An agent-based model of farmers' crop and best management practice (BMP) decisions is developed and linked to a hydrologic-agronomic model of a watershed, to examine farmer behavior, and the attendant effects on stream nitrate load, under the influence of markets for conventional crops, carbon allowances, and a second-generation biofuel crop. The agent-based approach introduces interactions among farmers about new technologies and market opportunities, and includes the updating of forecast expectations and uncertainties using Bayesian inference. The model is applied to a semi-hypothetical example case of farmers in the Salt Creek Watershed in Central Illinois, and a sensitivity analysis is performed to effect a first-order assessment of the plausibility of the results. The results show that the most influential factors affecting farmers' decisions are crop prices, production costs, and yields. The results also show that different farmer behavioral profiles can lead to different predictions of farmer decisions. The farmers who are predicted to be more likely to adopt new practices are those who interact more with other farmers, are less risk averse, quick to adjust their expectations, and slow to reduce their forecast confidence. The decisions of farmers have direct water quality consequences, especially those pertaining to the adoption of the second-generation biofuel crop, which are estimated to lead to reductions in stream nitrate load. The results, though empirically untested, appear plausible and consistent with general farmer behavior. The results demonstrate the usefulness of the coupled agent-based and hydrologic-agronomic models for normative research on watershed management on the water-energy nexus.
 Watersheds are coupled human-natural systems where human decisions affect the environment (e.g., water quality, streamflow), and in turn, are affected by it (e.g., resource quality, water availability). The reliable modeling of such systems for planning, management, and other purposes requires an approach that considers both the human and natural aspects. The primary objective of this study is to develop an agent-based model of a population of hypothetical farmers and to couple it to a watershed-scale hydrologic-agronomic model that is useful for normative research on watershed management, particularly on water quality management and the energy-water nexus. Normative research in these and other areas is valuable in providing insight on future possibilities for planning purposes [e.g., Barreteau et al., 2004; Schluter and Pahl-Wostl, 2007].
 The decision context concerns farmer choices about growing first- and second-generation biofuel crops and carbon trading, which are of interest because of climate change and energy independence concerns. Currently, in the United States, the dominant biofuel crop is corn, a first-generation biofuel crop, used for producing corn ethanol. However, with advances in cellulosic ethanol technology, perennial grasses representing second-generation biofuel crops are expected to become dominant as well. The model allows for the exploration of the potential effects of various public policies on these decisions and the resulting implications on water quality. In this study, the Salt Creek watershed in Central Illinois, a typical Midwestern agricultural watershed, is used as the study site, where markets are assumed to exist for conventional crops (including corn), carbon allowances, and a second-generation biofuel crop, Miscanthus x giganteus. Miscanthus is a perennial grass that possesses many characteristics desirable in an energy crop [Heaton et al., 2004]. These markets affect farmer crop and best management practice (BMP) decisions, and consequently, stream nitrate load.
 The model of farmer decision-making is developed using agent-based modeling (ABM). ABM differs from the conventional approach, which is to characterize a system by a least cost, or maximum utility, equilibrium. The latter can be thought of as a “top-down” approach, while ABM is a “bottom-up” approach where each agent is a discrete autonomous entity with distinct goals and actions within a particular social context [Bonabeau, 2002]. An “equilibrium” can emerge in an agent-based model, but it is not imposed. This makes ABM an arguably more realistic approach, with the potential to capture at least partially, the complexities of human decision-making.
 The agent-based model of farmers incorporates farmer adaptation and interactions. The famers adapt as they update their forecasts of future prices, costs, yields, and weather on the basis of new observations, and they interact by sharing information with one another. Bayesian inference is used to model the farmers' adaptation of their expectations and uncertainties of those variables, which can be thought of as a form of learning. It is a statistical method to weight existing expectations against observed values and to adjust uncertainties as functions of differences between expected and actual outcomes. By manipulating relevant Bayesian parameters, together with assumptions of different degrees of risk aversion, two behavioral types are defined: cautious and bold.
 A hydrologic-agronomic model of the watershed is used to compute crop yields and predict water quality (specifically, stream nitrate load) at the watershed outlet as a consequence of the farmers' decisions. The crop yields feed back to influence decisions in the agent-based model. Together, the models provide a complete representation of the whole system. This coupled modeling approach reveals how farmers' behaviors, characterized by their attitudes to risk and Bayesian adaptation processes at the local level, affect outcomes (i.e., crop acreages and stream nitrate load) at the watershed level. Also of interest are the diffusion of new practices, namely, miscanthus cultivation, carbon trading, and conservation tillage, which are assumed to be new to the farmers at the start of the simulation period.
 The unique micro-to-macro approach of ABM provides a high degree of flexibility to specify agents' decision-making rules. However, as discussed by Zenobia et al. , a major challenge in ABM lies in the construction of credible and accurate decision rules. While ABM provides a framework for incorporating real behavior, it is not straightforward to define what that behavior might be. The data requirements for such a task are significantly greater than those for defining behavior deemed economically rational, that is, profit- or utility-maximizing. The definition of plausible behavioral is an active area of research in ABM and elsewhere, including behavioral economics [Loewenstein, 2007].
 The flexibility of ABM also enables the incorporation into the model, without any sacrifice to the ease of solution, transaction costs that are often partly intangible and may not be described as simple, continuous differentiable functions of direct cost or other factors [e.g., Wilhite, 2001]. ABM is also able to accommodate dynamic conditions, where the environment and the agents themselves change with time as a result of experience and learning [e.g., Barr and Saraceno, 2005; Lettau, 1997; Palmer et al., 1994]. The dynamic nature of ABM, together with the bottom-up approach, makes it suitable for modeling interactions between agents. For example, ABM has been widely applied to modeling markets whose outcomes depend strongly on agent interactions [e.g., Ehrentreich, 2008; Sueyoshi and Tadiparthi, 2008].
2.1. ABM in Watershed Management
 A major difference between watershed systems and the economic and social systems where ABM has most commonly been applied is the strong interdependence between human agents and the environment. In the context of an agricultural watershed, as envisioned in this study, the link between human agents, i.e., farmers and the environment, is in the formation of the farmers' perceptions of crop yields and weather. Those perceptions are based on past observations and decisions. They affect the farmers' current decisions, which in turn, affect runoff and consequently, stream water quality. A complete model of the system consists of a physical model of the environment in addition to an agent-based model of the human agents.
 An early work on ABM in watershed management is by Lansing and Kremer , who developed an agent-based model of the cropping decisions of Balinese farmers. In Bali, rice paddy fields are flooded and drained according to the crop being planted and the date of planting. A farmer's decisions not only affect the amount of water available to downstream users but also the condition of pest populations. The authors found that farmers' actions to maximize individual yields without any conscious effort to coordinate with neighbors sufficient to lead to the emergence of complex synchronized cropping patterns that are observable among real Balinese farmers.
 Also, many of the aforementioned studies are based on relatively simple models of the environment. The present study is based on a complex hydrologic-agronomic model that takes into account spatial and temporal differences in soil type, soil nutrients, crop yields, climate, etc. The added complexity of the environmental modeling is necessary because this study requires modeling not only of the water budget, but also the nitrogen budget (which affects and is affected by crop growth) and associated water quality.
 In this study, farmers are assumed to be unfamiliar with miscanthus crop cultivation, carbon trading, and conservation tillage. The adoption of these practices over time, the effects of public policies and their adoption, and the implications for water quality are the foci of this study. ABM has been used in several studies of innovation diffusion. Galán et al.  adapted the diffusion model of Bass  and combined it with a behavior diffusion model to model the adoption of water-saving technology by agents in a metropolitan area. The agents' decisions to reduce water consumption were based on probabilities of behavior change as defined by current behaviors, as well as exogenous terms representing water availability and/or public awareness of the importance of water conservation.
Berger  used ABM to model technology diffusion in an agricultural system. The author used linear programing to model the decision-making processes of individual farmers as functions of expectations of each ones own costs of adoption, which decreased as more farmers adopted the technology. The diffusion of the technology was also characterized by dividing the farmers into groups, with each group willing to adopt the technology only after a characteristic percentage of the population had done so.
 Other examples of ABM applied to innovation diffusion include Schwarz and Ernst , Delre et al. , and Alkemade and Castaldi . In the present study, innovation diffusion is modeled by considering the agents' variances of net return as barriers to adoption. The variances associated with new practices decrease as neighbors and the agents themselves adopt the practices. Diffusion occurs when an agent's variance of a new practice is sufficiently reduced, making the expected return from adopting the new practice worth the risk of adoption. This process is described in sections 3.3 and 3.4.
3. Data and Methods
 The agent-based model in this paper is based on individual farmers who are modeled as economic optimizers. The farmers make their decisions as functions of risk aversion, and forecasts of future prices, costs, yields, and weather, which they update according to a Bayesian algorithm as new observations become available from public sources or interactions with neighbors. The agent-based model is linked to a hydrologic-agronomic model of the system (see Figure 1). The hydrologic-agronomic model generates crop yield data as input to the agent-based model, which, in turn, computes the farmers' management decisions. Those decisions feed back to the hydrologic-agronomic model to estimate stream nitrate load. Crop acreages and nitrate load data are the so-called “emergent outcomes” resulting from the interactions and decisions of the individual agents.
 The Salt Creek watershed in Central Illinois serves as an example case. The watershed drains to the Illinois River and eventually to the Gulf of Mexico. It is predominantly agricultural; its primary crops are corn and soybeans planted in rotation. The watershed is typical of many agricultural watersheds in the Midwest in losing significant quantities of agricultural nutrients used for corn production, in particular, to surface waters. A new perennial biofuel crop, miscanthus, is assumed to be available as an alternative to corn and soybeans. Miscanthus uses relatively little nitrogen, but its production cost, yields, and market demand are not well known to farmers.
3.1. Hydrologic-Agronomic Model
 The hydrologic-agronomic model of the Salt Creek watershed was developed by Ng et al. [2010a]. The model was developed using the Soil and Water Assessment Tool (SWAT) [Arnold and Fohrer, 2005], and calibrated and validated using historical data for daily streamflow at four locations within the watershed, annual corn and soybean yields, and monthly nitrate load at the watershed outlet. To calibrate the model, 22 parameters affecting surface runoff, nutrient transformation, evapotranspiration, groundwater flow, and crop growth were adjusted. SWAT includes a database of default parameters for a number of crops (e.g., corn, soybeans, wheat, etc.), but such values are unavailable for miscanthus as it is a relatively new crop. To model miscanthus in SWAT, crop growth parameters from Ng et al. [2010b] are applied.
3.2. Deterministic Model of an Individual Farmer
 The decisions of an individual farmer are represented by a deterministic economic model where choices are based on crop, fertilizer and carbon allowance prices, and crop production costs and yields, Figure 2 provides a flow diagram of the different factors affecting the farmer. The model forms the basis for the development of a stochastic model of an individual farmer, which together with the deterministic model, forms the basis for the agent-based model of multiple farmers. Here due to space limitations, a qualitative description of the decision model, with just a few key equations, is provided. Ng  presents the full model.
 Given current and future prices, costs, yields and weather, a farmer must decide the best combination of crops and best management practices (BMPs) for a given year. In this study, two BMPs (conservation tillage and uncultivated grass planting) and three crops (corn, soybeans, and miscanthus) are considered. The farmer's objective is to maximize the sum of discounted returns over a planning horizon of N years. Assuming economically rational behavior and perfect information, the farmer's objective function is
where TP, total profit, is the sum of discounted returns over N years, Rn is yearly net returns, and is a discount factor between zero and one to represent the farmer's time valuation of money. In this study, is assumed to fall between 0.92 and 0.98 and differs from farmer to farmer; CSn is the farmer's gain from crop sales in year, n and is a function of crop prices and yields; FCn is the cost of nitrogen fertilizer in year n and is a function of fertilizer price and usage; OCn is the total cost of crop production (excluding nitrogen fertilizer cost) in year n and is a function of crop choice and tillage type; and CCn is income from carbon trading in year n and is a function of carbon price, crop choice, and tillage type.
 A yearly time step for farmers' cropping decisions is assumed and should suffice for the scale of this study, in which simulations are run for periods of 15 yrs and the primary results of interest are those at the watershed level. Further, a yearly basis is consistent with historical cropping patterns in Illinois, as well as current crop budgeting and risk management models developed by the University of Illinois Extension to aid local farmers (available at http://www.farmdoc.illinois.edu). The farmer is assumed to be responsible for three plots of land, one of which is marginal land for purposes of crop production, and the other two are of standard quality for the watershed. This approach allows farmers to grow multiple crops simultaneously as a means of risk management [Falco and Perrings, 2005]. The plot of marginal land consists of low-lying riparian land, and when compared to the plots of land of standard quality, has a higher frequency of flooding. The fraction of total cropland that is marginal is farmer-specific.
3.2.1. Crop Yields
 The hydrologic-agronomic model is used to compute yields, which, because of variations in temperature, precipitation, and soil type, are different for different points in time and space. Corn and soybean yields depend on weather and flooding. Miscanthus yield is also weather-dependent, but is assumed to be relatively insensitive to brief occasional flooding. Miscanthus yield is age-dependent. Typically, there is insufficient biomass for a harvest in the first year of establishment, about 50% of the maximum yield in the second year, and maximum yields in the third year and onward [Heaton et al., 2004]. A miscanthus crop can continue yielding up to 10 to 20 years before replanting is required [Lewandowski et al., 2003; Vleeshouwers, 2002]. In this study, miscanthus yields are assumed to start declining after year 15.
3.2.2. Prices and Costs
 In the model, crop, fertilizer, and carbon allowance prices are time-varying exogenous variables. Historical corn, soybean (available at http://www.farmdoc.illinois.edu) and fertilizer (available at http://www.ers.usda.gov) prices are used. Since miscanthus is not yet grown commercially in the U.S., there are no price data to draw upon. Historical carbon allowance prices are available, but records are too short for use here. Therefore, for miscanthus and carbon allowances, artificial prices based on possible future scenarios are assumed.
 Nonfertilizer production costs (e.g., energy, labor, chemical, and equipment costs) for corn and soybean cultivation under typical conditions (i.e., conventional tillage and a 1:1 corn-soybean rotation schedule) are drawn from historical values [Schnitkey and Gupta, 2007; Schnitkey and Lattz, 2007]. Actual nonfertilizer production costs are then adjusted for conservation tillage, alternative crop rotation schedules, and frost damage.
 Nonfertilizer production costs for miscanthus are taken from Khanna et al. , and depend on the age of the crop. In the first year of establishment, the costs of planting and chemicals are significant but not other costs. In the second year and onward, there is no cost of planting and a reduced cost of chemicals, but harvesting and transportation costs can be significant. Miscanthus is assumed to be relatively insensitive to frost damage, and therefore, there is no additional cost of replanting due to frost. It also does not require annual tillage (except in the first year of establishment), so there are no conservation tillage savings.
3.2.3. Carbon Allowance Trading
 Carbon allowance trading provides an incentive for farmers to adopt conservation tillage, convert existing cropland to uncultivated grassland and/or cultivate miscanthus. (In this study, miscanthus is differentiated from “uncultivated grass.” When cultivating miscanthus, there is fertilization (albeit, at moderate levels), harvesting, and other activities to maximize profit; when planting uncultivated grass, there are no such activities beyond initial planting.) The rates of carbon offset that may be claimed for sale from adopting uncultivated grass, conservation tillage, or miscanthus are assumed to be 0.247, 0.148, and 0.198 kg CO2e/km2, respectively, per year. Except for the rate for miscanthus, these rates are the same as those used by the Chicago Climate Exchange (CCX) (available at http://www.chicagoclimateexchange.com). A miscanthus carbon offset rate of 0.198 kg CO2e/km2/yr is the average of the other two rates, and is assumed here because miscanthus can be thought of as somewhere between uncultivated grass and conservation tillage in terms of intensity of crop cultivation. (There is nitrogen fertilization of miscanthus but not of uncultivated grass. Therefore, it is assumed here that miscanthus has a smaller effective carbon sequestration potential than uncultivated grass, since nitrogen fertilization has been shown to increase soil emissions of nitrous oxide, a greenhouse gas, emissions [Bouwman, 1996].) In line with CCX regulations, to qualify for carbon offsets by adopting conservation tillage or converting to uncultivated grassland, the farmer must commit to doing so for five continuous years. However, the same commitment is not explicitly required for miscanthus cultivation since the farmer is likely to maintain the miscanthus crop for at least that long once it is established and initial planting costs are covered.
3.3. Stochastic Model of an Individual Farmer
 When the system is deterministic, all price, cost, yield, and weather variables are time-varying but known. However, in reality, the system is stochastic; these variables are unknown to the farmer, who instead forms perceptions of them, which can be represented as probability distributions. Those perceptions are updated with new observations using Bayesian inference, as described in section 3.3. The farmer's objective is to maximize total utility, TU, over the planning horizon where utility is a function of his perceptions of future conditions. Therefore, assuming a time-separable form of utility, the farmer's objective function becomes
where U(·) is the farmer's utility as a function of return in period n, Rn. In this study, a mean-variance type of utility is assumed, where utility is proportional to expected return and inversely proportional to variance of return. For all n,
E(·) is expectation and Var(·) is variance. E(Rn) and Var(Rn) can be readily computed from basic statistical relationships if the expectations, variances, and covariances of the independent random variables defining Rn are known. Parameter r is a coefficient representing the farmer's risk aversion. The greater is r, the greater his aversion to risk. When r is zero, the farmer is risk-neutral and the objective function simplifies to one of maximizing the sum of discounted expected returns.
 Bayesian inference is a statistical method of updating existing knowledge. Knowledge is characterized by probability distributions over variables of interest. The farmer begins with so-called “prior” distributions and uses new observations to revise the distributions to produce so-called “posterior” distributions. In this study, Bayesian inference is used to model the farmer's adaptation of forecasts of future conditions as he makes new observations on yield, price, cost, and weather variables. The resulting posteriors then become the priors for the next time period.
 To represent agent heterogeneity, the Bayesian parameters affecting the updating of the priors to the posteriors are set differently for different farmers (see section 3.4.1). The farmer's observations are drawn from publicly available information, personal experience, and interactions with neighbors (see section 3.4.2). Bayesian inference, as applied here, does not account for the influence of information outside of the time series data used in the model which, in reality, can be significant [Brown and Rozeff, 1979]. The inclusion of such information is beyond the scope of this study, particularly because it is often difficult to characterize and its effects are not easily defined in a mathematical sense.
 This Bayesian approach is consistent with the adaptive expectations model [e.g., Schmalensee, 1976; Williams, 1987], which has been shown to explain adequately expectation formation by real agents. In the adaptive expectations model, as in Bayesian inference, an agent's forecast of a future event is a function of his past forecasts and forecast errors. The adaptive expectations model, however, unlike Bayesian inference, does not compute the agent's forecast confidence, which, as shown by Schmalensee , is affected by the accuracy of his past predictions. Other studies using Bayesian inference to model the beliefs of individual farmers include Feder and O'Mara  and Foltz .
 A farmer's perceptions of the random variables of yields, prices, costs, and weather are assumed in this study to be normally distributed, with the exception of two weather-related variables (to describe the occurrences of flood and frost events) that are assumed to be binomially distributed. For variables that are correlated, their correlation coefficients are assumed fixed at predetermined constants estimated from historical data.
 For a normally distributed variable, hH, assume it has a prior distribution characterized by the normal-inverse- distribution, that is, h is normally distributed with a variance of and a mean of , where follows a scaled-inverse- distribution, and is conditional on and is normally distributed. The normal-inverse- distribution is defined by four parameters, , , , and . is the mean of , while is the variance of . and are the degrees of freedom and scale of , respectively. The normal-inverse- distribution results in a posterior of the same form with the parameters , , , and [Gelman et al., 1995; Press, 2002]:
where nh is the number of data points in the current observation, sh is the observed standard deviation, and mh is the observed mean. As given in equation (4a), the posterior mean is simply the weighted average of the prior and observed mean; and in equation (4d), the posterior sum of squares is the total of the prior sum of squares, the observed sum of squares, and an additional uncertainty term to account for the difference between the observed and prior means. In this specification, the newly observed data dominate the posterior distribution when nh is large, and the prior distribution dominates when nh is small.
 The normal-inverse- distribution yields the predictive distribution of h as a Student-t distribution with degrees of freedom and the following expectation and variance:
 For a binomially distributed variable, gG, assume it has a prior characterized by a binomial- distribution such that g is binomially distributed with an uncertain probability of success that, in turn, follows a distribution with the shape parameters and . The binomial- distribution results in a posterior distribution of the same form with the updated shape parameters and [Press, 2002]:
where tg is the total number of new observations of g, out of which sg is the observed number of successes. The -binomial posterior yields the predictive mean and variance of ng future observations as [Bi, 2006]
where xg is the number of successes out of ng total future observations.
3.4. Agent-Based Model of Multiple Farmers
 An agent-based model of the crop and BMP decisions of multiple agents are developed on the basis of the deterministic and stochastic individual-farmer models presented above. Figure 3 presents an overview of the model. The agent-based model consists of 50 farmers. (In reality, there are more than 50 farmers in the study site. However, to limit to computational runtime of the agent-based model, some aggregation is assumed.) The farmers differ in terms of their land areas, fractions of marginal land, crop yields, time discount rates, ranges of foresight, and risk aversions. Crop yields differ from farmer to farmer because of differences in soil type and from year to year due to differences in weather.
 Before the start of the first year in the simulation period, the farmers' initial perceptions of future prices, costs, yields, and weather are set somewhere in the range of historical expectations and variances based on data for the 15 to 20 yrs prior to the first year of the simulation period. However, such historical data are unavailable for the production costs of miscanthus, as well as carbon allowance and miscanthus prices. Further, historical costs of and savings from conservation tillage, even though known, are not easily observable to farmers. In these cases, the initial perceptions are set to have large variances to represent the farmers' initial unfamiliarity with miscanthus cultivation, carbon allowance trading, and conservation tillage (which are assumed in this study to be activities with which the farmers, at the start of the simulation period, have no prior experience). The large variances have the effect of decreasing the farmers' confidence in predicting future profits from those activities. These expectations and variances are updated with time and experience using Bayesian inference as described in section 3.3 and as follows.
3.4.1. Farmers Adaptation of Forecasts
 Once the farmers' initial perceptions are set, simulations are then carried out iteratively, year by year, for the entire simulation period. At the beginning and end of each growing season (before planting and after harvest), the farmers' perceptions are updated with new observations according to the Bayesian algorithm in section 3.3. At the start of each new growing season, the farmers update their perceptions of corn and soybean prices as their futures prices become known. However, the farmers' perceptions of other prices, costs, yields, and weather are assumed unchanged at this point, presumably due to a lack of reliable forecast information for those items. The crop and BMP decisions of the farmers for that particular year are then determined by solving the deterministic or stochastic model for each farmer. After the farmers' decisions are computed and weather conditions are simulated, it is considered to be the end of the growing season. At this point, the farmers' perceptions of prices (including those of corn and soybeans) and also of weather are updated as their actual values for the year become known. This updating of the farmers' perceptions of incipient prices and weather (but not of costs and yields) is done by all farmers based on publicly available information that is accessible to all. Exceptions to the updating process are the farmers' perceptions of carbon allowance and miscanthus prices. As discussed in section 3.4.2, those perceptions (and perceptions of costs and yields) are updated by selected farmers depending on their own as well as their neighbors' decisions.
 The updating of a farmer's perceptions of variables that are assumed normally distributed (i.e., prices, costs, yields, and certain weather-related variables) depends on the parameters and in equations (4a)–(4d). For a normally distributed variable h, the two parameters (together with nh as described in section 3.3) define the weight that is given to the farmer's existing perception of h against new information (or in Bayesian terminology, the weight of the prior against new observations), as well as his reaction toward unexpected changes. A greater weight is given to the farmer's existing expectation of h when is large and vice versa. Further, a large together with a small means a relatively large increase in his level of uncertainty when an actual outcome is vastly different from his expectation. In this study, and are constantly readjusted to keep them constant so that the influence of the farmer's existing perception (prior) of h on his updated perception (posterior) is more or less unchanging with time. (If there is no readjustment of and , their values will increase with time, thereby increasing the influence of the existing perception of h and reducing the influence of new observations.) Different farmers have different values of and . Note, however, that they are assumed to be the same for all h for a given farmer.
 Similarly, the updating of a farmer's perceptions of variables that are assumed binomially distributed (i.e., certain weather-related variables) depends on the parameters and in equations (6a) and (6b). For a binomial variable g, the sum of and can be thought of as the sum of the farmer's experiences with g, out of which is the number of “successful” experiences, and the number of “unsuccessful” ones. In this study, and are rescaled after each new update so their sum is constant with time. As in the case of the normal variables, this method fixes the influence of the farmer's existing perception (prior) of g on his updated perception (posterior). The larger the sum of the two parameters, the greater is the influence of the existing perception (prior) versus new information. A larger sum of the two parameters can also be thought of as the farmer acting with a longer memory. In this study, the sum of and is different for different farmers, though it is assumed to be the same for all g for the same farmer.
 For the purposes of this study, two opposing behavior types are defined: cautious and bold. Cautious farmers have relatively high (assumed here as (15–25) and low (10–25) values, and large sums of and (15–20). This means that they are slow to adjust their expectations in response to new observations but quick to reduce their forecast confidence when new observations do not match expectations. On the other hand, bold farmers have low (0.5–5) and high (50–100) values, and small sums of and (5–10), meaning that they are quick to adjust their expectations with new observations but slow to reduce their forecast confidence when there are unexpected changes in the system.
 Bold farmers are also less risk averse than cautious farmers, and are assigned r values between 0–0.0005, while cautious farmers are assigned r values between 0.0005–0.0010 (see equation (3)). These ranges have been selected to represent a spectrum of values and induce a variety of decisions. At the same time, limits are placed on r to exclude implausible decisions.
 To illustrate the differences between cautious and bold farmers, Figure 4 shows how new observations of corn prices affect the price expectations and forecast uncertainties of a cautious and a bold farmer. In this particular example, the former has a of 24.9 and of 15.5, while the latter has a of 2.7 and of 66.8. In Figure 4, the solid lines and gray bands represent the farmers' expectations and standard deviations of corn price, respectively. Since standard deviation is a measure of uncertainty, the narrower the gray bands, the smaller the farmers' forecast uncertainties, and vice versa.
3.4.2. Interactions Between Farmers
 The farmers interact with neighbors to acquire cost, yield, and price information. These new observations are used to update their perceptions of costs, yields, and certain prices. A farmer's neighbor(s) is (are) assigned according to their spatial locations. Each farmer has control over one farm, whose spatial location is assumed to correspond to one of 50 agricultural subbasins in the hydrologic-agronomic model of the Salt Creek watershed. The 50 agricultural subbasins are numbered from 1 to 50 such that subbasins that are close in number are also close in space. It is thus assumed that each farmer x (corresponding to subbasin x) has two neighbors, farmers x + 1 and x − 1. However, farmers 1 and 50, being at the upstream and downstream edges of the watershed, are assumed to have only one neighbor each, farmers 2 and 49, respectively.
 In each year, at the end of the growing season, the farmers' perceptions of costs and yields (like his perceptions of prices and weather as described in section 3.4.1) are updated. (At this point, the hydrologic-agronomic model of the watershed is run to simulate yields for the year as functions of the farmers' crop and BMP decisions, and the year's weather realization.) However, the updating of cost and yield perceptions is carried out only by certain farmers, and not necessarily for all costs and all yields. The updating of a farmer's cost and yield perceptions depends on his decisions for the year and interactions with neighbors, which determine the information that is available to him. In this study, it is assumed that cost and yield information are available only when related activities are performed. For example, the cost of and savings from conservation tillage can only be known to a farmer if the farmer or one of his neighbors practices it. Similarly, the production costs and yield of a crop are observable to the farmer only if he or his neighbor is cultivating the crop.
 In the same manner, a farmer's perceptions of carbon and miscanthus prices are updated only when related activities are performed by the farmer or one of his neighbors. Since carbon trading and miscanthus cultivation are considered to be activities that are relatively new to the farmers, carbon and miscanthus prices are assumed to be information that is not commonly known. Miscanthus prices only becomes known to the farmer when he or one of his neighbors grows it. Similarly, the price of a carbon allowance only becomes known when he or one of his neighbors adopts conservation tillage, plants uncultivated grass, or cultivates miscanthus for generating carbon credits. (In reality, such information may be available on the Internet and/or government and crop advisor reports. However, it is assumed here that a direct experience based on interactions with neighbors has a stronger effect than an indirect experience based on these sources, and that a farmer would only know where, or even be interested, to obtain such information if he himself or a neighbor is involved in a related activity. Moreover, the farmers defined here as “bold” are more likely to adopt changes in the absence of interactions with neighbors and the parameter values assigned to them arguably cover these aspects of their behavior.)
 In this manner, the farmers' interactions determine their likelihoods of taking unfamiliar actions. A risk-averse farmer is less likely than a risk-tolerant farmer to take unfamiliar actions. However, as neighbors, who may be more risk tolerant or may have more optimistic expectations, decide to experiment with heretofore unknown techniques, that farmer's misgivings are reduced. Thus, with time, the farmer becomes more willing to experiment himself and, in turn, to influence his neighbors.
3.5. Method of Solution
 As shown in Figure 3, the agent-based model consists of two loops, an outer loop that is time (year)-incremented and an inner loop that is farmer-incremented. The outer loop contains Bayesian procedures (as described in section 3.3) to update the farmers' perceptions of prices, costs, yields, and weather with time and new observations, as well as procedures to call the hydrologic-agronomic model of the Salt Creek watershed to calculate crop yields and stream nitrate load. The inner loop encompasses computations to predict the crop and BMP decisions of an individual farmer based on the deterministic or stochastic individual-farmer models presented in sections 3.2 and 3.3.
 For each iteration of the outer loop, the inner loop is repeated 50 times to estimate the decisions of all 50 farmers for the year. Parallel programming is applied at this point such that multiple executions of the deterministic or stochastic individual-farmer model are solved concurrently. Note that the farmers make their decisions independent of each other, i.e., they do not consider others' profits or the benefit of the system as a whole when making their decisions. Also, unlike an irrigation situation where the water use decision of a farmer affects downstream farmers, in the current situation, the decisions of a farmer do not physically affect other farmers in any way.
 The deterministic and stochastic individual-farmer models are solved using dynamic programming by backward induction [Bellman, 1957]. For a particular year, for a particular farmer, the solutions to the models depend on the farmers' perceptions of future conditions for that year. The model solutions also depend on the farmer's state for the year, which is a function of his state and decisions in the previous year. Specifically, the variables defining the farmer's state for the year are (1) his previous year's combination of crops, (2) the ages of his plots of miscanthus, if any, in the previous year, (3) the numbers of years left of mandatory conservation tillage and grass planting for each of his three plots of land, and (4) whether or not he is equipped for conservation tillage (if not, existing equipment will have to be modified at a cost should he decide to practice conservation tillage in the current year). Note that when solving the deterministic or stochastic models using dynamic programming, the complete sequence of optima for the entire period within the farmer's range of foresight is identified. However, only the optimum for the current year is relevant and kept.
 The agent-based model is developed and coupled to the hydrologic-agronomic model of the Salt Creek watershed using the C++ programming language. The parallel programming is implemented using the OpenMP Application Program Interface (API) (available at http://openmp.org/wp/).
4. Results and Discussion
4.1. Sensitivity Analysis of Agent-Based Model
 To assess the agent-based model, a sensitivity analysis is conducted to evaluate the behaviors of selected output variables in response to changes in 18 selected sets of input variables. Admittedly, such an approach does not prove the realism of the model, but, in the absence of empirical data, it is a common means of establishing model plausibility [e.g., Forrester and Senge, 1980; Miller, 1974].
 The sensitivity analysis is carried out by changing one set of input variables at a time and running the agent-based model each time for the resulting combination of input variables. The simulations are carried out for the simulation period 1985–2000 for a population of 26% bold–74% cautious farmers. The farmers adapt and interact as described in section 3.4. The sensitivity analysis is carried out to simulate the effects of the 18 sets of input variables in Table 1 on four output variables, namely the fractions of the Salt Creek watershed in corn, soybeans, miscanthus, conservation tillage, and enrolled in carbon trading averaged over time. (Although the agent-based model allows farmers to convert cropland to uncultivated grass for the purpose of producing carbon offsets, the fraction of watershed in uncultivated grass is not of concern since for all scenarios examined here and below, the fraction of the watershed in uncultivated grass is negligible (<1%).) To do so, the 18 sets of input variables are perturbed one set at a time from their “base” values by −50% to +50% and the resulting values of the four output variables recorded.
Table 1. 18 Sets of Input Variables Perturbed in the Sensitivity Analysis of the Agent-Based Model
Description of Input Variables in Set
Time series of corn prices and farmers' initial expectations of them
Time series of soybean prices and farmers' initial expectations of them
Time series of miscanthus prices and farmers' initial expectations of them
Time series of carbon allowance prices and farmers' initial expectations of them
Time series of fertilizer prices and farmers' initial expectations of them
Time series of corn production costs and farmers' initial expectations of them
Time series of soybean production costs and farmers' initial expectations of them
Time series of miscanthus production costs and farmers' initial expectations of them
Costs of converting existing equipment for conservation tillage and farmers' initial expectations of them
Savings in crop production from conservation tillage and farmers' initial expectations of them
Farmers' risk aversions
Farmers' time discount factors used in the valuation of the time value of money
Farmers' initial standard deviations of miscanthus production costs and prices
Farmers' initial standard deviations of conservation tillage costs and savings
Farmers' initial standard deviations of carbon allowance prices
Time series of corn yields and farmers' initial expectations of them
Time series of soybean yields and farmers' initial expectations of them
Time series of miscanthus yields and farmers' initial expectations of them
Table 2. Average Annual Nitrate Load (103 t N/yr) at the Salt Creek Watershed Outlet From 1985–2000a
Averages in table are for a population of farmers with perfect foresight, a population of 26% cautious–74% bold farmers that are adaptive, and a population of 26% cautious–74% bold farmers that are stationary (with zero adaptation of forecasts).
MPR is miscanthus price ratio, the ratio of miscanthus price to the average of corn and soybean prices.
 For the most part, the base values are set according to historical values. However, historical data are unavailable for miscanthus and carbon allowance prices. Therefore, base values for the former are set at 0.4 of the average of corn and soybean prices. Base values for the latter are constructed from a forecast of future prices by Pew Center on Global Climate Change , and are set at $17/t CO2e in 1985 and rising steadily to $40/t CO2e in 2000.
Figure 5 reports the results of the sensitivity analysis. From the sensitivity analysis, the most influential factors affecting a farmer's crop decisions are crop prices, production costs, and yields. This is in line with intuition, as well as the literature [Schnitkey and Batts, 2011]. Additionally, the output variables change in directions according to intuition. For example, the fractions of land in crops are positively correlated to crop prices and yields, but negatively correlated to production costs. Similarly, the fraction of land in conservation tillage (and with it, the fraction of land enrolled in carbon trading) increases with potential cost savings and also with carbon allowance prices, but decreases with the costs of converting existing equipment. Moreover, the fractions of land in miscanthus and conservation tillage, with which farmers are assumed to have no prior experience at the start of the simulation period, decrease with the farmers' initial uncertainties of associated prices, costs, and savings.
 To assess the agent-based model further, a second set of simulations is carried out to evaluate farmers' aggregate behavior for different assumptions of the proportions of bold and cautious farmers, and with and without farmer interactions. The simulations are run for the same simulation period, and with the same values of input price, cost, yield, and weather variables as for the sensitivity analysis described above. Figure 6 presents the results. Figure 6 compares the aggregates of farmers' crop and BMP decisions for five populations of farmers. In the first population, the entire population is 100% bold (0% cautious) and, in the second through the fifth, the percentages are 74% (26%), 50% (50%), 26% (74%), and 0% (100%), respectively. The results are presented as time series of fractions of the watershed in corn, soybeans, miscanthus, conservation tillage, and enrolled in carbon trading.
Figure 6 also compares the results when the farmers interact and share information with neighbors and when they do not. For comparison, Figure 6 presents predictions for a population of farmers with perfect foresight and another population of 26% bold–74% cautious farmers with zero adaptation of forecasts. For the farmers with perfect foresight, risk aversion and the bold-cautious distinction do not apply as there is no uncertainty of future prices, costs, yields, and weather, and therefore, no Bayesian adaptation of forecasts and no aversion to risk. As for the farmers with zero adaptation, their perceptions of future conditions do not change with new observations and are stationary at their initial values. For farmers with zero adaptation of forecasts, the definitions of bold and cautious still apply (in terms of risk aversion) even though there is no Bayesian updating, but there is still uncertainty, and therefore the potential for aversion to risk.
 The results in Figure 6 agree with intuition. Take, for example, the farmers' adoption of miscanthus cultivation, which is considered in this study as a new practice with which the farmers have no prior experience at the start of the simulation period. In general, the farmer populations that are interacting and with a higher proportion of bold farmers tend to adopt miscanthus cultivation quicker. Bold farmers, who are more risk tolerant, are more willing to adopt miscanthus cultivation. And, as they interact with their neighbors, who may be cautious and therefore, more risk averse, their neighbors' uncertainty about miscanthus cultivation is gradually reduced until it reaches a point where they too are willing to adopt it. In this manner, miscanthus cultivation is propagated from one farmer to the next, eventually to include the entire population. The same pattern can be observed for conservation tillage and carbon trading. As shown in Figure 6, the farmers' uptakes of the three new practices all follow an S-shape pattern, which is consistent with predictions of the Bass model of new product diffusion [Bass, 1969].
 A closer look at the individual decisions of the farmers (not shown here) reveals that when adopting miscanthus cultivation or conservation tillage for the first time, the farmers (whether bold or cautious) tend to do so first on one (out of three) plot of land, usually the marginal plot, which is the smallest and which has a higher flood frequency (which affects corn and soybean yields but not miscanthus yield). After some time, they will plant miscanthus on a second plot of land, and finally on the third plot. Due to their risk aversion, most of the farmers are not willing to commit all their land at once to the new practices but will do so only after they have gained some experience and reduced their uncertainties. This pattern is consistent with observations and discussions in the literature [e.g., Tisdell, 2000; Scherr, 1995].
 The observation that the results differ for the different farmer populations in terms of the consistency of their year-to-year decisions also agrees with intuition. Consider, in Figure 6, the time series of the fractions of the watershed in corn and soybeans. The time series for the populations with higher proportions of cautious farmers tend to be more stable with time. There are two reasons for this. First, cautious farmers are relatively slow to adjust their perceptions of prices, costs, yields, and weather with new observations, and therefore, slow to adjust their decisions too. Second, the higher levels of risk aversion ascribed to cautious farmers mean that they tend to cultivate mixed crops (as opposed to single crops) to reduce the uncertainties in their net returns. Most of the time, cautious farmers allocate roughly the same amount of land to corn as to soybeans. The results here are consistent with other studies, such as Falco and Perrings , which have found risk aversion to be a major reason for farmers to diversify their activities.
 In general, farmers in Central Illinois have been quite consistent in their crop decisions. For several decades, crop acreage patterns in that region have been fairly constant. (An exception is 2004–2008, when corn acreage showed a sudden increase in response to a growing demand for corn-based biofuels.) Perhaps it can thus be inferred that the historical behavior of real farmers tend more toward caution than boldness, and that bold farmers are a minority if they exist at all.
4.2. Example Application
 The agent-based model coupled to the hydrologic-agronomic model may be used to study farmer decisions and the environmental impact of those decisions under various assumptions of farmer adaptation of forecasts and interactions, and different scenarios of prices, costs, yields, and weather. For example, Figure 7 presents predictions of the time-average fractions of the watershed in corn, soybeans, miscanthus, conservation tillage, and enrolled in carbon trading over the simulation period 1985–2000 for multiple combinations of miscanthus and carbon allowance prices. The predictions are for the 26% bold–74% cautious population of adaptive and interacting farmers. Predictions for when they have perfect foresight and when they are stationary (i.e., when there is no adaptation of forecasts), are also provided for comparison.
 The results are for four scenarios of carbon allowance prices based on forecasts of future prices by the Pew Center on Global Climate Change . These scenarios differ in terms of their starting and ending prices but are similar in that they all show a steady rise in the price with time. In Figure 7 and below, they are identified by their starting prices in 1985, the first year of the simulation period. In the first scenario, the 1985 carbon allowance price is $13/t CO2e and in the second, third, and fourth, $17, $28, and $33/t CO2e, respectively. Note that the prices under the $17 scenario are the same as the prices used in the sensitivity analysis in section 4.1. Miscanthus price is assumed to be a fixed multiple of the average of corn and soybean prices. Here, two scenarios are examined: in the first, the ratio of miscanthus price to the average of corn and soybean prices is set at 0.4 (as assumed in section 4.1) and in the second, 0.5.
 As in Figure 6, it is apparent that different assumptions of farmers' behavioral profiles lead to very different predictions of their decisions and the environmental consequences of those decisions. For example, Figure 7 shows that when there is a positive miscanthus price, there are large differences in miscanthus acreage between farmers who are adapting their forecasts, those with perfect foresight, and those who do not adapt. The latter are very unlikely to cultivate miscanthus on a large scale. On the other hand, farmers who are adaptive are more likely to grow miscanthus if its price is favorable in relation to corn and soybean prices. Further, for a given set of prices, farmers with perfect foresight have an even greater likelihood of cultivating miscanthus. The implication of this is that a biofuel crop program (based on uncertain market prices) can succeed or fail depending on the individual behaviors of participating farmers.
 Similarly, as shown in Table 2, stream nitrate load, under the same set of prices, can be quite different between farmers with different adaptive tendencies. As before, the implication is that the success of a water quality protection program based on the cultivation of second-generation biofuel crops and/or carbon trading to stimulate conservation tillage depends on the individual behaviors of participating farmers.
5. Conclusions and Future Work
 An agent-based model of farmers' crop and BMP decisions under the influence of carbon allowance and miscanthus market prices is developed, and a sensitivity analysis is conducted to assess the model. The agent-based model is linked to a hydrologic-agronomic (SWAT) model of the Salt Creek watershed in Central Illinois to simulate the interactions between agents (farmers), and between agents and the environment. The coupled model is applied to 50 hypothetical farmers who are heterogeneous in terms of initial expectations and uncertainties of future prices, costs, yields, and weather, and the manner in which they update those perceptions with new observations, which in this study is modeled using Bayesian inference. They are also heterogeneous in terms of land area, fraction of marginal land, crop yields, time discount rate, range of foresight, and risk aversion. Further, the farmers are interacting in their knowledge of initially unfamiliar activities, i.e., carbon trading, conservation tillage, and miscanthus cultivation. Their uncertainties of the costs of and returns from those activities are reduced as they or their neighbors engage in the activities and gain experience.
 The agent-based model coupled to the hydrologic-agronomic model is useful for future normative research in a number of areas. The ABM framework allows great flexibility in modifying the model for different purposes. Further, the use of Bayesian inference to model farmers' adaptation of future variables allows for the explicit consideration of adjustments in farmers' expectations and uncertainties, which may be significant, depending on the rate and magnitude of change. These make the coupled models, with appropriate modifications, suitable for future work on water quality policy modeling, the bioenergy-water nexus, and climate change impacts on farmer behavior (and the resulting consequences for water quality and streamflow).
 The results show that the most influential factors affecting farmers' decisions are crop prices, production costs, and yields, and suggest that the historical behavior of real farmers tend more toward caution than boldness. These findings agree with intuition demonstrating the feasibility and robustness of the agent-based model. The results also show that different dispositions toward adaptation of forecasts and interaction with neighbors can lead to different predictions of farmers' decisions and the environmental consequences of those decisions. Generally, the farmers who are predicted to be more likely to adopt new practices are those who interact more with other farmers and who are defined as bold, i.e., who are less risk averse, are quick to adjust their expectations, and are slow to reduce their forecast confidence. The results, although impossible to empirically verify with current data, appear plausible and consistent with general farmer behavior. The implication of these results is that the outcome of a policy, such as a biofuel crop program or a water quality program, is subject to the individual behaviors of the human participants.
 The results demonstrate the potential of ABM to capture, at least partially, the complexities of human decision-making. However, although ABM allows for the incorporation of noneconomically rational behavior [Ariely, 2008], this work does not take advantage of that capability. This is primarily because of the lack of empirical data that can be used to deduce farmers' actual thought processes to develop more realistic rules that are able to account for the noneconomic factors behind their motivations. Examples of such noneconomic factors are the reasons why in reality, many farmers appear to be yield, rather than profit, maximizers, and do not practice conservation tillage and may not cultivate miscanthus in the future, even if these activities are economically optimal. It should be acknowledged that farmers face many barriers to adopting new practices that are not easily defined, e.g., conflicting information, increased complexity, incompatibility with other aspects of farm management and personal objectives, high implementation costs, and capital outlay, etc. [Vanclay and Lawrence, 1994].
 Thus, for future work, it would be worthwhile to conduct interviews and/or role-playing games involving real farmers to obtain empirical data on their decision-making and adaptation processes. This idea of conducting interviews and role-playing games to deduce real behavior has been implemented before [Castella et al., 2005]. Empirical data from historical records, if available, may also be useful for providing a retrospective view of the farmers' behavior. For example, historical data on farmers' adoption of hybrid seeds or other new technologies can be analyzed to gain insights, especially on risk, uncertainty, and updating of perceptions [e.g., Smale and Heisey, 1993; Sarkar, 1998]. This is an opportunity for interdisciplinary work between economists, psychologists, sociologists, and engineers.
 Empirical data are also useful for the field validation of the agent-based model, as done by Bianchi et al. . Provided suitable data are available, such an endeavor would entail the adjusting of model parameters to produce an acceptable fit between simulated and actual values of output variables of interest. Because of the large number of adjustable parameters, an autocalibration approach utilizing an appropriate optimization method may be required. From the sensitivity analysis in section 4.1, key data necessary for the field validation include price, yield, and cost data. Also of importance are the actual proportions of farmers that may be regarded as bold and cautious. For now, empirical validation remains a challenge due to the lack of data, but nonetheless, is a possible direction for future work when suitable data become available.
 This work is supported in part by project no. INT USGS 06HQGR0 of the IL Water Resources Center, EBI-2007-136 of the Energy Biosciences Institute (EBI), and project no. ILLU-470-316 of the U.S. Department of Agriculture (USDA)/National Institute of Food and Agriculture. The authors thank George F. Czapar, Jürgen Scheffran, and Gary D. Schnitkey for their contributions. The opinions, interpretations, conclusions, and recommendations are entirely the responsibility of the authors and do not necessarily reflect the views of the sponsors or these colleagues.