Regulation of relationships between heterogeneous farmers and an aquifer accounting for lag effects

Abstract

Many environmental problems are due to damage caused by pollutants that accumulate with a time lag following their emission. In this study, we focus on nitrates used in agriculture, which can pollute groundwater many years after their initial application. A dynamic optimal control problem with heterogeneous farmers is proposed. The usual structural parameters such as the discount rate, the natural clearing rate and the lagged time interval between the occurrence of soil‐level pollution and the impact on groundwater are taken into account. We also examine pollution as caused by a continuous set of farms characterised by their individual performance index and by their individual marginal contribution to the pollution. The issue is further investigated by taking account of change in the information context, successively related to perfect information and to asymmetric information. As a result, when the delay between the spreading of N‐fertilizer and the impact on the aquifer increases, that is, the longer the lag, the steady‐state pollution stock and the steady‐state shadow price of the stock both increase. Moreover, we show that the optimal regulation may require a decreasing amount of fertilizer over time, even in the case of initial underpollution.

Optimal pollution control with distributed delays

“…Delays have only quantitative but not qualitative effects on the optimal trajectory. An intuitive result, shown by Winkler (2008) and Bourgeois and Jayet (2011), and for which we present a full proof in A.2 (Lemma 5), is that an increase of the delay increases the optimal steady state pollution stock. The optimality conditions, derived and discussed in A.1 (equation 13), allow us to show that a higher delay leads to a higher emission level at the steady state.…”

“…This problem has been studied by Winkler (2008) and Bourgeois and Jayet (2011). As we recall in A with a full resolution, on the optimal path, c (t) depends only on p (t − τ ), for every t ≥ 0.…”

“…Winkler (2011) proves that a modified model with a discrete delay will display monotonic optimal paths if the objective function is separable in both stock and control variables, otherwise oscillatory paths may take place. Using such a separable objective function in a model with heterogeneous polluters, Bourgeois and Jayet (2011) show that higher time lags lead to a higher optimal pollution stock at the steady state and that this effect is amplified by asymmetric information. The common feature of these contributions is that they use a single discrete delay, assuming that an emission at time t will reach entirely and systematically the pollution stock at time t + τ .…”

Lag Time as an Indicator of the Link between Agricultural Pressure and Drinking Water Quality State

“…Lag times may be one of the key pieces of information for developing water protection programs and setting achievable goals [54][55][56][57]. Various models have been developed to estimate the lag times between mitigation measures and the water quality changes [55][56][57][58][59]. Most models are process-based and provide comprehensive views of the system.…”

Optimal Control for a Groundwater Pollution Ruled by a Convection–Diffusion–Reaction Problem

“…Using delay equations, Winkler [5] proves that optimal pollution emission paths are monotonous if the objective is separable into an emission component and a stock component, but that they may display damping oscillations if not. Bourgeois and Jayet [6] consider the impact of the time delay taken by the pollution to reach the aquifer on the optimal path, and they prove that this effect is amplified by asymmetric information. Augeraud and Leandri [7] prove that optimal path may be cyclic for some specific time delay.…”