Two-part pricing contract and competition between two water supply chains: a theoretical and empirical analysis of the South-to-North Water Transfer Project in China

Water scarcity is becoming acute along with economic development, population growth, and climate change (Kanakoudis et al. 2016, 2017; Wilson et al. 2017). In China, the average water availability is 2,259 m³ per capita, much less than the world average 8,036 m³ per capita (Berrittella et al. 2006). With the newly minted ‘two-child policy’, China's population will likely increase at an annual rate of 10–20 million in future years, which will decrease the average water availability. In addition, uneven distribution of available water resources further intensifies water scarcity. In Northern China, less than 10% of the national water resources irrigates more than 30% of the irrigated croplands (Fan et al. 2014). In particular, in some populated areas, for instance, Beijing and Tianjin metropolitan areas, the average water availability is less than 400 m³ per capita (Berrittella et al. 2006; Fan et al. 2018). Therefore, to solve water crisis in North China Plain, the Chinese Central Government launched the national South-to-North Water Transfer Project (SNWTP) to increase water supply in Northern China (Tian et al. 1989).

On October 30, 1952, the former China president Mao Zedong visited the South and stated, ‘There is more water in the South, while less in the North. If there is any means, we can transfer water from South to North’ (East Lake Hotel 2013). Later, under the leadership and supervision of the Central Government and the State Council, a scientific investigation team consisting of experts and researchers conducted many field investigations and on-site surveys. They initially proposed more than 50 planning options, and these were further compared and evaluated through feasibility analyses, cost–benefit analyses, environmental assessments, etc. Finally, the current SNWTP with Eastern, Middle, and Western Routes took its form (Liu et al. 1984). The Eastern Route withdraws water from Jiangdu Hydro-junction located in Sanjiangkou, Jiangsu province. Using existing waterways in Jiangsu, Shandong, and Hebei provinces, it lifts and pumps water all the way to Northern China. The Middle Route takes water from Danjiangkou Reservoir in Hubei province, goes north passing southwestern regions in Henan province, and finally reaches Beijing and Tianjin. The Eastern and Middle Routes are shown in Figure 1. With this national strategic project, emphasis is given to changing the situation ‘flood in the South and drought in the North’ and solve water scarcity in Northern China, thus achieving a balanced and sustainable development in both the south and north of China (Wilson et al. 2017).

Some research has been documented regarding the SNWTP. Theoretical support includes cooperation of water users and their welfare changes based on game theory (Wei et al. 2010; Du et al. 2016), impacts of SNWTP on China's economy based on decision support systems (Feng et al. 2007), implementation of water demand management projects (He et al. 2010), risk assessment (Gu et al. 2012), environmental monitoring and pollution control (Wang et al. 2006; Zhang 2009). Using game theory, Wei et al. (2010) analyzed the water resources conflicts of SNWTP. However, engineering optimization (Chen & Wang 2012; Chen et al. 2013) and operation management (Liu & Zheng 2002; Ha et al. 2011) are still under-studied. To the best of the authors' knowledge, research has not reported the influence of rainfall on the performance of a water supply chain from the perspective of quantity competition, nor applied to the SNWTP (Lou 2014; Du et al. 2018, 2019). Therefore, this study tries to answer the following three fundamental questions relating to water supply chain management.

1. How are the profits of water distributors and suppliers affected by the competition intensity in different contracts?

2. How do the water distributors choose different contracts under different competition intensities and utilization rates of the regional rainfall?

3. What are the effective management strategies implemented by government to deal with the fixed costs caused by different suppliers?

Figure 2 presents research design, methods, and analysis in this study. Specifically, two water supply chains including two water suppliers and two distributors constitute the whole water supply chain system. The two distributors have deterministic demands in the water supply chain system, whereas the two water suppliers can change their supply using various contracts (Wang & Hu 2005; Du et al. 2019). Based on inter-chain competition and wholesale price contract, a Stackelberg game is built, and specific expressions of performance under varying contracts are derived. Sensitivity analysis is conducted to explore the performance of the water supply chain system and its components under two-part pricing contracts and with varying intensities of quantity competition. Fixed fees are determined under alternating contract options. Finally, the Eastern and Middle Routes of the SNWTP are considered as two water suppliers who provide water to two competing water distributors in Tianjin City. Thus, a numerical analysis is conducted with real values of water prices, water use, rainfall, etc.

A wholesale price contract sets a fixed unit price for any amount of water purchased by a water distributor from a supplier. While a two-part pricing contract consists of a lump-sum fee and a per-unit price (Chen et al. 2013). The lump-sum fee is used to secure the transaction and stabilize the purchase price, and the per-unit charge reflects the value of water (Du et al. 2018). Taking the wholesale price contract as a benchmark, the study looks at the performance of the two-part pricing contracts between two suppliers (M1 and M2) and two distributors (S1 and S2) with water quantity competition between the water supply chains of the Eastern and Middle Routes. The two-part pricing contract takes the form of (⁠⁠, ⁠) (i = 1, 2), where is a fixed cost (lump-sum fee), and is a wholesale price for the supplier (i.e., a per-unit charge). The fixed cost is assumed to be exogenous to the supply chain, and it can be regulated and controlled by government at any time (Du et al. 2018). Assume that the suppliers can offer different wholesale prices to the two distributors regardless of the contract type. A Stackelberg game is played between the suppliers and distributors, with the suppliers as the leaders and distributors as the followers. The model is built as follows.

• First, the two suppliers offer each distributor a wholesale price contract or a two-part pricing contract.

• Second, the distributors independently choose the water quantity they want to order according to the contracts they accept.

• Third, the suppliers meet the demand of water orders made by the distributors and the distributors meet the water demand of end-user market.

To ease the analysis and better evaluate the performance of water supply chain system under competition intensity and local rainfall, other costs of the suppliers and distributors are assumed to be zero.

Proposition 1. In the case of ⁠, when and ⁠, then ⁠, ⁠, ⁠, ⁠.

The proofs of Proposition 1 and following propositions are provided in the Appendix, available with the online version of this paper. This proposition means that in the case of ⁠, both the wholesale prices and order quantities are monotonically decreasing functions of competition intensity and regional rainfall. The purchase decisions of water distributors are affected by competition intensity and rainfall. In regions with varying amounts of rainfall, a certain level of competition intensity can help control the water quantities ordered by the distributors and change the wholesale prices offered by the suppliers.

Proposition 2. In the case of ⁠, when and ⁠, then ⁠, ⁠, ⁠, ⁠, ⁠.

This proposition suggests that in the case of ⁠, the expected profits of the water suppliers, distributors and the whole supply chain system are monotonically decreasing functions of competition intensity and regional rainfall. Affected by the competition intensity and rainfall, the expected profits of all members and the whole system can be controlled at a certain level while given the competition intensity.

Proposition 3. In the case of ⁠, when and ⁠, then ⁠, ⁠, ⁠, ⁠.

This proposition shows that in the case of ⁠, the wholesale prices are monotonically increasing functions of competition intensity and rainfall, while the water quantities are monotonically decreasing functions of competition intensity and rainfall. The stronger the competition between water distributors, the higher prices set by the upstream water suppliers, and the less water ordered by the distributors (Ha & Tong 2008). In this situation, more rainfall is favorable to the distributors, so that they can satisfy the increasing water demand by terminal users. Similarly, if there is more rain in the location, the upstream water supplier can set a higher price. With more rain, the local water distributors can order less water from the suppliers, which indicates more rainfall benefits the distributors' decision-making.

Proposition 4. In the case of ⁠, when and ⁠, then ⁠, ⁠, ⁠, ⁠, ⁠, ⁠.

This proposition shows that in the case of ⁠, the expected profits of the water suppliers, distributors and the whole supply chain system are monotonically decreasing functions of competition intensity and regional rainfall. This indicates the expected profits of the supply chain decrease as the competition becomes stronger (Ha & Tong 2008). Similarly, more regional rainfall decreases the expected profits of water suppliers and distributors, and the expected profit of the supply chain system also decreases. In the situation of SNWTP, with more rainfall in regions the routes go through, there is more water in the distributaries and canals belonging to the two routes, thus the costs of transferring water can be reduced (Chen & Wang 2012).

Proposition 5. In the case of ⁠, when and ⁠, then ⁠, ⁠, ⁠, ⁠.

This proposition means that in the case of with different contracts, the price of the Eastern Route supply chain system under a two-part pricing contract is an increasing function of competition intensity and regional rainfall, while the price of the Middle Route supply chain system under a wholesale price contract is a decreasing function of competition intensity and rainfall. In this situation, for a given water quantity ordered by a distributor, a higher price set by the supplier will increase their expected profits.

Proposition 6. In the case of ⁠, (1) when and ⁠, then ⁠, ⁠, ⁠; (2) when and ⁠, then ⁠, ⁠, ⁠, ⁠.

This proposition states two facts given two sets of competition intensity values in the case of ⁠. First, when the competition intensity is greater than 0 and equal to or smaller than 0.3626, the water quantity ordered by the downstream distributor under a two-part pricing contract decreases as the competition intensity increases. Second, when the competition intensity is between 0.3626 and 1, the water quantity under a two-part pricing contract increases as the competition intensity increases. For both contracts, the water quantities in the whole supply chain are negatively correlated with the average rainfall, and thus the water price increases as more rainfall is received.

Proposition 7. In the case of ⁠, (1) when and ⁠, then ⁠; (2) when and ⁠, then ⁠; (3) when and ⁠, then ⁠, ⁠, ⁠, ⁠, ⁠; (4) when and ⁠, then ⁠, ⁠, ⁠, ⁠, ⁠.

This proposition suggests different performances of water distributors in the supply chain in the case of ⁠. First, when the competition intensity is between 0 and 0.3626, the expected profit of the water distributor under a two-part pricing contract decreases as the competition intensity increases. Second, with strong competition, indicated by any value between 0.3626 and 1, the expected profit of the water distributor under a two-part pricing contract increases as the competition intensity increases. Third, in the case of ⁠, the expected profits of the distributors are negatively correlated with rainfall. In addition, in the case of ww (Proposition 2), the expected profits of the distributors are negatively correlated with rainfall; in the case of tt (Proposition 4), the expected profits of the distributors are negatively correlated with rainfall as well. Furthermore, the expected profits of the distributors under two-part pricing contracts are negatively correlated with rainfall, while the expected profits of the distributors under wholesale price contracts are also negatively correlated with rainfall (Du et al. 2018).

Lemma 1. (1) ⁠; (2) when and ⁠, then ⁠, ⁠, and ⁠; (3) when and ⁠, then ⁠, ⁠, and ⁠.

Proposition 8. (1) when and ⁠, ⁠, then ⁠, ⁠; (2) when and ⁠, ⁠, then ⁠, ⁠.

This proposition suggests the fixed fee of supply chain system, ⁠, can be adjusted within a certain range. When the competition intensity is relatively weak, that is, ⁠, the fixed fee can have a value in the range with ⁠, and ⁠. The supply chain system offers two-part pricing contracts to realize Pareto improvement for the supplier and distributors at the same time. On the contrary, when competition intensity is relatively strong, i.e., ⁠, the fixed fee can be adjusted in the range ⁠, ⁠, and ⁠. The supply chain system offers two-part pricing contracts to realize Pareto improvement for the suppliers and distributors at the same time.

Proposition 9. When ⁠, then EM_1tw = ⁠, and ⁠.

This proposition suggests that if the competing supply chain offers a wholesale price contract, the other supply chain can adjust the fixed fee within the range ⁠. As the range highly depends on competition intensity and regional rainfall use efficiency, Pareto improvement can be achieved in terms of the performance of both the suppliers and distributors.

Lemma 3.⁠.

Proposition 10. When ⁠, then EM_2tt > ⁠, and ⁠.

This proposition indicates that if the competing supply chain offers a two-part pricing contract, the other supply chain can adjust the fixed fee within the range ⁠, which heavily depends on competition intensity and regional rainfall use efficiency. Using the two-part pricing contracts can realize Pareto improvement for the suppliers and distributor at the same time (Corbett & Karmarkar 2001; Du et al. 2016).

To better observe the effects of parameter changes on the supply chain system and the performance of individual members, this section presents a numerical analysis. Both the Eastern and Middle Routes transfer water to Northern China, in particular, both routes supply water for Tianjin City. The rainfall use efficiency is reported less than 5% in Tianjin (Liu et al. 2015). There are three water types according to which sectors the water is used, including (1) domestic, (2) industrial, municipal, and water for business purposes, and (3) water for special purposes. The domestic water use is charged based on a ladder-type water fee at 4.9/6.2/8.0 Yuan (1 USD = 6.80 Yuan)/m³; the industrial water price is 7.9 Yuan/m³; the water price for special purposes is 22.3 Yuan/m³ (Tianjin 2015, 2017). Thus, the average water price is 12 Yuan/m³. For simplicity, assume the local rainfall use efficiency is 0.03, and the water price a is 10 Yuan/m³. As the competition intensity shows differing effects when it is lower than 0.5723 and when higher, this section considers the performance of the supply chains separately. The study first presents results of the numerical analysis with varying competition intensity levels and constant regional rainfall and then presents the results of rainfall changes and constant competition intensity.

Results in Table 1 show under the three contract combinations, the parameters of water supply chain system change as the competition intensity changes, and some changes of the parameters are drastic. When both suppliers in the two competing water supply chains choose wholesale price contracts, there is no fixed fee in the supply chain system, thus the performance of the supply chain members and the entire system cannot be adjusted through changing the fixed fee. However, the parameters of all members and the system decrease as the competition intensity increases.

When mixed contracts are chosen (a wholesale price contract for one chain and a two-part pricing contract for the other chain), the supplier with the two-part pricing contract can change the expected profits through adjusting the fixed fee in a certain range given by government. In this case, the water supplier can maximize his profits. When the competition intensity equals 0.5723, the fixed fee can be determined by equaling the lower and upper bounds, i.e., ⁠. This indicates when the competition is weak, the fixed fee is in a range, and this range gets smaller as the competition intensity increases, and when the competition is strong, the range for the fixed fee gets larger as the competition intensity increases. These changes further increase or decrease the expected profits of the suppliers and the entire supply chain system. When the suppliers choose two-part pricing contracts at the same time, both can make a difference in the expected profits of all members and the systems through changing their fixed fees.

Interestingly, the results under the three contract combinations show the fixed costs in both Middle and Eastern Routes are correlated with the expected profits of both suppliers and distributors, rather than the performance of the entire supply chain. With the wholesale price contracts (ww), the performance of both Middle and Eastern Routes decreases as the competition intensity increases, but their expected profits are always equal. With one wholesale price contract and one two-part pricing contract (tw), the expected profit of the supply chain using the two-part pricing contract increases faster than that using the wholesale price contract. Thus, the total system profits are larger than that under the ww contracts. With the two-part pricing contracts (tt), the expected profit of each supply chain is greater than that under the ww contracts, and the total system profit is even greater than that under the tw contracts. In this case, both individual supply chains and the entire system achieve optimal performance, and the total system profit is greatest among the three contract combinations.

Table 2 shows the results of weak competition between water distributors under three different contract combinations. All parameters of the members and the entire system decrease as the average rainfall increases. In a similar vein, when the competition intensity is strong, as shown in Table 3, all parameters also decrease as the average rainfall increases. Table 3 shows with a strong competition intensity the range of decrease is much larger than that under a weak competition in Table 2. The range of fixed fee under a strong competition (Table 3) is much larger than that under a weak competition in Table 2. This suggests a stronger competition makes water suppliers adjust the fixed fee easier (Arshinder et al. 2008; Du et al. 2018), and the suppliers can have more flexibility as the regional rainfall increases (Chen et al. 2013; Du et al. 2016).

A comparison between Tables 2 and 3 shows the competition intensity presents varying effects on the supply chain members and the system under the three contract combinations. With wholesale price contracts (ww), the expected profits of the suppliers, distributors and each supply chain decrease as the competition intensity increases. A stronger competition intensity is not favorable for the performance of supply chains under the ww contracts, and it is a barrier for profit increase (Srdjevic et al. 2004). With a wholesale price contract and a two-part pricing contract (tw), a stronger competition intensity is favorable for the chain using the two-part pricing contract, while unfavorable for the other chain using the wholesale price contract (Corbett & Karmarkar 2001; Du et al. 2016). The profit of the total system, however, is still greater than that under the ww contracts. In this case, the competition intensity helps redistribute profit among supply chain members and increase the total system profit (Srdjevic et al. 2004; Wu 2013). With the two-part pricing contracts (tt), the expected profits of supply chain members are related to the fixed costs of each corresponding supply chain, the total system profit is independent of the fixed costs (Cachon 2003; Chen & Wang 2012). Therefore, the competition intensity is unfavorable for the two supply chains and the entire system (Du et al. 2016).

In addition to the effect of competition intensity on the entire supply chain system mentioned above, comparisons between tw and tt in Tables 1–3 indicate government can adjust the fixed costs to balance the profits of the supply chains (Du et al. 2018). The investments are more controllable by the government, and more effective as well to change the performance of local suppliers and distributors (Ai et al. 2012; Chen & Wang 2012; Wu 2013).

In this research, the Eastern and Middle Routes of SNWTP are considered competing water supply chains. Each supply chain consists of a risk-neutral supplier and a risk-neutral distributor. Taking regional rainfall into consideration, this study investigates the performance improvement under two-part pricing contracts compared to wholesale price contracts. Furthermore, this study analyzes the effects of competition intensity and rainfall use efficiency on contract choosing behaviors. The theoretical and empirical analyses show some main findings.

1. Under different contracts, the parameters of the supply chain members and the entire system decrease as the competition intensity increases. However, the suppliers can affect the expected profits of all members and the system through adjusting the range of the fixed fees.

2. Under different contracts, when the competition is weak, all parameters of the supply chains decrease as the average rainfall increases. While when the competition is strong, all parameters also show a decrease, but the decreasing trend is much larger than the range under a weak competition. Similarly, the adjusting range for the fixed fees under a strong competition is larger than that under a weak competition.

The limitation of our research is from the assumption of risk-neutral supply chain members and each supply chain consisting of one supplier and one distributor. One important extension of this work in the near future is to include stochastic demand, supply costs, and environmental costs (Wu 2013; Kanakoudis & Papadopoulou 2014; Kanakoudis 2015; Li 2017). Another future research can examine the structural models with multiple supply chains incorporating risk-averse water suppliers and distributors (Chen et al. 2013; Du et al. 2018).

This research is supported by the Youth Project of Natural Science Foundation of China (Grant number 71702067), the Key Project of Natural Science Foundation of China (Grant number 71531003), and the USDA Agricultural Research Service Initiative-Ogallala Aquifer Program (FY2017-2018).