due to highly dynamic environments by flows generated by all customers. To tackle this challenge, in this paper, we propose an end-to-end Price-Aware Congestion Control Protocol (PACCP) for cloud services. PACCP is a network utility maximization (NUM) based optimal congestion control protocol. It supports three different classes of services (CoSes), i.e., best effort service (BE), differentiated service (DS), and minimum rate guaranteed (MRG) service. In PACCP, the desired CoS or rate allocation for a given flow is enabled by properly setting a pair of control parameters, i.e., a minimum guaranteed rate and a utility weight, which in turn, determines the price paid by the user of the flow. Two pricing models, i.e., a coarse-grained VM-Based Pricing model (VBP) and a fine-grained Flow-Based Pricing model (FBP), are proposed. The optimality of PACCP is verified by both large scale simulation and small testbed implementation. The price-performance consistency of PACCP are evaluated using real datacenter workloads. The results demonstrate that PACCP provides minimum rate guarantee, high bandwidth utilization and fair rate allocation, commensurate with the pricing models. 
    more » « less
  3. ; ; (, Mathematics of Operations Research)
    In online sales, sellers usually offer each potential buyer a posted price in a take-it-or-leave fashion. Buyers can sometimes see posted prices faced by other buyers, and changing the price frequently could be considered unfair. The literature on posted-price mechanisms and prophet inequality problems has studied the two extremes of pricing policies, the fixed-price policy and fully dynamic pricing. The former is suboptimal in revenue but is perceived as fairer than the latter. This work examines the middle situation, where there are at most k distinct prices over the selling horizon. Using the framework of prophet inequalities with independent and identically distributed random variables, we propose a new prophet inequality for strategies that use at most k thresholds. We present asymptotic results in k and results for small values of k. For k = 2 prices, we show an improvement of at least 11% over the best fixed-price solution. Moreover, k = 5 prices suffice to guarantee almost 99% of the approximation factor obtained by a fully dynamic policy that uses an arbitrary number of prices. From a technical standpoint, we use an infinite-dimensional linear program in our analysis; this formulation could be of independent interest to other online selection problems. 
    more » « less
  4. ; ; (, Econometrica)
    We study the propagation of monetary shocks in a sticky‐price general equilibrium economy where the firms' pricing strategy features a complementarity with the decisions of other firms. In a dynamic equilibrium, the firm's price‐setting decisions depend on aggregates, which in turn depend on the firms' decisions. We cast this fixed‐point problem as a Mean Field Game and prove several analytic results. We establish existence and uniqueness of the equilibrium and characterize the impulse response function (IRF) of output following an aggregate shock. We prove that strategic complementarities make the IRF larger at each horizon. We establish that complementarities may give rise to an IRF with a hump‐shaped profile. As the complementarity becomes large enough, the IRF diverges, and at a critical point there is no equilibrium. Finally, we show that the amplification effect of the strategic interactions is similar across models: the Calvo model and the Golosov–Lucas model display a comparable amplification, in spite of the fact that the non‐neutrality in Calvo is much larger. 
    more » « less
  5. ; ; (, La Matematica)
    Abstract We show that a competitive equilibrium always exists in combinatorial auctions with anonymous graphical valuations and pricing, using discrete geometry. This is an intuitive and easy-to-construct class of valuations that can model both complementarity and substitutes, and to our knowledge, it is the first class besides gross substitutes that have guaranteed competitive equilibrium. We prove through counter-examples that our result is tight, and we give explicit algorithms for constructing competitive pricing vectors. We also give extensions to multi-unit combinatorial auctions (also known as product-mix auctions). Combined with theorems on graphical valuations and pricing equilibrium of Candogan, Ozdagar and Parrilo, our results indicate that quadratic pricing is a highly practical method to run combinatorial auctions. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email