We use the gcamland v2.0 software package in this study (Calvin et al., 2019a). gcamland separates the land allocation mechanism in GCAM (Calvin et al., 2019b) into an R package.

GCAM and gcamland are separate models. While gcamland replicates the land allocation mechanism in GCAM, it is not run within GCAM. Similarly, GCAM is not run as a part of gcamland. gcamland only includes a representation of land allocation. GCAM includes representations of agricultural supply and demand, land allocation, and other sectors (energy, water, economy, climate). The land allocation mechanism within gcamland uses price, yield, cost, subsidy, logit exponents, expectation parameters, and initial land area as exogenous inputs and endogenously determines land area in subsequent years. Changes in demand are explicitly represented in GCAM. In gcamland, changes in demand are captured through changes in price. For example, the increase in demand for corn and soybean due to biofuels policy is captured through changes in the prices of these goods.

Land allocation in gcamland (and GCAM) is determined based on relative profitability, using a nested logit approach (McFadden, 1981; Sands, 2003; Wise et al., 2014). The logit land supply is presented in Eq. (1). All else equal, an increase in the rental profit rate (ri) of one land type will result in an increase in the land area (Xi) allocated to that land type. The magnitude of the land supply response is dependent on the positive logit exponent (ρ) and share-weight parameters (λi). These parameters influence the land supply elasticity, which is non-constant (i.e., it varies depending on the relative profitability as described in Wise et al., 2014). Y is the total land supply, i.e., ∑iXi=Y. The logit formulation assumes that there is a distribution of profit rates for each land type, and the resulting land allocation for a given land type is the probability that land type has the highest profit (Zhao et al., 2020b). The logit share weights (the scale parameters in the distribution) are calculated to perfectly reproduce the data in a base year. The logit exponent (the shape parameter in the distribution which governs the magnitude of land transformation given relative profit shocks) is one of the parameters of interest in our study (see Sect. 2.2, Table S2 in the Supplement). 1Xi=(λiri)ρ∑j(λjrj)ρ⋅Y, The logit approach is advantageous compared with the constant elasticity of transformation (CET) approach widely used in computable general equilibrium (CGE) models as it can directly provide traceable physical land transformation. But like the CET function, the logit land sharing function is parsimonious and a nested structure can be used. In gcamland, all crops are nested under cropland. Cropland is nested with forest and then pasture; see Fig. S1 in the Supplement. In a nested logit, the area of a particular land type is determined by not just the logit of its nest, but also by the logit of the nests above that. Thus, there are three logit exponent parameters governing land transformation for crops in gcamland. In the nested version, land allocation at each of these nests is determined by Eq. (2) (a modified version of Eq. 1, where Y is replaced by the land allocated to that particular nest). The land allocated to a particular nest is dynamic and varies over time. In Eq. (2), dynamic variables are indicated with subscript t. 2XjtC=λjCrjtCρC∑jλjCrjtCρC⋅YtCfor C the cropland nest including all crops and other arable land (see footnote 1 in Table 2)XjtA=λjoArjtAρA∑jλjArjtAρA⋅YtAfor A the ag, forest and other nestXjtR=λjRrjtRρR∑jλjRrjtRρR⋅YtRfor R the gcamland dynamic modeling nest Profit rates (rj) at the lowest level of the nest are computed based on price, cost, yield, and subsidy (if included) for commercial land types (crops, pasture, commercial forest); profit rates for non-commercial land types are input into the model and are based on the value of land (see also Table S1). Profit rates for commercial lands evolve over time as price, cost, yield, and subsidy change. Profit rates for non-commercial lands are constant over time. The logit approach effectively depicts a supply curve for non-commercial land with the land supply elasticity implicitly determined by the logit exponent and the assumed rental profit rates (i.e., implying a cost of land transition). The supply curve approach, which views the amount of land available as endogenous, offers more modeling flexibilities with traceable results compared to approaches of assuming non-commercial lands to be inaccessible and fixed over time or aggregating non-commercial lands with commercial lands (Dixon et al., 2016). Profit rates for higher levels of the nest (rnode) are determined by 3rnode=∑j=1nλjrjρ1/ρ. gcamland tracks both physical area and harvested area for crops. Physical area is determined by the logit-based land allocation scheme described in this section. Harvested area is calculated using physical area and a fixed harvested-to-physical area ratio, estimated in the base year, and held constant in the future. Note that, since forestland is not an annually planted and harvested commodity, GCAM, gcamland, and other similar models assume that land must be set aside at every time step to ensure enough commercial forestland is available to meet harvest demand at the time the forest matures. To do this in gcamland, we assume that the amount of land allocated to forest depends on the harvest yield and the rotation length.

There are multiple means of forming expectations in the literature. With perfect foresight, the expected value of a given variable is equal to its realized value: 4Ext=xt. In an adaptive expectation approach (Nerlove, 1958), the expected value is a linear combination of the previous expectation and the new information acquired, with α being the coefficient of expectations: 5Ext=(1-α)xt-1+αExt-1. Finally, a linear expectation approach uses a linear extrapolation of previous information to form the expectation: 6Ext=Cov[x(n),year(n)]Var[year(n)]yeart, where n is a fixed number of previous years considered in forming the expectation, x(n) and year(n) are vectors of the variable and year index, respectively, with historical information from year t-1 to t-n. That is, instead of using all available historical information, forward-looking producers are assumed to rely on only information of the most recent n years.

In our study, we combine these basic approaches into four different expectation types, specifying the means of calculating expected price and expected yield (Table 1).

Note that other expectation types can be tested within gcamland, e.g., expectation types that are a hybrid of past and perfect information. Such expectations types can be useful for understanding the value of additional information. However, we exclude them in this paper as they are unlikely to explain past behavior and are not covered in the literature on land use decision making.

To initialize gcamland in this study, we started from the GCAM v4.3 agriculture and land use input data (see Table S1). The GCAM data processing reconciles land use data from FAO with land cover data, ensuring that total areas do not exceed the amount of land in a region. Thus, we chose to use this reconciled data instead of using FAO data directly. We have made two changes to the GCAM v4.3 initialization data.

First, since GCAM has a 5-year time step, it uses 5-year averages of land use and agricultural production for initialization. For this study, we have updated the input data to remove the averaging since we are primarily focused on annual time steps in gcamland; that is, the initialization data in gcamland for a particular year are the data for that year only and not a 5-year average around that year as it is in GCAM.

Second, GCAM models land use and land cover at the subnational level (v4.3 used Agro-Ecological Zones; v5.1 and subsequent versions use water basins). However, much of the comparison data are provided at national level. For this study, we aggregate the initialization data to the national level, representing the USA as a single region. The qualitative insights in this paper would not change if we disaggregated to subnational level, but the exact quantitative results would.

Third, GCAM uses constant costs over time. For this study, we have updated the costs to use time-evolving cost data (see next section).

We use data for producer price and yield from the U.N. Food and Agricultural Organization (FAO, 2018a, b, 2020b), with data available for all non-fodder commodities for 1961–2018. Data were aggregated from individual crops to the GCAM/gcamland commodity groups, weighting non-fodder crops by their production quantity. In some cases, data prior to 1961 are required to generate expectations for the model years (1975–2015); in these cases, we assume that prices and yields prior to 1961 are held constant at their 1961 values. For cost, we use data provided by the U.S. Department of Agriculture (USDA, 2020a), with data available for major crops from 1975–2018. We only include the variable costs as reported by USDA and exclude the allocated overhead costs. We use a representative crop from USDA for each GCAM/gcamland commodity group, as data do not exist for all crops (i.e., we use soybean cost from USDA as a proxy for the cost of OilCrop in gcamland). The producer prices used in gcamland are defined as “prices received by farmers…at the point of initial sale” or “prices paid at the farm-gate” (FAO, 2018a) and thus do not reflect subsidies. However, subsidies are a reality of crop agriculture in the United States. However, there are not continuous, complete, and consistent data sets for all types of subsidies paid to farmers. Additionally, crop-specific information (of the type needed for gcamland) is only available for direct payments, making the inclusion of other types of subsidies difficult. Therefore, for subsidies, we combine two different data sets from USDA: the federal government direct payments (USDA, 2020c) and the farm business income (USDA, 2020b). We only include direct payments from these two reports; thus, our subsidy data are missing many other forms of payment. Additionally, we only have data for a subset of crops and the categories reported change over time across the two data sets. Because these data are inconsistent and incomplete, we only use it as a sensitivity in this paper and do not include it in the primary analysis.

Note that our choice to use it as a sensitivity and not the default is because it does not improve NRMSE and did not alter the parameter set that minimized NRMSE between simulated and observed land allocation (as discussed in Sect. 4).

In total, gcamland has between 29 and 35 parameters (depending on the expectation type) that are used to calculate land allocation in each year (see Eq. 2). This study samples all six parameters used in the expectation calculation and three of the seven logit exponents. The remaining four logit exponents are specified exogenously, as these exponents have minimal impact on the outcomes of interest in this paper (see Sect. S1 and Table S2 in the Supplement). The values of those four logit exponents have not been obtained from an explicit statistical analysis and instead were selected based on authors' judgment (see Sect. S1). The remaining 22 parameters are share-weight parameters (λi in Eqs. 1 and 2). These parameters are calculated from the observed land allocation in the initial model year and the other specified parameters to ensure that land allocation in the initial year exactly matches observations.

Within this study, we vary a total of nine parameters (Table 2), including three logit exponents, the coefficient of expectations (α) for the adaptive expectation and the number of years (n) used in the linear expectation. In addition, we allow α and n to vary across commodity groups, resulting in three separate realizations for each parameter. We group the commodities to minimize the number of free parameters. The first group includes Corn and OilCrop, which are used for biofuels in the United States and have had shifts in the demand over time as a result of biofuel policies. The second group includes the other two large commodities produced in the United States, Wheat and OtherGrain. The third group includes all other crops. The range of values spanned in the ensemble was chosen to cover all plausible values of each parameter but avoid potential numerical instabilities. Those ranges and their justification are described in Table 2. We use a Latin hypercube sampling

We use the R “lhs” package for the sampling (Carnell, 2020; R Core Team, 2020).

Hindcast simulations are experiments where a model simulation is conducted for a time period in which observational data are available but in which the observational data are specifically not used in the model simulation. In the example of gcamland running a hindcast from 1990–2015, this would correspond to a gcamland forecast of land allocation from 1990–2015. When 1990 is used as the initial model year, observed data from 1991–2015 are not used at any point in the gcamland simulation of land allocation, and 1990 observed data are only used to initialize gcamland.

We use each of the 10 000 parameter sets to run a gcamland hindcast for each of the four expectation types described, resulting in 40 000 simulations. Each parameter set includes nine parameters (see Table 2); the three logit parameters are used for all expectation types, but the expectation parameters are only used in expectation types requiring them (e.g., perfect expectations only uses the three logit exponents; the hybrid linear adaptive expectation type uses all nine parameters).

We compare model outputs to observation data to evaluate the performance of gcamland under each expectation type and parameter set. Ideally, the observation data would be completely independent of the model. However, due to limited availability of data sets,

The only other data set we are aware of the provides a time series of cropland area by crop is the USDA. However, since FAO base their reporting for the United States on submissions from the USDA, these two data sets are identical.

Different measures of model performance are used to select parameter sets that optimize different aspects of model performance.

For this analysis, we use the R “stats” package (R Core Team, 2020).

Normalized root mean square error (NRMSE) considers all deviations between simulated and observed values and places them in the context of the variance seen in the observational data. For crop i, 7NRMSEi=meani(obsi,t-simi,t)2meani(obsi,t-obsi‾)2. One benefit of this measure is that it includes a natural benchmark of acceptable model performance. While NRMSE=0 corresponds to perfect model performance, any NRMSE<1 is considered acceptable model performance (e.g., Tebaldi et al., 2020, and the review of metrics in Legates and McCabe, 1999). Using the standard deviation of observation as an error baseline puts the deviations between simulation and observation for each crop in the context of that crop's historical variations. If errors in a 1990–2015 gcamland hindcast simulation are greater than the historic standard deviation, then by definition, simply using the 1990–2015 mean value of land allocation in every simulated year 1990–2015 would have resulted in better errors than the model under consideration. Note, however, that in a hindcast approach, one would not actually have access to the 1990–2015 mean observed land allocation to use as a model to simulation 1990–2015 land allocation; it is simply an easy conceptual counterfactual model. Even when comparing two different model results that each have NRMSE>1, the model with the smaller NRMSE value is considered better.

We also consider the root mean square error (RMSE), 8RMSEi=meani(obsi,t-simi,t)2, and bias, 9biasi=(obsi‾-simi‾). These un-normalized measures make no distinction between different crops; a bias or RMSE of 200 km2 means exactly the same for Corn as it does for Rice, despite the fact that Corn represents a larger proportion of harvested area in the United States in the historical period. While RMSE is concerned with all deviations between observation and simulation for a crop, bias simply compares the means between observation and simulation. While these means tend to be determined more by the smoothed trend in a time series than variations about the trend, it is important to note that bias specifically does not penalize volatility the way that RMSE and other measures may.

Finally, the Kling–Gupta efficiency score (Knoben et al., 2019) is also implemented for each crop: 10KGEi=1-(r-1)2+σsimσobs-12+μsimμobs-12, for correlation coefficient r, standard deviation σ, and mean μ. While a perfect simulation (NRMSE=RMSE=bias=0) would by definition give perfect KGE (KGE=1), KGE is defined by penalties between different time series summary statistics, as opposed to the penalties based on simple deviations between simulation and observation at each time point in the other error metrics considered here.

For a given error measurement, the metric is calculated for each crop in each ensemble member. For NRMSE and RMSE, the average value across crops is then minimized to select the ensemble member with the most optimal parameters for matching observation. For bias, it is the average across crops of the magnitude of bias that is minimized, to avoid cancellation of errors between crops. For KGE, it is the average across crops of the quantity 1-KGEi that is minimized so that the average across crops of KGEi is optimized as needed. As an additional sensitivity, the actual land types included in this average metric can be adjusted to include all crops, simply one individual crop, or any combination of land types of interest. By default, we include any land type where we have observations for the full time series of the simulation, which effectively means all crops excluding fodder crops (see Sect. 2.4.3 and Table S1); however, we include a sensitivity on the set of land types included in Sect. 5.2.2.

We calculate goodness of fit for each land type of the gcamland ensemble members and each metric of goodness of fit. We then choose the ensemble member that optimizes the average across land types of interest for each measure of goodness of fit for each expectation type. The parameter set used to generate that ensemble is considered the “best parameter” set for that expectation type. Our default is to use NRMSE as a measure of goodness of fit, but we discuss sensitivity to measure of goodness of fit in Sect. 4.2.

The previous step generates four parameter sets, one for each expectation type. In this step, we choose the expectation type and parameter set that optimizes average goodness of fit across all land types, resulting in a single “best model”.

The default ensemble analyzed in this paper uses 1990 as the initial model year, runs annually through 2015, excludes subsidies, and differentiates the expectation parameters (α and n) by crop groups. To test the sensitivity of the results to each of these assumptions, we re-run the ensemble with alternative specifications for each assumption (Table 3).

Over the last several decades, yields have increased in the United States; prices and profits are more variable (Fig. S2 in the Supplement). Changes in the area of a particular crop, however, are not always correlated with in year profit (Figs. S3 and S4 in the Supplement). There are several potential reasons for this:

Farmers do not know the profit at the time of planting and instead are basing their planting decisions on expectations.

The parameter sets (Table S6 in the Supplement) and cropland time series (Fig. 6) that are numerically optimal for KGE are somewhat similar to those of NRMSE and the parameter set that minimizes RMSE is identical to that of NRMSE.

Note that this is not true in general but is true for the default model. Other configurations of the model have different parameter sets that minimize NRMSE than those that minimize RMSE.

The parameter set that minimizes bias, however, is fundamentally different. The logit exponents dictating the substitution between crops and other land types are large (2.18 for the Dynamic Land nest; 1.38 for the Ag, Forest, and Other nest). The parameter set that minimizes bias also includes the lowest Cropland nest logit value of any objective function (0.28). The resulting simulations for bias exhibit large volatility in land area. Given that bias simply compares the model mean across time to the observation mean across time, this volatility is not penalized in the bias metric, whereas it is penalized for KGE, RMSE, and NRMSE. For example, the parameter sets that minimize bias result in an average simulated Corn area of 307×103 km2 compared to an average observed Corn area of 306×103 km2, resulting in a bias of less than 1×103 km2. This bias is much lower than the bias for Corn in the other objective functions (NRMSE and RMSE have a bias of 6×103 km2; KGE has a bias of 16×103 km2). Bias is effectively assessing whether the model is correct on average and not whether it captures the trends or volatility; such an objective function is less useful in systems where trends are significant or where the goal is to capture the volatility. From a mechanistic perspective, we hypothesize that the difference in the cropland area volatility when bias is minimized is due to the differences in the Ag, Forest, and Other logit.

Figure 7 shows the difference in the best models when we optimize for a particular set of land types or crops. As seen in this figure, gcamland can track land area for any given crop very well when the ensemble with optimal parameters is chosen specifically for that crop. However, matching all crops at once is more challenging. For example, the parameter sets that minimize NRMSE for Corn result in an excellent match between observations and model output for Corn; however, those parameters result in an overestimation of Wheat land by 250×103 km2 in 2015 (or ∼1/2 of the actual area). The insights from this figure are also confirmed numerically. The NRMSE for Corn is reduced from 1.88 to 0.72 when we go from minimizing NRMSE across all crops to minimizing NRMSE for Corn only. Similarly, the NRMSE for OilCrop is reduced from 1.67 to 0.54 when we go from minimizing NRMSE across all crops to minimizing NRMSE for OilCrop only. Optimizing for a single crop has less effect on the NRMSE for Wheat and OtherGrain (from 1.16 to 0.79 for Wheat, and from 0.7 to 0.43 for OtherGrain). Finally, including all dynamic land cover types where observations are available for any period of the simulation years (e.g., non-fodder crops, grassland, shrubland, and forest) in the calculation of NRMSE increases the NRMSE substantially (from 1.4 to 75) due to definitional differences in land cover types. The change in land area for land cover types is reasonably consistent with observations (Fig. S10); however, the absolute area for grassland and shrubland differs substantially (Fig. S9). Despite the increase in NRMSE, the inclusion of land cover types does not alter the parameter sets that minimize NRMSE.