Abstract
We use a random time parameter bivariate probit to estimate individual time preferences using exponential and hyperbolic discounting forms for agricultural producers making an irrigation investment. The standard deviation estimate of the random time parameter suggests that there is substantial unobserved heterogeneity in time preferences across farmers. We control for time preference heterogeneity by making the time parameter a function of farm and personal characteristics. Using multiple irrigation techniques correlates with producers who are less patient for the investment. Education or participation in a government cost-share program for a similar irrigation investment correlates with greater patience by producers.
1. Introduction
A capital investment by an agricultural producer represents an intertemporal trade-off between an up-front purchase price and the future flow of services from the purchase. The discount rate is what producers use to weigh current costs against future benefits. However, there has been no empirical measurement of an agricultural producer’s discount rate for investment decisions. Although the sensitivity of investment decisions to the discount rate is widely known, the literature provides little guidance about the time preferences that producers use for those decisions. Individual discount rates for investment could be lower than the rates associated with consumption because producers typically have access to stable capital markets. On the other hand, the discount rates may be high because of market barriers and failures such as imperfect information, principal-agent issues, and credit constraints (Gillingham and Palmer 2014).
We use stated preference data in a contingent valuation (CV) study to estimate the implicit discount rate for agricultural producers. Our work builds on the limited literature that endogenously estimates the discount rate in a valuation model (Lew 2018) through sample level variation in either the benefits horizon (Meyer 2013a, 2013b) or the payment horizon (Kovacs and Larson 2008; Bond, Cullen, and Larson 2009). We contribute to the recent studies that empirically estimate the hyperbolic time preferences, which allows the discount rate to decline over time, with stated preference data (e.g., Meyer 2013a, 2013b; Lew 2018; Vasquez-Lavin et al. 2019). This study estimates a random time parameter for both exponential and hyperbolic discounting forms to allow for unobserved heterogeneity in the time preferences, and only Meyer (2013a, 2013b) has done this before with stated preference data. Andersen et al. (2013) and Meyer (2015) consider unobserved heterogeneity in the time preferences with experimental economics data. Accounting for heterogeneity, observed or unobserved, in the discount rate matters because the experimental economics literature on intertemporal choices indicates significant variation in the discount rate among individuals (Harrison, Lau, and Williams 2002).
One study in the energy economics literature estimates a discount rate for business decisions outside a valuation model. From the adoption of energy audit recommendations of manufacturing plants, Anderson and Newell (2004) find that firms require at most a two-year payback and infer a discount rate of 50%–100% for projects that last at least 10 years. Their approach looks at the expected payback periods for energy efficiency opportunities and infers the time parameter in the discounting formula that equates the opportunities. Much of the research on business investments use simulation techniques to show how uncertainty and opportunities to delay investment represent significant hurdles that must be overcome before a producer will choose to adopt (Carey and Zilberman 2002; Baerenklau and Knapp 2007). Ex post studies indicate that agricultural investments depend on production risk, price uncertainty, and social learning, among other things (Schuck et al. 2005; Genius et al. 2014; Schoengold and Sunding 2014). This article assesses how farm operators discount the future payments for an on-farm water storage investment using data from an irrigation survey.
Past studies that estimate the discount rate in a valuation model find evidence of heterogeneity in time preferences for consumption goods among respondents. By using the benefit horizon for a water quality program in the Minnesota River Basin (MRB), Meyer (2013a, 2013b) estimates the random time parameter for several discounting forms and finds a significant estimate on the standard deviation of the time parameter in each case. Meyer (2013b) examines how personal characteristics relate to time preferences with the exponential discounting form, and he finds that being a resident of the MRB correlates with more patience, while education and the male gender correlates with less patience. Vasquez-Lavin et al. (2019) use the payment horizon for marine conservation actions in Chile to estimate time parameters for exponential and hyperbolic forms, and they find that familiarity with a conservation area correlates with more patience for all discounting forms. Using the exponential discounting form, Bond, Cullen, and Larson (2009) find that more education correlates with greater patience for sea lion protection; Kovacs and Larson (2008) find that respondents who are older and have children correlate with less patience for additional urban park land. Recent experimental studies indicate that young women with high education correlate with more patience for trade-offs involving money (Meier and Sprenger 2010, 2015).
Studies suggest that a person’s time preferences can vary over time (Meier and Sprenger 2015) or by the good (Ubfal 2016; Kumar and Kant 2019) or a proposed program (Vasquez-Lavin et al. 2019) under evaluation. Information about context of the intertemporal tradeoff is likely necessary before being able to identify the individual’s time preferences that will guide the trade-off. By knowing who is patient for the future benefits of a particular program, policy makers can provide individuals with the services from that program when they want them, and this will presumably make the programs more appealing. For example, an extension agent presenting conservation practices to farmers might increase adoption of those practices by emphasizing practices with short-term benefits to impatient farmers and the practices with long-term benefits to the patient ones. Another example is that farmers might have more patience for soil conservation than for water conservation, and a better way to increase adoption of long-run conservation practices could be to focus on programs for soil rather than water.
We present a brief explanation of discounting forms followed by a description of the empirical approach for estimating the time preferences and willingness to pay (WTP) for an agricultural investment. We follow this with information about the CV question in the farmer irrigation survey. Last, we present and discuss the results from the model estimation and offer concluding remarks.
2. Discounting Forms
The exponential discounting form of time preferences has a discount factor for year t of Ψt = (1 + ρ)−t, where ρ is the discount rate. Exponential discounting is analytically elegant, but this assumes that the discount rate is constant over time, and some empirical research shows this does not fit behavior (Frederick, Loewenstein, and O’Donoghue 2002). Instead, discount rates might decline over time (Coller and Williams 1999), and time preferences associated with hyperbolic discounting fit better fit with a hyperbolic function than an exponential function. A hyperbolic discounting form, using a single parameter to facilitate estimation, has a discount factor for year t of Ψt = (1 + t)−μ (Harvey 1986). Two other hyperbolic discounting forms, Ψt = (1 + ωt)−1 (Mazur 1987) and 
(Laibson 1997), appear in studies to estimate implied discount rates with stated preference data (Meyer 2013a, 2013b; Lew 2018; Vasquez-Lavin 2019). However, estimation with the Mazur (1987) hyperbolic form was never successful because of a highly uneven log-likelihood surface associated with our data, and the Laibson (1997) quasi-hyperbolic form requires a delay in the start of payments not available in our data.
Hyperbolic discounters place less emphasis on the near future and more emphasis on the distant future than do exponential discounters. A hyperbolic discounter is impatient in near-term temporal trade-offs but switches to greater patience in trade-offs in the distant future. While exponential discounters exhibit time consistency, the hyperbolic discounters do not. Those with hyperbolic time preferences consume more in the present than their future selves would recommend, and this has relevance for studies on procrastination and addiction, among other areas (Frederick, Loewenstein, and O’Donoghue 2002).
3. Empirical Model
An individual i responds to potentially two double-bounded CV questions. Each question a person sees has a different payment horizon. The utility of an individual i who responds positively to a question j in a double-bounded CV question k has a component xi related to the observed attributes of the individual and a component related to the payment for question j in a doubled-bounded CV question k for each period t, cikjt, with a payment horizon Tk, given by
[1]
where β, δ, and γ are parameters to be estimated that represent the marginal utilities of observed attributes, a structural shift in utility associated with the second question in a double-bounded CV, and the marginal utility of money, respectively (Alberini, Kanninen, and Carson 1997). The assumed utility of a negative response is zero. ψt(φi) is a discounting form that depends on a random parameter 
, which is unobserved for each i and varies in the population with density f(φi | θ* for θ* as the true parameters of the distribution. The error term, εikj = uik + vikj, is the sum of two normally distributed error components, one common to each double-bounded CV question k a person answers, uik, and the other a transitory error term, vikj. The error components are assumed to be independent of the parameters β, γ, φi and variables xij, cijt.
We assume the random parameter in the discounting function has the normal distribution, and the random coefficient can be expressed as 
, where zij are observed attributes to reflect observed heterogeneity in temporal tastes, and πφ are additional parameters. The 
represents the mean estimate of φi, and the standard deviation estimate for the unobserved heterogeneity of φi is σφ, where ηi is an independent standard normal deviation in the individual’s temporal tastes relative to the average tastes in the population.
Estimation of a model with random parameters for the marginal utilities of the observed characteristics for WTP was not possible. Our data set is not large enough to estimate the parameters twice, namely, the standard deviations for the random marginal utilities on the observed characteristics. We assume we are measuring patience for an irrigation investment through variation in the payment horizon, but variation in the benefit horizon could better represent the time preferences for the investment. Another assumption of equation [1] is risk neutrality because the utility is linear and additively separable in the utility from money. We assume a fixed price coefficient γ, and this allows for easy calculation of the distribution of the marginal WTP for the observed attributes (Train and Weeks 2005). The mean marginal WTP for an attribute m is −(βm / γ) where the marginal utility of attribute m βm, and the mean WTP is 
.
Designate the individual’s choice j as yij, and the sequence of Ji choices as yi = {yi1,…, yiJi}. The number of Ji choices depend on individual i because some individuals only respond to the first of possibly two double-bounded CV questions. Respondents who indicate at the end of the first question that they are not willing to spend $1 do not hear a second question with a different payment horizon. The log-likelihood function for a bivariate probit model, which assumes independent and identically distributed errors between the two double-bounded questions but not within each question, conditional on β, γ, φi, is
[2]
where the probabilities of each pair of responses are calculated from the bivariate normal c.d.f. Φ(zi1,zi2,τ) and the univariate normal c.d.f.s Φ(zi1) and Φ(zi2) as 
, 
, 
, and 
. The corre lation coefficient, τ, is between the standardized utilities zi1 and zi2 for the first and second questions in the double-bounded CV.1 The correlation coefficient allows for different utilities across the initial and follow-up responses, with the special case that these are the same when τ = (Alberini, Kanninen, and Carson 1997).
The probability conditional on β, γ, and θ* is the integral of L(yi | β, γ, φi for all values of φi weighted by its density:
[3]
The parameter φi associated with individual i represents that respondent’s temporal tastes, and these tastes vary over individuals. The aim is to determine the mean and variance of f(φi | θ*) that describe the temporal tastes.
An explicit form of the log-likelihood function is not possible because the integral in equation [3] does not have an analytical solution. However, there is approximation of the probability through simulation, and subsequently estimation of the parameters by the maximization of the simulated log-likelihood function. For a particular value of θ*, a value of 
is the rth draw from the distribution f(φi | θ*), and this allows for the calculation of 
. The simulated choice probability can be represented by the average of R randomized Halton draws, 
, where R is sufficiently large. The log of the simulated probability is a biased estimate of the log of the true probability, but the bias decreases with the number of random Halton draws. We opted to use 500 random Halton draws in all random parameter estimations. The discounting form around the time parameter makes utility a nonlinear function of parameters, and there is no guarantee of global concavity of the simulated log-likelihood function (Train 2009). To safeguard against the possibility of multiple local maximums, we use several initial values to find the appropriate parameter estimates. The standard errors for the parameters come from the square root of the diagonal of the inverted Hessian matrix. The estimation occurs in the R statistical computing software by coding the likelihood function and using the maxLik package (Henningsen and Toomet 2011) with maximization through the Broyden-Fletcher-Goldfarb-Shanno algorithm using a numerical gradient and Hessian.
4. Data
We apply the empirical model to CV data from Arkansas from a telephone survey designed by a team of agricultural research scientists to understand the irrigation practices by agricultural producers in the lower Mississippi River Valley. Arkansas is the largest user of groundwater in the region, and the state designation of several critical groundwater areas reflects concern about persistent groundwater overdraft (USDA-NASS 2012; ANRC 2017). The CV data allow us to measure farm operator’s preferences and values for the use of on-farm storage and tail-water recovery (OFS-TWR) systems to hold and recycle nutrient-rich surface water for irrigation. The OFS-TWR systems are typically permanent installations that require maintenance with periodic regrading and removing sediment from the storage facilities. The surface water from OFS-TWR is often less saline than groundwater and reduces reliance on scarce groundwater, but OFS-TWR are expensive to build and occupy productive land otherwise available for crops. State and federal programs provide cost-share and technical assistance for agricultural landowners who want OFS-TWR. The survey had a pilot and two focus groups with farm operators that irrigate and extension specialists that advise farmer operators.2
The farm operators answered multiple questions about irrigated crops and irrigation practices before responding to the CV question. Immediately before the CV question, the survey asks how the farm operator would irrigate without groundwater, and more than half indicated that they would construct OFS-TWR. More than one-third of the sample already makes use of an OFS-TWR. The use of surface water with OFS-TWR to address declining groundwater levels was a recommendation of the well-publicized 2014 Arkansas Water Plan (ANRC 2015). In potentially two double-bounded CV questions, farm operators choose whether to undertake a series of constant annual payments over a payment horizon for an additional acre of OFS-TWR.3 The wording of the question is as follows: “Would you be willing to invest $____ in reservoirs and/or tail water recovery systems per acre of reservoir each year for ____ years?” The survey concludes with sociodemographic questions about the farm operator and their household.
Each double-bounded CV question has a single payment horizon (10, 20, 25, or 30 years) and a set of three bids values (see details in Appendix Table A1). The first bid value in a set is for the initial question in the double-bounded format. A “yes” response to the initial question means the respondent sees the second bid value in the set, and a “no” response means the respondent sees the third bid value. The annual bid values are larger for shorter payment horizons and smaller for the longer payment horizons, and we modified the bid values after the pilot and focus groups to receive a similar spread of yes and no responses across the bid value sets and payment horizons. The bid values were set to reflect the actual cost of an acre of an OFS-TWR. The uneven coverage of the probability distribution was less severe in the pilot study and the focus groups. We could have adjusted the bid values down to get a more even coverage of the probability distribution, but we would have learned less about the WTP in the range most relevant for policy.
The telephone survey took place in 2016, and the Survey Research Laboratory at the Social Science Research Center with Mississippi State University administered it. Potential survey respondents come from the water user database managed by the ANRC and all commercial crop growers identified by Dun & Bradstreet records for Arkansas. There are about 150 questions in the survey, and respondents took 30–40 minutes to finish by phone. Among the farm operators eligible to complete the survey, 255 declined to begin the survey, 171 declined to complete the survey, and 199 completed the survey for a 31.8% response rate. Comparing our response rate to other studies, mail surveys to farmers in Missouri in 2017 (Burli et al. 2019) and farmers in Tennessee in 2005 (Jensen et al. 2007) had 23% and 15% response rates, respectively. However, response rates to mail surveys sent to farmers in Northern states such as Iowa in 2010 (Varble, Secchi, and Druschke 2016) and Michigan in 2014 (Lowry and Brainard 2017) were higher at 41% and 35%, respectively. The data for the analysis consist of 152 respondents after the removal of 43 individuals with missing data and 4 protest respondents.4 Because most respondents answer two double-bounded CV questions, we had 282 observations for the analysis.
A narrative review of the literature on the effects of the mode of survey administration on data quality is the basis for several remarks about the quality of telephone survey data (Bowling 2005). Question nonresponse is typically lower with an encouraging telephone interviewer than a self-administered (postal or electronic) survey, but self-administered surveys might give respondents more time to think about the questions to increase the response accuracy. Telephone interviewers control the order of the questions while self-administered survey respondents can preview the questions ahead of time and adjust their answers. Telephone interviewers can keep respondents on the topic of relevance, probe to retrieve information, and prompt memory. The presence of an interviewer can produce more positive and socially desirable responses and reduce the disclosure of sensitive information relative to self-administered surveys. The net effect of the telephone mode on the quality of the WTP and time preference data is therefore difficult to determine. We do not know how much the telephone interviewers helped respondents stay focused on the CV questions and retrieve memories.
Information about representativeness of the survey sample for an Arkansas producer of irrigated crops is in Appendix Table A2. Our survey respondents have more farming experience and cultivate more rice on their farms. We organize the explanatory variables for measuring the farm operator’s preferences and values for OFS-TWR into three categories: irrigation and farm characteristics, sociodemographics, and social learning. Appendix Table A3 has summary statistics for the explanatory variables drawn from the features of the sample.
More than a third of the respondents currently have at least one reservoir, and many producers have more than one (NUM_RES). Sprinkler irrigation is less common than gravity irrigation in Arkansas and less popular than in the past, but about one-half the respondents used a type of center pivot at one time (PIVOT). About one-third of the respondents use cover crops (COVER_CROP), and about one-half are aware of state tax credits for constructing storage reservoirs or land leveling (CRED_AWARE), but less than a fifth have used them (CRED_USE). Less than one-tenth have used the state tax credit for land leveling (LCRED_USE). More than one-quarter of respondents indicate that the groundwater level rose over the past five years (GW_RISE), and about two-fifths live in counties close to the Mississippi River, where natural recharge of the aquifer is greater (CROWLEY). Nearly three-fourths of farm operators indicate that there is a groundwater shortage in the state (GW_SHORT). Explanatory variables such as crop types and the amount of irrigated acres were not significant, although we predicted those variables might influence investment choices.
About three-fifths of respondents have a household income above $75,000 (D_INC_ MID_HIGH). This is similar to the average net cash farm income of $72,084 (USDA-NASS 2017) but higher than the 2016 median household income for Arkansas of $42,798 (U.S. Census Bureau 2017), although one-fifth of the respondents did not report a household income (D_INC_NA). Less than one-tenth have an education beyond a bachelor’s degree (D_ADV_EDU), and more than one-half of the respondents have a formal education related to agriculture (AGRI_EDU). About 28% of respondents have more than 40 years of farming experience (D_HIGHEXP), and less than 5% have 5 years or less of farming experience (D_ LOWEXP). About one-half of the producers indicate they belonged to an environmental organization, usually a duck hunting organization, at one time (DUCKORG).
Two variables in the social learning category relate to peer networks for particular irrigation practices. Producers who belong to a peer network of producers that use surge irrigation or precision leveling (PEER_DRAIN) can conserve groundwater, but the primary benefit of these practices is to improve drainage from furrow irrigation. The peer network of producers who use center pivot, zero grade leveling, or alternate wetting and drying (PEER_EFF) typically use these practices because of an interest in irrigation efficiency.
5. Results
Estimates of the time preference statistics for the bivariate probit models for the exponential and hyperbolic discounting forms are in Table 1. The estimates for the random time parameter without personal characteristics (RTPw/o) are first. The mean time parameters for discounting forms in the RTPw/o model are not statistically different from zero, but the estimated standard deviations are highly significant, indicating that the time parameter indeed varies in the population. This motivates looking at a model with the time parameter as a function of observed personal characteristics, allowing a random time parameter with eight observable characteristics in the RTPw model. A random time parameter model with more than eight observable characteristics could not provide recoverable standard errors. The mean time parameter in the RTPw model is significant in the exponential form and much closer to significance in the hyperbolic form. The estimated annual discount rate for the exponential form 
is 0.128, and for the hyperbolic form 
the mean time parameter is 0.269. The standard deviation for the unobserved heterogeneity decreases for both discounting forms, now that there are observable characteristics for the time parameter, but is still statistically significant. The hyperbolic form has a slightly better fit than the exponential based on the likelihood ratio index (LRI) and the “corrected” Akaike’s information criterion (AIC).5
Increasing the number of observable characteristics describing the time parameter from 8 to 17 increases the fit according to the LRI or the AIC. The difference in the fit between the exponential and hyperbolic forms is virtually indistinguishable in the FTPw model. The discount rate estimate in the exponential form increases to 0.195 and is significant at the 1% level, while the time parameter for the hyperbolic form rises slightly to 0.455 and is significant at the 10% level. In the FTPw model, the correlation coefficient (τ ) between the initial and follow-up questions is now positive and significant. The most inclusion of the observable characteristics switches the sign and the significance of the correlation coefficient compared to a model with no observable characteristics (RTPw). The estimate of the structural shift in utility with the follow-up question (δ ) is consistently negative and similar in magnitude across all the models.
Figure 1 indicates the discount factor and its 95% confidence interval for the average time parameter of the exponential and hyperbolic forms in the FTPw models for a 30-year horizon. The discount factor is higher with the exponential form than the hyperbolic form for the first 5 years and then the hyperbolic is higher for the last 25 years. The model fit for the discounting forms is similar, and this is likely because the discount factor regardless of the discounting form shows that most of the OFS-TWR value occurs within the first 10 years.
Comparison of Discount Factors (and 95% Confidence Bounds) from the FTPw Model: Exponential ψt = (1 + ρ)−t with 
and Harvey Hyperbolic ψt = (1 + t)−μ with 
The coefficient estimates for the observed characteristics to explain the fixed time parameters for the exponential and hyperbolic discounting forms are shown in Table 2. There are two specifications for the fixed time parameter model to investigate the sensitivity of the coefficient estimates. The first includes all 17 observable characteristics, and the second drops two variables that are not statistically significant in the first specification. The coefficient estimates for the exponential show sensitivity to the specification because there is a change in coefficient sign for five of the observable characteristics. However, there is no change in the coefficient signs across the two specifications for the hyperbolic form. The greater robustness of the hyperbolic may be evidence that hyperbolic discounting is a better fit for the time preferences. The coefficient signs are identical across the exponential and hyperbolic forms for the first specification with all the observable characteristics.
Respondent characteristics that significantly correlate with the time parameter relate to farm operations, peer networks in the irrigation community, and sociodemographics. Previous studies of discounting with stated choices or experiments use the general population, and the only observable characteristics were the sociodemographics. The intensity of OFS-TWR systems (NUM_RES) use, the previous or current use of alternative water conservation techniques such as a center pivot system (PIVOT) or cover crops (COVER_CROP), and the past use of a state tax credit for land leveling (LCRED_USE) correlates with greater impatience by respondents. However, previous use of a state tax credit for an OFS-TWR system is associated with greater patience. Respondents belonging to a peer network of irrigators that use old precision leveling techniques for drainage (PEER_DRAIN) correlates with greater patience for older irrigation approaches like the OFS-TWR system. However, respondents in a peer network of irrigators that use newer irrigation practices (PEER_EFF) correlates with more impatience.
Respondents with a groundwater level rising on the farm over the past five years (GW_ RISE) correlates with greater patience possibly because the returns from an OFS-TWR system can wait if water scarcity is less a concern. The opinion that there is ground water shortage some place in the state (not necessarily on the respondent’s farm) correlates with greater impatience by the respondent. Living in the water-abundant counties east of Crowley’s Ridge (CROWLEY) correlates with greater impatience likely because there is no serious consideration given to OFS-TWR use.
Higher income (D_INC_MIDHIGH) and more education related to agriculture (D_ AGRI_EDU) correlate with respondents who have greater patience. Field experiments that examine the sociodemographics of discounting find higher incomes correlate with greater patience in some cases (Tanaka, Camerer, and Nguyen 2010) and lower in others (Meier and Sprenger 2015); it can also have no influence (Harrison, Lau, and Williams 2002). Field experiments (Harrison, Lau, and Williams 2002; Tanaka, Camerer, and Nguyen 2010) and a stated preference study (Bond, Cullen, and Larson 2009) have found that education correlates with greater patience. However, we do not find evidence that additional general education (D_ADV_EDU) correlates with patience. More farming experience (D_ HIGHEXP) correlates with more impatience, which could be because these respondents are older and closer to retirement. Meier and Sprenger (2015) and Harrison, Lau, and Williams (2002) find in field experiments and Kovacs and Larson (2008) in a stated preference study that greater age correlates with less patience, but Tanaka, Camerer, and Nguyen (2010) find in a developing country field experiment that age correlate with greater patience. Participation in an environmental organization at one time, usually a duck hunting organization (DUCKORG), correlates with more patience.
To examine whether explaining the time parameter with observed characteristics influences the welfare estimates, Table 3 reports the marginal willingness to pay (MWTP) estimates for the exponential and hyperbolic forms for the RTPw and FTPw models. The marginal utility estimates 
, 
, and 
to derive the MWTP are shown in Appendix Table A4. The signs and magnitudes of the MWTP across the exponential and hyperbolic forms are similar whether the time parameter is random or fixed. The level of observed heterogeneity taken into account in the time parameter affects the optimization surface for the model solution due to joint estimation of the time parameter and WTP. This can lead to the MWTP results changing in sign and significance across a model with 8 observed characteristics for the time parameter (RTPw) versus 17 observed characteristics (FTPw). We see five coefficients change in sign and four coefficients change in significance within the 10% level as the number of observed characteristics describing the time parameter increases.
Looking at the estimates for MWTPs in the FTPw model, we see that respondents with more OFS-TWR systems (NUM_RES) and those who previously used a state tax credit for precision leveling (LCRED_USE) have much lower MWTP. Those who previously used a state tax credit for an OFS-TWR have a much higher MWTP. The farm operators who used a center pivot system (PIVOT), who live in water-abundant counties (CROWLEY), who think there is groundwater shortage problem in the state (GW_SHORT), or who have more years of farming experience (HIGH_EXPER) have a lower MWTP. Respondents with more income (D_INC_MIDHIGH), who have more education (D_ADV_EDU), and who belonged to a duck-hunting organization (DUCKORG) have a higher MWTP.
To compare the WTP for OFS-TWR and time parameter across the sample in the FTPw model (Table 4), we define subgroups of respondents (reservoir user, pivot user, low efficient peer, and high efficient peer) from the four clusters in a kmeans cluster analysis using a Euclidean similarity matrix. We name the clusters by comparing how much the mean values of the explanatory variables in a cluster differ from the mean values for the whole sample (Appendix Table A5). The reservoir user and pivot user subgroups have the highest standardized deviations above the whole sample mean for NUM_RES and PIVOT, respectively. The low efficient peer subgroup has the lowest standardized deviation below the whole sample mean for PEER_EFF, meaning that members of the subgroup were the least likely to have a family member, friend, or neighborhood producers who used center pivot, zero grading, or alternate wetting and drying for rice. The high efficient peer subgroup has the highest standardized deviation above the whole sample mean for PEER_EFF.
The MWTP is greater than zero and statistically significant for both discounting forms, and the value per acre for a farmer with average characteristics is above the per acre OFS-TWR cost of about $750 (Wailes et al. 2004). However, the reservoir user and pivot user subgroups have a WTP that is negative and not statistically significant. Farmers in the reservoir user subgroup already have many reservoirs and do not want more, and farmers who use pivots and land leveling to irrigate in the pivot user subgroup do not want OFS-TWR. The low efficient peer subgroup has the highest WTP. Farmers without peers who use pivots, zero grading, alternate wetting and drying for rice may not like or do not know about those irrigation techniques. The low efficient peer subgroup also has the most years of farming experience (Appendix Table A5). The attention toward OFS-TWR was significant after a late 1990s drought in the region, when many of the older agricultural producers would have been in their farming prime, and this subgroup may be the most familiar with the traditional OFS-TWR (Wailes et al. 2004). The high efficient peer subgroup has a WTP slightly higher than the MWTP. Because the high efficient peer subgroup uses fewer pivots and reservoirs than the average, the subgroup has slightly more value for the OFS-TWR.
The reservoir user and pivot user subgroups have the largest time parameters but for different reasons (Appendix Table A5). The greater use of reservoirs and cover crops principally explain the larger time parameter for the reservoir user subgroup, but the lower incomes and the greater belief that there is a groundwater shortage also increase the time parameter. For the pivot user subgroup, the greater use of pivots and a tax credit for land leveling, as well as frequently living in water-abundant counties east of Crowley’s Ridge and having high efficient peers, appear to be the main reasons for a large time parameter. The low time parameter of the low efficient peer subgroup is due to the lower use of pivots and the lower use of a tax credit for land leveling. The high efficient peer subgroup has a time parameter slightly lower than the mean because of the lower use of pivots, cover crops, and reservoirs.
Formal tests of differences in WTP between discounting forms and between subgroups are in Appendix Table A6 using the method of convolutions approach (Poe, Giraud, and Loomis 2005). There is statistically significant difference in WTP for both the discounting forms at the 5% level between the pivot user subgroup and the average farmer. The WTP of the pivot user subgroup has a significantly lower WTP than the other subgroups, except for the reservoir user subgroup. One other statistically significant difference at the 5% level is the lower WTP of the reservoir subgroup relative to the low efficient peer group. A similar pattern of statistically significant differences between the subgroups and the average farmer are evident in the time parameter.
Figure 2 shows the plots for the exponential and hyperbolic discounting forms of individual WTP in panel a and the instantaneous discount rate in panel b.6 The spread of respondent WTP is greater with the hyperbolic than exponential discounting, and the WTP centers around $1,000–$2,000 per acre. At the evaluation of the hyperbolic discount rate in the 10th year, the spread of the instantaneous discount rate is less for the hyperbolic than exponential, and there is a slight skew to the left in the distribution for the exponential rate.
Distribution of Mean Willingness to Pay and Instantaneous Discount Rate from the FTPw Model: Exponential 
and Harvey Hyperbolic ψt = (1 + t)−μ with 
Because the instantaneous discount rate declines over time for the hyperbolic form, in Table 5 we test for differences in the instantaneous discount rate across the discounting forms for seven points in time (Poe, Giraud, and Loomis 2005). For the average respondent, the instantaneous discount rate for the hyperbolic form is larger in the present but significantly less than the exponential rate at the 5% level for all later points in time. A similar pattern occurs for the reservoir user and pivot user subgroups, except that the instantaneous discount rate is not statistically different at the 5% level in the fifth year across the exponential and hyperbolic forms. There is no significant difference between the exponential and hyperbolic discount rates, less than the 5% level, for the low efficient peer subgroup beyond the present period.
6. Discussion
Studies of time preferences tend to focus on money or consumption goods rather than production decisions. How individual business owners or operators trade future benefits against current costs, and the discount rate that implies, is crucial for understanding production decisions. Here joint estimation of alternative models of discounting with CV data measure the value and preferences for an on-farm investment. Variation in the stream of future payments farmers would pay for an OFS-TWR system allows for the recovery of farmer discount rates and discounting behavior for this kind of investment. The choice model for estimation of the CV data accounts for unobserved and observed heterogeneity in time preferences.
The finding that exponential and hyperbolic discounting have nearly the same fit matches recent stated preference studies that jointly estimates time preferences and CE behavior (Lew 2018; Vasquez-Lavin et al. 2019), although Meyer (2013a, 2013b) finds that the exponential is the preferred discounting model. Vasquez-Lavin et al. (2019) show that the discounting model with the best fit depends on what ecosystem service program is under evaluation. Experimental data studies originally highlighted hyperbolic discounting (Coller and Williams 1999), but recent experimental analyses show a preference for exponential discounting (Andersen et al. 2014) or a preference for exponential discounting after a correction for subjective time perception (Bradford, Dolan, and Galizzi 2019). The different WTPs across discounting forms present a challenge to policy makers deciding on the most appropriate WTP to use for benefit-cost analysis. Although there is no preferred time preference model, the magnitudes of WTP across the same model specification but different discounting forms can be substantial. Further empirical research could examine how much discounting forms vary across individuals or across investment types for one individual.
The respondents appear to have lower discount rates than in many previous stated preference studies (Lew 2018), but there is a lot of variation in the magnitude of that discounting among the respondents. The estimate of the average discount rate with exponential discounting is 19.5% in the FTPw model using seventeen observed characteristics for the time parameter. The time preferences from other stated preference studies examine consumption goods rather than an investment one. Kovacs and Larson (2008), Bond, Cullen, Larson (2009), and Lew (2018) find annual rates between 23% and 227%. However, in the joint estimation of discount rates with choice experiment data through the benefits horizon, Meyer (2013a, 2013b) find exponential discount rates between 10% and 13% and hyperbolic time parameters between 0.39 and 0.45. Lew (2018) suggests caution in comparing the discount rate from variation in benefit horizon versus the payment horizon because the uncertainty differences between the levels of future benefits versus the size of future payments could explain the gap in the discount rate estimate. Many of the estimates of discount rates from experimental studies (see Frederick, Loewenstein, and O’Donoghue 2002, table 1) are below and well above the estimate in this article.
A contribution of this article is illustrating the importance of controlling for heterogeneity in time preferences for identifying those time preferences and the welfare estimates. The RTPw/o model does not find a statistically significant mean time parameter in either discounting form. The RTPw/o model does not incorporate observed heterogeneity in time preferences, and the estimate of the time parameter is highly imprecise. The RTPw model, which incorporates some of the observed heterogeneity for explaining the time parameter, has a better fit than RTPw/o and has a statistically significant exponential time parameter. Random parameter models often have a good fit, but the in-sample predictions for the welfare and discount rate estimates can be worse than for fixed parameter models (Klaiber and von Haefen 2018; Howard, Whitehead, and Hochard 2020). Accounting for all observed heterogeneity, the FTPw model has the best fit, and the exponential and hyperbolic time parameters are statistically significant.
The coefficient estimates on the observed characteristics explaining the time parameter suggest that farming practices and peer networks related to an agricultural investment correlate with the patience for that investment. Alternative irrigation techniques in use by a farmer or their peers other than OFS-TWR correlate with less patience for an investment in OFS-TWR. Perhaps any agricultural investment different from the techniques most familiar to a farmer receives less patient consideration. A rising groundwater level on the farm correlates with more patience for an OFS-TWR, possibly because the respondent views water scarcity as a less pressing concern. Similar to other studies (Harrison, Lau, and Williams 2002; Tanaka, Camerer, and Nguyen 2010), income and education correlate with greater patience. To our knowledge, no prior study considers whether participation in an environmental organization correlates with patience; for our respondents, the correlation is with greater patience.
Differences in the discount rate and WTP for OFS-TWR occur for alternative values of the observed characteristics, which define subgroups of respondents. The low efficient peer subgroup has a mean discount rate half that for the average farmer, while the pivot user subgroup has a discount rate nearly double that for the average farmer. The current farming practices and sociodemographics of respondents may be more important in the correlation with the patience for an agricultural investment. Government programs that promote stewardship on agricultural land often cost-share agricultural practices with the justification that the tax paying public receives environmental benefits. The taxpayers might have lower costs if extension personnel knew which producers are willing to wait for the benefits for a conservation investment. Additional research could validate our empirical findings for irrigation investments in other settings and consider other agricultural investments.
Policy analyses often require assessments that depend on the timing of future benefits and costs of a proposed investment. Because discount rates vary across individuals and could vary by types of investments for one person, researchers should consider better ways of eliciting time preferences in stated preference questions. We find that controlling for heterogeneity in the time parameter influences the MWTP and time parameter estimates. Unobserved heterogeneity in the time parameter, although raising challenges for the estimation, can reveal any remaining heterogeneity in the time parameter that could be a concern to policy makers deciding on a discount rate. Elicitation of discounting behavior separate from the CV question may not be adequate if the time preferences respondents have for money-money trade-offs differ from the time preferences for other goods and investments (Ubfal 2016; Kumar and Kant 2019; Vasquez-Lavin et al. 2019).
Acknowledgments
The authors thank two anonymous reviewers for insights that added robustness and literature to the study and enhanced the clarity of the article. Any errors are our own. Support for this research comes from the Arkansas Soybean Promotion Board and the Arkansas Rice Research and Promotion Board.
Footnotes
Appendix materials are freely available at http://le.uwpress.org and via the links in the electronic version of this article.
↵1 The two standardized utilities are
and , where σ is the standard deviation of the latent utility.↵2 There was also helpful input from the staff of the UA Division of Agriculture and the Arkansas Natural Resources Commission.
↵3 A respondent who indicates an unwillingness to invest at least $1 at the end of the first double-bounded CV question does not hear the second question. The average number of responses per individual is 3.71 out of 4, so most individuals respond to both questions.
↵4 An inquiry directly follows the first double-bounded CV to identify protest respondents, who indicate political reasons or lack of fairness as the explanation for the “no-no” response.
↵5 The likelihood ratio index is a measure of goodness-offit, defined as 1 − [LL(θ)/LL(0)], where LL(θ) is the value of the log-likelihood function at the estimated parameters, and LL(0) is the value with all parameters equal to zero.
↵6 An instantaneous discount rate for a discounting form ψt is
. The instantaneous discount rate for the exponential discounting form is constant and equal to ln(1 + ρ). For the hyperbolic discounting form, the instantaneous discount rate equals µ / (1 + t), which declines over time and we evaluate at t = 10.
