Theoretical Model

This analysis focuses on firm irrigation management decisions across a decision framework that includes allocation of irrigated land among crop choices and water application decisions. The firm's maximization problem is rooted in a simple total profit function for a multi-output irrigating firm (equation 1).

(1)$$\Pi \lpar {{\bi p}\comma \; \; b\comma \; \; N\comma \; {\bi x}} \rpar $$

Where p is a vector of crop prices, b is the cost of irrigation water, N is the land constraint, and x is a vector of other exogenous environmental variables (climate, weather, soil quality).

To develop a theoretical framework for the crop allocation decision, the total profit function is decomposed into a set of individual irrigated crop profit functions, where i indicates a particular crop:

(2)$$\pi _i\lpar {\,p_{\bi i}\comma \; \; b\comma \; \; n_i^{^\ast}\comma \; {\bi x}} \rpar $$

The optimization can be restated as a choice of irrigated acreage allocation for the individual crops, constrained by the total acreage under irrigation $N_{irr}^*$.

(3)$$\Pi \lpar {{\bi p}\comma \; \; b\comma \; \; N\comma \; {\bi x}} \rpar = \mathop {\max} \limits_{n_1 \ldots n_m} \left\{{\mathop \sum \limits_{i = 1}^m \pi_i\lpar {\,p_{\bi i}\comma \; \; b\comma \; \; n_i^{^\ast}\comma \; {\bi x}} \rpar {\bi \; } \colon {\bi \; } \mathop \sum \limits_{i = 1}^m n_i = N_{irr}^{^\ast} {\bi \; }} \right\}$$

The estimable forms for the crop allocation and water application decisions are derived from the crop level model of a multi-output irrigating firm. At the intensive margin, the specific management behavior of interest is the volume of water applied to a particular crop—corn, soybeans, or potatoes—given that a firm is growing the crop on a field with irrigation infrastructure in place.

Empirical Model

Assuming a normalized quadratic profit function, the estimable empirical functions are linear in the exogenous variables (Lau Reference Lau, Fuss and McFadden1978; Moore and Negri Reference Moore and Negri1992; Moore, Gollehon, and Carey Reference Moore, Gollehon and Carey1994). The equation for $n_i^*\comma$ acreage allocation for crop i, is presented as a function of crop prices, water cost, total cropland, and environmental conditions with effects varying by crop.

(4)$$n_i^{^\ast} = a^i + \mathop \sum \limits_{\,j = 1}^m \beta _j^i {\bi \; } p_j + \delta ^ib + \tau ^iN + \mathop \sum \limits_{s = 1}^t \eta _s^i x_s\; \; \; \; \; \; i = 1\comma \; \; \ldots\comma \; m$$

This function is intended to capture the indirect water use response observed as the change in the allocation of irrigated land among the m crops, each of which has unique water requirements and favors certain environmental conditions. In the crop acreage models, the environmental and price variables, x and p, include weather and price conditions lagged one year with additional controls for long-run climate conditions. The variables were chosen to reflect the information available to the firm in the winter of the survey year when planting decisions are made.

Application of Hotelling's lemma to the individual crop profit function produces the estimable intensive margin water demand function.

(5)$$-\displaystyle{{\delta \pi _i\lpar {\,p_{\bi i}\comma \; \; b\comma \; \; n_i^{^\ast}\comma \; {\bi x}} \rpar } \over {\delta b}} = {\bi \; } w_i\lpar {\,p_{\bi i}\comma \; \; b\comma \; \; n_i^{^\ast}\comma \; {\bi x}} \rpar \; \; \; \; \; \; \; \; \; \; \; \; \; i = 1\comma \; \; \ldots\comma \; m$$

(6)$$w_i = \alpha ^i + \beta ^ip_i + \delta ^ib + \tau ^iN + \mathop \sum \limits_{s = 1}^t \eta _s^i {\bi \; } x_s\; \; \; \; \; \; \; i = 1\comma \; \; \ldots\comma \; m$$

The general forms for the two estimations are similar, although cross prices do not appear in the empirical function for w. The price and environmental variables that appear in the water application models are selected to reflect the relevant information and conditions available to the firm during the irrigation season. The exclusion of cross prices from the water application model reflects an assumption that water application decisions are made independently for each crop.

FRIS

FRIS is a supplement to the Census of Agriculture (COA), a general farm management survey conducted on five-year cycles. The FRIS is collected in the years following the COA from a sample frame of firms that reported having participated in irrigation in the latest COA. The sample used in this article includes major irrigating states in the Great Lakes Region – Illinois, Indiana, Michigan, Minnesota, and Wisconsin, and it includes three survey years – 2003, 2008, and 2013.

In 2013, the national FRIS sample targeted 35,000 farms and obtained responses from 34,966. The targeted farms were selected via a stratification strategy. The major irrigators in each state were assigned to a certainty stratum (i.e., probability = 1). The remaining noncertainty strata (probability < 1) were sampled systematically by acreage. The boundaries of each strata were uniquely defined by state to reflect the distribution of farm size in each state measured as total acres irrigated. Of the survey responses, 2,095 responding farms were from the certainty stratum and the remaining 32,871 farms were from the various noncertainty strata (USDA 2013, Appendix A-1). This sampling strategy was also used for the 2003 and 2008 FRIS (USDA 2003, 2008, 2013, Farm and Ranch Irrigation Survey). The individual response data includes weights that are used to correct for non-randomness in the sample selection strategy.

The selected sample includes corn, soybean, and potato irrigators. These crops compose the majority of the irrigated acreage in the five states. Agricultural irrigation occurred on over 2.5 million acres across the five-state region in 2012. Figure 2 displays these acres by the share in each crop. The relative shares of irrigated acreage for each crop are similar across the states in the region with the exception of Wisconsin, where vegetables contribute a larger share. This study did not consider vegetable irrigation because the data does not allow for distinguishing among vegetable types. Additionally, individual vegetable types are grown by relatively few farms, and management practices are likely to vary by type. Potato irrigation occurs on a larger share of acreage in the northern part of the region, and potatoes are an important crop to evaluate because they generally require greater irrigation volume than corn or soybeans.

Figure 1. Irrigated Acres of Major Irrigated Crops, by State and Year

Source: FRIS summary reports 2003, 2008, 2013

Figure 2. Irrigated acres by county in western Great Lakes states, 2012

Note: NA indicates counties where data was suppressed in published USDA COA summary tables to protect survey respondent confidentiality.

Source: Census of Agriculture, 2012

Figure 3 displays the spatial distribution of irrigated acres as reported in the COA 2012. The figure highlights the presence of several key irrigation areas within the sample region. Most notably, the largest concentrations of irrigating farms are in Southwest Lower Michigan/Northern Indiana, Central Wisconsin, and Central Minnesota. Table 1 contains the number of farms by year, state, and crop as they appear in the final study sample. The 4,737 farms are relatively evenly distributed over the three sample years and five sample states. Summing the number of firms over the three crops in a given state and year does not sum to the reported total number of firms because many firms irrigate more than one of the studied crops.

Figure 3. Seasonal Variation in Water Withdrawals in Wisconsin

Source: Wisconsin Water Use Report 2013

http://dnr.wi.gov/topic/WaterUse/documents/WithdrawalReportDetail2013.pdf

Table 1. Number of Farms in Study Sample, by Year, State, and Crop

*S indicates values that are suppressed according to USDA NASS confidentiality requirements.

Some firms appear in multiple survey years. Approximately 10 percent of the total number of surveyed firms appear in all three survey years, comprising 20 percent of the observations in the sample. Approximately 18 percent of unique firms appear in two years of the survey, comprising 25 percent of the observations.

Water Use and Acres Irrigated

The FRIS questionnaire asks firms to report water applications to each irrigated crop as an annual per-acre value. These reported values were used directly as the dependent variable in the water application estimation. Potatoes are the most water intensive of the three crops, receiving an average of 9.6 inches per acre. The difference in water intensity provides the basis for the hypothesized effects of water cost in the crop allocation model. In response to higher water prices, firms are expected to substitute away from potatoes and toward corn and soybeans.

The dependent variables in the crop allocation models are the FRIS reported values for irrigated acreage of the specific crop. The mean irrigated potato acreage is significantly larger than the respective means for corn or soybean, indicating a greater degree of firm concentration in potato production. The vast majority of firms irrigated corn or both corn and soybeans in the observed years. This distribution is consistent with typical crop rotations where firms alternate between corn and soybeans on two- or three-year rotations. Similarly, a majority of the potato irrigators in the sample are also irrigating other crops. This is expected, as potatoes are also typically grown on a two- or three-year rotation. Considering the nature of typical crop rotations, it is likely that some, if not all, firms in the sample regularly participate in irrigation of at least two of the studied crops. Thus, substitution effects in the crop allocation parameters are expected to appear primarily as a decision to participate or not participate in growing irrigated potatoes. In the short run, potato production decisions are likely partially constrained by production contracts, but the FRIS lacks a useable identifier for firms operating with production contracts.

Measuring Water Cost

An explanatory variable of primary interest is the cost of irrigation water. An increase in the cost of water is hypothesized to cause a substitution away from water intensive crops and cause a reduction in water application rates. There are three general approaches to measuring irrigation cost in the irrigation demand literature when water itself is unpriced.

First, the energy price approach relies on variation in local energy prices applied as a proxy for the marginal water cost. Mieno and Brozović (Reference Mieno and Brozović2016) showed that “energy price elasticity is identical to the irrigation cost elasticity of groundwater use when groundwater itself is not priced” (423). The energy price approach is simple in construction, but it does not account for a number of firm technology characteristics that affect the cost of water (e.g., groundwater depth, pumping pressure, total dynamic head). Additionally, this method is only suitable if price varies sufficiently across the sample. In a variation of this approach, direct energy charges may be used rather than prices. When the data are available, this approach is ideal because it accounts for variations in irrigation technology directly. In the context of this study, the available measures of energy price do not provide sufficient variability to use the energy price approach and direct energy charges are not available.

Two general alternatives to the energy price approach, the engineering and average cost approaches, leverage the additional variation between firms with unique water delivery infrastructure. The engineering approach requires data on pump characteristics to impute cost parameters using engineering relationships (Gonzalez-Alvarez, Keeler, and Mullen Reference Gonzalez-Alvarez, Keeler and Mullen2006; Moore, Gollehon, and Carey Reference Moore, Gollehon and Carey1994; Hendricks and Peterson Reference Hendricks and Peterson2012). Common parameters used in the engineering approach include well depth, pump technology, pump system pressure, etc. A number of irrigation demand studies using FRIS data have applied the engineering approach to impute pumping costs (Moore, Gollehon, and Carey Reference Moore, Gollehon and Carey1994; Mullen, Yu, and Hoogenboom Reference Mullen, Yu and Hoogenboom2009; Hendricks and Peterson Reference Hendricks and Peterson2012). However, Mieno and Brozovic (Reference Mieno and Brozović2016) raised concern that correlations between unobserved characteristics of a well or pump can introduce bias. Further, depending on the direction of the correlation, the unobserved variables might introduce amplification bias.

Olen, Wu, and Langpap (Reference Olen, Wu and Langpap2016) used FRIS data and applied the average cost approach, which requires individually reported irrigation expenditure data and is distinct from the energy price and engineering approaches in that it may capture some irrigation costs that are fixed over the relevant interval. A profit maximizing firm with complete information would optimize water use as a function of the marginal cost of water, but firms in the context of this article may instead respond to average cost over the time scale of a regular billing cycle. This expectation applies particularly to firms that primarily use electricity as their energy source for pumping.

Findings from Ito (Reference Ito2014) suggest that electricity consumers may not effectively respond to marginal prices due to complicated signals from nonlinear pricing. Many utility rate plans include pricing structures that might obscure an irrigating firm's perception of marginal cost (e.g., demand charges, block rates). Thus, the average cost of water may be more salient than marginal cost for irrigating firms. For irrigators in this study, diesel expenditures composed a large portion of the total pumping expense, but electricity was the majority source in most state-years in the sample.

For this study, firm-level average cost of irrigation water was calculated as the total annual energy expenditures for pumping, E, divided by the total number of acre inches applied (calculated by summing the product of irrigated acreage, n, and water application, w, over m crops).

(7)$$b = \displaystyle{E \over {\mathop \sum \nolimits_{i = 1}^m w_in_i}}\; $$

The average cost of water variable, b, may approximate the marginal cost of water when there are no significant changes in energy prices during the irrigation season and firms do not conflate fixed and marginal costs. The majority of irrigation activity occurs over a relatively short period of time (see Figure 4), so large variation in within-season energy price is unlikely. The reported average cost might be a poor proxy for the marginal cost if irrigators were able to adjust a system's diesel vs. electric energy mix in response to within-season changes in energy price ratios. A subset of the sampled firms reports expenditures on multiple types of energy, primarily electricity and diesel. These irrigators might be potential candidates for energy switching behavior, except the nature of irrigation pump technology makes this behavior unlikely. Irrigation systems are relatively long-term investments for agricultural firms in the Great Lakes Region, and the presence of redundant pumping systems for energy switching is not a known practice (B. Russell, personal communication, February 1, 2018).

Climate and Weather Data

Precipitation and temperature data were obtained from the PRISM Climate Group. Daily precipitation and temperature records were available at a 4km grid resolution. Current year and lagged year temperature and precipitation variables were derived using daily precipitation, maximum temperature, and minimum temperature data. Additional degree day and precipitation variability measures were produced with modifications to the daily PRISM values. PRISM also publishes 30-year normal climate variables calculated as moving averages. Thirty-year average temperature and precipitation variables have been used in recent literature to estimate climate impacts on long-term irrigation management decisions (Bigelow and Zhang, Reference Bigelow and Zhang2018). These variables were included in the crop allocation models to control for gradual climate impacts on crop substitution decisions.

The specifications for the climate variables used in this article were guided by information from agricultural irrigation extension specialists at Michigan State University (MSU). Specifically, MSU irrigation specialists indicated that May 1^st –September 31^st is a sufficiently wide growing season window during which environmental conditions would affect irrigation decisions, with July and August being the heaviest irrigation months (S. Miller, personal communication, September 17, 2017). Irrigation would only occur outside the growing season window under exceptional circumstances (e.g., to “water-in” a cover crop). The seasonality of irrigation water demand is also apparent in Wisconsin water use reports presented in Figure 4.

Temperature is hypothesized to have a positive effect, and precipitation volume is hypothesized to have a negative effect on water application. Due to the relative sensitivity of potatoes, higher temperatures are hypothesized to cause a substitution away from potato production. With the growing season calendar in mind, the preferred climate specification includes variables for peak irrigation season precipitation volume and average temperature. Peak irrigation season is defined as the months of July and August. These variables were generated by converting the 4km cells in the raw daily PRISM data to their central points and then taking the mean of all points that fall within the county to generate a county level aggregate. The daily, county-level precipitation data was summed to generate the cumulative values over July and August. The mean of the daily, county-level temperature data gives the average daily max temperature over the same time period.

An additional measure was designed to account for nonlinear temperature effects. The measure, Extreme Heat Degree Days (EHDD), is similar to a growing degree day specification common in the crop yield literature. It was calculated as the count of degrees in excess of an extreme heat threshold (34°C) summed over days in the irrigation season, D. In the following equation, t _i is the maximum temperature on day i.

(8)$$EHDD = \; \mathop \sum \limits_i^D {\rm max}\lpar t_i\comma \; \; 34\rpar -34\; $$

The 34°C threshold has been identified as the threshold at which additional heat reduces crop yields (Deschenes and Greenstone Reference Deschenes and Greenstone2007; Ritchie and NeSmith Reference Ritchie, NeSmith, Hanks and Ritchie1991). Irrigation applications are hypothesized to be increasing in EHDD because irrigation is a potential strategy to mitigate heat stress.

Predictions of climate-related changes in precipitation in the Great Lakes Region are subject to a greater degree of uncertainty than predictions of climate-related temperature changes. The existing literature indicates that precipitation will become more variable across multiple time scales ranging from daily, to seasonal, annual, and even decadal (Pendergrass et al. Reference Pendergrass, Knutti, Lehner, Deser and Sanderson2017; Hatfield et al. Reference Hatfield, Brown, Andresen, Bidwell and Winkler2014). Common precipitation measures in the existing irrigation demand literature include seasonal and annual precipitation volume. These broad measures do not account for the importance of precipitation timing. Simply stated, between two locations that receive the same total precipitation over a given time period (e.g., one month), the location that receives that precipitation distributed most evenly throughout the month is expected to use less irrigation water.

A number of peak irrigation season precipitation variability measures were considered: the standard deviation of daily precipitation, a Shannon index measure of precipitation evenness, a count measure of 10-day drought events, and a count measure of 20-day drought events. The water price and main precipitation coefficients did not change significantly with the inclusion of any precipitation variability measure. None of the considered measures produced statistically significant effects, so they were excluded from the final models.

NRCS Soils Data

Soils data was obtained from the USDA STATSGO database. Variables for this analysis were generated from the Soil Capability Class data layer, which groups soils “according to their limitations for field crops, the risk of damage if they are used for crops, and the way they respond to management.” For this analysis, the soil capability class data was converted to create a county level soil quality variable. Soil quality was measured as the percentage of land that falls into either class 1 or class 2 in each county. Capability classes 1 and 2 have few to moderate limitations for crop production. This specification is similar to the approach used by Olen et al. (Reference Olen, Wu and Langpap2016). The expected effect of soil quality on water use at the intensive margin is negative, since higher quality soils that better retain moisture would reduce the need for irrigation. At the extensive margin, soil quality is hypothesized to have a positive effect on acreage allocations of water intensive crops (i.e., potatoes) and a negative effect on acreage allocations of less water-intensive crops (i.e., soybeans). The direction of this effect may be confounded by differences in soil types that are not captured by the capability class soil quality measure. Potato growers are generally expected to prefer sandy soils (or other soils with good drainage) because potatoes require careful control of soil moisture and can be easily damaged in overly wet or overly dry soils. Given this sensitivity, land with few impediments (as measured with the capability class data) may be a necessary, but not sufficient, condition for a typical firm to participate in potato production.

Crop Prices

Price data was obtained from the state-by-state monthly crop price database maintained by the USDA National Agricultural Statistics Service. In the water application estimation, a variable indicating same-year, July price was used to measure firm expectations at the time irrigation decisions are made. Lagged marketing-year prices were included in the crop allocation model to capture price expectations at the time planting and investment decisions are made. Importantly, spatial variation in the state-level price data is limited, so the estimation of price effects relies on variation between years. Corn and soybean prices are highly correlated in the sample (ρ=0.97), so their effects cannot be distinguished in the crop allocation models. To address the correlation between corn and soybean prices, a composite price was calculated as the average of the corn and soybean prices faced by each firm. This composite variable is used in the crop allocation model in place of separate corn and soybean prices.

Addressing Measurement Error

The distribution of marginal energy cost for the study sample is skewed with a number of extreme values in both tails of the distribution. The outliers with unexpectedly large average water cost values may be attributable to errors in the FRIS responses or data entry errors for either irrigation volume or total energy expenditures. Measurement errors in irrigation volume, w _i, would be especially problematic because that term also appears in the denominator of the formula for constructing the water cost variable. The water cost variable, b, calculated as shown in equation 9, is a primary covariate of interest in both the intensive and extensive margin estimations. Measurement error in w _i would introduce amplification bias in the estimated parameter on b, whereas measurement error in E, total energy expenditures, would introduce attenuation bias.

To test for amplification bias, two approaches for calculating total water use were applied. First, the reported responses for acre-inches applied by crop were aggregated across irrigated acreage, as shown in the denominator of equation 9. Second, FRIS responses on the volume applied by water source—ground water, on-farm surface water, or off-farm water—were aggregated. The first approach is preferable because the crop level questions are more narrowly focused, and their targeted nature reduces the likelihood of recollection error and other sources of survey response error (e.g., lack of clarity in reported units). As expected, the variance of b as calculated using this approach is significantly smaller.

The values of b calculated using the sum of by-crop applications were compared to the values of b calculated using the total of all water sources, and the sample was restricted to the subset of observations that reported consistent total water quantity values (difference between the two values < 5 percent). If measurement error is driving amplification bias in the full sample, the parameter of interest estimated with the reduced sample would be smaller in magnitude. However, when the general model was estimated with the limited sample, the estimated parameter on b was slightly larger in magnitude. This suggests the results are unlikely to be significantly biased by measurement error in w _i.

Energy expenditures, which are summed to produce E, may be misreported for a variety of reasons, including but not limited to a blurred differentiation between irrigation-related expenditures and other non-irrigation energy expenditures. Consider two illustrative examples. First, a firm using primarily electricity for irrigation may receive a single bill for irrigation-related and non-irrigation-related electricity use. A second firm using primarily diesel fuel for irrigation may buy diesel fuel in bulk for numerous uses. When asked to report total energy expenditures by energy source, firms may fail to accurately distinguish between these competing uses. In such cases, firms might underreport or overreport actual irrigation expenditures.

To address the measurement error in the numerator of the equation for b, firms above the 95^th percentile and below the 5^th percentile for average cost of water, b, were removed from the sample. This change had the expected effect; the magnitude of the estimated coefficient on b increased due to the reduction of attenuation bias from measurement error.

Regression Weights

All regressions are reported using the USDA-provided sample weights to correct for the non-randomness in the sampling method. The probability weights denote the inverse of the probability that a farm in the sample frame has been included in the sample. In a simple sense, probability weights can be interpreted as the number of unobserved firms of a similar size that are represented by a single firm in the sample. In this context, the farms selected into the certainty strata would receive a weight of 1. The farms from the non-certainty strata receive weights greater than 1. All models throughout the article are estimated using the provided weights interpreted in Stata as probability weights. The distribution of the sample weights within the group of observations culled from the sample to address attenuation bias closely mirrors the distribution of weights in the kept sample.