Optimal Land Use and Pollution Abatement under Discounting: A Spatiotemporal Analysis with State Constraints

Carmen Camacho (), Weihua Ruan () and Benteng Zou ()
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Carmen Camacho: PSE - Paris School of Economics - UP1 - Université Paris 1 Panthéon-Sorbonne - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris Sciences et Lettres - EHESS - École des hautes études en sciences sociales - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement - ENPC - École nationale des ponts et chaussées - IP Paris - Institut Polytechnique de Paris, PJSE - Paris Jourdan Sciences Economiques - UP1 - Université Paris 1 Panthéon-Sorbonne - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris Sciences et Lettres - EHESS - École des hautes études en sciences sociales - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement - ENPC - École nationale des ponts et chaussées - IP Paris - Institut Polytechnique de Paris
Weihua Ruan: Purdue University Northwest
Benteng Zou: uni.lu - Université du Luxembourg = University of Luxembourg = Universität Luxemburg

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Abstract: We develop a rigorous spatiotemporal framework that incorporates ecological state constraints and reversible pollution dynamics with fertile land as the sole bounded production input. Using an extended Pontryagin Maximum Principle and generalized Kuhn-Tucker conditions, we characterize the structure of optimal solutions across discounting regimes. Thus, this paper investigates optimal strategies for land use and consumption under varying time discount rates in spatial settings. Under low discounting, the system supports sustainable dynamics, avoiding excessive consumption and enabling full restoration of fertile land. Fertility restoration is local in the heterogeneous case and global in the homogeneous case. In contrast, high discounting leads to boundary behavior, partial degradation, or irreversible loss, depending on critical thresholds. We derive explicit solutions when space is homogeneous. In heterogeneous settings, we construct hybrid solutions where the system transitions from a finite-horizon control problem to a structured long-run regime. Our results provide analytical benchmarks and highlight the pivotal role of time preferences in shaping long-term environmental and economic outcomes. This work contributes to the literature on spatial growth, optimal control with state constraints, and sustainable resource management under ecological limits.

Keywords: Optimal Control; Limited resources; Soil Pollution; Diffusion; Economic growth (search for similar items in EconPapers)
Date: 2026-01
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