3.2.2 The maximization problem in Si

max Si

maximization of G Si ð Þ ;Ci as a function of Ci. We have

<sup>¼</sup> <sup>h</sup><sup>0</sup> i

> h0 i

<sup>i</sup> ð Þ Si is an increasing function of Si.

Si

h0 i

<sup>i</sup> the solution in Ci of

C�

The preceding remark, the concavity of hi, and the definition in (4) and (7) of

<sup>i</sup> ð Þ Si <C<sup>þ</sup>

<sup>i</sup> and of bSi:

<sup>i</sup> ðSiÞ=b

<sup>i</sup> =b

<sup>i</sup> ðSiÞ=b ≤ Si ≤C<sup>þ</sup>

<sup>i</sup> ðSiÞ if Si <C�

<sup>i</sup> if Si >C<sup>þ</sup>

the regulator and of the profit function hi of farmer i. When starting from small Si,

bSi if C�

We remark by the way that when bSi <Ci, we have

2 1ð Þ � <sup>a</sup> <sup>D</sup> Ci

We note that this solution does not depend on Si.

We can easily deduce that the optimal solution Csol

8 ><

>:

Csol <sup>i</sup> ¼

Note that this formula gives a relation Csol

<sup>i</sup> ð Þ Si <C<sup>þ</sup>

C�

C<sup>þ</sup>

With a simple derivation of this last equation, it is easy to see that

8 < :

h0 i

<sup>∂</sup>Gi Si ð Þ ;Ci ∂Ci

> C�0 <sup>i</sup> ð Þ¼ Si

3.2.1 The maximization problem in Ci

Drought - Detection and Solutions

• For bSi <Ci, we note C�

Therefore C�<sup>0</sup>

C�

34

<sup>i</sup> and of C<sup>þ</sup>

• For bSi >Ci, we call C<sup>þ</sup>

<sup>i</sup> imply that

the relative positions of C�

function.

max Ci

hið Þ� Ci F Si ð Þ ð Þ ;Ci � �

where the production function of farmer i, hi, is an increasing and concave

We note G Si ð Þ¼ ;Ci hið Þ� Ci F Si ð Þ ;Ci , and we calculate the solution of the

<sup>i</sup> ð Þ Si the solution in Ci of

ð Þ¼ Ci 2 1ð Þ � <sup>a</sup> <sup>D</sup> Ci

2 1ð Þ � a DSi � h

2 1ð Þ � a DCi

ð Þ� Ci 2 1ð Þ � <sup>a</sup> <sup>D</sup> Ci

Si

ð Þ� Ci ð Þ 1 � a bD if bSi >Ci

Si

00 ð Þ Ci <sup>S</sup><sup>2</sup> i

>2 1ð Þ � a bD >ð Þ 1 � a bD (6)

ð Þ¼ Ci ð Þ 1 � a bD (7)

<sup>i</sup> (8)

<sup>i</sup> =b

<sup>i</sup> ð Þ Si that depends on the parameters of

<sup>i</sup> of (2) for Si given depends on

(9)

if bSi <Ci

(2)

(3)

(4)

>0 (5)

When solving the maximization of our problem in Si, knowing the optimal value Ci, which is generally, as we saw, a function of Si, we must consider the relation between these two variables.

• If Si <C� <sup>i</sup> ð Þ Si =b, we must solve

$$\max\_{\mathbf{S}\_i} \left( h\_i \left( \mathbf{C}\_i^- \left( \mathbf{S}\_i \right) \right) - F \left( \mathbf{S}\_i, \mathbf{C}\_i^- \left( \mathbf{S}\_i \right) \right) \right) \tag{10}$$

and the first-order condition gives

$$h\_i' \left( \mathbf{C}\_i^- (\mathbf{S}\_i) \right) \mathbf{C}\_i'^- (\mathbf{S}\_i) = D \left[ a + (\mathbf{1} - a) \frac{2 \mathbf{C}\_i^- (\mathbf{S}\_i) \mathbf{C}\_i^{'-} (\mathbf{S}\_i) \mathbf{S}\_i - \left( \mathbf{C}\_i^- (\mathbf{S}\_i) \right)^2}{\mathbf{S}\_i^2} \right] \tag{11}$$

• If C� <sup>i</sup> ð Þ Si =b≤Si ≤C<sup>þ</sup> <sup>i</sup> =b we must solve

$$\max\_{\mathbf{S}\_i} \left( h\_i(b\mathbf{S}\_i) - D\left( a\mathbf{S}\_i + (\mathbf{1} - a)b^2 \mathbf{S}\_i \right) \right) \tag{12}$$

for which the first-order condition is

$$bh\_i'(b\mathbb{S}\_i) = D\left[a + (\mathbb{1} - a)b^2\right] \tag{13}$$

• and if Si >C<sup>þ</sup> <sup>i</sup> =b the maximization problem is:

$$\max\_{\mathbf{S}\_i} \left( h\_i(\mathbf{C}\_i^+) - F(\mathbf{S}\_i, \mathbf{C}\_i^+) \right) \tag{14}$$

that does not have a solution, which means that the farmer will at least consume bSi (as consuming less would decrease his production without decreasing his water bill; in other words we have bSi ≤ Ci).

To obtain the optimal solution of our problem in Si, we must analyze the admissibility of solutions of (11) and (13).

#### Theorem

The optimal strategy of farmer i is 1/Si ¼ Ci ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>1</sup> � <sup>a</sup> <sup>=</sup><sup>a</sup> <sup>p</sup> if 0<sup>&</sup>lt; <sup>b</sup><sup>2</sup> <sup>1</sup>þb<sup>2</sup> <sup>&</sup>lt;a<1, where Ci is the solution of

$$h\_i'(\mathbf{C}\_i) = \mathbf{2D}\sqrt{a(1-a)}\tag{15}$$

2/Si <sup>¼</sup> Ci=<sup>b</sup> if 0<a<sup>&</sup>lt; <sup>b</sup><sup>2</sup> <sup>1</sup>þb<sup>2</sup> <sup>&</sup>lt; 1, where Ci is the solution of

$$h\_i'(\mathbf{C}\_i) = \frac{D\left(a + (\mathbf{1} - a)b^2\right)}{b} \tag{16}$$

#### Proof of the theorem

We start by introducing a Lemma.

#### Lemma

$$\max\_{\mathbf{S}\_i} \left[ \max\_{\mathbf{C}\_i} \left( h\_i(\mathbf{C}\_i) - F(\mathbf{S}\_i, \mathbf{C}\_i) \right) \right] = \max\_{\mathbf{S}\_i, \mathbf{C}\_i} \left[ h\_i(\mathbf{C}\_i) - F(\mathbf{S}\_i, \mathbf{C}\_i) \right] \tag{17}$$

#### Proof of the Lemma

In Section 3.2.1 of this chapter, we have shown that the solution of

$$\max\_{\mathbf{S}\_i} \left[ \max\_{\mathbf{C}\_i} \left( h\_i(\mathbf{C}\_i) - F(\mathbf{S}\_i, \mathbf{C}\_i) \right) \right] \tag{18}$$

And the first order conditions give

DOI: http://dx.doi.org/10.5772/intechopen.86529

<sup>∂</sup> hið Þ� Ci D aSi <sup>þ</sup> ð Þ <sup>1</sup> � <sup>a</sup>

We remark that this implies, as bSi <Ci, that

<sup>∂</sup> hið Þ� Ci D aSi <sup>þ</sup> ð Þ <sup>1</sup> � <sup>a</sup>

h i � �

∂Ci

h0

with μ the dual variable associated to the constraint bSi ≥Ci.

<sup>∂</sup> hið Þ� Ci D aS <sup>ð</sup> <sup>i</sup> <sup>þ</sup> ð Þ <sup>1</sup> � <sup>a</sup> bCiÞ þ <sup>μ</sup>ð Þ bSi � Ci ½ � ∂Si

h i � �

Can Nonlinear Water Pricing Help to Mitigate Drought Effects in Temperate Countries?

∂Si

Si ¼

b

r

C2 i Si

C2 i Si

ffiffiffiffiffiffiffiffiffiffiffi 1 � a a

r

ffiffiffiffiffiffiffiffiffiffiffi 1 � a a

¼ h<sup>0</sup>

ð Þ¼ Ci <sup>2</sup><sup>D</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>¼</sup> <sup>a</sup> � ð Þ <sup>1</sup> � <sup>a</sup> <sup>C</sup><sup>2</sup>

ð Þ� Ci <sup>D</sup> ð Þ <sup>1</sup> � <sup>a</sup> <sup>2</sup>Ci

ðhið Þ� Ci D aS ð Þ <sup>i</sup> þ ð Þ 1 � a bCi Þ þ μð Þ bSi � Ci (30)

¼ h<sup>0</sup>

a b

h i <sup>¼</sup> <sup>D</sup> ð Þ <sup>1</sup> � <sup>a</sup> <sup>b</sup><sup>2</sup> <sup>þ</sup> <sup>a</sup>

ð Þ¼ Ci D ð Þ 1 � a b þ

The optimal solution is a continuous function of parameters a and b. Moreover, the regulator can choose parameters a and b in order to enforce an interior solution

Si

<sup>a</sup>ð Þ <sup>1</sup> � <sup>a</sup> <sup>p</sup> (29)

¼ �Da þ μb ¼ 0 (31)

ð Þ� Ci Dð Þ 1 � a b � μ ¼ 0 (32)

b " #

(33)

S2 i

i

Ci (26)

<1 (27)

¼ 0 (25)

¼ 0 (28)

a=

which is equivalent to

and

b=

• If bSi ≥Ci,

a=

Ci <sup>¼</sup> bSi, <sup>μ</sup> <sup>¼</sup> Da

Remark 1

or a border solution.

and

b=

37

which is equivalent to

then (23) can be written as

max Si,Ci

The first-order conditions give here:

The solution of these equations is.

<sup>∂</sup> hið Þ� Ci D aS <sup>ð</sup> <sup>i</sup> <sup>þ</sup> ð Þ <sup>1</sup> � <sup>a</sup> bCiÞ þ <sup>μ</sup>ð Þ bSi � Ci ½ � ∂Ci

<sup>b</sup> <sup>&</sup>gt;0 and <sup>h</sup><sup>0</sup>

is given either if bSi = Ci (border solution) by the maximization in Si of G Si ð Þ ;Ci or if bSi < Ci (interior solution) by h<sup>0</sup> i ð Þ¼ Ci 2 1ð Þ � <sup>a</sup> <sup>D</sup> Ci Si (Eq. (4)).

Now if we solve the problem

$$\max\_{\mathbf{S}\_{i\bullet}\mathbf{C}\_{i}} \left[ h\_{i}(\mathbf{C}\_{i}) - F(\mathbf{S}\_{i}, \mathbf{C}\_{i}) \right] \tag{19}$$

its interior solution (with bSi < Ci), knowing that (the first equivalence being due to the fact that G(Si,Ci) depends on Si only through F(Si,Ci) and not through hi)

$$\frac{\partial G(\mathbb{S}\_i, \mathbb{C}\_i)}{\partial \mathbb{S}\_i} = \mathbf{0} \Leftrightarrow \frac{\partial F(\mathbb{S}\_i, \mathbb{C}\_i)}{\partial \mathbb{S}\_i} = \mathbf{0} \Leftrightarrow \mathbb{S}\_i = \sqrt{\frac{1-a}{a}} \, \mathbb{C}\_i \text{ and } b\sqrt{\frac{1-a}{a}} < \mathbf{1} \tag{20}$$

is given by

$$\frac{\partial G(\mathbf{S}\_i, \mathbf{C}\_i)}{\partial \mathbf{S}\_i} = \mathbf{0} \iff h'(\mathbf{C}\_i) = \mathbf{2}(\mathbf{1} - a)D \frac{\mathbf{C}\_i}{\mathbf{S}\_i} \tag{21}$$

which coincides with (4). Replacing this equation in (11), we obtain

$$2(1-a)D\frac{C\_i}{S\_i}C\_i' = D\left(a + \frac{(1-a)2C\_iC\_i'}{S\_i} - (1-a)\frac{C\_i^2}{S\_i^2}\right) \tag{22}$$

Simplifying we find Eq. (18) so that the interior solution coincides with the solution of (2). The border solutions are also the same (i.e., bSi = Ci) for (2) and for (19). Finally, both solutions are the same.

We now return to the demonstration of the Theorem itself. Thanks to the Lemma, we can now compute the solution of (2), by computing the solution of

$$\max\_{\mathbf{C}\_{\mathbf{S}\_i}, \mathbf{C}\_i} \left[ h\_i(\mathbf{C}\_i) - F(\mathbf{S}\_i, \mathbf{C}\_i) \right] \tag{23}$$

• If bSi <Ci,

then (23) can be written as

$$\max\_{\mathbf{S}\_{\mathbf{S}\_{i}}, \mathbf{C}\_{i}} \left( h\_{i}(\mathbf{C}\_{i}) - D \left( a \mathbf{S}\_{i} + (\mathbf{1} - a) \frac{\mathbf{C}\_{i}^{2}}{\mathbf{S}\_{i}} \right) \right) \tag{24}$$

Can Nonlinear Water Pricing Help to Mitigate Drought Effects in Temperate Countries? DOI: http://dx.doi.org/10.5772/intechopen.86529

And the first order conditions give

$$\mathbf{a} / \frac{\partial \left[ h\_i(\mathbf{C}\_i) - D \left( a \mathbf{S}\_i + (\mathbf{1} - a) \frac{\mathbf{C}\_i^2}{\mathbf{S}\_i} \right) \right]}{\partial \mathbf{S}\_i} = \mathbf{a} - \frac{(\mathbf{1} - a) \mathbf{C}\_i^2}{\mathbf{S}\_i^2} = \mathbf{0} \tag{25}$$

which is equivalent to

$$\mathbf{S}\_{i} = \sqrt{\frac{1-a}{a}} \mathbf{C}\_{i} \tag{26}$$

We remark that this implies, as bSi <Ci, that

$$b\sqrt{\frac{1-a}{a}} < 1\tag{27}$$

and

Proof of the theorem

Drought - Detection and Solutions

max Si

or if bSi < Ci (interior solution) by h<sup>0</sup>

Now if we solve the problem

¼ 0 ⇔

Proof of the Lemma

Lemma

not through hi)

<sup>∂</sup>G Si ð Þ ;Ci ∂Si

is given by

• If bSi <Ci,

36

then (23) can be written as

max Si,Ci

We start by introducing a Lemma.

max Ci

hið Þ� Ci F Si ð Þ ð Þ ;Ci � �

> max Si

<sup>∂</sup>F Si ð Þ ;Ci ∂Si

<sup>∂</sup>G Si ð Þ ;Ci ∂Si

> Si C0

2 1ð Þ � <sup>a</sup> <sup>D</sup> Ci

(19). Finally, both solutions are the same.

In Section 3.2.1 of this chapter, we have shown that the solution of

i

its interior solution (with bSi < Ci), knowing that (the first equivalence being due to the fact that G(Si,Ci) depends on Si only through F(Si,Ci) and

¼ 0 ⇔ Si ¼

¼ 0 ⇔ h<sup>0</sup>

which coincides with (4). Replacing this equation in (11), we obtain

<sup>i</sup> <sup>¼</sup> D a <sup>þ</sup> ð Þ <sup>1</sup> � <sup>a</sup> <sup>2</sup>CiC<sup>0</sup>

Simplifying we find Eq. (18) so that the interior solution coincides with the solution of (2). The border solutions are also the same (i.e., bSi = Ci) for (2) and for

We now return to the demonstration of the Theorem itself. Thanks to the Lemma, we can now compute the solution of (2), by computing the solution of

hið Þ� Ci D aSi þ ð Þ 1 � a

� � � �

max Si,Ci

Si

max Ci

max Si,Ci

¼ max Si,Ci

hið Þ� Ci F Si ð Þ ð Þ ;Ci � �

ð Þ¼ Ci 2 1ð Þ � <sup>a</sup> <sup>D</sup> Ci

ffiffiffiffiffiffiffiffiffiffiffi 1 � a a

ð Þ¼ Ci 2 1ð Þ � <sup>a</sup> <sup>D</sup> Ci

i

!

r

is given either if bSi = Ci (border solution) by the maximization in Si of G Si ð Þ ;Ci

hið Þ� Ci F Si ð Þ ;Ci ½ � (17)

Si (Eq. (4)).

hið Þ� Ci F Si ð Þ ;Ci ½ � (19)

Ci and b

Si

� ð Þ 1 � a

hið Þ� Ci F Si ð Þ ;Ci ½ � (23)

C2 i Si

C2 i S2 i

ffiffiffiffiffiffiffiffiffiffiffi 1 � a a

<1 (20)

(21)

(22)

(24)

r

(18)

$$\text{b/} \frac{\partial \left[ h\_i(\mathbf{C}\_i) - D \left( a \mathbf{S}\_i + (\mathbf{1} - a) \frac{\mathbf{C}\_i^2}{\mathbf{S}\_i} \right) \right]}{\partial \mathbf{C}\_i} = h'(\mathbf{C}\_i) - D \frac{(\mathbf{1} - a) 2 \mathbf{C}\_i}{\mathbf{S}\_i} = \mathbf{0} \tag{28}$$

which is equivalent to

$$h'(\mathbf{C}\_i) = \mathfrak{D}\sqrt{a(1-a)}\tag{29}$$

• If bSi ≥Ci,

then (23) can be written as

$$\max\_{\mathbf{S}\_{\mathbf{i}t}, \mathbf{C}\_{i}} \left( h\_{i}(\mathbf{C}\_{i}) - D(a\mathbf{S}\_{i} + (\mathbf{1} - a)b\mathbf{C}\_{i}) \right) + \mu(b\mathbf{S}\_{i} - \mathbf{C}\_{i}) \tag{30}$$

with μ the dual variable associated to the constraint bSi ≥Ci. The first-order conditions give here:

$$\mathbf{a} / \frac{\partial [h\_i(\mathbf{C}\_i) - D(a\mathbf{S}\_i + (1 - a)b\mathbf{C}\_i) + \mu(b\mathbf{S}\_i - \mathbf{C}\_i)]}{\partial \mathbf{S}\_i} = -Da + \mu b = \mathbf{0} \tag{31}$$

and

$$\text{b/}s\frac{\partial[h\_i(\mathbf{C}\_i) - D(a\mathbf{S}\_i + (\mathbf{1} - a)b\mathbf{C}\_i) + \mu(b\mathbf{S}\_i - \mathbf{C}\_i)]}{\partial \mathbf{C}\_i} = h'(\mathbf{C}\_i) - D(\mathbf{1} - a)b - \mu = 0 \quad \text{(32)}$$

The solution of these equations is.

$$\mathbf{C}\_{i} = b\mathbf{S}\_{i}, \boldsymbol{\mu} = \frac{\mathbf{D}a}{b} > \mathbf{0} \text{ and } h'(\mathbf{C}\_{i}) = D\left[ (\mathbf{1} - a)b + \frac{a}{b} \right] = D\left[ \frac{(\mathbf{1} - a)b^{2} + a}{b} \right] \tag{33}$$

#### Remark 1

The optimal solution is a continuous function of parameters a and b. Moreover, the regulator can choose parameters a and b in order to enforce an interior solution or a border solution.

Remark 2 Note that in 2/ of the theorem

$$\lim\_{b \to 0} \left( \frac{D\left(a + (1 - a)b^2\right)}{b} \right) = \infty \tag{34}$$

∑i

<sup>1</sup>þb<sup>2</sup> <sup>&</sup>lt;a<1 and then Si <sup>¼</sup> Ci

2λ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>1</sup> � <sup>a</sup> <sup>=</sup><sup>a</sup> <sup>p</sup> <sup>∑</sup><sup>i</sup>

� ��<sup>1</sup> D a <sup>þ</sup> ð Þ <sup>1</sup> � <sup>a</sup> <sup>b</sup><sup>2</sup> � �

b !

<sup>b</sup> and Ni <sup>≕</sup> ð Þ BA <sup>1</sup>

� ��<sup>1</sup> 2D ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

and the budget equilibrium constraint (35) must be written as

Ci <sup>¼</sup> <sup>2</sup>λ<sup>B</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>f</sup>ð Þ<sup>λ</sup> <sup>≕</sup> <sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ci ¼ h<sup>0</sup> i

DOI: http://dx.doi.org/10.5772/intechopen.86529

1/ either 0< <sup>b</sup><sup>2</sup>

2λB ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>1</sup> � <sup>a</sup> <sup>=</sup><sup>a</sup> <sup>p</sup> <sup>∑</sup><sup>i</sup>

we know then that

which gives

Cαi i αi

tion, this last equation becomes

Noting that since <sup>α</sup><sup>i</sup>

Previously we showed that

Assuming here too that hið Þ¼ Ci

Ci ¼ h<sup>0</sup> i

with <sup>A</sup> <sup>≕</sup> D aþð Þ <sup>1</sup>�<sup>a</sup> <sup>b</sup><sup>2</sup> ð Þ

by trials and errors.

39

2/ or 0< a< <sup>b</sup><sup>2</sup>

be written as

hið Þ¼ Ci

As f 0

The WUA manager may choose parameters a and b in such a way that

Can Nonlinear Water Pricing Help to Mitigate Drought Effects in Temperate Countries?

að Þ 1 � a

ð Þ <sup>1</sup> � <sup>a</sup> <sup>=</sup><sup>a</sup> <sup>p</sup> <sup>∑</sup><sup>i</sup>

If we assume in order to facilitate the presentation of the demonstration that

ð Þ <sup>1</sup> � <sup>a</sup> <sup>=</sup><sup>a</sup> <sup>p</sup> <sup>∑</sup><sup>i</sup>

where Mi ≕ 2B ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð Þλ <0, we deduce that there exists a unique λ which verifies (39).

Cαi i αi

gð Þλ ≕ A∑<sup>i</sup>

<sup>1</sup>þb<sup>2</sup> <sup>&</sup>lt;1 and then bSi <sup>¼</sup> Ci, according to the Theorem.

≕ ki

λ αi αi�1

So, once parameters a and b are chosen for considerations of water savings, the WUA manager can force the system to be in budgetary equilibrium with the choice of the parameter λ value. Of course, not knowing the true value of αiparameters, or more generally ignoring the precise form of the hi(Ci) functions, he will not be able to compute directly the optimal value of λ, but the existence result on a unique λ value and the monotonicity of f(λ) and of g(λ) allows him to find the correct value

F Si ð Þ¼ ;Ci B (35)

að Þ 1 � a

� � <sup>p</sup> <sup>¼</sup> <sup>1</sup> (38)

gi <sup>2</sup>λ<sup>B</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi að Þ 1 � a

ð Þ <sup>1</sup> � <sup>a</sup> <sup>=</sup><sup>a</sup> <sup>p</sup> according to the Theorem;

� � <sup>p</sup> (36)

� � <sup>p</sup> <sup>¼</sup> <sup>B</sup> (37)

<sup>α</sup><sup>i</sup> <sup>α</sup>i�<sup>1</sup>Mi <sup>¼</sup> <sup>1</sup> (39)

(40)

(41)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� � <sup>p</sup> <sup>≕</sup> gi <sup>2</sup><sup>D</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

gi <sup>2</sup>λ<sup>B</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi að Þ 1 � a

, with 0 <α<sup>i</sup> <1, which reminds us of a Cobb–Douglas production func-

λ

að Þ 1 � a � � <sup>p</sup> <sup>α</sup><sup>i</sup> <sup>α</sup>i�<sup>1</sup>

<sup>α</sup>i�<sup>1</sup> <sup>&</sup>lt;0, we have lim<sup>λ</sup>!<sup>0</sup> <sup>f</sup>ð Þ¼þ <sup>λ</sup> <sup>∞</sup> and also lim<sup>λ</sup>!<sup>∞</sup> <sup>f</sup>ð Þ¼ <sup>λ</sup> 0.

D a <sup>þ</sup> ð Þ <sup>1</sup> � <sup>a</sup> <sup>b</sup><sup>2</sup> � � b !

, the budget equilibrium constraint (39) can

Ni ¼ 1 (42)

<sup>α</sup>i�1, and we obtain the same conclusion as in 1/.

If we choose b small enough and therefore a small enough to remain in case 2/, (20) incites the farmers to use less water, that is, Ci ! 0 when b ! 0. But in general we cannot draw any conclusion on the value of Si.

In conclusion the WUA manager may use these two parameters a and b in order to decrease the water consumption, but he cannot make water decrease at discretion since as in our example he might decrease also the reserved volume and at the end the budget equilibrium would not be satisfied.

Note also that in 1/ of the Theorem, we cannot make the consumption Ci decrease at will, since the maximum value of h<sup>0</sup> i ð Þ Ci is equal to D according to Eq. (15).

Figure 1 shows how Si must be tightly correlated to Ci by the farmer in order to obtain a good remuneration Gi Si ð Þ¼ ;Ci hið Þ� Ci F Si ð Þ ;Ci for his activities. (Numerically, it is computed with the following functions and values: hið Þ¼ Ci <sup>2</sup>:C<sup>0</sup>:<sup>5</sup> <sup>i</sup> ; a = 1/3; b = 0.7; D = 2; negative values have been replaced by 0). A slight deviation from the optimum value of Si at the reservation time, and of the optimal consumption Ci, once Si is chosen, will diminish considerably the value of the gain G. This means that from the value of Si, the WUA manager is able to predict accurately the level of the water demand.

#### 3.3 The budget equilibrium constraint

In this section, we study the conditions in which the budget equilibrium may be obtained, or in other terms, in which

Figure 1. Representation of G(Ci, Si) as a function of Ci and of Si. Numerical values: see text.

Can Nonlinear Water Pricing Help to Mitigate Drought Effects in Temperate Countries? DOI: http://dx.doi.org/10.5772/intechopen.86529

$$\sum\_{i} F(\mathbf{S}\_{i}, \mathbf{C}\_{i}) = B \tag{35}$$

The WUA manager may choose parameters a and b in such a way that

1/ either 0< <sup>b</sup><sup>2</sup> <sup>1</sup>þb<sup>2</sup> <sup>&</sup>lt;a<1 and then Si <sup>¼</sup> Ci ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>1</sup> � <sup>a</sup> <sup>=</sup><sup>a</sup> <sup>p</sup> according to the Theorem; we know then that

$$\mathbf{C}\_{i} = \left[ h\_{i}^{\prime} \right]^{-1} \left( 2D \sqrt{a(1-a)} \right) =: \mathbf{g}\_{i} \left( 2D \sqrt{a(1-a)} \right) \tag{36}$$

and the budget equilibrium constraint (35) must be written as

$$2\lambda B\sqrt{(1-a)/a}\sum\_{i}\mathbf{C}\_{i} = 2\lambda B\sqrt{(1-a)/a}\sum\_{i}\mathbf{g}\_{i}\Big(2\lambda B\sqrt{a(1-a)}\Big) = B \tag{37}$$

which gives

Remark 2

the water demand.

Figure 1.

38

Note that in 2/ of the theorem

Drought - Detection and Solutions

lim b!0

general we cannot draw any conclusion on the value of Si.

the budget equilibrium would not be satisfied.

at will, since the maximum value of h<sup>0</sup>

3.3 The budget equilibrium constraint

obtained, or in other terms, in which

D a <sup>þ</sup> ð Þ <sup>1</sup> � <sup>a</sup> <sup>b</sup><sup>2</sup> � � b !

If we choose b small enough and therefore a small enough to remain in case 2/, (20) incites the farmers to use less water, that is, Ci ! 0 when b ! 0. But in

In conclusion the WUA manager may use these two parameters a and b in order to decrease the water consumption, but he cannot make water decrease at discretion since as in our example he might decrease also the reserved volume and at the end

Note also that in 1/ of the Theorem, we cannot make the consumption Ci decrease

Figure 1 shows how Si must be tightly correlated to Ci by the farmer in order to obtain a good remuneration Gi Si ð Þ¼ ;Ci hið Þ� Ci F Si ð Þ ;Ci for his activities. (Numeri-

b = 0.7; D = 2; negative values have been replaced by 0). A slight deviation from the optimum value of Si at the reservation time, and of the optimal consumption Ci, once Si is chosen, will diminish considerably the value of the gain G. This means that from the value of Si, the WUA manager is able to predict accurately the level of

In this section, we study the conditions in which the budget equilibrium may be

ð Þ Ci is equal to D according to Eq. (15).

i

cally, it is computed with the following functions and values: hið Þ¼ Ci <sup>2</sup>:C<sup>0</sup>:<sup>5</sup>

Representation of G(Ci, Si) as a function of Ci and of Si. Numerical values: see text.

¼ ∞ (34)

<sup>i</sup> ; a = 1/3;

$$2\lambda\sqrt{(1-a)/a}\sum\_{i}\mathbf{g}\_{i}\left(2\lambda B\sqrt{a(1-a)}\right) = \mathbf{1} \tag{38}$$

If we assume in order to facilitate the presentation of the demonstration that hið Þ¼ Ci Cαi i αi , with 0 <α<sup>i</sup> <1, which reminds us of a Cobb–Douglas production function, this last equation becomes

$$f(\lambda) = 2\sqrt{(1-a)/a} \sum\_{i} \lambda^{\frac{a\_i}{a\_i - 1}} \mathbf{M}\_i = \mathbf{1} \tag{39}$$

$$\text{where } M\_i \coloneqq \left( 2B \sqrt{a(1-a)} \right)^{\frac{a\_i}{a\_i - 1}} \tag{40}$$

Noting that since <sup>α</sup><sup>i</sup> <sup>α</sup>i�<sup>1</sup> <sup>&</sup>lt;0, we have lim<sup>λ</sup>!<sup>0</sup> <sup>f</sup>ð Þ¼þ <sup>λ</sup> <sup>∞</sup> and also lim<sup>λ</sup>!<sup>∞</sup> <sup>f</sup>ð Þ¼ <sup>λ</sup> 0. As f 0 ð Þλ <0, we deduce that there exists a unique λ which verifies (39).

2/ or 0< a< <sup>b</sup><sup>2</sup> <sup>1</sup>þb<sup>2</sup> <sup>&</sup>lt;1 and then bSi <sup>¼</sup> Ci, according to the Theorem. Previously we showed that

$$\mathbf{C}\_{i} = \left[h\_{i}^{\prime}\right]^{-1} \left(\frac{D\left(a + (1-a)b^{2}\right)}{b}\right) =: k\_{i} \left(\frac{D\left(a + (1-a)b^{2}\right)}{b}\right) \tag{41}$$

Assuming here too that hið Þ¼ Ci Cαi i αi , the budget equilibrium constraint (39) can be written as

$$\mathbf{g}(\boldsymbol{\lambda}) = \mathbf{:} \boldsymbol{A} \sum\_{i} \boldsymbol{\lambda}^{\frac{a\_i}{a\_i - 1}} \mathbf{N}\_i = \mathbf{1} \tag{42}$$

with <sup>A</sup> <sup>≕</sup> D aþð Þ <sup>1</sup>�<sup>a</sup> <sup>b</sup><sup>2</sup> ð Þ <sup>b</sup> and Ni <sup>≕</sup> ð Þ BA <sup>1</sup> <sup>α</sup>i�1, and we obtain the same conclusion as in 1/.

So, once parameters a and b are chosen for considerations of water savings, the WUA manager can force the system to be in budgetary equilibrium with the choice of the parameter λ value. Of course, not knowing the true value of αiparameters, or more generally ignoring the precise form of the hi(Ci) functions, he will not be able to compute directly the optimal value of λ, but the existence result on a unique λ value and the monotonicity of f(λ) and of g(λ) allows him to find the correct value by trials and errors.
