Gisser-Sánchez revisited: A model of optimal groundwater withdrawal under irrigation including surface–groundwater interaction

We revisit the classic problem of determining economically optimal groundwater withdrawal rates for irrigation. The novelty compared to previous mathematical analyses is the inclusion of non-linear groundwater-surface water interaction that allows for incorporating the impact of capture, i.e. the fact that all or part of the pumped groundwater comes out of reduced surface water flow or increased recharge. We additionally included the option to internalize environmental externalities (e.g. streamflow depletion) and maximize social welfare rather than farmer's profit. This analysis results in a fixed optimal groundwater withdrawal rate q_opt when withdrawal q remains smaller than some critical withdrawal rate (maximum capture) q_crit and provides depletion trajectories, either under competition or optimal control, if q is larger than q_crit. Based on the relative value of q, q_crit and q_opt it also yields four quadrants of distinct withdrawal strategies. Using global hydrogeological and hydroeconomic datasets we map the global occurrence of these four quadrants and provide global estimates of optimal groundwater withdrawal rates and depletion trajectories. For the quadrants with groundwater depletion (q > q_crit) we derive and compare depletion trajectories under competition, optimal control and optimal control including environmental externalities, and assessed globally where the differences between these depletion modes are small, which is known as the Gisser-Sánchez effect. We find that the Gisser-Sánchez effect is globally ubiquitous, but only if environmental externalities are ignored. The inclusion of environmental externalities in optimal control withdrawal result in notably reduced groundwater decline and larger values of social welfare in many of the major depletion areas.