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## Chilean journal of agricultural research

##
*versión On-line* ISSN 0718-5839

### Chilean J. Agric. Res. v.68 n.2 Chillán jun. 2008

#### http://dx.doi.org/10.4067/S0718-58392008000200007

Chilean Journal of Agricultural Research 68:175-182 (April-June 2008)
^{1}* and Alejandra Engler P.^{2 }^{1} Instituto de Investigaciones Agropecuarias, Centro Regional de Investigación Quilamapu, Casilla 426, Chillán, Chile. E-mail: rtoledo@inia.cl *Author for correspondence. ^{2} Universidad de Talca, Facultad de Ciencias Agrarias, Casilla 747, Talca, Chile. E-mail: mengler@utalca.cl. Rubus idaeus L.) and the production function associated with their production system in the Bio-Bio Region of Chile. Under a mean-variance approach, the estimation procedure uses a flexible utility function to incorporate a variety of risk preference alternatives. Three different estimation procedures were used: Least Squares Estimation, Seemingly Unrelated Regression and Full Information Maximum Likelihood, which revealed the same conclusions. Results showed that small farmers are risk averse (γ = 0.104) and present increasing relative and absolute aversion to risk (θ = 0.099 < 1 and θ < γ, respectively). The hypotheses of risk neutrality (γ = 0) and constant absolute risk aversion (θ = 1) were rejected with 94% and 99% confidence, respectively. The chosen function of production is the Cobb Douglas type, because it presents a better adjustment, and the relevant factors are fertilizer quantity per hectare, the experience of the producer and the planted area. This function presents decreasing returns to scale, then β_{2} + β_{3} + β_{4} is equal to 0.18. The hypothesis of constant returns to scale is rejected with 99% confidence. Key words: risk aversion, mean-variance utility function, production function.
Rubus idaeus L.), y la función de producción asociada a este sistema productivo, en la Región del Bío-Bío, Chile. Utilizando un modelo de media-varianza, se estima una función de utilidad flexible de manera de incorporar diferentes alternativas de preferencias por riesgo. Para la estimación se utilizaron tres procedimientos: Mínimos Cuadrados Ordinarios, Sistemas de Ecuaciones Aparentemente no Relacionadas y Máxima Verosimilitud con Información Completa, arrojando similares resultados. Los resultados revelan que los productores son aversos al riesgo (γ = 0,104) y tienen aversión absoluta y relativa creciente (θ = 0,099 < 1 y θ < γ, respectivamente). Se rechazan las hipótesis de neutralidad (γ = 0) y aversión absoluta constante al riesgo (θ = 1), con un 94% y un 99% de confianza, respectivamente. La función de producción elegida es del tipo Cobb-Douglas, por presentar un mejor ajuste, y los factores relevantes para esta función son la cantidad de fertilizante por hectárea, la experiencia del productor y la superficie plantada. Esta función presenta rendimientos decrecientes a escala, pues β_{2} + β_{3} + β_{4} es igual a 0,18. Se rechaza la hipótesis de rendimientos constantes a escala con un 99% de confianza. Palabras clave: aversión al riesgo, función de utilidad media-varianza, función de producción.
et al., 1994a; Saha, 1997; Isik and Khanna, 2003; Abadi et al., 2005; Lusk and Coble, 2005). There are only few studies in Chile about risk aversion of agricultural producers. Nevertheless, this information would allow for understanding how farmers make their decisions and thus provide support for formulating development policies in specific areas. It is of special interest to know the degree of risk aversion among small producers in order predict the probability of adoption of riskier production alternatives, or technologies that could present a higher level of uncertainty than those already known by the producers. Raspberries production is one of the profitable activities that small producers can develop on their farms. The high labor requirements and low mechanization of the crop make this an attractive alternative for this segment, and consequently of special interest for institutions that promote the development of small-scale agriculture in the country. In the Bio-Bio Region, the second region in terms of surface area planted with raspberries after the Maule Region, there are 1671 raspberry producers and 1752 ha of raspberry gardens (Jorge Vargas. 2007. Good Agricultural Practices, Servicio Agrícola y Ganadero (SAG). Personal communication), which means an average area of 1.05 ha per producer. The main destination of raspberries is the international market, in which Chile enjoys the advantage of being an off-season exporter for the northern hemisphere. To maintain and improve this position requires increasing the surface area dedicated to production, and incorporating environmental friendly technologies and practices that contribute to the safety of the exported fruit, both conditions that are necessary for international trade. In this sense, the risk preferences of the producers is an additional input in designing an adequate strategy for the adoption of technologies associated with this fruit crop. The decisions of agents under uncertainty have traditionally been modeled based on the Expected Utility (EU) model suggested by Von-Neumann and Morgenstern (1944). Nevertheless, EU models have not been without criticism. For example, Kahneman and Tversky (1979) using experimental choice, concluded that decisions can be inconsistent under an EU approach, which encouraged the search for new alternatives. Separately, Tobin (1958) and Markowitz (1959) presented the mean – variance approach, which considered that the utility from random prospects can be described as a function of the first two moments of the distribution around a mean outcome. One common difficulty of estimation has been the lack of flexible utility functions that allow for representing different structures of risk preference (Saha, 1993). Saha (1997) proposed a function that solves this restriction, incorporating all the possible risk preferences, in other words, decreasing, constant and increasing absolute aversion, as well as decreasing, constant and increasing relative aversion. This new functional form is linked to the mean-variance model and has proven to be an alternative that allows a high power of prediction of farmers` preferences (Saha, 1997; Isik and Khanna, 2003). The central objective of this study was to determine the structure of preferences for risk of small producers of raspberries in the Bio-Bio Region. As specific objectives, it was intended to test the behavior of the expected utility function, and estimate the production function of raspberries for small producers of this region. MATERIALS AND METHODS Model The proposed utility function for this study depends on the first two moments of the distribution, mean and variance of the income. Let U be the utility function: U (μ, σ) [1]where µ is the mean of income and σ is the standard deviation. where the sub-indices indicate partial derivatives. The functional form of the utility function proponed by Saha (1997) is shown in Equation [4]: U (M, S) = M^{θ} - S, θ > 0 [4] ^{y}where θ and γ are parameters that determine the type of risk preferences. If Meyers results (1987) are applied to the function proposed by Saha (1997), the distinct measurements of risk preferences can be defined based on Equation [5]: where it can be observed that the MV function exhibits: a) aversion, neutrality and affinity to risk when γ > 0, γ = 0, γ < 0, respectively; b) decreasing absolute aversion to risk, constant and increasing aversion to risk, when θ > 1, θ = 1, θ < 1, respectively; and c) relative decreasing, constant and increasing aversion to risk, when θ > γ, θ = γ, θ < γ, respectively. Assuming that the wealth of the raspberry producer is given by the following expression: where: Ŵ defines random wealth of the producer, is the random price of raspberries, Q is raspberry production level, C(r,Q) denotes the cost function defined by input prices (r) and production level, and w is the initial exogenous wealth of the producer. Thus, random wealth and its deviation can be estimated based on the variables M and S, defined in Equations [7] and [8]. where the income of the season is determined by the average price and the production. where σ _{p} is the standard deviation of prices. Assuming that the producer maximizes his utility derived from wealth, we can write the producers` optimization problem as Equation [9]: The first order condition (FOC) for this function is: [10] where C _{q }is marginal cost. If γ = 0, that is, if the producer were neutral to risk, Equation [10] would simplify to price equals marginal cost condition. Equation [10] proposes that the difference between the average price and marginal cost (left side of the equation), can be explained by the risk preferences of the producer. To estimate Equation [10], we apply natural logarithm, obtaining Equation [11]: The sub-index i denotes the ith observation, ε _{i} corresponds to the estimation error, and α represents the technological parameters of the cost function, whose form is assumed to be known. Equation [11] considers the differential between average price and marginal cost as an independent variable, that is, the risk premium that the producer expects, given his risk preferences. The right-hand side of Equation [11] corresponds to a production function whose formal function shall be defined based on empirical estimations. This equation allows estimating the parameters θ and g to identify the risk preferences of raspberry producers. Saha et al. (1994b) conclude that for a more efficient estimation, the technological and risk preferences parameters should be estimated jointly. For the production function, two functional forms were estimated: quadratic and Cobb-Douglas. Quadratic function Where the sub-index i indicates the ith individual and ε _{i} is the error term of the equation. The x_{ij} are explanatory variables that should be sought among a wide set. In the case of the Cobb-Douglas function, the values of the β_{j} parameters reflect the contribution of production factors in the profit and А represents exogenous technological progress. Data source The data for estimating the equations was gathered through a survey applied to a sample of 62 producers in the Bio Bio Region. The survey was applied during the months of March and April of 2007. The sample was taken in the three counties of the region that have the highest concentration of raspberry producers. The information with regard to the raspberry producers and the total surface area planted at county level was provided by Jorge Vargas (2007), in charge of Good Agricultural Practices, Servicio Agrícola Ganadero (SAG), personal communication. The number of individuals surveyed in each county was: Coihueco (43 surveys), San Carlos (3 surveys) and Ñiquén (16 surveys). The information gathered was the following: amount of fertilizer used (F), fuel used (E), labor contracted for harvesting (T), area planted with raspberry per producer (size), number of years that the producer has cultivated raspberries (experience), production of raspberries (Q), gross income (M), average selling price of raspberries (p), price variation (σ_{p}) and marginal production cost (C_{q}). The information gathered was from the 2006-2007 season. The estimations were made with the econometric program Eviews 5.1 (Quantitative Micro Software, 2004). RESULTS AND DISCUSSION The sample used is of small producers, whose data presents a high variability (Table 1). The average area of the raspberry crop was 0.6 ha. Total income per hectare fluctuates between $23 000 and $4 760 000, with an average earning of $1401 536. This variability in incomes is explained by the variability of surface planted, production costs and productivity. The average yield was 8270 kg ha^{-1}, with a range between 1467 and 24 000 kg ha^{-1}.
In terms of the producers` characteristics, the average age was 51 years, and the majority of them have a low level of schooling: 6.4% were illiterate, and 61.3% had only basic level education, either partial or completed. The surveys were taken to the member of the family responsible of the raspberry orchard, in which, 55% of the sample was male and 45% female. As well, the survey subjects were asked about their main occupation. Only 55% of the sample declared themselves as farmers. A high percentage, 37% of the total sample, declared housewife as their main occupation (which correspond to 82% of the total of women). Table 1 presents the mean, minimum and maximum values of the variables of the system of estimated equations. ^{2}), the test for model specification, Akaike Information Criteria, Schwarz criteria and Log Likelihood, indicated a better adjustment of the Cobb-Douglas function. The model does not present heteroscedasticity or self-correlation, and the errors exhibit a normal distribution.
Analyzing the results under the three methods, it is possible to conclude that the estimations by SUR and FIML are not more efficient than the ordinary least square estimation, such as suggested by Saha et al. (1946), given that the sum of squared errors of the estimation is not significantly reduced with the joint estimation. As well, it was possible to verify that the errors of Equation [11] do not have a normal distribution, so it does not comply with the supposition of joint normality for the errors, considered in the FIML estimation. Table 3 summarizes the results of the estimation by ordinary least squares of Equations [11] and [13].
In the production function, the variables experience, size of planted area and quantity of fertilizer used resulted significant (Table 3). According to the results, the experience acquired in the cultivation of raspberries measured by the number of years that the producers has been involved in cultivating this berry, allows for explaining differences in the levels of production among producers. According to these results, two additional years of experience implies an average increase in production of 1.21 kg ha
Saha (1997) applied the flexible utility function to estimate risk preferences for a set of 15 wheat producers in the state of Kansas, USA, from whom he obtained information for four years, totaling 60 observations. Later, the sample was sub-divided into two sub-samples with the goal of evaluating the scale of the producers on the structure of risk preferences. The results of Saha (1997) indicated that agricultural producers of a lower scale are less adverse to risk (lower estimated value for the γ parameter). The parameters reported in Table 4 were for the smaller producers.
These tests of hypothesis allow for verifying that returns to scale are decreasing, given that the sum of β γ > 0), that they have an increasing absolute (θ < 1) and relative aversion (θ < γ). That is, the risk aversion increases as the risk of an activity is greater and the income of the producer is higher. The results of the estimation of the raspberry production using a Cobb-Douglas function reveals that the returns to scale are decreasing (returns to scale = 0.18) and that the relevant factors in determining yield are the experience of the producer, the size of the plantation and fertilizers dosage. LITERATURE CITED Abadi, A.K., D.J. Pannell, and M.P. Burton. 2005. Risk, uncertainty, and learning in adoption of a crop innovation. Agric. Econ. 33:1-9. [ Links ]Arrow, K.J. 1964. The role of securities in the optimal allocation of risk bearing. Rev. Econ. Stud. 31(2):91-96. [ Links ] Isik, M., and M. Khanna. 2003. Stochastic technology, risk preferences, and adoption of site-specific technologies. Am. J. Agric. Econ. 85:305-17. [ Links ] Kahneman, D., and A. Tversky. 1979. Prospect theory: An analysis of decision under risk. Econometrica 47:263-92. [ Links ] Lusk, J.L., and K.H. Coble. 2005. Risk perceptions, risk preference, and acceptance of risky food Am. J. Agric. Econ. 87:393-405. [ Links ] Markowitz, H.M. 1959. Portfolio selection: Efficient diversification of investment. 344 p. John Wiley & Sons, New York, USA. [ Links ] Meyer, J. 1987. Two-moment decision models and expected utility maximization. Am. Econ. Rev. 77:421-30. [ Links ] Pratt, J.W. 1964. Risk aversion in the small and in the large. Econometrica 32(1-2):122-136. [ Links ] Quantitative Micro Software. 2004. Eviews 5.1. User guide. Quantitative Micro Software. Irvine, California, USA. [ Links ] Saha, A. 1993. Expo-Power Utility: A flexible form for absolute and relative risk aversion. Am. J. Agric. Econ. 75:905-13. [ Links ] Saha, A. 1997. Risk preference estimation in the nonlinear mean standard deviation approach. Econ. Inquiry 35:770-82. [ Links ] Saha, A., H.A. Love, and R. Schwart. 1994a. Adoption of emerging technologies under output uncertainty. Am. J. Agric. Econ. 76:836-46. [ Links ] Saha, A., R. Shumway, and H. Talpaz. 1994b. Joint estimation of risk preference structure and technology using Expo-Power Utility. Am. J. Agric. Econ. 76:173-184. [ Links ] Tobin, J. 1958. Liquidity preferences as behavior towards risk. Rev. Econ. Stud. 25:65-86. [ Links ] Von Neumann, J., and O. Morgenstern. 1944. Theory of games and economic behaviour. 776 p. Princeton University Press, Princeton, UK. [ Links ] Received: 21 June de 2007. |