Model Setup

We model a generic BMP adoption decision set in a generic performance-based nutrient reduction payment program. We start by simulating a watershed landscape of farms with heterogeneous soil productivity indicator s (0 < s < 1) where s is the normalized indicator for the productivity of the farm field for this BMP. This indicator represents BMP productivity or the nutrient reduction performance of adopting the BMP. Higher s represents higher productivity or reduction performance and thus higher payoff for participating in the performance-based programs. Such indicators can be derived from a hydrologic model such as SWAT or APEX that take into account physical characteristics including soil type and location, agricultural decisions including BMP adoption and fertilizing timing, and climate information such as rainfall. Each cell in our model represents one decision-making farmer with a farm field. We first randomly draw the soil productivity indices from a uniform distribution then smooth this landscape, making adjacent fields more similar to each other and reducing discontinuity, which is often the case in real landscapes. This also reflects the historical environmental outcome externality, where high/low runoff fields tend to increase/decrease the runoff of the adjacent fields (e.g., USDA NRCS soils, FAO of the UN). Figure 1 provides one example of such landscape of soil productivity indices, where the brighter cells correspond to higher s.

Figure 1. Example of Simulated Landscape

We consider a generic BMP that reduces nutrient runoff and improves the receiving waterbody's water quality. Farmer i at time period t who adopts this BMP has x _i,t of 1 (and 0 otherwise), indicating that they participate in the performance-based payment program and get a payoff corresponding to their nutrient reduction outcome R(s _i, s _j, x _i,t, x_j,t), which also depends on their neighbors’ participation through the binary vector x_j,t representing all neighbors’ participation. The model setup thus captures the literature's established fact that BMPs do not generally provide enough private benefit for farmers to adopt them, leading to low participation in performance-based programs.

There are costs to getting to know how the program works—getting certified and monitored, learning how to adopt the BMP, and more. We assume this transaction cost has two parts, a one-time cost (y _i,t) when trying the BMP and participating in the program for the first time and an annual cost (c _a) for machinery, labor, paperwork, and other such recurring costs. In addition, farmers have a willingness to adopt (w _i) that reflects different attitudes towards new BMP technology and participating in government programs. Borrowing from Schwarz and Ernst (Reference Schwarz and Ernst2009), we assume there are three types of farmers, which we call social leader or innovator (w_h), mainstream (w_m), and traditional (w_l) types.

As described above, we do not impose rational profit maximization or forward-looking ability on the farmers in our ABM. Farmers make decisions year by year while observing their own results, possibly learning from their neighbors, and being affected by their neighbors’ decisions. Farmers first consider adopting the BMP, comparing their willingness-to-pay to participate with the transaction cost, and then after a year of doing so they calculate their profit from the program to see if it has improved compared to the previous year,

(1)$$V_{i\comma t}\lpar {x_{i\comma t}\comma \;x_{\,j\comma t}\;s_i} \rpar = \lpar p_R\,\ast\, R\lpar s_i\comma \;s_j\comma \;x_{i\comma t}\comma \;x_{\,j\comma t}\rpar -y_{i\comma t}-c_a\rpar {\rm \ast }\;x_{i\comma t}$$

(2)$$x_{i\comma t} = 1\;{\rm if}\;w_i \ge y_{i\comma t} + c_a$$

where p _R is the price for a nutrient reduction credit. We have located farmers on a two-dimensional planar grid, so each farmer i has four neighbors from whom they can learn and whose BMP adoption decisions also affect i's performance. We test a wider neighborhood definition below.

Positive Spillover Effects

There are two positive spillover effects between each farm and its neighbors. One is the knowledge spillover effect, which is represented in the one-time information cost part of the transaction cost,

(3)$$y_{i\comma t} = c_I-\theta \,\ast\, I_{\,j\comma t-1}$$

for the first time participating and 0 in any later year. I_j,t-1 =1 if at least one of the neighbors has participated in the program prior to year t (and 0 otherwise). This captures the learning from experienced neighbors that reduces the information cost, thus lowering the hurdle to participate. The parameter c _I > 0 is the representative information cost, and θ> 0 is the information sharing coefficient.

The other externality is the environmental outcome externality from neighbors’ adoption of BMP and is represented by the nutrient reduction outcome function

(4)$$R\lpar s_i\comma \;s_j\comma \;x_{i\comma t}x_{\,j\comma t}\rpar = \alpha + \beta s_i + \gamma \sum\limits_{J = 1}^4 {s_j} \,\ast\, x_{\,j\comma t}$$

where α, β, and γ are positive parameters. Higher soil productivity leads to higher nutrient reduction and the positive externality from neighbors can be additive through γ, the conservation spillover coefficient, increasing the performance. This environmental outcome externality provides an incentive for farmers to share their knowledge to make it easier for their neighbors to participate.

The hypotheses that we investigate with this model are:

1. 1) Communities with a higher percentage of innovators will result in a higher participation rate, and in a non-linear fashion.

2. 2) Communities with a higher percentage of innovators will arrive at the equilibrium faster.

3. 3) Communities with a higher percentage of high-productivity land will result in higher participation rate.

4. 4) Communities with a higher percentage of “traditionals” can create regions where no one participates in the program.

Analytical Baselines

Before presenting the simulation results, we derive some mathematical relationships. The first component of a farmer's decision to adopt the BMP and participate in the performance-based payment program is the interaction between their WTP to participate and the participation cost. Combining Eq. [2] and Eq. [3], the farmer will participate (set x _i,t =1) if w _i ≥ c _a + c _I − θ*I _j,t-1. We start at t = 0 with no participation to simulate the beginning of a payment program, so in the first year that constraint becomes w _i ≥ c _a + c _I. This is thus the lower bound on WTP to get participation started. Once at least one of farmer i's neighbors have adopted, the farmer's necessary WTP becomes w _i ≥ c _a + c _I − θ, as i benefits from the knowledge spillover effect. Individuals with w _i < c _a + c _I − θ, or w _i < c _a + c _I without any participating neighbors will never participate, which explains why in reality there are areas where no one participates.

Farmers also condition their adoption with the goal of increasing profits as described in the profitability Eq. [1] and the nutrient reduction Eq. [4], where the latter captures the external effects of nutrient reduction by neighbors j = 1 to 4 on farmer i's profit. After participation at t, farmer i observes their realized return p _R*R(s _i, s_j, x_i,t, x _j,t), henceforth simply p_RR _i,t, and use this to determine whether to participate in the next period, thus setting x _i,t+1 before the beginning of period t + 1. Because choosing not to participate results in a return of zero—farmer i receives no payment at no cost if they don't adopt the BMP—Eq. [1] and [2] combine with Eq. [3] to form the condition p _RR _i,t ≥ c _a + c _I − θ*I _j,t-1 for farmer i to participate at time t + 1 (set x _i,t+1 =1)

Because farmers in our simulation do not initially participate, V _i,0 = 0. At the beginning of t = 1 anyone who decided to participate had w _i ≥ c _a + c _I, but now at the end of t = 1 they are considering whether p_RR _i,t ≥ c_a + c_I to determine whether to participate in period t = 2. (Note that I_j,t-1 is 0 for all farmers at t = 1.)

From t = 2 forward, any farmer who has participated in period t falls into one of the following four cases.

1. 1) If farmer i participated at t for the first time, then the condition to participate at t + 1 becomes p_RR _i,t ≥ c_a + c_I − θ*I _j,t-1

2. 2) If farmer i didn't participate at t – 1 and participated at t but the farmer has participated before t – 1, there is no information cost, so we have p_RR _i,t ≥ c_a

3. 3) If farmer i participated at t – 1 for the first time and also participated at t, then to continue participating in t + 1, the return needs to be higher:

   (5)$$p_RR_{i\comma t}-c_a \ge p_RR_{i\comma t\ndash 1}-c_I + \theta \,\ast\, I_{\,j\comma t-2}-c_a$$

Recall Eq. [4] and the soil productivity doesn't change, so this case is equivalent to:

(6)$$\gamma s_j\,\ast\, \left({\mathop \sum \limits_{\,j = 1}^4 x_{\,j\comma t}-\mathop \sum \limits_{\,j = 1}^4 x_{\,j\comma t-1}} \right)\ge -c_I + \theta \,\ast\, I_{\,j\comma t-2}\rpar $$

1. If farmer i participated at t – 1 and t and has participated at any time before t – 1, the derivation follows Case 3 except that the information cost is zero in both periods, so the condition becomes:

   (7)$$\mathop \sum \limits_{\,j = 1}^4 x_{\,j\comma t}-\mathop \sum \limits_{\,j = 1}^4 x_{\,j\comma t-1} \ge 0$$

That is, the number of BMP-adoptiing neighbors did not decrease from t – 1 to t.

Parameterization

We set the initial parameters based on data from Duke et al. (Reference Duke, Liu, Monteith, McGrath and Fiorellino2020). Information for 196 fields located on the Eastern Shore of Maryland was collected and processed through Maryland's Nutrient Trading Tool (MDNTT) to predict their participation in the Maryland water quality trading (MWQT) program. We set our parameters to match the productivity information and associated costs for these fields. The unit price for nutrient reduction is p_R, ranging from $20 to $100 per pound of phosphorus (P) reduction. We assume the information cost y _i,t to be between $0 and $30/lb/year, representing a range of complexity and length of WQT contracts (DeBoe and Stephenson Reference DeBoe and Stephenson2016).The adoption cost of a BMP per year (c_a) ranges between $10 to $250 per acre per year. The productivity of BMP adoption in the sample ranges from 1 to 8 lbs per acre in terms of phosphorus reduction. We set the information sharing cost θ to be half of the information cost. Table 1 summarizes the model's parameters.

Table 1. Parameters and Descriptions

Simulation Results

We simulate a 33 by 33 landscape, which has 1089 fields. The following figures show an example run. Figure 2 shows the landscape and soil qualities and the farmers’ WTP values (randomly distributed among the grid). Figure 3 shows the process of BMP adoption arriving at a steady state equilibrium in the landscape and community of Figure 2.

Figure 2. Example Landscape: Soil Productivity (left) and Farmer Types (right). Green: social Leader; Red: Mainstream; Yellow: Traditional

Figure 3. Participation Process in the Landscape, with Time Moving from Left to Right. Black: Nonparticipants; Blue: Participants

To test hypotheses 1 and 2, we fix the percentage of traditional types and run through varying percentages of the mainstream and social leader types. In the plots of Figure 4, each line represents the evolution of a different type of community, averaged over 30 simulations, with the number of years on the horizontal axis. The panels from left to right have 30%, 40%, and 50% of the farmers classified as traditional. The lines in each graph vary from blue to red as we increase the representation of social leaders from 2% to 20% in steps of 2 percentage points. The blue lines thus are also those lowest and to the right in the diagram and the red lines the highest and to the left.

Figure 4. Adoption Process, Fixing the Percentage of Traditional Types

Next we fix the percentage of the mainstream types in the community and consider varying levels of the traditional and social leader types in each simulation. Figure 5 shows, from left to right, 30%, 40%, and 50% of farmers set to the mainstream type. Each line again shows an average across 30 simulations ranging from 2% (blue, bottom right) to 20% (red, top left) innovators.

Figure 5. Adoption Process, Fixing the Percentage of Mainstream Types

These simulation results support hypotheses 1 and 2: Communities with a higher percentage of innovators result in higher participation and arrive at such steady states faster. The non-linear nature of that relationship is apparent in Figures 4 and 5, as well as in Figure 6 where we group the mainstream and traditional types as “non-innovators”.

Figure 6. Number of Participats and Time to Arrive at Steady State Given % of Innovators in a Community

Because there are many possible topologies of agent connectivity other than the one we are investigating, we also considered farms on the same two-dimensional layout but with a larger number of neighbors who can be information sources and spreaders of the environmental externality for the focal farm. We explained in above that we have thus far treated farm i's neighbors as those to the North, South, East, and West of i (a von Neumann neighborhood). To investigate one other possible topology, we increased the neighbors to 8 by additionally including those to i's Northeast, Southeast, Southwest, and Northwest (a Moore neighborhood). The results are similar to what we have shown above, and as a comparison to Figure 5's four neighbors to farmer i we show the results in Figure 7 for eight neighbors to i. We can see that the steady-state results and speed to reach them are generally higher and faster—because farmers with eight neighbors can learn from more possible sources. We thus have further support of our results, although an in-depth exploration of other topologies is beyond the scope of this paper.

Figure 7. Adoption Process When There are More Neighbors

To test our third hypothesis, we fix the percentage of farmer types and test different distributions of soil productivity. In Figure 8 we observe no evidence to support this hypothesis (the curves with differing soil productivity overlap each other), which is counterintuitive. It appears that with these parameter settings, the farmer type drives adoption dynamics more than the soil productivity does. This is consistent with previous findings in which farmers’ perceived efficacy is the most important determent of their BMP adoption decisions (Zhang et al. Reference Zhang, Wilson, Burnett, Irwin and Martin2016), but future research is needed to explore the mechanism behind this result.

Figure 8. Participation Process with Varying Soil Productivity Distributions

The last hypothesis can be shown in an example steady-state adoption map as in Figure 9, where we observe blocks of black (non-adopter) areas. Without enough neighbors participating (blue), there is not enough incentive for participants who are not innovators.

Figure 9. Example of Steady State Adoption Map