Theoretical foundations
This study develops a methodology for assessing the conditional adoption (complementarity) of alternative farm practices using a joint adoption framework. Suppose a farmer can choose from adopting r possible practices on the farm. These practices can form M = 2r conservation bundles or conservation management plans. Let \(\delta_{m} ,\) m = 0, 1, …, M, be a specific bundle, where \(\delta_{m}\) is a (R × 1) vector of indicator variables, Yr, r = 1, …, R, equal to 1 if the rth practice is part of bundle m. Under the assumption of utility maximization, a farmer i derives utility from choosing bundle m with a given set of attributes/factors Xi that maximizes his or her utility umi. The utility for adopting bundle m can be represented as:
$$u_{mi } = U\left\{ {E\left[ {R\left( {{\varvec{X}}_{i} } \right)} \right]; {\varvec{Z}}_{i} ; {\varvec{\beta}}_{m} } \right\}$$
(1)
where E[R(\({\varvec{X}}_{i}\))] is expected profit from adopting the given bundle of conservation practices, \({\varvec{X}}_{i}\) is a vector of individual specific explanatory variables affecting the profitability of bundle m, \({\varvec{Z}}_{i}\) is a vector of other variables that impact the utility for bundle m, and \({\varvec{\beta}}_{m}\) is a vector of parameters specific to the utility received for adoption of bundle m. Farmers’ decisions on adoption of conservation practices are often influenced and motivated by other factors rather than profit related factors under a utility framework (Skaggs et al. 1994). Thus, it is necessary to distinguish and separate those profit related variables, \({\varvec{X}}_{i}\), and nonprofit but utility related variables, \({\varvec{Z}}_{i}\), including farming experience, education and employment (Ervin and Ervin 1982; D’Souza et al. 1993), age and other demographics (Skaggs et al. 1994). A farmer will adopt bundle m if:
$$u_{mi} = max\left( {u_{1i} , \ldots ,u_{mi} , \ldots ,u_{Mi} } \right).$$
(2)
Empirical model
A researcher only observes the choice of plan or practice bundle adopted. Thus, the theoretical model represented by Eqs. (1) and (2) can be viewed in a random utility framework:
$$u_{mi } = V_{m } \left\{ {E\left[ {R\left( {{\varvec{X}}_{i} } \right)} \right]; {\varvec{Z}}_{i} ; {\varvec{\beta}}_{m} } \right\} + \varepsilon_{mi}$$
where \(V_{m }\) is the deterministic component of utility and \(\varepsilon_{mi}\) is the random or unobserved component of utility (Louviere et al. 2000).
Following the methods in Bergtold and Molnar (2010) and Wu and Babcock (1998), a polychotomous-choice selectivity model of joint adoption is employed. If the residuals,\(\varepsilon_{mi}\), m = 0,1, …, M are independently distributed with extreme value distribution (type 1), then the probability of a farmer choosing bundle m, \(\delta_{m}\), can be written as:
$$\pi_{m} = \Pr \left( {I = m} \right) = \frac{{\exp (V_{m } [E(R\left( {{\varvec{X}}_{i} ));{\varvec{Z}}_{i} ; {\varvec{\beta}}_{m} } \right)}}{{\mathop \sum \nolimits_{s = 0}^{M} {\text{exp}}(V_{s } \left[ {E\left( {R\left( {{\varvec{X}}_{{\varvec{i}}} } \right)} \right); {\varvec{Z}}_{i} ; {\varvec{\beta}}_{s} } \right)}},$$
(3)
where I is a polychotomous index equal to m if bundle m is chosen. The adoption of a particular bundle of conservation practices is conditional on a number of explanatory factors, including experience, farm sales, land tenure, participation in conservation programs, farmer perceptions, use of insurance and a number of demographic variables.
For those farm characteristics, it is expected that farm size and rental percentage could have a positive effect on conservation practices. With larger farm size and more rented land, it is likely farmers will adopt conservation practices which provide net positive returns in the short term, but avoid or delay adopting conservation practices that are likely to only provide net positive returns over a longer time horizon. Farmers participating in EQIP and/or CSP can obtain financial incentives to adopt, as well as obtain more information and gain additional experience, increasing the probability of adopting conservation practices. Compared with risk-averse farmers, risk-neutral or risk-loving farmers may choose higher return crops or conservation practices regardless of time and regional constraints.
For farmer demographics and characteristics, we expect farmer experience, off-farm employment, crop insurance and farmers’ education level could increase the likelihood of adopting conservation practices. The weather, region and landscape attributes will likely affect adopting conservation practices differently across the various management plans.
With the limited number of observations for conservation management plan bundles CM and NCM listed in Table 2, it is assumed that P(I = CM) = 0, and P(I = NCM) = 0 (i.e., the probability of adopting these bundles is equal to zero), such that they will have no direct effect on the estimation of the model. Given the limited number of observations, the effects of the explanatory variables on the adoption of these management plans cannot be reliably identified. To not bias results, the observations with these associated conservation management plans are removed from the dataset, leaving 2091 observations for estimation of the model. While there are limited observations for other management plans, there does exist at least a 2:1 ratio of observations to parameters for each of the remaining conservation management plans (bundles) in the empirical model, providing enough degrees of freedom for estimation.
Marginal effects and measures of practice complementarity
Following Eq. (3) a multinomial logistic model is used to estimate the joint adoption of bundles of conservation practices or management plans. The model estimates the probability of adopting a bundle given a set of explanatory factors, but allows one to estimate the marginal probability of adopting a single practice and conditional probability of adopting a practice given other practice adoption. Marginal effects can be derived for all of these types of probabilities.
It is difficult to interpret the meaning of coefficients in the multinomial logistic model. The marginal effects of explanatory variables on the probability of adopting a bundle of practices provide a measure to assess the impact of specific explanatory factors. The marginal effects provide both a sign and magnitude for the marginal change in an explanatory variable on the probability of adoption. The marginal effect for a given explanatory variable,\(x_{k}\), is given by (Greene 2012):
$$\frac{{\partial \pi_{m} }}{{\partial x_{k} }} = \pi_{m} \left[ {\beta_{m,k} - \mathop \sum \limits_{s = 0}^{M} \pi_{s} \beta_{s,k} } \right]$$
(4)
It should be noted that the sign of the marginal effect may not follow the sign of \(\beta_{m,k}\) for m = 0, 1,…, M.
Wu and Babcock (1998) emphasize that the unconditional marginal probability of adopting a practice or single element of a conservation bundle sequence may be of interest. The marginal probability of adopting a single practice can be derived from the joint modeling framework as:
$$P_{s} = \mathop \sum \limits_{{m\varepsilon \left\{ {\delta_{m} : Y_{s} = 1} \right\}}} \pi_{m}$$
(5)
where s is the index for the single practice of interest and YS is an indicator variable equal to 1 when practice s is included in bundle m. The associated marginal effects for the marginal probabilities can be expressed as (Wu and Babcock 1998):
$$\frac{{\partial P_{s} }}{{\partial x_{k} }} = \mathop \sum \limits_{{m\varepsilon \left\{ {\delta_{m} : Y_{s} = 1} \right\}}} \frac{{\partial \pi_{m} }}{{\partial x_{k} }}$$
(6)
Joint probabilities of adopting two or more practices can be derived, as well. For example, the probability that a farmer jointly adopts two conservation practices is:
$$P_{rs} = \mathop \sum \limits_{{m\varepsilon \left\{ {\delta_{m} : Y_{r} = 1, Y_{s} = 1} \right\}}} \pi_{m,}$$
(7)
which can be useful when examining complementarity between conservation practices. The associated marginal effect for the bivariate probability given by Eq. (7) is:
$$\frac{{\partial P_{rs} }}{{\partial X_{k} }} = \mathop \sum \limits_{{m\varepsilon \left\{ {\delta_{m} : Y_{r} = 1, Y_{s} = 1} \right\}}} \frac{{\partial \pi_{m} }}{{\partial X_{k} }}$$
(8)
The joint adoption or multinomial model estimated allows for the estimation of conditional probabilities, which can be interpreted here as a measure of complementarity between two practices. For example, one could estimate the adoption of cover crops, given the no-tillage adoption decision. Given the cross-sectional nature of the data though, it should be cautioned that the conditional probabilities should not be interpreted as sequential adoption. The use as a measure of complementarity may assist in examining what factors affect farmers’ choices to intensify conservation efforts on-farm in order to help develop outreach strategies and incentive mechanisms. Using this framework, the adoption of practice s given practice r can be represented as:
$${\text{P}}_{{{\text{s}}|{\text{r}}}} = { }\frac{{{\text{P}}_{{{\text{sr}}}} }}{{{\text{P}}_{{\text{r}}} }}$$
(9)
where the marginal and bivariate probabilities are given by Eq. (5) and (7). Equation (9) may be interpreted as a measure of complementarity that ranges from 0 to 1. The larger the value of the conditional probability given by Eq. (9) the more likely two conservation practices will be adopted together over time, which may be simultaneous or sequential. Of additional interest is the estimation of marginal effects of the explanatory variables for the conditional probabilities assessed. These marginal effects allow for the examination of how different agronomic, economic, ecological and social factors may impact the complementarity between two given practices. These can be obtained by differentiating the conditional probability with respect to an explanatory variable of interest (k):
$$\frac{{\partial P_{s|r} }}{{\partial X_{k} }} = \frac{{\frac{{\partial P_{sr} }}{{\partial X_{k} }}*P_{r} - P_{sr} * \frac{{\partial P_{r} }}{{\partial X_{k} }}}}{{P_{r}^{2} }}$$
(10)
where the associated marginal effects for the marginal and bivariate probabilities are given by Eqs. (6) and (8). It should be emphasized that all the marginal effects estimated can be done using the joint probabilities and marginal effects estimated using the joint multinomial logistic model given by Eqs. (3) and (4). That is, the joint framework inherently captures the complementarities (dependencies) between adopting different practices.
To test for the significance of marginal effects, asymptotic estimates of the standard errors are required. Given the complexity of some of the equations of the marginal effects above, the method of Krinsky and Robb (1986) is utilized to estimate the asymptotic standard errors for the calculation of asymptotic z-statistics (see Greene 2012, as well). All marginal effects were calculated as partial averages following Greene (2012).
