4.1. Historical Yields and Prices

For each representative farm we simulate, we require a yield history which will be used to characterize the joint distribution of crop yields, ERP payments, and crop price realizations. For the yields, we use the National Agricultural Statistics Service (NASS) county yield data as a starting point and use a similar methodology as Goodwin (Reference Goodwin2009) to construct individual representative producer yields for determining production history for each state-county-crop-type-practice (SCCTP) combination. Goodwin (Reference Goodwin2009) assumes that crop yields follow a linear trend and the deviations from trend are proportional to the trends. Specifically, we regress NASS county yields on a time trend and recover the predicted values, Ⓨ _t , and associated residuals, ε _t for year t. Then, we use RMA’s 2022 reference yield ${\bar Y}$ ₂₀₂₂ for the relevant SCCTP combination to recenter the yields.Footnote ²

It is widely accepted that farm-level yields present more variability than county-level yields (Claassen and Just, Reference Claassen and Just2011; Cooper et al., Reference Cooper2012; Gerlt et al., Reference Gerlt, Thompson and Miller2014; Goodwin, Reference Goodwin2009; Maisashvili et al., Reference Maisashvili, Bryant, Kpnakep and Raulston2019; Schnitkey et al., Reference Schnitkey, Sherrick and Irwin2002; Ubilava et al., Reference Ubilava, Barnett, Coble and Harri2011). Therefore, when generating a random historical representative producer’s yield, we follow Goodwin (Reference Goodwin2009) by adding an additional, zero-mean, normally distributed shock to the recentered county yields. We specify a standard deviation for this additional shock equal to 75% of that of the residuals recovered from detrending county yields (0.75σ _ε ). The complete process for generating a random representative historical realized yield (Y _t ) for time t is then

(6) $$ Y_t = \left (1 + {\epsilon _t \over \hat{Y}_t}\right ) \times \bar{Y}_{2022} + \eta $$

$$ \eta _t \sim N(0, 0.75 \sigma _\epsilon ) $$

For each SCCTP combination, two types of historical price series were constructed in accordance with the current Commodity Exchange Price Provisions guidelines (USDA Risk Management Agency, 2022b). Projected price (P) values were obtained as the average daily settlement price from the harvest futures contract during the projected price discovery period. Harvest price (H) values were obtained as the average daily settlement price from the harvest futures contract during the harvest price discovery period. The relevant underlying historical futures prices were purchased from CME Group. For recent years, pre-calculated P and H values were collected directly from RMA’s Actuarial Data Master (USDA Risk Management Agency, 2022a).

4.2. Starting Wealth

Starting wealth is an important determinant of the magnitude of ending wealth values, and consequently, the extent of curvature of the utility function in the neighborhood is relevant for evaluating simulation outcomes. Given that data are unavailable for a producer’s starting wealth, we follow Maisashvili et al. (Reference Maisashvili, Bryant and Jones2020) in obtaining estimates of starting wealth using USDA’s Agricultural Resource Management Survey Farm Financial and Crop Practices data (USDA Agricultural Resource Management Survey, 2021). Based on reported income statements and balance sheets for years 2015 through 2018, and using farm equity values as a proxy for initial wealth, they found that the value of farm equity generally was between 3.8 to 5.2 times higher than expected crop value (the product of the county reference yield, the insured price, and number of acres, A). We follow this same approach and draw a random wealth parameter for each representative farmer for a given SCCTP combination using a uniform distribution. That is, starting wealth S is given as

(7) $$ S= u \times P\times \bar{Y}_{2022} \times A$$

where u is a stochastic wealth parameter that is uniformly distributed over the interval [3.8,5.2].

4.3. Production Costs

Individual production costs are nearly impossible to obtain from publicly available data. Some studies have simply excluded the cost data from their analysis (Maisashvili et al., Reference Maisashvili, Bryant and Jones2020; Paulson et al., Reference Paulson, Schnitkey and Kelly2016; Ubilava et al., Reference Ubilava, Barnett, Coble and Harri2011). Goodwin (Reference Goodwin2009) argues that production costs are important factors influencing production decisions.

We collected, where available, state-level crop production costs published by the respective state extension services. If production cost data were unavailable from an extension service for a given state, we consulted USDA’s Economic Research Service commodity cost and return aggregated data for an approximation (USDA Economic Research Service, 2021).

We acknowledge that production costs are highly likely to substantially vary across individual producers and using state-level, and when necessary region-level, data can substantially deviate from the true fixed and variable expenses at the individual level. What is more, these budgets are usually prepared for a “representative” producer using recommended production practices and, as discussed by Goodwin (Reference Goodwin2009), crop budget and production costs are difficult to measure considering they are subject to individual judgment on the part of the specialist constructing the measures. However, considering that the individual cost of production data is unavailable, using state-level extension service data is a reasonable approach. We acknowledge that many individual producers have cost advantages and use varying production practices. We therefore follow Goodwin (Reference Goodwin2009) in assuming that costs are 60% of the levels implied by enterprise budgets.

We calculate crop insurance premiums using RMA’s official 2022 Actuarial Data Master (USDA Risk Management Agency, 2022a) and in accordance with the M-13 handbook (USDA Risk Management Agency, 2022c) for each SCCTP.Footnote ³ By construction of the historical yield series (as described in subsection 4.1), a representative producer’s approved yield ${\bar Y}$ used in premium calculations will be equal to the county reference yield ${\bar Y}$ ₂₀₂₂.

Because this research aims to explain potential changes in coverage level choices, for tractability, hypothetical corn, and soybean farms are assumed to buy revenue protection (RP). That is, we do not analyze the choice of policy type. In 2021, more than 95% of insured corn and soybean acres were insured with RP on an individual farm basis (USDA Risk Management Agency, 2021b).

4.4. ERP Related Losses

To conduct the analysis, we rely on the publicly available Cause of Loss (COL) database, which is maintained by RMA (USDA Risk Management Agency, 2021a). We collected the COL annual data from 1991 to 2021, which contains 932,084 and 808,963 total entries over 1,881 and 1,737 counties for corn and soybeans, respectively. In addition, the COL data are disaggregated by type of cause, which is determined by a claims adjuster. In total, we observe up to 50 reported causes. From the list of reported causes from the COL database, the following causes could potentially qualify for the ERP payments: cyclone, excess moisture, drought, freeze, fire, flood, hurricane/tropical depression, storm surge, tidal wave/tsunami, tornado, and volcanic eruption.Footnote ⁴ Figure 1 presents the distribution of indemnities by cause of loss for corn and soybeans during the 1991-2021 period. For brevity, we use a similar representation as described by Perry et al. (Reference Perry, Yu and Tack2020).

Figure 1. Crop insurance indemnity payment shares by cause of loss, 1991–2021.

For both crops, the three largest causes of indemnity payments are drought, excess moisture, and decline in price. In the case of corn, drought, excess moisture and decline in price accounted for about 38, 30 and 8.5% of payments, respectively. For soybeans, drought, excess moisture, and decline in price accounted for 33, 37, and 7.5% of payments, respectively. Considering that in a given county not all drought categories qualify for ERP, and given that the COL data do not provide further details when the specific cause is cited as drought, we queried data from the National Drought Mitigation Center at the University of Nebraska-Lincoln (2022) to categorize the COL drought data as ERP-eligible only if it was categorized as D2 for eight consecutive weeks or D3 or higher at any time during the calendar year. Specifically, the drought data identify general areas of drought and labels them by intensity, with D0 being the least intense level and D4 the most intense. The source also allows users to filter specific drought levels by consecutive weeks or by specific time spans. By merging the drought monitor data with the COL data, Figure 2 shows combined historical corn and soybean drought-related indemnities among the five drought levels (D0 through D4) in those periods when the COL was reported as drought. The figure shows that over 60% of historical corn and soybean drought-related indemnities are at the level of D2 or above.

Figure 2. Drought-related combined historical corn and soybean crop insurance indemnity payment shares by drought intensity level.

For each SCCTP combination, we calculate, for 1991 through 2021, the share of the indemnified acres that could qualify for ERP payments compared to the total insured acres by creating a new variable, ranging in value between 0 and 1 for each SCCTP in given year (hereafter x _q ).Footnote ⁵ In the simulations, this x _q is employed as a directly drawn stochastic variable, as described in the following subsection.

4.5. Simulation and Dependence of Random Variables

We assume that producers optimize over probabilistic forecasts of harvest time market outcomes, which we characterize using stochastic simulation. In each simulation, we draw stochastic values for the natural log changes of harvest price with respect to observed projected prices, x _p . We thus use the relation

(8) $$ \ln \left ( {H \over P} \right ) = x_p $$

to recover a stochastic value for the harvest time price H given the observed (at planting time) value for the known projected price P and a random draw for x _p .

For crop yields, we stochastically draw x _y , the natural logarithm of the realized yield relative to the producer’s approved yield. We therefore recover simulated realized yields Y using the relation

(9) $$ \ln \left ( {Y \over \bar{Y}} \right ) = x_y $$

given the known value ${\bar Y}$ of the approved yield and a stochastic draw for x _y .Footnote ⁶ We constrain random yield draws at 0 and 150% of the RMA’s reported reference yield for relevant SCCTP combinations, similar to the approach in Goodwin (Reference Goodwin2009).

We use a multivariate kernel density estimation (MVKDE) approach to generate stochastic draws for x _q , x _p , and x _y . This approach accommodates both arbitrary marginal densities and arbitrary non-linear relationships among variables. The multivariate kernel density estimate is

(10) $$ \hat{f}({\bf{{x}}}|{{\bf{B}}}, {\bf{\breve{{X}}}}) = {1 \over M} \sum _{m=1}^M \vert {{\bf{B}}} \vert ^{-1/2} \kappa \left ( {{\bf{B}}}^{-1/2} \left ( {{\bf{x}}} - \bf{\breve{{X}}}_m \right ) \right ) $$

where ${{\bf{x}}}$ is a vector of values associated with the three random variables, ${{\bf{B}}}$ is a 3 × 3 symmetric, positive definite bandwidth matrix, $\bf{\breve{{X}}}$ is a sample of M historical observations for the three variables, κ is a symmetric multivariate density function whose mean is a vector of three zeros, and ${\bf{\breve{{X}}}}_m$ reflects the m ^th observation in the data sample.

We use a Gaussian PDF for κ, and the kernel density then becomes

(11) $$ \hat{f}({{\bf{x}}}|{{\bf{B}}}, {\bf{\breve{{X}}}}) = {1 \over M} \sum _{m=1}^M \vert {{\bf{B}}} \vert ^{-1/2} (2\pi )^{-1/2K} e^{-1/2} ({\bf{x}} - {\bf{\breve{{X}}}}_m)^T {{\bf{B}}}^{-1} ({{\bf{x}}} - {\bf{\breve{{X}}}}_m) $$

where ${{\bf{B}}}$ in this context would be the covariance matrix for a multivariate (non-standard) normal distribution. Consistent with Gerlt et al. (Reference Gerlt, Thompson and Miller2014), we apply Silverman’s rule for the bandwidth selection (Silverman, Reference Silverman1986), where the off-diagonal elements of ${{\bf{B}}}$ are zero and the k ^th diagonal element is, for this three variable application, given by

(12) $$ \sqrt{{{\bf{B}}}_{kk}} = \sigma _k \left ({{4M \over 5}}\right )^{-1/ 7} $$

where σ _k is the sample standard deviation for the k ^th variable.Footnote ⁷ Alternatively, we could have employed copula-based methods, which provide a means to model dependency among random variables with mixed marginal distributions. Copula-based models have been used in various applications (Fei et al., Reference Fei, Vedenov, Stevens and Anderson2021; Nadolnyak and Vedenov, Reference Nadolnyak and Vedenov2013; Power et al., Reference Power, Vedenov, Anderson and Klose2013; Woodard et al., Reference Woodard, Paulson, Vedenov and Power2011), including crop insurance (Goodwin and Hungerford, Reference Goodwin and Hungerford2015; Ramsey et al., Reference Ramsey, Goodwin and Ghosh2019). We chose to use the MVKDE approach, however, due to its ability to represent any arbitrary multivariate dependence structure.

4.6. Indemnity Calculations

Given stochastic realizations for the harvest price, H, and a representative producer’s individual yield, Y, we follow RMA’s procedures (USDA Risk Management Agency 2022d) for crop insurance indemnity (I) calculations:

(13) $$ I_C = max\{0, [(C \times \bar{Y} \times max\{P, H\}) - Y\times H]\} \times A$$

where C is a crop insurance coverage level for a given scenario, between 50 and 85% in 5% increments, and A is the insured acreage.

4.7. Ending Wealth

For the baseline scenario, a producer’s ending wealth E for trial i for a given coverage level C is calculated as follows:

(14) $$ E_{C, i} = S+ (H_i \times Y_i + I_{C, i} - R_C - V) \times A$$

where S is producer’s starting net worth (as described in subsection 4.2), H _i is the stochastic harvest time price, Y _i is the stochastic realization of the crop yield; I _{C, i} is crop insurance indemnity (a function of Y, H, and C), R _C is the producer premium for coverage level C, V is deterministic variable expenses other than insurance premiums (as described in subsection 4.3), and A is insured acreage. Here, we abstract from basis considerations, so that market revenue is simply H × Y.

For counterfactual scenarios that include an ERP payment, ending net worth is

(15) $$ E_{C, i} = S+ (H_i \times Y_i + I_{C, i} - R_C - V+ x_{q, i} \times W) \times A$$

where W is a per-acre ERP payment per equation 1.

4.8. Modified Expected Utility Framework

To analyze the impacts of the ERP program on coverage level decisions, we assume that producers’ decisions are dependent on the distribution of ending net worth (E). An expected utility approach for ranking risky alternatives has been used widely in the crop insurance literature (Babcock, Reference Babcock2015; Coble et al., Reference Coble, Knight, Pope and Williams1996; Du et al., Reference Du, Ifft, Lu and Zilberman2015, Reference Du, Feng and Hennessy2017; Du et al., Reference Du, Feng and Hennessy2017; Goodwin, Reference Goodwin1993; Goodwin, Reference Goodwin2009; Hardaker, Reference Hardaker2004; Hennessy, Reference Hennessy1998; Mahul, Reference Mahul1999; Maisashvili et al., Reference Maisashvili, Bryant and Richardson2016; Maisashvili et al., Reference Maisashvili, Bryant and Jones2020; Paulson et al., Reference Paulson, Schnitkey and Kelly2016; Richardson et al., Reference Richardson, Herbst, Outlaw and Gill2007; Ubilava et al., Reference Ubilava, Barnett, Coble and Harri2011).

Preliminarily, suppose that producers exhibit constant relative risk aversion (CRRA) (following Goodwin, Reference Goodwin2009; Paulson et al., Reference Paulson, Schnitkey and Kelly2016; Ubilava et al., Reference Ubilava, Barnett, Coble and Harri2011). There is mounting evidence directly supporting CRRA under the basic expected utility framework (Brunnermeier and Nagel, Reference Brunnermeier and Nagel2008; Chiappori and Paiella, Reference Chiappori and Paiella2011; Sahm, Reference Sahm2012). CRRA implies an isoelastic utility function:

(16) \begin{align}\psi (E) = \left\{ {\matrix{ {{{{E^{(1 - \alpha )}} - 1} \over {1 - \alpha }}} & {{\rm{for}}\;\alpha \in [0,\infty )\backslash \{ 1\} } \cr {\ln (E)} & {{\rm{for}}\;\alpha = 1} \cr } } \right.\end{align}

where α is a producer’s coefficient of CRRA.

Then, a producer’s certainty equivalent income across N simulated trials for a given coverage level C is:

(17) $$ \psi _\alpha ^{-1} \left ( {1 \over N} \sum _{i=1}^N \psi _\alpha (E_{C, i}) \right ) $$

where ψ _α ⁻¹ is the inverse utility function conditioned on the CRRA coefficient α.

However, our preliminary investigation revealed that a basic expected utility approach cannot reasonably reproduce producers’ observed crop insurance choices. In many instances, under the modeling approach laid out above, no non-negative coefficient of CRRA will result in lower coverage levels, such as 75%, having the highest certainty equivalent, even though we observe a non-trivial proportion of producers choosing 75% coverage. Therefore, simple producer heterogeneity cannot salvage the basic expected utility framework under our modeling approach. Many previous studies have observed this shortcoming of the basic expected utility approach, as applied to producers’ crop insurance choices (Babcock and Hart, Reference Babcock and Hart2005; Babcock, Reference Babcock2015; Du et al., Reference Du, Hennessy and Feng2013; Du et al., Reference Du, Feng and Hennessy2017; Luckstead and Devadoss, Reference Luckstead and Devadoss2019; Michaud, Reference Michaud2021). This motivates us to pursue either an alternative theory of decision-making under risk or modifications to the expected utility approach.

We do, however, wish to acknowledge that producers are likely heterogeneous. The SOB data show that for a given SCCTP combination, even though there may be a dominant coverage level which producers mostly purchase, a non-trivial portion of farmers opt to purchase other coverage levels. As an example, the 2021 RMA Summary of Business (SOB) data show that in Boone County, Iowa, a large portion of corn and soybean acres were insured under 80 and 75% coverage levels, even though the 85% coverage level is the most popular (Figure 3). Likewise, in many counties where 70 and 75% coverage levels are the most popular choices, a non-trivial portion of producers buy 80 and 85% coverage. We require our approach to accommodate this feature of the observed data.

Figure 3. Actual coverage level choices observed in 2021 for corn and soybean producers in Boone County, Iowa.

Du et al., (Reference Du, Feng and Hennessy2017) suggest that farmer preferences for crop insurance coverage might be captured by cumulative prospect theory initially proposed by Tversky and Kahneman (Reference Tversky and Kahneman1992). Babcock (Reference Babcock2015) found that cumulative prospect theory does explain coverage level choices that are consistent with the observed SOB choices only if crop insurance is a lottery considered in isolation. Babcock finds, however, that if a producer’s overall portfolio (market revenue, in addition to crop insurance premiums and indemnities) is the lottery under consideration, then the optimal choices suggested by cumulative prospect theory are not consistent with observed choices. Optimal choices for producers’ overall portfolios are exactly the problem we consider here, and we therefore do not apply cumulative prospect theory.

If farmers do not make choices according to straight expected utility or cumulative prospect theory, one potential explanation could be that they select the coverage level that is simply “good enough.” This may be due to one or more of several possible reasons: cognitive constraints (Simon, Reference Simon1955, Reference Simon1956; Tversky and Kahneman Reference Tversky2013), non-negligible computational burden (Newell and Simon, Reference Newell and Simon1972; Simon, Reference Simon1997; Van Rooij Reference Van Rooij2008; Lieder and Griffiths, Reference Lieder and Griffiths2020), or information acquisition costs (Bogacz et al., Reference Bogacz2006; Gabaix et al., Reference Gabaix, Laibson, Moloche and Weinberg2006; Sims, Reference Sims2003; Sims, Reference Sims2006). Analysis whereby agents select alternatives that are “good enough” has been conducted in the economic literature as well (see Dickhaut et al., Reference Dickhaut, Rustichini and Smith2009; Gabaix et al., Reference Gabaix, Laibson, Moloche and Weinberg2006; Sims, Reference Sims2003; Verrecchia, Reference Verrecchia1982; Vulkan, Reference Vulkan2000; Woodford, Reference Woodford2014).

The challenge in implementing such an approach is defining what “good enough” is, and how an agent chooses among a set of multiple options that are good enough. We adopt a simple and intuitive approach. First, we assume a fixed reasonable value for the relative risk aversion parameter. Then, we specify a neighborhood of certainty equivalent values that are approximately maximal (including the one exactly maximal certainty equivalent). We then define the coverage level choices corresponding to the certainty equivalents in this neighborhood as the satisficing set. We then assume that producers choose from this set at random.

Many different values for the relative risk aversion coefficient have been employed in the literature. For example, Anderson and Dillon (Reference Anderson and Dillon1992) propose the values between 0 and 4, where 0 implies a risk-neutral individual and 4 corresponds to an extremely risk-averse agent. Hennessy (Reference Hennessy1998) used values in the range of 4.1 and 9.7. Goodwin (Reference Goodwin2009) employed values from 0 to 15. Paulson et al. (Reference Paulson, Schnitkey and Kelly2016) selected coefficients between 0 and 12; McSweeny and Kramer (Reference McSweeny and Kramer1986) selected values between 2.5 and 11.5. Michaud (Reference Michaud2021) used a coefficient of 0.12 to represent a mildly risk-averse individual. Here, we employ a relative risk aversion coefficient of 2.5 to represent a moderately risk-averse decision maker, as suggested by some of the aforementioned literature, in addition to other studies (Lien and Hardaker, Reference Lien and Hardaker2001; Lusk and Coble, Reference Lusk and Coble2005; Maisashvili et al., Reference Maisashvili, Bryant and Jones2020; Ubilava et al., Reference Ubilava, Barnett, Coble and Harri2011).

For each SCCTP combination, the specific steps in our approach are as follows.

• Step 1: We generate 20 representative producers, each with a random starting wealth (per subsection 4.2).

• Step 2: For each representative producer, we simulate ending wealth and calculate certainty equivalents for each of the eight available coverage levels using equation 17 with α = 2.5.

• Step 3: For each representative producer, we randomly draw a satisficing threshold parameter between 0 and 2% from a uniform distribution. We then determine the set of coverage level choices with certainty equivalent values that are less than the maximum certainty equivalent by no more than this percentage. This is the satisficing set.

• Step 4 (baseline scenario): In the context of no ERP, for each coverage level in the satisficing set, we assume that producers select at random, with probabilities equal to the relative proportions of acres covered at that level in the SOB data for the relevant SCCTP.

• Step 4 (ERP scenario): For the counterfactual scenario, we assume that a representative producer changes to a new coverage level if that coverage level both newly enters (vis-à-vis the baseline scenario) the satisficing set and has a certainty equivalent that is greater than their previous choice.

This last rule regarding changing coverage level choice under the ERP scenario is clearly not the only criterion that could be specified. Indeed, one might argue that this rule violates the spirit of the satisficing set. We also considered an alternative rule whereby producers changed coverage level only if their original choice fell out of the satisficing set. This resulted in very nearly zero changes in coverage level choice across all SCCTPs we consider. Even though there is little point in presenting those results in detail, we acknowledge that essentially zero changes in crop insurance choices may indeed be reasonably interpreted as the correct answer. We present results below that reflect the change rule presented above and interpret the resulting projected changes as the maximum extent of coverage level changes that might reasonably be expected to result from permanent authorization of the ERP program.

4.9. Model Validation

To validate the model, we ran the baseline scenario (no ERP) and compared it with observed choices in the SOB data. In total, we analyzed 67 million acres of dryland corn and 63 million acres of dryland soybeans. Results for 3,622 individual SCCTP combinations were aggregated and are summarized in Figure 4, which shows coverage level choices, measured in acres.

Figure 4. Validation results (simulated vs. summary of business) for the baseline scenario.

The two most popular coverage levels for both crops are 75 and 80%, whereas using a direct expected utility maximization framework would have over-selected an 85% coverage level in a substantial portion of counties for the reasons described above.

The validation result shows that the model slightly under-selects 85% coverage levels and moderately over-selects the 80% coverage levels. The simulation results showed that combined insured acres for corn and soybeans under the 75, 80, and 85% coverage levels were 58 and 53 million acres, when compared to the SOB data of 56 and 54 million acres, respectively.