# A Study on the Mechanism and Pricing of Drainage Rights Trading Based on the Bilateral Call Auction Model and Wealth Utility Function

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Study on the Transaction Mechanism of Bilateral Call Auction of Drainage Rights

#### 2.1. Basic Assumptions

**Assumption**

**1.**

_{i}and sellers S

_{j}in a given area. B

_{i}denotes the ith drainage right purchaser in the region, S

_{j}denotes the jth drainage right seller in the region, m denotes the number of purchasers, and n denotes the number of sellers. Generally, a subject unit tends to be a seller of drainage rights when its flood losses are low or when the drainage power generates a surplus, and conversely tends to be a purchaser [7].

**Assumption**

**2.**

_{bi}and C

_{sj}. Unit flood economic loss is abbreviated as unit flood loss (hereinafter), which indicates the direct economic loss formed by a unit of flooding suffered by a trading entity. The unit flood loss is closely related to the economic level of the region [30].

**Assumption**

**3.**

_{i}and y

_{j}, and the expected transaction prices as o

_{i}and f

_{j}. x

_{i}and y

_{j}reflect the difference between the total demand for flood discharge of the subject and the number of drainage rights received in the initial allocation; o

_{i}reflects the ith drainage right purchaser’s assessment of the value of the drainage rights existing in the market (let the ith drainage right purchaser consider the lowest value of the drainage rights existing in the market to be f

_{min}), and o

_{i}is less than or equal to the unit flood loss (C

_{bi}) of the ith drainage right purchaser, then we have f

_{min}≤ 2264o

_{i}≤ C

_{bi}; f

_{j}should be greater than or equal to the unit flood loss of the jth drainage rights seller (C

_{sj}) in the case of non-power surplus out of rational economic man assumption, i.e., f

_{j}≥ C

_{sj}.

**Assumption**

**4.**

_{i}and the jth drainage right seller S

_{j}as W

_{i}(C

_{bi}) and U

_{j}(C

_{sj}). The revenue expected by the buyer and seller of drainage rights in a transaction can be understood as the product of the expected unit profit and the expected number of transactions. In this study, the expected unit profit of the drainage rights purchaser in the transaction is the difference between the flood losses that the purchaser can avoid through the transaction and the expected purchase price, i.e., C

_{bi}-o

_{i}; The expected unit profit of the seller of drainage rights in a transaction is the difference between the seller’s expected sales price and the additional flood losses sustained as a result of the transaction, i.e., f

_{j}-C

_{sj}; Then W

_{i}(C

_{bi}) = x

_{i}∗ (C

_{bi}− o

_{i}) and U

_{j}(C

_{sj}) = y

_{j}∗ (f

_{j}− C

_{sj}).

#### 2.2. Model Construction

_{i}≥ f

_{j}, both parties to the transaction have the expected profit margin and are satisfied with each other’s offers, and the transaction holds, at which time the equilibrium price P

^{0}= (o

_{i}+ f

_{j})/2. When o

_{i}< f

_{j}, the seller fails to obtain the entire expected profit margin, and the transaction does not hold. In the state of incomplete information, multiple solutions of Bayesian Nash equilibrium exist, requiring the design of trading rules and trading systems for bilateral call auctions, in which we consider that market participants expect to maximize not only their returns in centralized bidding but also the social returns. Expressing the above expectations in the functional form will result in the following objective function.

#### 2.2.1. Rational Price Constraints

#### 2.2.2. Macroscopic Control Total Quantity Constraints

_{i}and y

_{j}denote the trading demand of the ith drainage right buyer and the jth drainage right seller, respectively.

#### 2.3. Clearance Rules

^{*}and the seller’s offer set s

^{*}, and the buyer’s offer set b

^{*}is arranged in descending order to get b

^{*}= {o

_{1}, o

_{2}, o

_{3}..., o

_{g},... o

_{m}}, the set of seller’s offers s

^{*}is arranged in ascending order to get s

^{*}= {f

_{1}, f

_{2}, f

_{3}...,f

_{h},... f

_{n}}, where o

_{1}≥ o

_{2}≥ o

_{3}≥ o

_{g}≥ o

_{m}and f

_{1}≤ f

_{2}≤ f

_{3}≤ f

_{h}≤ f

_{n}. Assuming f

_{h}≤ o

_{g}< f

_{h+1}and o

_{g}< f

_{h}≤ o

_{g+1}, the top g subjects in the buyer’s offer set and the top h subjects in the seller’s offer set enter the transaction set from the offer set, and this rule is called the clearing rule for bilateral call auctions [32]. At this point, we can write the transaction set of the buying side as B

^{*}= {B

_{1}, B

_{2}, B

_{3}..., B

_{g}} and the transaction set of the selling side as S

^{*}= {S

_{1}, S

_{2,}S

_{3}..., S

_{h}}.

#### 2.4. Matching Rules

_{1}in the buyer’s transaction set will get the first choice, and its selection process is more complicated, with the main reference factors being the seller’s expected sales price, sales demand, related taxes and fees, political considerations, etc. Therefore, B

_{1}does not necessarily consider only the lowest priced S

_{1}, so it brings the following picture: (1) B

_{1}fails to choose S

_{1}as counterparty, B

_{2}still enjoys the option to trade with S

_{1}; (2) B

_{1}chooses S

_{1}as counterparty, but S

_{1}’s target trading volume does not fully satisfy B

_{1}’s demand, so B

_{1}chooses another seller as second counterparty; (3) B

_{1}chooses S

_{1}as counterparty, but S

_{1}still has spare volume for other purchasers to choose after trading with B

_{1}.

## 3. Transaction Pricing Study of Bilateral Call Auction for Drainage Rights

#### 3.1. Wealth Utility Function Construction

^{0}= (o

_{i}+ f

_{j})/2 for the drainage rights transaction. The equilibrium price at this point is not affected by the parameters and is the transaction price in perfect equilibrium. The transaction price in perfect equilibrium can distribute the difference between the expected transaction prices of the two parties fairly in terms of quantity. However, the pricing of drainage rights transactions should not only take the equal distribution of proceeds between the buyer and the seller as the whole concept, but the satisfaction of both parties to the transaction with the proceeds received and the fairness of resource allocation should also be reflected [33]. Therefore, we adopt the wealth utility concept to construct the function, apply the social welfare function to associate the wealth utility functions of both sides of the transaction, and draw on the environmental Gini coefficient concept to construct a pricing model for drainage rights transactions based on parallel equity and efficiency.

^{*}denotes the actual transaction price, which should be between the desired prices of the two sides of the transaction, i.e., f

_{j}≤ p

^{*}≤ o

_{i}; o

_{i}-p

^{*}denotes the utility space of the buyer, and p

^{*}-f

_{j}denotes the utility space of the seller, i.e., the buyer and seller split the total utility space o

_{i}-f

_{j}; Z

_{b}(p

^{*}) denotes the degree of utility acquisition of the buyer, and Z

_{s}(p

^{*}) denotes the degree of utility acquisition of the seller. When the actual transaction price exceeds the buyer’s expected purchase price, i.e., p

^{*}> o

_{i}, the buyer’s willingness to trade is 0. When the actual transaction price is less than the seller’s expected sales price, i.e., p

^{*}< f

_{j}, the seller’s willingness to trade is 0. Then in both cases, the transaction is not valid.

#### 3.2. Social Welfare Function Construction

_{b}(p

^{*}) denotes the degree of utility acquisition of the buyer, Z

_{s}(p

^{*}) denotes the degree of utility acquisition of the seller, T denotes the value of the social welfare function, and α denotes the inequity pullback coefficient. The unfair pullback coefficient reflects the loss of flooding carried per unit of a given indicator. Based on existing literature, we argue that such a flood distribution is equitable when the loss of flooding per unit of GDP, area, and population carried by two regions is close [38]. Thus, this study is based on the three dimensions of economic development level, land area, and population size to calculate the unfair pullback coefficient. Considering that the direct economic loss of flooding in the study year of the drainage rights seller may be zero, it will lead to the unfair pullback coefficient of the drainage rights buyer is 1, which loses the significance of the setting, so we chose to extend the data for three years. In summary, the inequitable pullback factor is calculated as follows.

_{ae}represents the arithmetic mean of the number of indicators in region e in the ath three previous years, y

_{ae}represents the arithmetic mean of the direct economic losses from flooding in the ath region in the previous three years, and e represents the number of the three indicators: GDP, land area, and population.

#### 3.3. Solving for the Transaction Price

^{*}is derived. The derivation process is as follows. First, organize Equation (8) to derive $T=\frac{{({o}_{i}-{p}^{\ast})}^{\partial}}{{({o}_{i}-{f}_{j})}^{\partial}}\ast \frac{{({p}^{\ast}-{f}_{j})}^{1-\partial}}{{({o}_{i}-{f}_{j})}^{1-\partial}}=\frac{{({o}_{i}-{p}^{\ast})}^{\partial}{({p}^{\ast}-{f}_{j})}^{1-\partial}}{{o}_{i}-{f}_{j}}$; second, find the derivative ${T}^{\u2019}=-\partial {({o}_{i}-{p}^{\ast})}^{\partial -1}{({p}^{\ast}-{f}_{j})}^{1-\partial}+(1-\partial ){({o}_{i}-{p}^{\ast})}^{\partial}{({p}^{\ast}-{f}_{j})}^{-\partial}$ of the social welfare function T; third, extract the common factorization ${({o}_{i}-{p}^{\ast})}^{\partial}{({p}^{\ast}-{f}_{j})}^{-\partial}$ to obtain ${T}^{\u2019}={({o}_{i}-{p}^{\ast})}^{\partial}{({p}^{\ast}-{f}_{j})}^{-\partial}[\frac{-\partial ({p}^{\ast}-{f}_{j})}{{o}_{i}-{p}^{\ast}}+1-\partial ]$; fourth, in order to find the extreme value point of the social welfare function T, so that the derivative function T

^{’}is zero, i.e., ${T}^{\u2019}={({o}_{i}-{p}^{\ast})}^{\partial}{({p}^{\ast}-{f}_{j})}^{-\partial}[\frac{-\partial ({p}^{\ast}-{f}_{j})}{{o}_{i}-{p}^{\ast}}+1-\partial ]=0$, and because in general p

^{*}is not equal to o

_{i}and f

_{j}(the expected transaction price between the two sides of the transaction), there exists the equation $\frac{-\partial ({p}^{\ast}-{f}_{j})}{{o}_{i}-{p}^{\ast}}+1-\partial =0$, by which ${p}^{\ast}=\partial (({f}_{j}-{o}_{i})+{o}_{i}$ can be found; finally, as $\frac{-\partial ({p}^{\ast}-{f}_{j})}{{o}_{i}-{p}^{\ast}}+1-\partial =0$ is a monotonically decreasing primary linear function, indicating that there is only one solution to the equation, and the original function first increases and then decreases, that is, to find the p

^{*}for the original function of the only extreme value point and the extreme value of the maximum.

## 4. Simulation of Calculations for Drainage Rights Trading in Jiangsu Section of Huaihe River Basin

#### 4.1. Study Area

#### 4.2. Simulation of Drainage Rights Trading Mechanism

_{1}(8.14, 100), B

_{2}(5.38, 50), B

_{3}(7.21, 100), and B

_{4}(5.2, 62.5), and the quoted and traded quantities on the selling side are S

_{1}(9.45, 100), S

_{2}(6.8, 75), S

_{3}(6.47, 50), and S

_{4}(4.38, 37.5), so the buying and selling total demand is 312.5 m

^{3}and 262.5 m

^{3}respectively. Let the total limit of drainage rights trading by the higher government be 250 m

^{3}, which indicates that there will be 62.5 m

^{3}of total demand for drainage rights purchase and 12.5 m

^{3}of total demand for drainage rights sale that cannot be satisfied.

#### 4.2.1. Clearance Rule

^{*}is arranged in descending order to get b

^{*}= {8.14, 7.21, 5.83, 5.20}, the set of sellers’ offers s

^{*}is arranged in ascending order to get s

^{*}= {4.38, 6.47, 6.80, 9.45}, and the final results are shown in Table 1. Since only 250 m

^{3}of drainage rights are allowed to be traded, purchaser B

_{4}will be strictly excluded from the transaction set, and the expected sales price f

_{1}of seller S

_{1}is greater than any value in the purchaser’s offer set b

^{*}, indicating that it is not possible to match and reach a deal with either purchaser. Therefore, we can write the transaction set for the buyer as B

^{*}= {B

_{1}, B

_{3}, B

_{2}} and the transaction set as S

^{*}= {S

_{4}, S

_{3}, S

_{2}}.

#### 4.2.2. Matching Transaction Sets

_{1}in the buyer’s transaction set will get the first choice, and its selection process is more complicated, with the main reference factors being the seller’s offer, the seller’s expected transaction volume, and relevant taxes, political considerations, etc. In this case, only the expected sales price of the seller and the expected transaction volume are used as reference factors. The consideration process is: (1) B

_{1}’s acquisition demand x

_{1}= 100, which is a large amount, gives priority to the full drainage rights of seller S

_{4}and considers the remaining 62.5 m

^{3}. However, the expected sales price of seller S

_{3}is slightly smaller than the expected sales price of seller S

_{4}, but it does not fully satisfy B

_{1}’s remaining demand, and B

_{1}tends to choose seller S

_{2}for the 62.5 m

^{3}transaction. (2) After all of B

_{1}’s acquisition demand is satisfied, B

_{3}will be given the preference, and B

_{3}’s acquisition demand x

_{3}= 100, which is also a large amount. At this time, the total expected sales volume remaining on the selling side is 62.5 m

^{3}(50 m

^{3}remaining on the selling side S

_{3}and 12.5 m

^{3}remaining on the selling side S

_{2}), then the purchasing side B

_{3}will purchase the entire remaining volume of 62.5 m

^{3}. By observing Table 1, it can be found that the transaction demands of both drainage right buyer B

_{2}and drainage right buyer B

_{4}are not satisfied due to the low bids.

^{3}(less than the total restricted amount of drainage rights trading), and four transactions were formed, as follows: (1) buyer B

_{1}and seller S

_{4}find the optimal transaction price between 4.38 RMB/m

^{3}and 8.14 RMB/m

^{3}for 37.5 m

^{3}of flooding and reach a deal; (2) purchaser B

_{1}and seller S

_{2}find the optimal transaction price between 6.80 RMB/m

^{3}and 8.14 RMB/m

^{3}for 62.5 m

^{3}of flood discharge and reach a deal; (3) purchaser B

_{3}and seller S

_{3}find the optimal transaction price between 6.47 RMB/m

^{3}and 7.21 RMB/m

^{3}for a 50 m

^{3}flood discharge and reach a deal; (4) purchaser B

_{3}and seller S

_{2}find the optimal transaction price between 6.80 RMB/m

^{3}and 7.21 RMB/m

^{3}for a 12.5 m

^{3}flood discharge and reach a deal. In the following, the optimal transaction price will be selected for these four transactions.

#### 4.3. Simulation of the Pricing of Drainage Rights Transactions

#### 4.3.1. Wealth Utility Function

^{3}, 7.47 RMB/m

^{3}, 6.84 RMB/m

^{3}, and 7.01 RMB/m

^{3}, respectively; meanwhile, the wealth utility function of Transaction 1 can be obtained according to Equations (6) and (7).

_{1}is as follows.

_{4}is as follows.

#### 4.3.2. Social Welfare Function

_{1}and seller S

_{4}in Transaction 1 (Table 3) as an example. According to Equation (9), the inequity pullback coefficient of purchaser B

_{1}at the gross regional product level is calculated as follows.

_{1}, which can also be calculated by Equation (9), as follows.

_{1}in the past three years is much larger than that of seller S

_{4}. The inequity pullback coefficient of purchaser B

_{1}in these three aspects is much smaller than that of seller S

_{4}, as shown in Table 4. The inequity pullback coefficients of purchaser B

_{1}and seller S

_{4}are 0.03 and 0.97, indicating that the market focuses more on the seller’s utility in the process of trading drainage rights, i.e., the increase in social welfare per unit increase in utility on the seller’s side is much greater than that on the buyer’s side.

#### 4.3.3. Optimal Transaction Price

^{3}and 8.14 RMB/m

^{3}(as shown in Table 2). According to the previous section, the arithmetic means of the fair pullback coefficients of purchaser B

_{1}and seller S

_{4}are 0.03 and 0.97, respectively, indicating that the market pays more attention to the utility of the seller, i.e., expanding the expected profit margin of the seller, in the process of drainage rights trading. The optimal transaction price will occur well above the equilibrium trading price (6.26 RMB/m

^{3}).

^{3}is 0.07 RMB/m

^{3}lower than the purchaser’s expected transaction price of 8.14 RMB/m

^{3}and 3.65 RMB/m

^{3}higher than the seller’s expected transaction price of 4.38 RMB/m

^{3}. From the perspective of the wealth utility function, it can be understood that the total wealth utility space of both sides of the transaction is 3.76 RMB/m

^{3}, among which the wealth utility space of the buyer is 0.07 RMB/m

^{3}, and the wealth utility space of the seller is 3.69 RMB/m

^{3}. When the optimal transaction price is 8.03 RMB/m

^{3}, the wealth utility acquisition degree of the buyer is 1.86% and the wealth utility acquisition degree of the seller is 98.14%, the wealth utility acquisition degree of the selling side is significantly higher than that of the buying side due to its larger fair pullback coefficient. From the perspective of the transaction itself, it can be understood that the buyer pays 0.07 RMB/m

^{3}less for 1 m

^{3}of drainage rights purchased, and the seller gets 3.65 RMB/m

^{3}more for 1 m

^{3}of drainage rights sold, and both sides of the transaction obtain certain wealth utility from the transaction and make the social utility at its maximum.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Buyer B_{i} | Seller S_{j} | ||||
---|---|---|---|---|---|

Buyer | Expected Acquisition Price o _{i}(yuan/m ^{3}) | Expected Acquisition Volume x_{i}(m ^{3}) | Seller | Expected Sales Price f _{j}(yuan/m ^{3}) | Expected Sales Volume y_{j}(m ^{3}) |

B_{1} | 8.14 | 100 | S_{4} | 4.38 | 37.5 |

B_{3} | 7.21 | 100 | S_{3} | 6.47 | 50 |

B_{2} | 5.83 | 50 | S_{2} | 6.80 | 75 |

B_{4} | 5.20 | 62.5 | S_{1} | 9.45 | 100 |

Transaction Serial Number | Buyer B _{i} | Seller S _{j} | Expected Trading Volume (m^{3}) | Expected Acquisition Price o _{i}(yuan/m ^{3}) | Expected Sales Price f _{j}(yuan/m ^{3}) |
---|---|---|---|---|---|

I | B_{1} | S_{4} | 37.5 | 8.14 | 4.38 |

II | B_{1} | S_{2} | 62.5 | 8.14 | 6.80 |

III | B_{3} | S_{3} | 50 | 7.21 | 6.47 |

IV | B_{3} | S_{2} | 12.5 | 7.21 | 6.80 |

Indicators (Arithmetic Average of the Previous Three Years) | B_{1} | S_{4} |
---|---|---|

GDP (100 million yuan) | 5911.45 | 2392.48 |

Land area (km^{2}) | 3063 | 3012 |

Population (10,000 people) | 336.2 | 222.01 |

Flooding direct economic loss (100 million yuan) | 1.44 | 0.03 |

Indicators | Transaction I | Transaction II | Transaction III | Transaction IV | ||||
---|---|---|---|---|---|---|---|---|

B_{1} | S_{4} | B_{1} | S_{2} | B_{3} | S_{3} | B_{3} | S_{2} | |

GDP | 0.05 | 0.95 | 0.36 | 0.64 | 0.58 | 0.42 | 0.1 | 0.9 |

Land | 0.02 | 0.98 | 0.37 | 0.63 | 0.80 | 0.20 | 0.73 | 0.27 |

Population | 0.03 | 0.97 | 0.39 | 0.61 | 0.65 | 0.35 | 0.56 | 0.44 |

Average | 0.03 | 0.97 | 0.37 | 0.63 | 0.67 | 0.33 | 0.46 | 0.54 |

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## Share and Cite

**MDPI and ACS Style**

Shen, J.; Zhu, T.; Sun, F.
A Study on the Mechanism and Pricing of Drainage Rights Trading Based on the Bilateral Call Auction Model and Wealth Utility Function. *Water* **2022**, *14*, 2269.
https://doi.org/10.3390/w14142269

**AMA Style**

Shen J, Zhu T, Sun F.
A Study on the Mechanism and Pricing of Drainage Rights Trading Based on the Bilateral Call Auction Model and Wealth Utility Function. *Water*. 2022; 14(14):2269.
https://doi.org/10.3390/w14142269

**Chicago/Turabian Style**

Shen, Juqin, Tingting Zhu, and Fuhua Sun.
2022. "A Study on the Mechanism and Pricing of Drainage Rights Trading Based on the Bilateral Call Auction Model and Wealth Utility Function" *Water* 14, no. 14: 2269.
https://doi.org/10.3390/w14142269