Evaluating the Effects of Organic vs. Conventional Farming on Aquifer Water Quality

TL;DR

This work addresses sustainable groundwater management under fertilizer-driven agricultural activity by developing two infinite-horizon optimal-control formulations that couple aquifer height $h_t$, water quality $ extPsi_t$, and fertilizer input $ abla_t$ via linear and concave price-discount structures. The authors extend the Martin–Stahn framework to include pollution dynamics with $ frac{dC_t}{dt}= eta f_t abla_t - delta C_t$ and $ extPsi_t=1-C_t$, solved using a current-value Hamiltonian and Pontryagin’s Maximum Principle, yielding a bang-bang policy in the Linear Model and an interior optimum in the Non-Linear Model. Key findings show that when fertilizer price discounts are negative ($eta<0$), optimal fertilizer use vanishes in the Linear Model, while for larger $eta$ a threshold determines full use; the Non-Linear Model permits intermediate fertilizer use with $ abla_e$ in $[rac{eta^2}{4eta^2},1]$, and stability is established via Hartman’s theorem. The results highlight a trade-off between long-term welfare and water quality, suggesting policy levers such as narrowing the price gap between organic and conventional products and investing in pollution-reduction technologies to raise achievable welfare under MAR contexts. These insights provide a structured, policy-relevant lens for balancing agricultural productivity with aquifer sustainability in water-scarce regions.

Abstract

This study analyzes water quality dynamics and aquifer recharge through irrigated agriculture, contributing to the literature on Managed Aquifer Recharge (MAR) amidst growing water scarcity. We develop two optimal control models-- a linear and a non-linear extension of (Martin and Stahn, 2013) --that incorporate the impact of fertilizers on aquifer water quality, distinguishing between organic and conventional farming practices. The linear model applies a constant rebate mechanism, whereas the non-linear model employs a concave rebate scheme. Our results show that, depending on climate change scenarios, fertilizer-induced food price discounts, and pollution levels, a socially optimal equilibrium in fertilizer use can be attained. Policy implications are discussed, emphasizing the trade-off between environmental sustainability and social welfare.

Figures (9)

• Figure 1: In this scenario, where $f^{e}(0) \approx 0.761 < \frac{\beta \delta}{\eta^2} \approx 2.64$, full fertilizer application leads to improved equilibrium values, indicating that fertilizer-induced food price discounts, together with the system's natural degradation capacity, effectively mitigate pollutant generation.
• Figure 2: In this case, we find that $f^{e}(0) \approx 1.625$, which is greater than $\frac{\beta \delta}{\eta^2} \approx 0.562$. Full fertilizer application does not improve equilibrium outcomes, indicating that the benefits of reduced food prices are insufficient to offset the negative environmental impacts of pollution.
• Figure 3: If $\delta > 0.25$, then $f(\eta) < 1$ for all $\eta \in [0,1]$ while, if $0< \delta < 0.25$, $f(\eta)>1 \ \forall \eta \in [\frac{1-\sqrt{1-4\delta}}{2},\frac{1+\sqrt{1-4\delta}}{2}]$
• Figure 4:
• Figure 5: Simulated optimal paths for the three models with $\bar{h}=0.41$, $\rho=0.04$, $\beta= 0.4$, $\eta=0.77$, $b=0.16$, and $d=1.54$. A high $\eta$ significantly impacts the optimal policy, as large food discounts no longer support full fertilizer use and negatively affect water purity. Increasing $\eta$ raises the critical $\beta$.

Theorems & Definitions (11)

• Theorem 1: Pontryagin’s Maximum Principle (PMP)
• Remark 2
• Theorem 3: Sufficiency Conditions for Discounted Infinite-Horizon Problems
• Proposition 4
• Proposition 5
• Proposition 6
• Proposition 7: Optimal equilibrium under concave fertilizer model
• proof
• proof
• proof