Congestion and Penalization in Optimal Transport

TL;DR

This paper extends classical discrete optimal transport by introducing heterogeneous quadratic congestion costs and weighted penalties for deviations from target allocations, enabling modeling of excess demand and misallocation common in developing economies. The authors develop both analytical and computational tools, including a Neumann-series-based interior-solution method and an $O((N+L)N^2L^2)$ algorithm, along with special-case closed-form solutions and comparative statics. They establish existence and uniqueness results for the penalized problem and demonstrate the framework on Peru’s health and education sectors, where congestion and mismatching are pronounced. The approach provides a flexible, non-integer matching framework that captures real-world frictions, with potential extensions to dynamic settings and empirical validation for policy design.

Abstract

We introduce a novel model based on the discrete optimal transport problem that incorporates congestion costs and replaces traditional constraints with weighted penalization terms. This approach better captures real-world scenarios characterized by demand-supply imbalances and heterogeneous congestion costs. We develop an analytical method for computing interior solutions, which proves particularly useful under specific conditions. Additionally, we propose an $O((N+L)N^2 L^2)$ algorithm to compute the optimal interior solution. For certain cases, we derive a closed-form solution and conduct a comparative statics analysis. Finally, we present examples demonstrating how our model yields solutions distinct from classical approaches, leading to more accurate outcomes in specific contexts, such as Peru's health and education sectors.

Theorems & Definitions (25)

• Lemma 3.1
• Lemma 3.2
• Theorem 3.3
• Theorem 3.4
• Corollary 3.5
• proof
• Lemma 3.6
• Lemma 3.7
• Lemma 3.8
• Theorem 3.9