**4. Methods**

In this section, we adopt a model of profit maximization [54] and then turn to the maximization of economic irrigation water use efficiency to deal with market failure in water management. In multi-crop irrigated agriculture, producers make decisions on land allocation to each crop, and the amount of water for irrigation [55,56]. Choosing from common crops, a typical producer may plant two or more crops on a farm. Then decisions on land allocation and water supply can be made to maximize the expected total profit [57].

Following a multi-crop production model by Moore et al. [54], the expected profit functions of the multi-crop system and specific crop *<sup>i</sup>* can be represented by <sup>Π</sup>(*p*, *<sup>r</sup>*, *<sup>b</sup>*, *<sup>N</sup>*; *<sup>x</sup>*) and *<sup>π</sup>i*(*pi*, *<sup>r</sup>*, *<sup>b</sup>*, *ni*; *<sup>x</sup>*), respectively. *<sup>p</sup>* is a vector of crop prices; *pi* is the price of crop *<sup>i</sup>*, *<sup>i</sup>* <sup>=</sup> 1, ... , *<sup>m</sup>*; *<sup>r</sup>* is a vector of variable input prices excluding water prices; *b* is the water prices; *N* is the total farming area as a constraint; *ni* is the land allocation for crop *<sup>i</sup>*; *x* represents other exogenous variables including land characteristics, water sources, the adoption of various irrigation systems, and climate perceptions. Each crop-specific profit function *π<sup>i</sup>* is assumed to be convex and homogeneous of degree one in output prices, water price, and other prices of variable inputs, nondecreasing in output price and land allocation, and non-increasing in water prices and other variable input prices.

We extend the model of Moore et al. [54] by adding crop irrigation water use efficiency. A single producer makes production and irrigation decisions to maximize profits. While to achieve sustainability of the water resource, the total profit function of the whole society needs to consider the marginal user cost and higher pumping cost externality of extracting water by every farmer. Thus, in addition to the decision-making on conserving water use and increasing crop yield, the way to achieve higher crop irrigation water use efficiency should be explored. Following the discussion on indicators of water use performance and productivity by Pereira et al. [58], the following definition can be used to calculate the farm-level crop-specific economics irrigation water use efficiency.

$$EINIL = \frac{\text{Crop yield} \times P}{\text{Total amount of rejection water applied}} \tag{1}$$

where *EIWUE* is the economic irrigation water use efficiency, crop yield is the marketable grain yield, *P* is the crop price, and irrigation water application is measured based on all irrigation water sources, including well, on- and off-farm surface water. The greater the *EIWUE* value [59], the higher the efficiency due to irrigation water application.

To analyze the effects, *EIWUE* can be a function of the exogenous variables affecting both yield and water application. *EIWUEi* <sup>=</sup> *hi*(*p*, *<sup>r</sup>*, *<sup>b</sup>*, *<sup>N</sup>*; *<sup>x</sup>*) *<sup>i</sup>* <sup>=</sup> 1, . . . , *<sup>m</sup>* (2)

$$EINULE\_i = h\_i(\mathbf{p}, \mathbf{r}, \ b, \ N; \mathbf{x}) \; i = 1, \ldots, m \tag{2}$$

In addition, the farm-level water application can be decomposed to analyze the role of water price on production decisions regarding each crop [54]. The crop-specific water application can be decomposed into an extensive margin of water use (an indirect effect on water use due to land allocation change) and an intensive margin of water use (a direct effect on water use due to water application).

The farm-level total water application (*W*) equals the sum of water application for each crop grown on the farm with the optimal land allocation [54,60]:

$$\mathcal{W} = \sum\_{i=1}^{m} w\_i(p\_{i\prime}, r, b, \; n\_i^\*(p, r, b, \; \mathcal{N}; \mathbf{x}); \mathbf{x}) \; i = 1, \ldots, m \tag{3}$$

Taking the derivative of the equation with respect to water price gives

$$\frac{\partial \mathcal{W}}{\partial b} = \sum\_{i=1}^{m} \left( \frac{\partial w\_i}{\partial b} + \frac{\partial w\_i}{\partial n\_i^\*} \times \frac{\partial n\_i^\*}{\partial b} \right) \tag{4}$$

where *<sup>∂</sup>wi <sup>∂</sup><sup>b</sup>* is the intensive margin, and *<sup>∂</sup>wi ∂n*∗ *i ∂n*∗ *i <sup>∂</sup><sup>b</sup>* is the extensive margin. The total effect can be obtained by summing the effects on all the crops. The intensive margin will decrease in price and *<sup>∂</sup>wi <sup>∂</sup><sup>b</sup>* should have a negative sign for each crop. The sign of the extensive margin depends on *<sup>∂</sup>n*<sup>∗</sup> *i <sup>∂</sup><sup>b</sup>* . The total farm-level effect on water use should be negative, which indicates a decreasing water application as water price increases. This decomposition of the total marginal effect has been lately employed by Hendricks and Peterson [61], and Pfeiffer and Lin [62].
