Risk, risk aversion, and agricultural technology adoption ─ A novel valuation method based on real options and inverse stochastic dominance

Risk and risk preferences belong to the key determinants of investment-based technology adoption in agriculture. We develop and apply a novel approach in which an inverse second order stochastic dominance approach is integrated into a stochastic dynamic farm-level model to quantify the effect of both risk and risk aversion on the timing and scale of agricultural technology adoption. Our illustrative example on short rotation coppice adoption shows that risk aversion leads to technology adoption that takes place earlier, but to a smaller extent. In contrast, higher levels of risk exposure lead to postponed but overall larger adoption. These effects would be obscured if technology adoption is not analyzed in a farm-scale context or considered as a now-or-never decision ⸺ the still dominant approach in the literature. combines the real options approach and risk aversion in a programming setting. Literature provides no established approach for this yet (Homem-de-Mello and Pagnoncelli 2016). In the following, we discuss the advantages and disadvantages of the dominant approaches suggested by the literature, namely the expected utility function, using a risk-adjusted discount rate, and the concept of stochastic dominance. We show that they are particularly limited when not only optimal time and scale of technology adoption are considered, but also competition among different farm activities for limited resources. Our empirical results demonstrate that higher risk aversion leads to lower optimal scale of adoption. This is consistent with previous research findings 2013; Trujillo-Barrera Winsen We also find that risk aversion accelerates technology adoption. The effect is not apparent at very low or very high risk levels in our case study. A similar result was obtained by Truong and Trück (2016), who found that risk aversion encourages earlier investment in those climate change adaptation projects that are designed to reduce risk. Our results can be explained by the fact that the incentives are higher for a risk averse farmer to exploit the natural hedging effect of diversification, by adding novel to the established activities. The lower (or even the more negative) the correlation coefficient between both activities, the higher is the potential effect of natural hedging. Consequently, the effect of risk aversion on the timing of technology adoption might be different or obscured in other settings, especially if technology adoption is analyzed under different assumptions that do not imply natural hedging, such as stand-alone. these findings are ambiguous regarding their effect on Our results show due managerial flexibility, higher risk greater risk The risks risk our analysis A study


Introduction
Decisions to take up new activities and/or adopt new technologies are of crucial relevance for farm success (Blandford and Hill 2006, p.43;Kumar and Joshi 2014) and reflect production, market, technological and institutional risks as inherent properties of agriculture (e.g. Chavas 2004), as farmers are often risk averse (e.g. Iyer et al. 2020). This is confirmed in empirical studies which find risk exposure, risk perception (Marra et al. 2003;Liu 2013) and risk preferences (Liu 2013) to be among key determinants for the timing and scale of technology adoption. Thus, all three should be considered in dynamic investment analysis (Iyer et al. 2020). 2020) demonstrated that stochastic dynamic programming can be efficiently combined with Monte-Carlo simulations of stochastic variables followed by a scenario tree reduction technique to study the effect of risk on timing and scale of technology adoption in the context of a policy analysis. We extend this approach in two directions. First, we explicitly address the risk level based on lack of knowledge and experience as a crucial determinant of technology adoption (Marra et al. 2003;Karni 2006) by conducting sensitivity analysis. Second and more importantly, we relax the assumption of risk neutrality underlying 2020) and hence explicitly model the effect of risk preferences on technology adoption.
The frequently used expected risk utility theory (Morgenstern and von Neumann 1953) and prospect theory (Kahneman and Tversky 1979) provide a straightforward way to operationalize (perceived) risk exposure, risk perception and risk preferences to investigate effects of both on optimal scale investment. In particular, they reveal that decision makers with a higher risk aversion tend to adopt a new technology at smaller scales (Liu 2013;Trujillo-Barrera et al. 2016;van Winsen et al. 2016). Yet, effects of risk and risk preferences on optimal timing remain often unexplored (van Winsen et al. 2014;Meijer et al. 2015), partly because to date there is no well-established approach to incorporate risk preferences approaches (Krokhmal et al. 2011;Homem-de-Mello and Pagnoncelli 2016). However, most of them require a risk aversion coefficient or a risk utility function, which is difficult to determine empirically (see e.g., Charness et al. 2013;Just and Just 2017;Iyer et al. 2020).
Furthermore, the computation of these approaches can become quite demanding if the programming model comprises integers, necessary to capture indivisibilities of specific assets and economies of scale. To cope with this, we employ the concept of second-order stochastic dominance instead, namely partial ordering of alternative stochastic distributions in terms of their superiority for a risk-averse decision maker. We consider this promising as it requires limited assumptions on risk preferences and can be efficiently incorporated into stochastic dynamic programming (Nie et al. 2012). Specifically, a set of additional constraints ensures that a new technology or activity is only adopted at a scale (or not at all) at which it stochastically dominates a risk benchmark given by the current farm program. There are a few examples of introducing stochastic dominance constraints into optimization models in financial applications (El Karoui and Meziou 2006;Roman et al. 2006;Luedtke 2008;Nie et al. 2012). Although these models are concerned with the optimal shares in a portfolio of risky assets, they are not considering resource (inequality) constraints or indivisibilities relevant for farm-scale optimization, which implies different approaches to numerical optimization.
We here contribute to close this gap by developing a novel farm-level stochastic dynamic programming 1 approach that quantifies the effects of risk and risk preferences on optimal scale and timing of investment-based technology adoption. In particular, we embed the concept of inverse stochastic dominance into the real options framework and demonstrate how the proposed approach can reflect risk levels and risk preferences in an empirical example of adopting a new investment-based activity. We call the approach DIASS -Dynamic programming and Inverse Approximated Second-order Stochastic dominance.
Using the designed model and applying it to an empirical case study, we test for this specific case whether risk aversion (vis-à-vis risk neutrality) leads to earlier technology adoption at a lower scale. Moreover, we test whether higher associated risk levels ceteris paribus lead to later technology adoption at a lower scale. We also quantify the economic relevance of these effects. Findings underline that the DIASS approach allows to simulate farmers' decisions more precisely and thus to better inform policy makers about expected adoption of targeted 1 In the following we use the term ‖stochastic dynamic programming‖ to emphasize that we refer to a long-term problem and solve for an optimal steady-state decision, in contrast to stochastic programming that provides a transient solution (King and Wallace 2012).
investment-based technologies, for instance, regarding contributing to environment protection, animal welfare, or digitalization.
Our case study features introduction of short-rotation coppice (SRC) biomass energy production systems as an investment-based new technology 2 on a typical arable farm in northern Germany. Setting up an SRC plantation with its typical production cycle of approximately 20 years implies significant sunk costs for planting, coppicing and final reconversion to arable land. Reconversion is considered as the plantation will be otherwise considered legally as a forest, which prevents future re-conversion to arable land and claiming of farming subsidies. SRC binds land for a longer period than other currently observed land uses in that type of farms and competes with annual crops for limited farm resources such as land and labour. Both SRC and annual crops imply stochastic returns; the observed distribution of returns from annual crops constitutes an observed risk benchmark. The case study thus encompasses the elements mentioned above as inherent for investment-based technology adoption in agriculture, such as sunk cost, uncertain future returns, and competition with existing activities. It hence perfectly fits to demonstrate how the effects of risk level and risk preferences on timing and scale of adoption can be quantified and analyzed.
To this end, we provide insights in both a generic modelling approach and in a specific case study. The DIASS approach can be applied to any other case study by adjusting the underlying stochastic processes and their mutual correlation, the number of investment and disinvestment decisions considered, or the time horizon. We provide the code, data, all the related documentation, as well as a graphical user interface, in Spiegel et al. (2017), in order to facilitate use of the proposed approach for other case studies in and beyond agriculture.
Our results show that risk aversion leads to technology adoption that takes place earlier, but to a smaller extent. In contrast, higher levels of risk exposure lead to postponed but overall larger adoption. These complex interdependencies between risk, risk preference and technology adoption would be obscured if technology adoption is not analyzed in a farm-scale context or considered as a now-or-never decision, i.e., according to the still dominant approach in the literature. based on stochastic dynamic programming where risk is captured by a scenario tree (Beraldi et al. 2013;Alonso-Ayuso et al. 2014;Simoglou et al. 2014). This is usually based on binomial scenario trees or lattices (Schulmerich 2010;Beraldi et al. 2013;Alonso-Ayuso et al. 2014) where model size increases quadratically or even exponentially 3 with the number of time points, which limits model complexity and timescale. These restrictions can be partly overcome with more advanced approaches such as Monte Carlo simulation followed by scenario tree reduction (Dempster 2006;Heitsch and Römisch 2008;Spiegel et al. , 2020.
The real options approach can be applied under different assumptions with regard to risk preferences. Incentives to postpone a managerial decision, e.g., technology adoption, might exist regardless of risk attitude (Dixit and Pindyck 1994, p.153).  demonstrated that in the risk-neutral context, decreasing or eliminating a risk might lead to earlier adoption at a lower scale. However, risk preferences can influence the optimal timing and scale of technology adoption as well (Marra et al. 2003;Liu 2013). Empirical results highlight that European farmers tend to be risk averse (Menapace et al. 2013;Meraner and Finger 2017;Iyer et al. 2020 combines the real options approach and risk aversion in a programming setting. Literature provides no established approach for this yet (Homem-de-Mello and Pagnoncelli 2016). In the following, we discuss the advantages and disadvantages of the dominant approaches suggested by the literature, namely the expected utility function, using a risk-adjusted discount rate, and the concept of stochastic dominance. We show that they are particularly limited when not only optimal time and scale of technology adoption are considered, but also competition among different farm activities for limited resources.
Introducing and maximizing a utility function is a straightforward approach and often used in empirical applications (Hugonnier and Morellec 2007;Shapiro 2012). Obviously, results are sensitive to functional choice and parameterization, found as empirically challenging (Lence 2009;Crosetto and Filippin 2016). In the context of programming models, expected meanvariance analysis initiated by Markowitz (1952)  Computational limits can be overcome by maximizing the certainty equivalent instead and using an approximation (e.g., see Henderson and Hobson 2002). The approach requires assuming a coefficient of risk aversion only, rather than formulating a risk utility function. Meyer and Meyer (2005) rather than a stochastic process (e.g., Chavas and Shi 2015). This automatically implies a potential natural hedging effect when accumulated over years, namely reduction of total risk exposure due to imperfect correlation of multiple stochastic processes. Capturing risk preferences by a Risk-Adjusted Discount Rate represents a conceptually different approach, not affecting computational feasibility. In contrast to a risk-free discount rate, a Risk-Adjusted Discount Rate reflects both the level of risk and the decision maker's attitude towards this risk. Therefore, it should be adjusted as the level of risk changes over time. More specifically, the adjustment would be specific for each farm activity, which is characterized by a different level of risk, and at each node of the scenario tree, since the risk decreases when approaching

2.2.Inverse stochastic dominance and stochastic dynamic programming
According to SSD, a random variable dominates a random variable (i.e., ) if the expected utility of is at least as high as that of , (i.e., ) (Dentcheva and Ruszczyński 2006). In general terms, the condition of SSD for a discrete case can be formulated as follows, as long as the underlying utility function is monotone and concave (Chavas 2004): Where stays for second-order stochastic dominance; and are stochastic variables with possible realizations ; and are their cumulative distribution functions; is the minimum possible realization higher than .
The incorporation of SSD as a constraint into an optimization model will typically imply a substantial increase in computational complexity, since it requires the introduction of additional binary variables (Gollmer et al. 2007;2008). Alternative (approximate) formulations of stochastic dominance are proposed to deal with this. In particular, Dentcheva and Ruszczynski (2003)  More specifically, for a probability space we first introduce the following definitions (Ogryczak and Ruszczynski 2002): where ̅ is the second quantile function 4 ; { } is the expectation operator; are realizations of a random variable; and is the so-called target value. It is shown that SSD of over is equivalent to the expected realization of being greater than or equal to the expected realization of at all intervals (Ogryczak and Ruszczynski 2002): 4 In the following, remains for the set of natural numbers for the set of real numbers, and ̅ for the set of real numbers extended by positive infinity and negative infinity ( The approach does not require ordering realizations beforehand; the target value is defined for each and all are multiplied with the respective probabilities to define { } without being ordered. We first derive the distribution of returns of a farm under the observed benchmark farm program as representing a revealed optimal choice given the farmer's risk preferences. Next, we solve for . More specifically, we determine a program with optimal timing and scale for the new technology or activity under the condition that it (approximately, using ISSD) stochastically second-order dominates the given benchmark .
instead of considering the annual distribution of cash inflows and outflows. This concept implies that an agent only considers the distribution of his/her (discounted) terminal wealth after the lifetime of a project. The literature suggests using a normative portfolio characterized by a tolerable distribution (Bailey 1992;Kuosmanen 2007) if alternatives are evaluated. In the farm context, a farmer's observed production activities and related realizations can be considered as such a benchmark (Musshoff and Hirschauer 2007). We find it straightforward to include the initial farming activity as the benchmark and optimize an alternative one considering the adoption of a new technology, using constraints to ensure that it stochastically dominates the status quo. Assuming that the status-quo is not based on rational behavior obscures the simulation outcome, since differences between the benchmark and the optimal solution would not only reflect the opportunities arising from considering the new technology, but also different behavior.

2.3.Risk analysis and hypotheses
With . If opportunity costs are also stochastic and correlated with the investment option to be exercised (as in our settings), there is a potential opportunity for hedging and a risk averse decision maker may be more willing to exploit this by investing earlier (Henderson and Hobson 2002;Truong and Trück 2016;Chronopoulos and Lumbreras 2017). Therefore, we hypothesize that risk aversion leads to a smaller scale and earlier adoption (Table 1). Previous studies often revealed differences between the ex-ante risk perception of investment projects and actual risk levels derived ex-post (Liu 2013;Menapace et al. 2013;Bocquého et al. 2014), suggesting that investment decisions are based on subjective beliefs (Savage 1972;Marra et al. 2003;Karni 2006). Empirical research identifies a number of factors that affect risk perception, including age (Menapace et al. 2013 hypothesize that a higher perceived risk level of a new technology leads to a smaller scale of technology adoption and postponement (Table 1).

3.1.General layout
We develop a model based on the stochastic-dynamic programming approach where decision variables are state-contingent. We allow the farmer to introduce a new venture which competes with established activities for (quasi-fixed) resources such as farm land and labor.
The adoption of the new activity requires investments subject to indivisibilities of assets and returns to scale. Per unit returns from the new venture are risky and follow a stochastic process, the same applies to established activities. This implies that opportunity costs for the new activity are not known beforehand, but depend on the interaction of the level of adoption and the states of nature. We also assume that the initial farm program before adopting a new technology constitutes an optimal portfolio under the given stochastic returns; it serves as our risk benchmark. With a set of additional constraints, we ensure that a new venture is only adopted at a scale (or not at all) at which it second-order stochastically dominates this benchmark. Furthermore, the decision maker has the flexibility to postpone the adoption of the new activity, motivating the use of a real option approach. To this end, we assume that the decision about optimal time and scale of a new technology adoption is based on an NPV maximization; subject to existing resource endowments and other farm-level constraints; conditional to possible future developments of stochastic variables; while ISSD constraints approximate inverse second order stochastic dominance over the endogenously simulated distribution of discounted terminal wealth. The optimization problem can then be expressed as follows:

3.2.Solution process
The mixed-integer model is solved with stochastic dynamic programming by using a scenario tree to represent uncertainty with a predetermined number of ̅ leaves. This tree stems from first running a Monte Carlo simulation with = 10,000 draws and subsequently reducing the resulting tree to the desired size by applying a scenario tree reduction technique according to Heitsch and Römisch (2008) 5 . This allows control over overall model size, while keeping the values assigned to each node within a certain plausible range and hence gaining a computational advantage. Since there are multiple stochastic variables in the model, a vector of values is assigned to each node of the scenario tree. The optimal decision with respect to technology adoption for each node of the tree is conditional to decisions made prior to this node and the possible follow-up scenarios (Fig. 2).
The effect of risk aversion is quantified by adding the ISSD constraints to the model, and then comparing resulting outcomes without these constraints, namely the risk-neutral case. As mentioned, we measure risk levels based on the final distribution of NPVs and use the currently observed behavior as the benchmark for tolerable risk. The additional ISSD constraints ensure (approximately) that, giving due consideration to the new activity, the distribution of NPVs under the benchmark is dominated second-order stochastically by the final distribution of NPVs (Fig. 2).
We conduct a sensitivity analysis to capture different risk levels associated with technology adoption by considering different parameters of the related stochastic process (Fig. 2), however without changing its long-term mean nor the expected mean in each year.
Consequently, results under a now-or-never risk neutral decision (or the classical risk neutral NPV approach) would not change. Draws of the other stochastic processes which relate to the activities present at the farm prior to technology adoption are obtained once and fixed.

3.3.Case study and model specification
We illustrate the DIASS approach using the example of potential introduction of a perennial Pe'er at al. 2016). SRC competes with annual crop production for land resources, while the setting up and harvesting of SRC are usually outsourced, so that little or no farm labor is required (Musshoff 2012). Economic considerations of introducing SRC are thus as follows.
On the one hand, SRC requires significant and irreversible investments for establishment and final reconversion and binds land for a long time, while its price is assumed to be stochastic.
On the other hand, SRC reduces the amount of idling land or catch crops required for the Ecological Focus Area requirement, while labor is saved due to use of contracted services (Musshoff 2012). Consequently, labor previously used on a plot now devoted to SRC can be reallocated to the more profitable and labor-intensive annual crop.
In our setting, the farmer considers introducing SRC immediately or within the next three where is a combination of time period and node of the scenario tree assigned to each where is the ecological focus area weighting coefficient for SRC.

ISSD constraints
where is a set of decision variables; and denote scenarios after and before SRC adoption respectively; { } denotes cumulative probability of ; set is a set of predefined intervals of cumulated distribution.
As explained below, various relationships in the model need integer variables. Thus, in order to avoid a mixed non-linear integer programming problem, we keep the model linear by predefining plots of certain sizes to be potentially converted into SRC plantation in 5-hectare increments (i.e., providing 0, 5, 10, …, 100 ha of SRC plantation). Each plot can be converted to SRC, coppiced, or clear-cut independently from the others, but partial coppicing on an individual plot is not possible. Two equations linked to either a positive (0 in -1 to 1 in ) or a negative (1 in -1 to 0 in ) change in SRC on a plot are used to describe set-up and reconversion costs respectively (nodes indices are left out for simplicity):  (Table 2). Appendix A provides further details and also compares our data assumptions with similar ones from the literature.  (2016) Gross margins of annual crops Stochastic, see Table 3 Land

3.4.Stochastic component
We assume that the natural logarithm of each stochastic variable follows a mean-reverting process. This choice is based on the premise that the farmer is a price-taker in an environment where market forces cause prices and gross margins to fluctuate around constant long-term levels, for instance, under the assumption that there is no monopolistic power (Metcalf and Hassett 1995) and/or technology is constant ). An mean-reverting process is characterized by a long-term mean, speed of reversion and standard deviation (Dixit and Pindyck 1994). We estimate the parameters of the process for annual crops using data 7 on gross margins of an average hectare of arable land in Germany between 1993-2012 from the CAPRI (2017) model data base following the procedure described in Musshoff and Hirschauer (2004). Appendix B provides more details on this estimation. The process for SRC biomass prices is based on Musshoff (2012).  (Pindyck and Rubinfeld 1997). Since there are no long time series of sufficiently lengthy duration, as in our example, the choice of a stochastic process must be supported by theoretical considerations, rather than statistical tests. (Dixit and Pindyck 1994). Therefore, some researchers argue for this process, because it allows a long term equilibrium level accompanied by temporal fluctuations that is plausible for many economic variables (Musshoff 2012).
a Starting value are stet equal to the long term mean to exclude any possible effect of a trend b The assumption is based on ambiguous evidence in the literature about sign and magnitude of the correlation (Musshoff and Hirschauer 2004;Du et al. 2011;Diekmann et al. 2014). c Multiplicative coefficients are assumed for draws converted back from natural logarithm into euro per hectare.
The literature provides ambiguous evidence regarding the correlation coefficient between SRC biomass price and annual crop gross margins (Musshoff and Hirschauer 2004;Du et al. 2011;Diekmann et al. 2014), while the effect the coefficient on farmers' behavior has been found to be limited . This lets us assume a zero correlation between the biomass price and annual crop gross margins, reflecting that gross margins of SRC and annual crops are not driven by similar market and climatic influences. In contrast, we assume that the gross margins of the two annual crops are perfectly correlated. Therefore, we use one process for both gross margins and then adjust the draw at each node of the scenario tree with multiplicative coefficients to derive gross margin levels (see Table 3). The correlation coefficient enters stochastic processes as follows: where is the time period; indicates short rotation coppice; index indicates both arable crops; is the natural logarithm for the price of SRC biomass; is speed of reversion of the stochastic process for SRC biomass price; is long-term logarithmic average price of SRC biomass; is standard deviation of logarithmic SRC biomass price; is standard Brownian motion independent from ; is correlation coefficient between two Brownian motions.
Further research might specify the alternative portfolio in greater detail, including different correlation coefficients between gross margins of annual crops. The DIASS approach does not imply any restrictions in this regard, but it is beyond the scope of our illustrative purposes.
We obtain draws (see Fig. 2) from the Monte Carlo simulation. In order to select the number of leaves in the reduced scenario tree, we performed multiple runs of the model gradually increasing the number of leaves and noticed that the expected area under SRC stabilizes beginning at 200 leaves ( ̅ on Fig. 2). For ISSD constraints, we consider 100 intervals 8 with a 1%-step ( in Fig. 2 and in Eq.8), which should render the impact of the approximation negligible. We performed the risk analysis by gradually increasing the standard deviation and decreasing the speed of reversion in the stochastic process for the SRC output price. The higher the standard deviation and the lower the speed of reversion, the more volatile the stochastic process becomes, reaching a higher spread and reverting to the long-term mean more slowly.

Results
The key results under risk neutrality and baseline risk levels are presented in Table 4. Note that introducing SRC immediately (in ) is not optimal, meaning that an option value exists even for a risk-neutral farmer. Accordingly, the investment decision is postponed and exercised later, or not at all, depending on future developments. In 39.43% of the simulated cases, we find that SRC would never be introduced. The expected area under SRC is 7.97 ha, which mainly stems from substituting the less profitable crop. The SRC would not fulfill the Ecological Focus Area requirement (16.67 ha if the total land endowment is 100 ha) and thus set-aside land and catch crops remain in the farm portfolio (1.70 ha and 6.69 ha respectively).

ha after introduction of SRC). This reallocation of resources creates an additional
incentive for adoption, which would be neglected when the technology adoption would be analyzed as a stand-alone investment and not in the farm context. The optimization under risk neutrality introduces SRC in some scenarios, such that the expected NPV must increase compared to the benchmark. However, this also implies substantially higher risk as seen in Fig. 3. The distribution of NPVs with SRC simulated under risk neutrality (i.e., without the ISSD constraints, black solid curve) does not stochastically dominate the benchmark (red curve): its lowest NPV realization undercuts the lowest one in the benchmark. Enforcing SSD by introducing the ISSD constraints turns the NPV distribution function with SRC in a counterclockwise direction, cutting the left-hand-side tail (black dashed curve). That also reduces the probability of larger NPVs compared to the higher adoption rates of SRC under the risk neutral case, underlining the tradeoff between a higher mean and a higher risk.
We now demonstrate the effect of risk aversion and changes in risk levels on the scale of technology adoption (i.e., the expected acreage of the farm under SRC). Fig. 4 combines the effects of adjusting the standard deviation and mean of reversion of the stochastic process for the SRC biomass price with and without the ISSD constraints. Our analysis shows that risk aversion (under the ISSD constraint) does indeed lead to a smaller expected area under SRC, which is consistent with the null hypothesis. Indeed, many white dots (risk aversion) on Fig. 4 lie substantially below the respective black dots (risk neutrality). The ISSD constraints cut off the lower tail of NPV distribution as discussed above such that no SRC adoption is observed in some leaves where it would be realized under risk neutrality. This reduces the overall expected scale of SRC adoption. These differences can reach up to 20 hectares or around 50% for the extreme cases, and are found to react more sensitive to changes in the speed of reversion.
In contrast, the null hypothesis on the effect of higher risk levels on optimal scale is rejected in our example. Results show that even for a risk averse decision maker, a higher risk level stems from the established annual crops, only, but also shifts up the expected SRC price for the nodes where the threshold price is exceeded, which triggers a larger scale of the investment project for these nodes. In our application, the expected mean area under SRC, which measures the scale of adoption, increases at higher risk levels for both risk neutral and risk-averse decision makers, even though the respective trigger price increases. For instance, the expected mean area under SRC for a risk-neutral decision maker increases from around 22 to 54 hectares when decreasing speed of reversion from 0.22 to 0.02. However, this effect of increasing risk levels is smoothed by risk-aversion, especially when adjusting the speed of reversion (Fig. 4).
Next, our results reveal a U-parabolic relationship between risk levels of SRC and incentives for earlier SRC introduction (Fig. 5). Lower standard deviation values limit incentives to postpone SRC introduction by reducing risk and the related option value, reflecting that the decision problem moves towards a classical NPV analysis. A similar U-parabolic relationship can be observed between SRC risk levels and the probability that SRC will never be adopted: there is a level of risk that implies the highest probability of never adopting SRC, which can be quantified using our approach (for instance, in our application it is associated with the standard deviation of around 0.64 on the left-hand side of Fig. 5). Therefore, the null hypothesis is confirmed in our settings for lower levels of risk and rejected for greater ones.
A comparison of the timing of SRC introduction in the case of risk neutral and risk averse decision makers (Fig. 5) reveals that risk aversion might lead to earlier SRC introduction (for instance, for standard deviation of 0.64 the probability of adopting in the second year increases from 13% to 22% when risk aversion is considered). This is due to the fact that risk averse decision makers exploit the hedging effect between the uncorrelated stochastic returns of annual crops and SRC. A risk averse farmer is predicted to introduce SRC earlier in order to reduce overall farm risk, although on average the area of SRC adopted is smaller compared to a risk neutral farmer. This effect would be obscured if the alternative land use portfolio is assumed to be deterministic or if technology adoption is considered stand-alone. The effect of risk preferences on timing of SRC adoption is highest at mid standard deviation values (e.g., standard deviation of 0.64, Fig. 5). Regardless of risk preferences, there is no incentive to postpone adoption for low levels of risk. Increasing risk levels imply that a trigger price that stimulates SRC adoption is reached sooner. In contrast, SRC adoption is not attractive for very high-risk levels. Therefore, we cannot reject the null hypothesis.
A decision maker who perceives SRC as quite risky (implying larger deviations of its price below and above expectation levels) tends to commit a larger area to SRC earlier, but not immediately. A respective trigger price must be reached in order to initiate SRC introduction, otherwise an investment decision will be postponed indefinitely. Furthermore, the negative effect of risk aversion on the scale of adoption rises as risk levels increase. The major findings are presented in Table 5.

Discussion and conclusion
The development of efficient policies and forecasting of targets demands a well-informed understanding of farmers' motives with respect to technology adoption. However, the analysis of joint effects of explanatory factors on technology adoption is still limited, especially with regard to risk and risk preferences (van Winsen et al. 2014;Meijer et al. 2015). Our empirical results demonstrate that higher risk aversion leads to lower optimal scale of technology adoption. This is consistent with previous research findings (Liu 2013;Trujillo-Barrera et al. 2016;van Winsen et al. 2016). We also find that risk aversion accelerates technology adoption. The effect is not apparent at very low or very high risk levels in our case study. A similar result was obtained by Truong and Trück (2016), who found that risk aversion encourages earlier investment in those climate change adaptation projects that are designed to reduce risk. Our results can be explained by the fact that the incentives are higher for a risk averse farmer to exploit the natural hedging effect of diversification, by adding novel to the established activities. The lower (or even the more negative) the correlation coefficient between both activities, the higher is the potential effect of natural hedging. The DIASS approach is subject to the following limitations. Firstly, it does not allow any differentiation between varying levels of risk aversion. This exceeded the scope of our study.
If the purpose is to differentiate between levels of risk aversion, the certainty equivalent approach could be used instead of inverse second order stochastic dominance, but it must be borne in mind that this involves another limitation, namely computational feasibility. Finally, we explicitly assume that the constructed scenario tree is assigned to the farmer and fixed. It would be more realistic to assume a process of learning for the true scenario tree and adjust behavior accordingly. However, stochastic dynamic programming does not provide any established approach to consider learning processes (Sunding and Zilberman 2001;Guthrie 2009 where represents dry matter yields (t ha -1 ); stands for density of trees (ha -1 ); is possible intermediate harvesting interval: 2, 3, 4, or 5 (y); is average amount of precipitation in May-June (mm); is soil quality index; is average temperature in April-July (°C); and is available ground water capacity (mm). We fixed all the variables except for the interval between harvests (Table A1) and fitted the values obtained to a linear function of available biomass in the previous year: (A2) where represents dry matter yields in the previous year (t ha -1 ). The model allows the interval between harvests to be adjusted or even transformed into a decision variable. In the latter case, tests revealed that a 5-year interval is usually the best. We used a fixed 5-year interval between harvests, in order to increase computational speed.

Resources endowments
Labor endowment and labor requirements only cover fieldwork and exclude management work, which is assumed to be fixed per farm and thus has no effect on resource distribution.
The total land endowment of 100 ha is representative for northern Germany: for instance, in

Appendix B: Estimation of a mean reverting process for gross margins of annual crops
The following data for gross margins of arable land were used CAPRI (2017):