AK growth with a bounded productive factor

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.