Introduction

Hardaker et al. (2007) suggest two main sources of risk substantially affecting farmers’ decisions: business and financial risks. Business risks are combinations of various sources, generated by the unpredictable nature of the weather causing the uncertain performance of farmers’ incomes. Price risks refer to the price volatility of farm inputs and outputs (Rosa et al. 2014). In a survey conducted in the Netherlands, the price risks were perceived as the most important, while financial risks were ranked least relevant (Meuwissen et al. 2001). The lesson is that farmers should be better informed about the risk sources and react by combining their farm enterprises more efficiently. This topic has been explored by many researchers in farm management problems. Farmers usually elaborate their production plan in a context of limited information that biases the outcomes of their decisions, even in a shorter time horizon (Bernstein, 1996; Chen et al. 1999; OECD 2000; Shapiro et al., 1993; Stewart 1991).Footnote 1 These decisions affect crop supply and incomes in the short term, while the rigidity of the assets invested in longer periods and higher fixed costs increase the difficulty in adjusting the farmers’ decisions about risky crop prospects (Anderson et al. 1977, Anderson and Hardaker 2003, Anderson et al. 2004; Hardaker et al. 2007; Markowitz 1952). Price volatility is an external source of risk, whose importance is growing due to changes in the farming environment and strengthening of the integration between agricultural and commodity markets (Rosa et al. 2014). One example is the high price volatility of the oil markets due to financial speculation, transmitted to the agricultural input markets through prices of fertilizer, feedstock, pesticides, machinery, etc. Table 1 summarizes some sources of risk affecting farmers’ decisions.

Table 1 Risk sources in the agricultural sector
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The farm situation in Italy is represented by 1.5 million units with average UAS of 10 ha; smaller farmers are the majority of the rural population, and in most cases, they are unable to deal with the complexity of the agricultural sector (Salvioni et al. 2009; Dries et al. 2012). Limits in entrepreneurial skills could be a factor that influences the risk avoider attitude.Footnote 2 Due to fear of losing all their assets in the case of adverse events, risk aversion could address preferences towards “risk minimization and safety-first” instead of a profit maximizing strategy (Kaine et al. 1994; Oguntade and Bamire 1994; Anderson et al. 1977; Knight 1971; Matlon 1991; Chavas and Holt, 1990; Martin and Shadbolt 2000; Anderson and Hardaker 2003, Anderson et al. 2004; Manfredo and Leuthold 1999; Sulewski and Kłoczko-Gajewska 2014).Footnote 3 In a survey to test farmers’ preferences for risk in the Netherlands, Meuwissen et al. (2001) found that risk aversion was correlated to changes in crop yield and suggested that “higher expected level of farmers’ incomes in certain conditions do not balance the risk of losing their assetsFootnote 4.” Tobin (1965) says that under risky conditions (variance), averse decision makers formulate their optimal portfolio with crop diversification (risky prospects), while for risk preferring decision makers, the optimal portfolio will be reduced to few or a single risky prospect. Crop diversification in farm planning is then usually adopted to hedge against risk; as the average of the sum of variance and covariance of different price-yield combinations compensate for each other if the risk factors are similar and are generally inferior to the variance of a single crop (Mishra et al. 2004; Backus et al. 1977).Footnote 5 The optimal combination of risky prospects about crops generates the efficient E-V frontier (Fig. 1), representing the locus of efficient solution with maximum E value for a given variance (or s.d = V 0.5) or minimum variance for a given E value. The literature reports different methods to assess farmers’ risk attitudes with computable models of the E-V frontier (Hardaker et al. 1988; Anderson and Hardaker 2003; Backus et al. 1997). The approach proposed in this research is based on (i) the variance of total gross income (GI) in the crop combination portfolio, using quadratic risk programming in the linear version to approximate the quadratic E-V frontier and (ii) introduction in the subjective expected utility (SEU) of individual beliefs and preferences about risk (Pannell and Nordblom 1988; Backus et al. 1997; Bazerman 2006). The subjective probabilities about changes in frequency of risky prospects are justified by the different farmers’ knowledge about external events (climate, market, finance, policy, etc.) that affect the SEU of farmers’ beliefs about future events (Hardaker et al. 1991, 2007; Pannell and Nordblom 1988). With these premises, the objective is to simulate optimal crop combinations portfolio of risky prospects about crops with an “ad hoc” utility function to parameterize the risk for specific traits of it.Footnote 6 The solutions are used to draw the E-V frontier and the tangency between utility and E-V frontier will give the optimal solution for portfolio crop combination. The risk is assessed with the mean and variance moments of some diffuse crops in Italy. Four cereal crops: maize, barley, wheat, sorghum, and three oilseed crops: soybean, rapeseed, and sunflower, are selected for the crop portfolio solution.Footnote 7 The stochastic condition is assessed by using the yield and price fluctuations of these crops, assuming that the annual observations are drawn from a stochastic distribution of events (Markowitz 1952; Stovall, 1966; Taylor 1990; Lien and Hardaker 2001; Pope 1982; Hardaker et al. 1988).Footnote 8 A second observation concerns linear segmentation of the utility function, which is a quadratic approximation to facilitate application of the parametric linear programming method, without losing the stochastic nature of the activity (Hardaker et al. 1988). The problem is formulated as the minimization of risk to a range of possible levels of expected GIFootnote 9 subject to farm constraints and other restrictions (Hazell and Scandizzo 1974; Anderson et al. 1977, p 207).

Fig. 1
figure 1

Efficient E-V frontier for whole farm planning

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This research is organized as follows: the “The utility efficient programming and linearization with MOTAD” section is a discussion on the properties of the sumex utility function (SUF) (Fig. 2) and risk modulation for its specific traits, the parametric programming method, and linearization; the “Empirical analysis: the risky prospects ” section discusses the sample of risky prospects used for empirical analysis; the “Computation of utility efficient frontier ” section is dedicated to analysis of the utility efficient frontier; the “Conclusions” section analyzes the results obtained with evidence of the initial hypothesis about crop diversification and other suggestions to implement these results.

Fig. 2
figure 2

Sumex utility (left) and coefficient of risk aversion

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The utility efficient programming and linearization with MOTAD

In this section, we examine the stochastic production frontier obtained with the efficient programming method (Hanoch and Levy 1970; Hardaker et al. 1991; Ghodake and Hardaker 1981; Meyer 1977; Lee et al. 1985; Kaiser and Apland 1989; Hossain et al. 2002; Lien and Hardaker 2001; Park et al. 2004). The efficient set of portfolio solutions is obtained with a formulation of appropriate objective function to modulate risk, and the MOTAD method is used to linearize the function to obtain an efficient set of portfolio plans.

The sumex utility function (SUF) proposed by Patten modulates the risk with λ

$$ \mathrm{U}\ (z)=\mathrm{G}(z)+\lambda \left(\mathrm{H}(z)\right) $$
(1)

where Z is the wealth variable in the G and H polynomials of the sumex function that satisfies the desired properties of separability and concavity conditions obtained with first and second derivatives, while the risk aversion is simulated by varying λ parametrically in the range between a and b (Hardaker et al. 1988; Nakamura 1996). The SUF with first- and second-order derivatives is reported below:

$$ \mathrm{U}(z)=-\exp \left(-\mathrm{a}z\right)-{\lambda}^{\ast}\exp \left(-\mathrm{b}z\right)\kern1em \mathrm{with}\;\uplambda, \mathrm{a},\mathrm{b}\ge 0\kern3em \mathrm{SUF}\;\mathrm{function} $$
(2)
$$ {\mathrm{U}}^{'}(z)=-\mathrm{a}\ \exp \left(-\mathrm{a}z\right)+\lambda \mathrm{b}\ \exp \left(-\mathrm{b}z\right)>0\kern7em 1\mathrm{st}\;\mathrm{derivative} $$
(3)
$$ {\mathrm{U}}^{^{{\prime\prime}}}(z)=-{\mathrm{a}}^2\exp \left(-\mathrm{a}z\right)-\lambda {\mathrm{b}}^2\exp \left(-\mathrm{b}z\right)<0\kern6em 2\mathrm{nd}\ \mathrm{derivative} $$
(4)

The SUF implies decreasing risk aversion as z increases, assuming “normal behaviour.” The coefficient of absolute risk aversion λ ranges between the extreme limits a and b, with the following parametric restrictions on a and b: a (minimum risk) for λ = 0; b (maximum risk) for λ = max. The linear segmentation of SUF allows to find efficient solutions using a parametric linear programming method for the efficient farm plan. Since G(z) and H(z) satisfy the concavity condition, the Duloy and Norton procedure can be used to find the linear approximation (McCarl and Önal 1989). The efficient programming is based on the linearized quadratic risk programming (QRP) method that uses a matrix of crop GI deviations from the mean value. The E-V combinations are the set of efficient solutions obtained with the minimization of variance subjected to the parametric constraint of expected income (GI). Hence, the MOTAD linear approximation of the QRP and E-M combinations obtained with MOTAD are an acceptable proxy for the E-V combinations obtained from quadratic function (McCarl and Önal 1989; Lambert and McCarl 1986). The method proposed by Hardaker et al. (1988) is based on a separable utility function in an efficient programming model:

$$ MaxE(U)=\sum {\rho}_k\left(G(z)+\lambda {\rho}_kH(z)\right)\kern0.2em with\kern0.2em \lambda \kern0.2em varied\kern0.34em parametrically $$
(5)
$$ \mathrm{subject}\ \mathrm{to}\ \mathrm{R}-{}^{\hbox{'}}\mathrm{x}=e\ \mathrm{and}\ \mathrm{Ax}\le \mathrm{b};\mathrm{for}\ \mathrm{all}\ \mathrm{x}\ge 0 $$
(6)

where E(U) is the sumex utility function; G and H are the two polynomial risk components of the SUF, pk is the probability of state k and λ is a non-negative parameter varied parametrically between 0 and maximum value in a defined interval of the risk aversion; R̅ = E(GI) is an unbiased estimator of GI and R̅′x is the estimated vector of expected GI generated from the activities x included in the crop portfolio; and x is the vector of activity level (positive values), e is the maximum acceptable limit of risk by farmers, and A is the matrix of technical coefficients of resources used in activity x, constrained to resource limit b (Hardaker et al. 1988; Pope 1982).

The relation between utility function and the efficient frontier has a solid empirical basis when there is exhaustive knowledge of the production coefficients. In this case, the objective function will capture the utility dimension embedded in the expected GI function (Hardaker et al. 2007; Ignizio 1982; McCarl and Önal 1989). A relevant question concerns the optimal size of the sample GI to reduce bias (Pope 1982). Many authors have suggested that the normal distribution of states is not required, as any observation about a risky prospect can be assumed to be a representative random event of equally likely outcomes or states with subjectively assessed probabilities. Table 1 reports the moments generating risk, mean, SD, and CV referring to the yield and prices of the sample of crops representing the risky prospects of the commodities used in the analysis. The standard deviation values of yields (s.d.) fluctuate in the range between 0.18 for sunflower and 0.59 for maize; these values are much lower than the s.d. of prices that fluctuate in the range between 1.84 for barley and 10.22 for rapeseed. The covariance values show the risk generated by the combination of price and yield changes affecting the farmers’ GI; the lowest absolute value is for barley (− 0.01), the highest for soybean (− 0.71). The price variability compared to yield variability ranges from moderately high (less than 10 fold for rapeseed, sorghum, maize, and wheat) to very high (for barley, soybean, and sunflower).

The sample mean activity GI includes 14-year observations (1995–2008) about the states of nature assumed to have equal probability;

$$ {\mathrm{p}}_{\mathrm{r}},\mathrm{for}\;\mathrm{r}=\mathrm{1..14}\;\mathrm{with}\;\sum \limits_{r=1}^{14}\Pr =1\;\mathrm{and}\;\mathrm{p}1=\mathrm{p}2=\dots \mathrm{p}14=1/14=0.07=7\% $$
(7)

The optimal portfolio combination of farm crops is computed using the stochastic utility efficiency frontier (UEF) and solved with the linear programming MOTAD procedure to minimize the total absolute deviations (MOTAD) of GI from mean value instead of using the variance as a measure of risk for the estimation of (E, V) efficient production frontiers. This method, first proposed by Hazell (1971) as a linear alternative approach to quadratic programming, has achieved substantial popularity due to its efficiency (Hazell and Scandizzo 1974). Computationally, the method has the important advantage of offering solutions to the problem by using a conventional linear programming approach. Also, in spite of the additional constraints and variables required, the formulation is highly tractable with respect to computing requirements. The MOTAD applies ordinary linear programming to simulate the farmers’ planning decisions (Brink and McCarl 1978). MOTAD in target version has the important advantage that solutions are second degree stochastically dominant, meaning that they are stochastically efficient for risk-averse decision makers (Hardaker et al. 1991). With this method, it is possible to obtain reliable results in situations where farmers’ decisions become increasingly complicated due to the instability of prices and yields affecting the crop combinations. The solution proposed by Hazell consists of risk computation using the LP-MOTAD approach to minimize the minimum absolute deviation (MAD) for a given level of expected GI. The assumptions of utility efficient programming are the linearity condition, and ahj, bhj, and c*j to be known and constant. The objective function is the SUF with λ varied parametrically between 0 and a very high value and pk = 1/14 = 0.07 = 7%.

E(U) is the expected value of SUF, K = 1..14 indicates the equally probable state of nature of all crops. E(U) is non-linear, then a procedure to approximate the SUF has been used (McCarl and Önal 1989). zk is the expected gross income deviation given by (crj–c̅j) to be maximized with a parametric constraint on the sum of negative deviations.

$$ \sum \limits_{j=1}^7\left({a_{hj}}^{\ast}\;{x}_j\le {b}_h\right)\; for\;h=1..n $$
(8)
$$ \sum \limits_{j=1}^7\left({c}_{rj}-{\overline{c}}_j\right){x}_j+{y}_r\ge 0\; for\;r=1..s $$
(9)
$$ \sum \limits_{r=1}^{14}\left({y}_r\le \lambda \right)\; for\;a=0\le \lambda \le b $$
(10)

The assumption for the validity of this computation is that all the risky prospects about crops, crj for j = 1..7 the crop indicator and r = 1..14 the year indicators are equally probable and all the crj are random extractions from a distribution of random natural events (Pope 1982).

All x and y are non-negative variables, and λ is varied parametrically from 0 to a very high value to determine the change in crop composition to a change in risk level.

Constraint description:

Constraint 8 specifies the restrictions of land, labor, and working capital resources bh; for h = 1..m.

Constraint 9 is specific to MOTAD and imposes the non-negative condition that all deviations of a specific crop crj–c*j for all activities j (j = 1..7) and year r (r = 1..14) multiplied by their use x in the crop activity. The variable Yr measures the negative deviations of GI for all states of nature r = 1..s. If the sum of total deviation for each state is positive, Yr is equal to 0.

Constraint 10 accounts for the sum of total negative deviations for the 14 states crj–c*j for r (r = 1..14) inferior to λ value. It is responsible for the basis change of specific critical lambda values.

Empirical analysis: the risky prospects

Data about gross farm income (GI) per hectare are collected for a period of 14 years (source Eurostat) for the following crops: wheat, maize, barley, sorghum, rapeseed, soybean, and sunflower.Footnote 10 Every year represents a state of nature, and it is assumed that each one is randomly extracted from a set of events with the same probability of occurring.Footnote 11 It is also assumed that the decisions to sow these crops are not constrained by the need to produce a minimum quantity to feed livestock or a rotation that would require additional constraints.

Table 2 reports the crop vectors of GI, which is the information disclosed to farmers about risky crop prospects and is used for their planning decisions.Footnote 12 It is a multivariate normal distribution, generated by yield and price variations that are stochastically independent as shown by the unconditional or marginal distribution of the two discrete random variablesFootnote 13 (Anderson et al. 1977). The risk is generated by the yearly changes of GI measured with the following: mean, standard deviation, and coefficient of variation.Footnote 14 A wider range of CV fluctuation is observed for rapeseed (0.41) and barley (0.51); for the other crops, values range between the minimum 0.12 (sunflower) to the maximum 0.22 (wheat).

Table 2 Crop vectors of GI (€/Ha) representing the risky prospects of seven crops for 14 years of observation
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For the activity sample reported in Table 2, the unbiased estimates of the mean absolute deviations of expected GI are calculated with the following formula:

$$ \mathrm{M}..=\kern0.37em {s}^{-1}\sum \limits_{r=1}^s\left|\sum \limits_{j=1}^n\left( crj-\overline{cj}\right) xj\right| $$
(11)

where s is the sample size composed of seven vectors (crops), each of which includes 14 observations, for a total of 98 observations about GI;

Crj is a single observation of the GI sample, referred to the jth activity and rth year;

\( \overline{cj} \) is the sample mean of GI referred to the jth activity, the mean, and standard deviation are the moments calculated on discrete values of the risky prospects, and PDF are the values to test for the normality distribution of the risky prospects.

The values reported in Table 2 are used to compute the mean absolute deviation values reported in Table 3.

Table 3 Mean absolute deviations of a single observation crj from mean state r: crj–c̅j (see Table 4)
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The MOTAD method requires a measure of the minimum absolute deviation of GI from its mean value; these values are reported in Table 3.

Computation of utility efficient frontier

The points on the efficient frontier correspond to the corner solutions; at this point, the efficient combination of risky crop prospects (efficient portfolio solutions) with the optimal GI values constrained by the level of risk aversion is obtained with the parametric change of λ (Hardaker et al. 2007; Ray et al. 1998; Hazell and Scandizzo 1974; Richardson et al. 2000). Starting with the initial value λ = 0 equivalent to a, the value of λ is progressively augmented to find the next corner solution; at this point, the value is the optimal combination of risky crop prospects compatible with the level of GI risk chosen. The simulation continues by changing λ parametrically to find the next corner solution and continuing to reach the superior limit b. After an exhaustive number of corner point solutions is found, it is possible to draw a figure that indicates the optimal crop portfolio combinations for a given level of risk. Table 3 reports ten corner point solutions that signal the optimal values of the objective function U(z), for λ varied in the range between 1480 and 12,000. These solutions suggest the crop portfolio combinations according to farmers’ preference for risk. Minimum and maximum values of U(z) are reported in the first and last line of Table 4; the risk aversion is given by λ/U that is a weighted measure of the relative risk aversion λ. The results confirm the Tobin hypothesis of increasing crop diversification in portfolio as the risk perception increases. Solution seven gives one of the highest values of weighted risk λ/U = 2.53 corresponding to the highest GI values (1539.64) obtained with the inclusion of seven crops in the portfolio and land use of 98%. Half of the land is cultivated with maize and wheat. The rest is equally divided among sorghum, rapeseed, soybean, sunflower, and lastly barley on 8%. The last line reports the solution corresponding to the highest risk with respect to U with value 2.86. This is the solution of a risk taker who has adopted the most specialized plan represented by maize covering 85% of the available land, with no other crops included in the portfolio. The utility is at its maximum absolute value with the maximum absolute risk aversion with λ = 120,000, the ratio λ/U is equal to 2.86 (the slope of the utility frontier shows maximum risk aversion) (Anderson et al. 1977). The minimum value of the U(z) and minimum risk coefficient λ = 1480.47 are reported on line 1 with land use limited to 62%; the farm plan includes three crops: maize is the most representative covering 40% of the available land, followed by rapeseed (10%) and barley (2%). With risk coefficients λ/U in the range between 2.31 and 2.34, there are three similar solutions for crop distribution with wheat and maize covering 60% of the land available; the difference is in land use that increases with a risk-taking farmer, but these solutions are not favorable economically. The coefficient of risk aversion λ/U = 2.35 causes the highest crop diversification and a greater land use that is 99% of the total. This is the best portfolio solution in terms of land use and crop diversification: with wheat, maize, and barley covering 20, 40, and 20%, respectively; soybean 10%; and sunflower 7%. The group of solutions with risk λ/U between 2.36 and 2.42 shows similar results in crop diversification and distribution. The economic results suggest that the risk induces farmers to diversify their crop portfolio and land use when risk aversion increases; in this case, land use also increases consistently.

Table 4 Corner solutions and crop diversification
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The plan reported on line 7 is the best one because it includes all crops (maximum diversification), the most land use (98%), the highest economic value of GI, and a moderate risk aversion.

The hypothesis that farmers’ decisions can be affected by the level of risk causing the diversification of the crop portfolio is tested with a linear regression where the dependent variable is the level of U(z) and the three independent variables are risk level λ, land use, and crop diversification. The U(z) is regressed with three independent variables: lambda, land use and crop diversification in single regression: U(z) = aXi for i = 1..3. The regression results are reported in Table 5.

Table 5 Regression results
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Comments: The size of λ coefficient means that a 1% increase in λ will cause an increase of 0.37% in the utility function. The R2 is 0.99, meaning that the correlation between the two variables is very tight and the high t statistic value suggests the likelihood that the coefficient of λ is significantly different from zero.

The crop diversification coefficient suggests a very high direct response of utility change to crop diversification. However, the low values of R2 and t statistic suggest that the diversification effect on the utility value is not so strong. The land use coefficient means that 1% change in land use will cause a 20.3% change in utility. The other statistical values are satisfactory, suggesting that the regression results can be accepted as statistically significant.

The regression between λ and crop diversification is also statistically significant, suggesting that the increase of risk will induce farmers to diversify the crop portfolio. This result is the answer to the initial hypothesis that the response to increasing risk perception by farmers is the crop diversification decision.

Conclusion

Risk is an important factor to be considered in farm decisions. The risk aversion is one of the main research questions in farm management, and evaluation of the consequences of risk aversion on farm decision making is crucial to improve the economic results and also has social and political implications. Although risk management strategies in agriculture are of interest to many authors, the problem of risk aversion and risk management strategies have usually not been analyzed jointly (Meuwissen et al. 2001). In this study, we have tried to implement the farm management decisions by testing the risk aversion in planning caused by fluctuations in yield and prices affecting the wealth measured by GI. Understanding the relationship between farmers’ risk attitude and choice of management strategies is important for two reasons. Firstly, the literature suggests that most farmers are averse to risk when faced with risky outcomes. Someone who is risk averse is willing to accept a lower average return for less uncertainty, with the trade-off depending on the individual level of risk aversion. This means that strategies cannot be evaluated solely in terms of average or expected return, but the risk must be accounted in decisions. Secondly, the knowledge of small-scale farmers’ attitudes to risk and its management is important in determining the type of strategies. Operationally, this problem has been tackled by using a sumex utility function allowing the risk for specific traits of the function to be modeled and solved with a utility efficient programming in linearized version. A sample of observations about GI for a period of 14 years and seven crops has been collected in Italy (Eurostat source). The same probability distribution of risky prospects generated by fluctuations in yield and prices is assumed. The MOTAD linear approximation method of a quadratic function is used to simulate the crop portfolios at different risk aversion levels. Our results are coherent with those obtained by other authors and suggest that the optimal portfolio solutions depend on both risky prospects generated by a mix of market (price fluctuation) and production (yield variation), with the higher risk generated by the market. The risk generated by price fluctuations in the last years seems to be more important and negatively affects farmers’ behavior, causing higher risk aversion than climate variability whose consequences are better controlled by technologies. In fact, commodity prices tend to fluctuate in a range amplified by speculative activities performed in future and financial markets. In the EU, the relative security assured in the past to farmers with the first pillar policies and now enforced with second pillar provisions may have limited the negative impact of risky prospects. Emergency measures are also blamed for undermining existing risk management systems to the extent that farmers are relying on ad hoc government intervention in the case of a crisis rather than risk management with portfolio diversification strategies. Reduced prices support and other protection mechanisms are progressively being replaced by new financial instruments to spread the risk with farm insurance subsidies and market regulation to help farmers to cope with risk sources. A last notation is for non-use of land that occurred in some solutions. This could be interpreted in terms of opportunity cost for pluriactivity that incorporates crop diversification, part-time farming, use of labor in an on- and off-farm mix, and saving land and capital resources. When the risk becomes excessive, it becomes more convenient to take the pluriactivity option, and employment in off-farm occupations implies less labor available for crop cultivation. This is coherent with the strategy to hedge against risk.