3. The model

#### 3.1 Notations

We consider a WUA composed of n farmers and which provides them irrigation water at a cost. Each farmer i has a production function we note hi(Ci) which is a

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

function of the volume Ci of the water he consumes. This production function is private information, known only by the farmer himself.

Each year, each farmer firstly reserves a water volume Si, for example, before choosing his planting and then consumes another volume Ci for the field irrigation, Ci being either inferior or superior to Si. The pricing formula is designed in order to take into account these two variables and to display some properties.

The notations we use are the following:

demand in the high blocks will be more elastic than demand in the low blocks, resulting in a net decrease in water use when compared to a uniform pricing. Although there is widespread consensus that IBT have many advantages, this type of tariff still deserves more careful examination since an incorrect structure of the IBTs leads to several shortcomings as argued in [19]. Some of them are difficulties to set the initial block; mismatch between prices and marginal costs; conflict between revenue sufficiency and economic efficiency; absence of simplicity, transparency,

The decreasing block tariff (DBT) is, unlike the preceding one, in accordance with the proposition that high-value goods "should" be bought at higher price than low-value goods. Water will be first purchased for uses with high values and then only for uses which will lead to less welfare increases. Concerning equity, this type of tariff is "not advisable". "The consumers who acquire smaller amounts of the good and/or service because of their low incomes would be bearing a higher price than those who can afford to consume greater amounts" (see [20]). But it can be

• When users have very different levels of consumption. A consumer hundred times bigger than the average consumer does not create costs hundred times higher, because there is only one pipe line, one billing process, etc. And, since cost per volume is lower with large consumers, it is justifiable to propose DBT

• In order to incite users to stay in the WUA: as we have explained above, IBT (e.g., see [21]) might encourage users who have access to alternative water sources to quit (partly at least) the network, stopping to contribute to the recovery of the costs. This can lead to cost recovery problems for the water supplier and besides might lead to negative environmental consequences. DBT

A two-part tariff combines a fixed and a volumetric rate (or a mix of fixed and variable elements). "Under this system, consumers must pay an entry charge that entitles them to consume the good. Subsequently they will pay an additional smaller amount for each extra unit consumed." "Two-part tariff is easy to explain and easy to understand" is mentioned in [20]. But in practice, it fails to reach the efficiency objective and suffer from the fact that it does not allow to reveal information on water demand, which may be at the origin of sudden discrepancy between water

In the following sections, we study the properties of a different pricing structure, in which farmers make a water reservation (e.g., during wet period or before planting) and then pay a water bill which is an increasing function of water reservation and of real water consumption (e.g., during dry period or during peak vegetation). This allows the WUA manager to forecast disequilibrium between water demand and supply. The water pricing is parameterized, in order to adapt the

We consider a WUA composed of n farmers and which provides them irrigation water at a cost. Each farmer i has a production function we note hi(Ci) which is a

price to the actual WUA situation and to the available water supply.

and implementation; incapacity of solving shared connections; etc.

justified in the following circumstances:

Drought - Detection and Solutions

in the case of heterogeneous users.

does not have this negative incentive.

supply and demand.

3. The model

3.1 Notations

32


For each agent i, the pricing formula is

$$F(\mathbf{S}\_i, \mathbf{C}\_i) = D\left(a\mathbf{S}\_i + (\mathbf{1} - a) \frac{\mathbf{C}\_i \max(\mathbf{C}\_i, b\mathbf{S}\_i)}{\mathbf{S}\_i}\right) \tag{1}$$

with a ∈ [0, 1] and b ∈ [0, 1].

The pricing scheme is a common knowledge for all farmers and is the same for all of them. Parameter a represents a kind of sharing of the price between, on the first hand, the reservation part and, on the other hand, the consumption part. As D = λB, the role of parameter λ is to ensure a balanced budget, under the financial conditions of the WUA, for example, by adjusting the value of this parameter by trial and error year after year, if a temporary budget imbalance is permissible. We will return to this point in Section 3.3, examining in particular the case where a minimum of revenue each year is required. The role of parameter b is to incite to reserve at least the forecasted consumption divided by b. For a Si given, when Ci > bSi, the C<sup>2</sup> <sup>i</sup> which appears in the pricing formula incites to diminish water consumption.

A deterministic approach, without acquisition of information between the reservation date and the consumption date, is sufficient in order to study some of the properties of this pricing. Of course other properties directly linked to stochastic variables (as the climate) cannot be studied here and are the object of further researches.

But we must note that since Ci depends on Si, we must take this relationship into account in optimizing the volumes reserved and consumed by the farmer i.

#### 3.2 The maximization problem of farmer i

When choosing the values of his control variables Si and Ci, the farmer must decide of the optimal value of Ci knowing the optimal value of Si previously announced. Therefore each farmer must solve

$$\max\_{\mathbf{S}\_i} \left[ \max\_{\mathbf{C}\_i} \left( h\_i(\mathbf{C}\_i) - F(\mathbf{S}\_i, \mathbf{C}\_i) \right) \right] \tag{2}$$

Csol

finally it is a constant.

3.2.2 The maximization problem in Si

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

<sup>i</sup> ð Þ Si =b, we must solve

and the first-order condition gives

<sup>i</sup> ð Þ Si =b≤Si ≤C<sup>þ</sup>

max Si � hi � C� <sup>i</sup> ðSiÞ � � F � Si, C� <sup>i</sup> ðSiÞ

<sup>i</sup> ðSiÞ ¼ D a þ ð1 � aÞ

max Si

> bh<sup>0</sup> i

for which the first-order condition is

bill; in other words we have bSi ≤ Ci).

sibility of solutions of (11) and (13).

The optimal strategy of farmer i is

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>

2/Si <sup>¼</sup> Ci=<sup>b</sup> if 0<a<sup>&</sup>lt; <sup>b</sup><sup>2</sup>

<sup>i</sup> =b we must solve

<sup>i</sup> =b the maximization problem is:

max Si � hiðC<sup>þ</sup>

> h0 i

h0 i ð Þ¼ Ci

2 6 4

between these two variables.

• If Si <C�

h0 <sup>i</sup> C� <sup>i</sup> <sup>ð</sup>Si<sup>Þ</sup> � �C0�

• If C�

• and if Si >C<sup>þ</sup>

Theorem

1/Si ¼ Ci

35

<sup>i</sup> ð Þ Si is first an increasing function of Si, then it is a linear function in Si, and

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

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

> 2C� <sup>i</sup> <sup>ð</sup>SiÞC<sup>0</sup> � <sup>i</sup> ðSiÞSi �

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

<sup>i</sup> Þ � FðSi, C<sup>þ</sup>

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

To obtain the optimal solution of our problem in Si, we must analyze the admis-

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

<sup>1</sup>þb<sup>2</sup> <sup>&</sup>lt; 1, where Ci is the solution of

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

i Þ �

<sup>1</sup>þb<sup>2</sup> <sup>&</sup>lt;a<1, where Ci is the solution of

<sup>a</sup>ð Þ <sup>1</sup> � <sup>a</sup> <sup>p</sup> (15)

<sup>b</sup> (16)

�� (10)

3 7

<sup>5</sup> (11)

(14)

� C� <sup>i</sup> ðSiÞ �2

S2 i

Si � � � � (12)

ð Þ¼ bSi D a <sup>þ</sup> ð Þ <sup>1</sup> � <sup>a</sup> <sup>b</sup><sup>2</sup> � � (13)

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

## 3.2.1 The maximization problem in Ci

We note G Si ð Þ¼ ;Ci hið Þ� Ci F Si ð Þ ;Ci , and we calculate the solution of the maximization of G Si ð Þ ;Ci as a function of Ci. We have

$$\frac{\partial \mathbf{G}\_i(\mathbf{S}\_i, \mathbf{C}\_i)}{\partial \mathbf{C}\_i} = \begin{cases} h\_i'(\mathbf{C}\_i) - 2(\mathbf{1} - a)D \frac{\mathbf{C}\_i}{\mathbf{S}\_i} & \text{if } b\mathbf{S}\_i < \mathbf{C}\_i \\\ h\_i'(\mathbf{C}\_i) - (\mathbf{1} - a)bD & \text{if } b\mathbf{S}\_i > \mathbf{C}\_i \end{cases} \tag{3}$$

• For bSi <Ci, we note C� <sup>i</sup> ð Þ Si the solution in Ci of

$$h\_i'(\mathbf{C}\_i) = \mathbf{2}(\mathbf{1} - a)D\frac{\mathbf{C}\_i}{\mathbf{S}\_i} \tag{4}$$

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

$$\mathbf{C}\_{i}^{-\prime}(\mathbf{S}\_{i}) = \frac{\mathbf{2}(\mathbf{1} - a)\mathbf{D}\mathbf{C}\_{i}}{\mathbf{2}(\mathbf{1} - a)\mathbf{D}\mathbf{S}\_{i} - h^{\prime}(\mathbf{C}\_{i})\mathbf{S}\_{i}^{2}} > \mathbf{0} \tag{5}$$

Therefore C�<sup>0</sup> <sup>i</sup> ð Þ Si is an increasing function of Si. We remark by the way that when bSi <Ci, we have

$$2(\mathbf{1} - a)D\frac{C\_i}{S\_i} > 2(\mathbf{1} - a)bD > (\mathbf{1} - a)bD \tag{6}$$

• For bSi >Ci, we call C<sup>þ</sup> <sup>i</sup> the solution in Ci of

$$h\_i'(\mathbf{C}\_i) = (\mathbf{1} - \mathbf{a})bD \tag{7}$$

We note that this solution does not depend on Si.

The preceding remark, the concavity of hi, and the definition in (4) and (7) of C� <sup>i</sup> and of C<sup>þ</sup> <sup>i</sup> imply that

$$\mathbf{C}\_{i}^{-}(\mathbf{S}\_{i}) < \mathbf{C}\_{i}^{+} \tag{8}$$

We can easily deduce that the optimal solution Csol <sup>i</sup> of (2) for Si given depends on the relative positions of C� <sup>i</sup> ð Þ Si <C<sup>þ</sup> <sup>i</sup> and of bSi:

$$\mathbf{C}\_{i}^{sol} = \begin{cases} \mathbf{C}\_{i}^{-} (\mathbf{S}\_{i}) & \text{if} \quad \mathbf{S}\_{i} < \mathbf{C}\_{i}^{-} (\mathbf{S}\_{i}) / b \\ b \mathbf{S}\_{i} & \text{if} \quad \mathbf{C}\_{i}^{-} (\mathbf{S}\_{i}) / b \le \mathbf{S}\_{i} \le \mathbf{C}\_{i}^{+} / b \\ \mathbf{C}\_{i}^{+} & \text{if} \quad \mathbf{S}\_{i} > \mathbf{C}\_{i}^{+} / b \end{cases} \tag{9}$$

Note that this formula gives a relation Csol <sup>i</sup> ð Þ Si that depends on the parameters of the regulator and of the profit function hi of farmer i. When starting from small Si,

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

Csol <sup>i</sup> ð Þ Si is first an increasing function of Si, then it is a linear function in Si, and finally it is a constant.
