Abstract
Governments are increasingly reliant on the reacquisition of water rights as a mechanism for recovering overexploited basins. Yet, serious concerns have recently been raised about the efficacy and operational dimensions of existing programs. Water buyback is typically implemented as the purchase of a fixed quantity of water rights from the agricultural sector at the price set by the Water Authority. This paper seeks to analyze whether the use of water buyback in its current form represents a sensible means of recovering overexploited basins. The results—which are particularly relevant to contexts characterised by poor enforcement regimes and widespread illegal water use—highlight the need for greater scrutiny of current programs and call for additional work to improve the design of reacquisition policies in the context of water resource management.
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Notes
According to recent estimates (CHG 2005), total legal rights over groundwater amount to about 600 Mm^{3}/yr. This contrasts with the aquifer’s estimated 300 Mm^{3}/yr renewable resources.
This tendency has been observed in some istances within the context of the MurrayDarling basin where irrigators have been encouraged to construct ‘winter fill’ storage which can then be accessed in summer to offset the limitations on extraction resulting from the sell of water rights (Crase and O’Keefe 2009).
This assumption seems to reflect fairly well the structure of the fine system in the Guadiana region, as well as in many other basins worldwide (see, for example, Stratton et al. 2008).
In the remainder of the paper, we will talk about increasing (decreasing) the enforcement severity without specifying whether we are increasing (decreasing) σ or φ.
It can be easily shown that the main results of the paper do not crucially depend on this particular assumption and that the incentivemechanism created by the introduction of a buyback policy is still in place when \( \overline{w}\geq w^{\ast}.\)
In order to keep the picture simple, we have assumed that both the net benefits from farming [pw ^{F} − C(w ^{F})], and the expected fine ρg(w ^{IL}) are quadratic, so that their slopes are linear. This is the case, for example, when \(C\left( w^{F}\right) =c\left( w^{F}\right) {}^{2}\) and \( g(w^{IL})=\left( w^{IL}\right) ^{2}\), as in the example previously introduced.
In a different setting, Dixon et al. (2011) similarly point to the potential problems of defining buyback schemes on the basis of a fixed quantity of water rights.
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Appendices
Appendices
1.1 A.1 Farmer’s optimal consumption of illegal water
This appendix shows that, when the initial endowment of water rights, \( \overline{w}\), is smaller than \(w^{\ast }\equiv \underset{w^{F}}{\arg \max } [pw^{F}C(w^{F})]\), it is never optimal for a farmer to set her consumption of illegal water at w ^{IL} ≤ 0.
To prove this, let us study farmer’s optimization problem under the case w ^{IL} ≤ 0. From Eq. 4, this can be written as:
The Lagrange function for the above problem is:
The KuhnTucker conditions are:

(a1)
\(L_{w^{F}}^{\prime }=0\) \(\Rightarrow \) p − C ^{′}(w ^{F}) = 0

(a2)
\(L_{w^{S}}^{\prime }=0\) \(\Rightarrow \) \(p^{w}\lambda _{1}+\lambda _{2}=0\)

(b1)
λ _{1} ≥ 0, \(w^{S}\overset{\_}{w}\) ≤ 0, and \(\ \lambda _{1}\times (w^{S}\overset{\_}{w})=0\)

(b2)
λ _{2} ≥ 0, w ^{S} ≥ 0, and \(\ \lambda _{2}\times w^{S}=0\)
Looking for the ’active constraint’, the following four cases can be identified:

(I)
\(w^{S}\overset{\_}{w}\) = 0, and w ^{S} = 0, which implies: \(\overset{\_} {w}\) = 0. This, however, cannot be since by definition the farmer is endowed with a strictly positive amount of water rights, that is: \(\overset{\_}{w}\) > 0.

(II)
\(w^{S}\overset{\_}{w}\) = 0, and w ^{S} > 0. Under this case, and from the definition of \(w^{IL}=w^{F}\overline{w}+w^{S}\), we have: w ^{IL} = w ^{F} . Since we are considering the hypothesis w ^{IL} ≤ 0, the previous equality implies: w ^{F} ≤ 0. This violates condition (a1).

(III)
\(w^{S}\overset{\_}{w}\) < 0, and w ^{S} = 0. The complementary slackness condition implies: λ _{1} = 0. Consequently, (a2) becomes: \( p^{w}=\lambda _{2}\). From (b2), we have: λ _{2} ≥ 0. Since water price cannot be negative, it must be: p ^{w} = 0. This corresponds to the status quo case when the farmer does not use any positive amount of illegal water. However, due to assumption AII, this case does not represent an optimal solution when \(\overline{w}<w^{\ast }\).

(IV)
\(w^{S}\overset{\_}{w}\) < 0, and w ^{S} > 0, so that: λ _{1} = λ _{2} = 0 . From (a2), we have: p ^{w} = 0. However, if the price offered by the Water Authority is zero, a farmer will not sell any positive amount of water rights. Therefore, it cannot be w ^{S} > 0.
From the above analysis, we can conclude that, when \(\overline{w}<w^{\ast }\) there is no solution to problem 4 such that at the optimum w ^{IL} ≤ 0.
1.2 A.2 Resolution of farmer’s optimization problem under a buyback scheme
Appendix A.1 showed that, under a buyback policy and for \(\overline{w} <w^{\ast }\), it is never optimal for a farmer to set her consumption of illegal water at w ^{IL} ≤ 0. This allows us to write G(w ^{IL}) = g(w ^{IL}) and to define the Lagrangian for farmer’s optimization problem as:
The KuhnTucker conditions are:

(a1)
\(L_{w^{F}}^{\prime }=0\) \(\Rightarrow \) \(pC^{\prime }(w^{F})\rho g_{w^{F}}^{\prime }(w^{IL})=0\)

(a2)
\(L_{w^{S}}^{\prime }=0\) \(\Rightarrow \) \(p^{w}\rho g_{w^{S}}^{\prime }(w^{IL})\lambda _{1}+\lambda _{2}=0\)

(b1)
λ _{1} ≥ 0, \(w^{S}\overset{\_}{w}\) ≤ 0, and \(\ \lambda _{1}\times (w^{S}\overset{\_}{w})=0\)

(b2)
λ _{2} ≥ 0, w ^{S} ≥ 0, and \(\ \lambda _{2}\times w^{S}=0\)
Conditions \(\lambda _{1}\times (w^{S}\overset{\_}{w})=0\) and \(\lambda_{2}\times w^{S}=0\) in (b1) and (b2) yield the following four cases:

(I)
\(w^{S}\overset{\_}{w}\) = 0, and w ^{S} = 0. This case can, in fact, be disregarded because it implies \(\overset{\_}{w}\) = 0, while a farmer’s initial endowment of water rights is, by definition, strictly positive.

(II)
\(w^{S}\overset{\_}{w}\) = 0, and w ^{S} > 0, so that λ _{2} = 0. Condition (a2) becomes as follows:
$$ p^{w}\rho g_{w^{S}}^{^{\prime }}(w^{IL}\left\vert w^{S}=\overline{w}\right. )=\lambda _{1} $$Therefore, for λ _{1} to be greater than or equal to zero as (b1) requires, it must be:
$$ p^{w}\geq \rho g_{w^{S}}^{^{\prime }}(w^{IL}\left\vert w^{S}=\overline{w} \right. ) $$Under this condition, the system admits the following solution:
$$ \left\{ \begin{array}{c} pC^{^{\prime }}(w^{F})=\rho g_{w^{{\scriptsize F}}}^{^{\prime }}(w^{IL}) \\ w^{S}=\overline{w} \end{array} \right. $$ 
(III)
\(w^{S}\overset{\_}{w}\) < 0, and w ^{S} = 0, so that λ _{1} = 0. In this case, condition (a2) is as follows:
$$ p^{w}\rho g_{w^{S}}^{^{\prime }}(w^{IL}\left\vert w^{S}=0\right. )=\lambda _{2} $$Therefore, λ _{2} ≥ 0 if and only if the following holds:
$$ p^{w}\leq \rho g_{w^{S}}^{^{\prime }}(w^{IL}\left\vert w^{S}=0\right. ) $$Under the above condition on p ^{w}, the whole system is satisfied and the solution to farmer’s optimization problem is:
$$ \left\{ \begin{array}{c} pC^{^{\prime }}(w^{F})=\rho g_{{\scriptsize w}^{{\scriptsize F}}}^{^{\prime }}(w^{IL}) \\ w^{S}=0 \end{array} \right. $$ 
(IV)
Finally, \(w^{S}\overset{\_}{w}\) < 0, and w ^{S} > 0, so that λ _{1} = λ _{2} = 0. From (a1) and (a2), we have that the solution to the problem is defined by the following conditions:
$$ \left\{ \begin{array}{c} pC^{^{\prime }}(w^{F})=\rho g_{w^{{\scriptsize F}}}^{^{\prime }}(w^{IL}) \\ p^{w}=\rho g_{w^{{\scriptsize S}}}^{^{\prime }}(w^{IL}) \end{array} \right. $$
It can be easily observed that the above results correspond to the solution summarized in Eq. 5.
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Marchiori, C., Sayre, S.S. & Simon, L.K. On the Implementation and Performance of Water Rights Buyback Schemes. Water Resour Manage 26, 2799–2816 (2012). https://doi.org/10.1007/s1126901200478
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DOI: https://doi.org/10.1007/s1126901200478