Abstract
We study the occurrence of shocks in a common groundwater resource problem using a differential game. In particular, we use Rubio and Casino’s adaptation of the Gisser and Sánchez model where we introduce a sudden change in the dynamics of the resource, namely a decrease in the recharge rate of the aquifer. We compare the pareto optimal solution with open-loop and feedback equilibria. First, we show analytically how different solutions, at the steady state, depend on the intensity of the shock. Moreover, we show that the cost and the strategic effects are decreasing functions of the intensity of the shock, i.e. that all the solutions get closer at the steady state for more intense shocks. We finally apply the game to the particular case of the Western La Mancha aquifer. The aim of this application is to estimate how shocks influence the inefficiency of open loop and feedback strategies in terms of welfare. We show that this inefficiency decreases the earlier the shock occurs or the higher the intensity of the shock.
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Notes
On the other hand, we call non-adaptation behaviour, when farmers do not have information about the shock until it happens and then change their extraction decisions just from the date of occurrence.
In Rubio and Casino (2001), the storage capacity of the aquifer corresponds to the expression G/H, where G is the volume of water and H is the water-table height of the aquifer.
We remind that inefficiency in terms of welfare is defined as the difference between gains obtained from the pareto optimal solution and the different private equilibria over the whole time horizon.
This choice is motivated by the fact that the decrease of the water-table obtained in \(t_a=20\) can be compared to the estimated drop of 3000 Mm\(^3\) over the last 30 years reported by López-Gunn.
This does not mean that there is an optimal time in order to implement policy instruments. Policy measures should be implemented over the whole planning horizon, as shown in Proposition 4, in order to reach the PO path.
These numbers are obtained by deducting total extractions before \(t_a\) for a shock of 210 Mm\(^3\)/year and a shock of 70 Mm\(^3\)/year, more specifically, by computing the differences 9757 − 9672, 8810 − 8383 and 6474 − 6044 for the feedback, open loop and respectively, pareto optimal solutions.
We thank an anonymous referee for indicating this possibility.
Solution of \(D_3\) is not detailed here, but they are available from authors request.
We remind that in this type of problem with a finite horizon planning, the value function has to be described as a function that depends on G and t independently.
We find that the expression \(\phi (ta,G_{ta})\) does not have the independent term \(t_a\). In what follows, we write the scrap value function, \(\phi (G_{ta})\).
We do not detail expression of \(\epsilon \) because it is not necessary for the resolution of the problem, but it is available from the authors upon request
We do not provide detailed solutions of \(B_i\)\((i=1 \ldots 3)\) because the equations are too long and they are not necessary for the proofs, however, they are available from the authors upon request.
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Acknowledgements
This work was supported by the ANR GREEN-Econ research project (ANR-16-CE03-0005).
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de Frutos Cachorro, J., Erdlenbruch, K. & Tidball, M. Sharing a Groundwater Resource in a Context of Regime Shifts. Environ Resource Econ 72, 913–940 (2019). https://doi.org/10.1007/s10640-018-0233-0
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DOI: https://doi.org/10.1007/s10640-018-0233-0