The impact of yield uncertainty on planting and water-saving decisions for high water consumption crops Open Access

China is a great agricultural country, the number of rural population is more than two-thirds of the total population. Agriculture plays an important role in China's economy. Most of the agricultural crops grown in China are high water consumption crops. According to the statistical results from the Ministry of Agriculture and Rural Affairs of China (MARAC 2019), the corn planting area in 2019 is 41.28 million hectares, accounting for 35.6 of the whole grain planting area, while the rice planting area in 2019 is 29.69 million hectares, a 25.6 share. The planting of so many high water consumption crops consumes a lot of water resources.

Insufficient water resources, uneven temporal and spatial distribution and low utilization rate have become the main restrictive factors of grain production in China. Hence, promoting water-saving in agriculture is an important measure to realize cost savings and efficiency increase in agriculture, promote sustainable agricultural development and increase farmers’ income under the increasingly prominent contradiction between supply and demand of water resources (Deng & Guo 2016; Cao et al. 2020; Yang et al. 2022). From the perspective of sustainable development, a growth input to water-saving is necessary for sustainable development of modern agriculture (Cheng et al. 2021). Vigorously promoting and utilizing agricultural water-saving technology can help improve farmers’ income and promote the sustainable development of agriculture (Yao et al. 2021).

Farmers’ revenues from planting crops are not only limited by high water consumption, but also affected by yield uncertainty caused by climate, technology or other factors. Sherrick (2012) states that the variability of farmers’ revenues is mainly attributed to crop yield variablity, as a result of investigating farmers’ revenue variations from 1975 to 2012. Low and uncertain yields have seriously affected farmers’ revenues from planting crops and damaged farmers’ enthusiasm for planting. Moreover, it can also affect the supply of grains in the market, leads to higher grain prices and thus damages the utility of consumers and social welfare.

To help farmers overcome the yield uncertainty and protect farmers’ revenues, the government is usually implementing various agricultural subsidy programs to motivate farmers to plant more acreages of crops and implement water-saving efforts. For example, in recent years the government has offered ‘four agricultural subsidies’ to farmers in China (Zhu & Jiang 2016), the implementation of these subsidy programs has made significant contributions to China's agricultural development. In recent years, government subsidies in agriculture industry have also attracted extensive interest in academia. Alizamir et al. (2019) investigated two subsidy programs offered by the U.S. government. One is the Price Loss Coverage (PLC) program; another is the Agriculture Risk Coverage (ARC) program. The authors provided guidelines for farmers on how to select the subsidy programs, they also studied the impact of the subsidy programs on consumer surplus and social welfare. Chen et al. (2017) investigated the impacts of agricultural subsidy on agricultural pollution, they found that innovation subsidy can help to alleviate agricultural pollution. Zhang et al. (2021) proposed a game theorical model to study the influences of three subsidy programs in an agricultural supply chain. They showed that no subsidy scheme alone can solve the conflict between ecological protection and agricultural development.

Government subsidies for water-saving efforts have also been widely studied in the literature. Chen et al. (2020) studied the subsidy impact on the water-saving supply chain, their results indicate that implementing subsidy programs and introducing water-saving service can increase the profits, consumer surplus and social welfare for high water consumption supply chains. Ma & Li (2022) considered a high water consumption manufacturer that takes efforts to save water by itself or through outsourcing. The government provides subsidies to it according to water-saving efforts or water-saving costs. The authors derived the equilibrium decisions for all the stakeholders and discussed the impact of subsidies on the stakeholder's profits. Chen & Wang (2020) studied how should the government provide subsidies to water-saving service supply chains to maximize social welfare, and compared equilibrium and cooperative models. They derived the equilibrium decisions for all the scenarios and obtained many managerial insights.

In this study, we propose a unified framework to study two practical problems faced by the farmer who plants a high water consumption crop such as rice, corn, among others. The problems that will be discussed are as follows: (1) how much effort should be taken by the farmer to save water in planting crops? (2) what is the optimal planting acreage of the crop under yield uncertainty? To promote farmers to plant more acreages of crops and take water-saving efforts under yield uncertainty, the government should implement some subsidy programs, such as the planting acreage subsidy (PAS) and water-saving efforts subsidy (WES). Hence, the third problem that will be explored is: (3) how does the yield uncertainty affect the implementation of subsidy programs?

We intended to perform this study by an optimization-simulation approach, which first established three optimization models for theoretical research, then based on these theoretical results and the estimated data, several simulation experiments were conducted.

Figure 1 presents the analytical framework, which illustrates the research steps and methods used in this study.

Long-grain rice is one of the typical high water consumption crops, which has a long history of cultivation and consumption. Half of the world's people eat rice, mainly in Asia, southern Europe, tropical America and parts of Africa. The total yield of rice ranks third in the world's crop yield. So we take long-grain rice for an example to perform this study.

The basic data for long-grain rice used in this study are listed in Table 1. All of these data are estimated by analyzing statistic information (NBSC 2021) or quoted from the literature (Alizamir et al. 2019; MARAC 2019).

Suppose that the water price is c and does not vary over all the growing season, and the initial water consumption per acre for planting the crop is ⁠. Let water-saving effort (WSE) taken by the farmer be e. following the literature (Chen et al. 2020; Chen & Wang 2020; Ma & Li 2022), we assume that the cost caused by water-saving effort is ⁠, where is the water-saving cost coefficient. The planting acreage decided by the farmer is denoted by a, and the corresponding planting cost is ⁠, where is the planting cost per acre other than water consumption cost. Quadratic planting cost functions are widely used in agricultural literature (Wickens & Greenfield 1973; Parikh 1979; Agbo et al. 2015; Alizamir et al. 2019).

The integral takes the expectation over all possible realizations of crop yield, and then obtains the expected revenues of the farmer by selling the crop. The terms ⁠, and indicate the planting cost, water consumption cost and water-saving cost, respectively. Especially, the term is the cost reduced by water-saving effort.

The result (3) indicates that the expected consumer surplus is positively correlated to the planting acreage.

In this study, social welfare is defined as the sum of consumer surplus and the farmer's expected profit minus the government expenditure caused by implementing the subsidy programs. The objective of the government is to maximize the social welfare by implementing subsidy policies.

Finally, to ensure our proposed model is feasible and has nontrivial solutions, we impose two assumptions on the settings of the parameters as follows:

• ASSUMPTION 1. The conditions and hold.

• ASSUMPTION 2. The condition holds.

The optimization models are formulated as follows:

And the social welfare equals ⁠, with given planting acreage a and water-saving effort e.

And the social welfare equals ⁠, with given planting acreage a and water-saving effort e.

By solving the optimization models proposed in section 2.4, we have derived the optimal decisions for the farmer and the government under three different scenarios. The theoretical results are summarized in Table 2.

The results in Table 2 show that, both the PAS and WES programs can promote the farmer to plant more acreages of high water consumption crop and take more effort to save waters. It turns out that, the subsidy programs can improve the farmer's expected revenue, as well as the consumer surplus and social welfare. Therefore, the subsidy programs implemented by the government can bring not only economic benefits, but also social welfares.

Table 3 reports the results of numerical analysis of the decision models under three different scenarios. For each scenario, we get the numerical results on planting acreage, water-saving effort, subsidy rate, then derive the resulting revenue of the farmer, the consumer surplus and social welfare.

Comparing the results among different scenarios, we observe that: (1) the farmer has been motivated to plant more acreage of long-grain rice by both the PAS program and the WES program, and the incentive effect of WES program is greater, under which the farmer plants the largest acreage of long-grain rice; (2) a similar phenomenon occurs for the decisions on water-saving efforts: the PAS project not only promotes the farmer to plant the largest area of rice, but also prompts him to take the highest water-saving efforts. Furthermore, we can observe that: (1) the revenue of the farmer is highest under the scenario with PAS, followed by the scenario with WES, and lowest if no subsidy is offered; (2) the comparison results of consumer surplus and social welfare are similar to that for the farmer's revenue; (3) though the improvements of the farmer's revenue, consumer surplus and social welfare brought the PAS are greatly higher than those brought by WES, the government expenditure used to pay for PAS is also greatly higher than that paying for WES.

Figure 2 shows the impact of the yield variability on the subsidy rates and the government expenditures, where varies from 0 to 4, while all other parameters keep unchanged as stated in Table 1. It can be seen from Figure 2(a) that, as the yield variability increases, both and are increasing. If ⁠, the optimal PAS subsidy rate equals ⁠; otherwise, equals ⁠, which is decreasing in ⁠. Similarly, if ⁠, the optimal WES subsidy rate equals ⁠; otherwise, equals ⁠, which is also decreasing in ⁠. On the other hand, the Figure illustrates that ⁠.

In addition, we can see from Figure 2(b) that, the government expenditure paying for PAS is greatly higher than that paying for WES, for any between 0 and 4. This gives an explanation to why the farmer's optimal revenue and the social welfare under the scenario with PAS are more than those under the scenario with WES, as presented in Table 3. Furthermore, the government expenditure paying for PAS is increasing in and decreasing in ⁠. A similar phenomenon occurs for the government expenditure paying for WES.

Figure 3 shows the impact of the yield variability on the farmer's revenues under three different scenarios. No matter what value the yield variability is, the farmer's optimal revenue under the scenario with PAS is higher than that under the scenario with WES, which is higher than that under the scenario without subsidy. This is because the subsidy fee paid by the government under PAS is greatly higher than that under WES, as illustrated in Figure 2. The PAS subsidy program promotes the farmer to plant more acreage of long-grain rice with lower cost, and thus the revenue is improved.

Furthermore, it can be seen from the figure that, the optimal revenue is decreasing in ⁠, while both and are first increasing and then decreasing in ⁠. The phenomenon is not difficult to be explained. As increases, the yield uncertainty increases, and thus the farmer's revenue is reduced if no subsidy is offered. If lies at a small level, the risk brought by the yield uncertainty can be overcomed by the government's subsidy, hence the farmer's revenues can be improved since the government's subsidy rate is increasing in ⁠. However, if the lies in a high level, the risk brought by the yield uncertainty is higher and can not be overcomed by the government's subsidy, the farmer's revenue will be reduced.

Figure 4 illustrates the impact of yield variablity on the consumer surplus and social welfare. One can see from Figure 4(a) that, the consumer surplus under PAS is highest while that the consumer surplus under scenario without subsidy is lowest, for any lies in the range ⁠. This is because the farmer is motivated to plant the highest acreage of long-grain rice if PAS is offered. Furthermore, we can see that ⁠, the consumer surplus under scenario with PAS, is increasing in and decreasing in ⁠. A similar phenomenon appears for the consumer surplus under scenario with WES. Differently, the consumer surplus under scenario without subsidy is always decreasing in ⁠.

Finally, the social welfare is always decreasing in under all the scenarios. The comparison results among three scenarios are the same as those for consumer surplus, that is, the social welfare under scenario with PAS is highest while the social welfare under scenario without subsidy is lowest, no matter what value the yield variablity is.

This study develops optimization models to investigate the impact of yield uncertainty on the famer's decisions on planting crops and water-saving efforts, as well as the impact on the implementation of the government's subsidy programs. Several theoretical results are derived, and based on which we have conducted simulation experiments to explore the answers to the problems faced by farmer and the government.

• (1)

  The optimal planting acreage and water-saving effort for the farmer are greatly affected by the yield uncertainty and the subsidy programs implemented by the government. Compared with the scenario with no subsidy, under both the PAS program and the WES program, the farmer is motivated to plant more acreages of the crop and take more effort in water-saving.

• (2)

  The farmer will plant more acreage of crop and take more effort in water-saving if he is offered the PAS rather than the WES. However, we also find that implementing the PAS program needs more government expenditure than implementing the WES program.

• (3)

  The uncertainty of the crop yield has a great influence on decision-makings of the farmer. The greater the yield variability, the less acreage of the crop the farmer will plant, and the lower effort to be taken in water-saving. This indicates that as the crop yield variability increases, the farmer's expected revenue is decreasing, as well as the consumer surplus and social welfare. Therefore, to help the farmer address the negative effect brought by the yield uncertainty, the government should provide subsidies to the farmer, according to acreages planted or water-saving efforts.

This work was funded by the National Natural Science Foundation of China (Grant No. 42001250), and the Fundamental Research Funds for the Central Universities (Grant No. B200201032), Changzhou Introduction and Cultivation of Leading Innovative Talents Program (Grant No. CQ20210095). We are grateful to the reviewers, editors for their insightful comments and suggestions for the improvement and expansion of the work.

All relevant data are included in the paper or its Supplementary Information.