# A Differential Game for Optimal Water Price Management

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Model

- The agency announces the tariff scheme of water $Z(t,\xb7)$.
- The buyer, having observed the tariff scheme, but not $S\left(t\right)$, decides the quantity of water $D\left(t\right)$ to demand, maximizing the present value of its net income.
- The agency supplies the buyer with the quantity of water $W\left(t\right)\le D\left(t\right)$.

## 3. Tariff Scheme and Water Demand

#### Special Pricing Schemes

- Linear pricing scheme. If water scarcity is not taken into account, then the most reasonable choice appears to be charging the same price for each unit of volume of water purchased. In this case$$Z(t,D(t\left)\right)=p\left(t\right)D\left(t\right),$$$$\varphi \left(t\right){\omega}^{\prime}\left(D\left(t\right)\right)-p\left(t\right)=0.$$Note that, as expected, $\widehat{D}$ decreases as the unitary price $p\left(t\right)$ increases.
- Block tariffs. Block tariffs are applied to discourage excessive use of a scarce resource, such as water. Let us consider the simplest case with two blocks. At any time t there is a threshold $B\left(t\right)$, such that the buyer pays a unit price ${p}_{1}\left(t\right)$ for each unit of water below threshold $B\left(t\right)$, while paying a higher unit price ${p}_{2}\left(t\right)$ for water consumption in excess of threshold $B\left(t\right)$.$$Z(t,D\left(t\right))=\left\{\begin{array}{cc}{p}_{1}\left(t\right)D\left(t\right)\hfill & \mathit{if}\phantom{\rule{0.277778em}{0ex}}D\left(t\right)\le B\left(t\right);\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & \\ {p}_{1}\left(t\right)B\left(t\right)+{p}_{2}\left(t\right)\left(D\left(t\right)-B\left(t\right)\right)\hfill & \mathit{if}\phantom{\rule{0.277778em}{0ex}}D\left(t\right)>B\left(t\right);\hfill \end{array}\right.$$The optimal demand is:
- (a)
- $\widehat{D}\left(t\right)<B\left(t\right)$ where $\widehat{D}\left(t\right)$ solves the equation $\varphi \left(t\right){\omega}^{\prime}\left(D\left(t\right)\right)-{p}_{1}\left(t\right)=0$, if ${\omega}^{\prime}\left(B\left(t\right)\right)<{\displaystyle \frac{{p}_{1}\left(t\right)}{\varphi \left(t\right)}}$;
- (b)
- $\widehat{D}\left(t\right)=B\left(t\right)$, if $\frac{{p}_{1}\left(t\right)}{\varphi \left(t\right)}}<{\omega}^{\prime}\left(B\left(t\right)\right)<{\displaystyle \frac{{p}_{2}\left(t\right)}{\varphi \left(t\right)}$;
- (c)
- $\widehat{D}\left(t\right)>B\left(t\right)$ and $\widehat{D}\left(t\right)$ solves the equation $\varphi \left(t\right){\omega}^{\prime}\left(D\left(t\right)\right)-{p}_{2}\left(t\right)=0$, if ${\omega}^{\prime}\left(B\left(t\right)\right)>{\displaystyle \frac{{p}_{2}\left(t\right)}{\varphi \left(t\right)}}$.

**Remark**

**1.**

- If the block threshold $B\left(t\right)$ is large enough, then it is optimal to buy a quantity of water $\widehat{D}\left(t\right)<B\left(t\right)$ at the cheapest price ${p}_{1}\left(t\right)$ and the block tariff gives exactly the same solution as the linear pricing scheme.
- If the block threshold $B\left(t\right)$ is neither too high nor too low, then it is optimal to buy a quantity of water equal to $B\left(t\right)$ at the cheapest price ${p}_{1}\left(t\right)$ and the block tariff reduces the demand for water compared with the linear pricing case.
- If the block threshold $B\left(t\right)$ is small enough, then it is optimal to buy a quantity of water $\widehat{D}\left(t\right)>B\left(t\right)$ paying the quantity $B\left(t\right)$ at the lower price and the excess quantity at the higher price; moreover, the demand is lower compared with the linear pricing case.

- 3.
- Convex tariffs. A more general setting to account for scarcity is to introduce increasing marginal prices through the convex function4$$Z(t,D\left(t\right))=q\left(t\right)D{\left(t\right)}^{\alpha};\phantom{\rule{1.em}{0ex}}\alpha >1.$$The optimal demand $\widehat{D}\left(t\right)$ solves the equation$$\varphi \left(t\right){\omega}^{\prime}\left(D\left(t\right)\right)-\alpha q\left(t\right)D{\left(t\right)}^{\alpha -1}=0.$$$\widehat{D}\left(t\right)$ decreases as $\alpha $ or $q\left(t\right)$ increase.

**Remark**

**2.**

- 1.
- Linear pricing scheme.$$\widehat{D}\left(t\right)={\left[{\displaystyle \frac{\gamma \varphi \left(t\right)}{p\left(t\right)}}\right]}^{{\displaystyle \frac{1}{1-\gamma}}}.$$
- 2.
- Block tariffs.$$\widehat{D}\left(t\right)=\left\{\begin{array}{cc}{\left[{\displaystyle \frac{\gamma \varphi \left(t\right)}{{p}_{1}\left(t\right)}}\right]}^{{\displaystyle \frac{1}{1-\gamma}}}<B\left(t\right),\hfill & \mathit{if}\phantom{\rule{0.277778em}{0ex}}B\left(t\right)>{\left[{\displaystyle \frac{\gamma \varphi \left(t\right)}{{p}_{1}\left(t\right)}}\right]}^{{\displaystyle \frac{1}{1-\gamma}}};\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & \\ B\left(t\right),\hfill & \mathit{if}\phantom{\rule{0.277778em}{0ex}}{\left[{\displaystyle \frac{\gamma \varphi \left(t\right)}{{p}_{2}\left(t\right)}}\right]}^{{\displaystyle \frac{1}{1-\gamma}}}\le B\left(t\right)\le {\left[{\displaystyle \frac{\gamma \varphi \left(t\right)}{{p}_{1}\left(t\right)}}\right]}^{{\displaystyle \frac{1}{1-\gamma}}};\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & \\ {\left[{\displaystyle \frac{\gamma \varphi \left(t\right)}{{p}_{2}\left(t\right)}}\right]}^{{\displaystyle \frac{1}{1-\gamma}}}>B\left(t\right),\hfill & \mathit{if}\phantom{\rule{0.277778em}{0ex}}B\left(t\right)<{\left[{\displaystyle \frac{\gamma \varphi \left(t\right)}{{p}_{2}\left(t\right)}}\right]}^{{\displaystyle \frac{1}{1-\gamma}}}.\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & \end{array}\right.$$Equivalently$$\widehat{D}\left(t\right)=\left\{\begin{array}{cc}{\left[{\displaystyle \frac{\gamma \varphi \left(t\right)}{{p}_{2}\left(t\right)}}\right]}^{{\displaystyle \frac{1}{1-\gamma}}}>B\left(t\right),\hfill & \mathit{if}\phantom{\rule{0.277778em}{0ex}}{p}_{1}\left(t\right)\le {p}_{2}\left(t\right)<\sigma \left(B\left(t\right)\right);\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & \\ B\left(t\right),\hfill & \mathit{if}\phantom{\rule{0.277778em}{0ex}}{p}_{1}\left(t\right)\le \sigma \left(B\left(t\right)\right)\le {p}_{2}\left(t\right);\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & \\ {\left[{\displaystyle \frac{\gamma \varphi \left(t\right)}{{p}_{1}\left(t\right)}}\right]}^{{\displaystyle \frac{1}{1-\gamma}}}<B\left(t\right),\hfill & \mathit{if}\phantom{\rule{0.277778em}{0ex}}\sigma \left(B\left(t\right)\right)<{p}_{1}\left(t\right)\le {p}_{2}\left(t\right);\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & \end{array}\right.$$$$\sigma \left(B\left(t\right)\right):={\displaystyle \frac{\gamma \varphi \left(t\right)}{{\left[B\left(t\right)\right]}^{1-\gamma}}}.$$
- 3.
- Convex tariffs.$$\widehat{D}\left(t\right)={\left[{\displaystyle \frac{\gamma \varphi \left(t\right)}{\alpha q\left(t\right)}}\right]}^{{\displaystyle \frac{1}{\alpha -\gamma}}}.$$

## 4. The Open-Loop Stackelberg Equilibrium

**Proposition**

**1.**

#### 4.1. Linear Pricing Schemes

**Proposition**

**2.**

**Proof.**

**Remark**

**3.**

#### 4.2. Block Tariffs

**Proposition**

**3.**

**Proof.**

**Remark**

**4.**

#### 4.3. Convex Tariffs

**Proposition**

**4.**

**Remark**

**5.**

## 5. Numerical Results

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Proof**

**of Proposition 2.**

**Proof**

**of Proposition 3.**

**Proof**

**of Proposition 4.**

## Notes

1 | In order to be more precise, companies purchase water as an employable factor in their production processes, while private individuals require water for primary activities, such as drinking, food preparation, bathing, washing clothes and dishes, flushing toilets, watering lawns and gardens, and maintaining pools. |

2 | The linear cost function, described by Gisser and Sanchez (1980), is utilized. However, unlike the article cited, in which costs linearly depend on the total lift in meters, we consider a cost function that linearly depends on the difference between the capacity and the current volume of the aquifer. Further, it can be immediately noticed that the two approaches are equivalent. |

3 | In this model, we assume the presence of seasonal cycles in defining the preferences of the buyer with respect to water. This modeling choice is in line with the strand of literature that studies the presence of seasonality in consumers’ preferences toward all kinds of purchasable goods, and the related reflections at both the microeconomic and macroeconomic levels. In this regard, see [22,23,24]. |

4 | Note that the coefficients q should not be confused with the unitary prices p appearing in the linear pricing scheme. In the convex tariffs case, if you buy a quantity of water, x, then the average cost of a unit volume of water is $q{x}^{\alpha -1}$ and it increases as x increases. |

5 | |

6 | The function defined in (20) has period $T=1$, while the function in (21) has period $\tau $. |

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**Figure 1.**Different scenarios for the dynamics of ${p}^{*}\left(t\right)$, ${S}^{*}\left(t\right)$ and ${W}^{*}\left(t\right)$. The continuous lines refer to the case with a linear price scheme, while the dashed lines refer to the case with a convex tariff scheme ($\alpha =2$).

**Figure 2.**Different behaviors of the payoffs J (continuous curve) and ${\int}_{0}^{1}{\pi}_{B}\left(t\right)dt$ (dashed curve), as $\alpha $ increases.

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**MDPI and ACS Style**

Caravaggio, A.; De Cesare, L.; Di Liddo, A.
A Differential Game for Optimal Water Price Management. *Games* **2023**, *14*, 33.
https://doi.org/10.3390/g14020033

**AMA Style**

Caravaggio A, De Cesare L, Di Liddo A.
A Differential Game for Optimal Water Price Management. *Games*. 2023; 14(2):33.
https://doi.org/10.3390/g14020033

**Chicago/Turabian Style**

Caravaggio, Andrea, Luigi De Cesare, and Andrea Di Liddo.
2023. "A Differential Game for Optimal Water Price Management" *Games* 14, no. 2: 33.
https://doi.org/10.3390/g14020033