*6.2. Decomposition of Farm-Level Water Application*

To decompose the effect of water cost on farm-level water application, extensive and intensive margins are provided in Table 3. This paper takes corn and soybeans as examples [74]. The estimated coefficients on crop acreage and water costs in the water application equation suggest a change in water use given a change in land use ( *<sup>∂</sup>wi ∂ni* ), and a marginal change in water use given a change in water cost ( *<sup>∂</sup>wi <sup>∂</sup><sup>b</sup>* ). The estimated coefficients on water cost in the land allocation equation represent a change in land use given a change in water cost ( *<sup>∂</sup>ni <sup>∂</sup><sup>b</sup>* ). The intensive margin can be obtained with *<sup>∂</sup>wi <sup>∂</sup><sup>b</sup>* while adjusting for the estimated probability that the crop is grown. The extensive margin can be calculated using *<sup>∂</sup>wi ∂ni ∂ni <sup>∂</sup><sup>b</sup>* . Summing the intensive and extensive margins for each crop gives the total effect of a change in water cost. Further summing the effects on all crops gives the total effect on a typical farm growing both crops.

Margins on both on-surface water costs and energy costs are calculated. Only water from off-farm surface sources is priced and investigated in the survey. Energy expenses on groundwater pumping are considered as the proxy of water price for groundwater. The results show that only *<sup>∂</sup>ni <sup>∂</sup><sup>b</sup>* decreases in energy expenses for soybeans, and other values of *<sup>∂</sup>ni <sup>∂</sup><sup>b</sup>* and *<sup>∂</sup>wi <sup>∂</sup><sup>b</sup>* are positive, which is contradictory to expectations. This indicates more water is used as water prices increases. This is probably true in practice when the adoption of enhanced irrigation systems increase acreage under irrigation and thus increase the amount of irrigation water, as reported in Kansas [75]. There are many debates regarding the empirical changes in water use as a result of changing prices and increasing the adoption of agricultural irrigation technologies [53,61,75]. A numerical illustration can help understand the effects of water prices. A 1 USD increase in groundwater costs (energy expenses) (Δ*b* = \$1) would lead to a decrease of 0.109 acre-feet of water application per acre of soybeans, and an increase of 0.0737 acre-feet of water per acre of corn. In a multi-crop system, a typical farm growing both corn and soybeans would decrease water application by −10.87 acre-feet as a result of a \$1 increase in energy expenses. These results show water use is highly inelastic in water cost [54]. While this may be different for regions/states with varying availability of water resources, an in-depth analysis of regional or state effect of water costs on water use can be helpful.



Following the definitions by Moore et al. [54], *<sup>∂</sup>wi <sup>∂</sup>ni* is the estimated coefficient of crop acreage in the water application equations, where *wi* is the acre-feet of irrigation water on crop *<sup>i</sup>* and *ni* is the acres of growing crop *<sup>i</sup>*. *<sup>∂</sup>ni <sup>∂</sup><sup>b</sup>* is the estimated coefficient of the water price in the land allocation equations, with b being the water price. *<sup>∂</sup>wi <sup>∂</sup><sup>b</sup>* is the estimated coefficients of the water price in the water application equation. The calculation of both intensive and extension margin should be adjusted by the share of the crop planted.
