AK growth with a bounded productive factor

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 study an AK-type growth model in which production relies on fertile land, an irreplaceable factor that is physically bounded. Unlike the standard AK framework, productive capacity cannot expand without limit because production generates pollution that degrades fertile land, while abatement can restore it. Using Pontryagin's state-constrained optimal control approach, we transform physical state constraints into control constraints and obtain a complete analytical characterization of the optimal solution. The analysis identifies three regimes determined by the discount rate. For sufficiently patient planners, fertile land reaches its maximum feasible level in finite time and consumption converges to its highest sustainable level. For intermediate discount rates, land and consumption decline gradually, while very impatient planners choose no abatement and rapidly deplete productive capacity. The results highlight how physical limits reshape optimal growth and resource-management policies and provide new insights for environmental and resource economics.

Date: 2026
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Published in Journal of Mathematical Economics, In press, ⟨10.1016/j.jmateco.2026.103322⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:pseptp:halshs-05837207

DOI: 10.1016/j.jmateco.2026.103322

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