Pro-rata mechanisms in groundwater markets

2026-08-01

AI Summary

This study addresses price formation and rationing challenges in groundwater markets arising from supply–demand imbalances by introducing a proportional rationing mechanism. Under trading constraints or regulatory objectives, the price-generation process is modeled using both Nash equilibrium and Stackelberg leader–follower frameworks. Theoretically, the mechanism ensures that any resulting price constitutes a Nash equilibrium, yielding a unique and equitable resolution to market imbalance. The analysis further elucidates the distinct impacts of unilateral and bilateral trading caps on Pareto-efficient prices and develops a regulatory optimization framework that jointly considers social welfare, transaction volume, and fairness in farmers’ revenue distribution. Numerical experiments on a four-farmer case validate the mechanism’s efficacy and systematically compare the efficiency and equity of various asymmetric rationing rules.

Abstract

We introduce a pro-rata rationing mechanism for resolving supply-demand imbalances in groundwater markets, extending the price-formation model of Cialenco and Ludkovski (2025). We show that under the pro-rata distribution, every price is a Nash equilibrium, thereby pro-rata approach provides a rationing device whenever supply and demand fail to match. By the very nature of the pro-rata mechanism, the resolution of supply-demand imbalances is unique, and the proportional rationing approach is fair. First, we consider markets with exogenous restrictions on the amounts each agent may buy and/or sell, deriving closed-form first-best consumption and characterizing how one-sided caps monotonically shift the Pareto price, while two-sided caps have an ambiguous effect. These results give the market-maker (or regulator) a tool for studying the impact of trading restrictions on price formation. Second, we study a leader-follower setting in which a regulator (the leader) sets the trading price by optimizing first its own objective, such as balancing social welfare against a target traded volume or a fairness objective such as Gini-type disparity measure across farmers' profitability, while farmers (the `followers') respond via pro-rata trading. We further compare the proposed pro-rata approach to a family of asymmetric rationing schemes (seniority-based, excess-based, uniform, and mixed pro-rata rules) that trade off proportional fairness against protections for small or senior water-rights holders. Throughout, we illustrate the theoretical results with a numerical case study calibrated to a stylized four-farmer groundwater market.

Problem

• groundwater markets
• supply-demand imbalance
• pro-rata rationing
• fairness
• resource allocation

Innovation

• pro-rata rationing
• groundwater markets
• Nash equilibrium
• leader-follower game
• trading restrictions