Application of Non-normally Distributed Stochastic Plateau Function in Determining the Optimal Economic Level of Chemical Fertilizer Inputs Usage in Irrigated Cereals (Irrigated Wheat and Barley Crops)

Agriculture is one of the basic sectors of any country and is very important in creating employment and production of industrial raw materials. Although the most important role of agriculture in any country is to provide the food security. The world's population is growing, and resources are dwindling. Therefore, feeding the growing population of the world requires more agricultural production. One of the ways to increase agricultural production is to increase yield per hectare. Chemical fertilizers significantly increase production per hectare. But excessive use of chemical fertilizers can also lead to environmentally externalities such as groundwater pollution, reduced quality of agricultural products and endanger human health and the environment. Therefore, the optimal use of production inputs in the agricultural sector is essential. Unfortunately, despite the emphasis of agricultural economists on the optimal use of production inputs, this issue has been taken for granted by farmers and policymakers in the agricultural sector. The purpose of this study is to determine the optimal economic level of use of chemical fertilizers (nitrogen, phosphate and potash) in the production of irrigated wheat and barley.

In order to determine the optimal economic level of chemical fertilizer inputs (nitrogen, phosphate and potash) in the production of irrigated wheat and barley in Iran, Bayesian approach and non-normally distributed stochastic plateau function, based on the developed Von Liebig algorithm were used. The estimation of the optimal amount of input usage depends on the functional form and the distribution assumptions based on the production data. The stochastic plateau function is one of the functions has been used to determine the optimal amount of inputs (especially chemical fertilizers). The stochastic plateau function provides insight into why farmers may over-use inputs. The efficiency of the linear stochastic plateau function is better than nonlinear and polynomial functions, and it estimates a more realistic pattern of farmers' expected profits, because the function is stochastic. For simple model estimation, only the input of chemical fertilizers (nitrogen, phosphate and potash) is considered as the limiting resource. If it is assumed that the threshold point is related to the intercept, which represents the yield of crops without input consumption, the equation of the stochastic plateau function is written as the following relation: (1)   Where the yield of the crops in Iran, K is the amount of input in the crop production,  and θ are the coefficients of the yield function that must be estimated, and  is the transmitter intercept that represents all random variables. The used data in this study were collected from agricultural statistics and the production cost database of the Agriculture Ministry. The panel data were collected during 2007-2017 period.

Based on the results of the study, the average optimal consumption of nitrogen fertilizer in the production of irrigated wheat and irrigated barley in Iran was estimated 117.05 and 29.00 kg/ha, respectively, while the current average consumption of nitrogen fertilizer in the production of irrigated wheat and barley is 163.626 and 38.75 kg/ha, respectively. In other words, during the years 2007 to 2017, the amount of nitrogen fertilizer used in the production of irrigated wheat was 46.576 kg/ha (equivalent to 28.46%) and in the production of irrigated barley was 9.75 kg/ha (equivalent to 25.16%) more than the optimal level. Also, the potential yield of irrigated wheat and barley with respect to nitrogen fertilizer input was estimated 2754.5 and 2549.80 kg/ha, respectively, in the Bayesian method. The average optimal use of phosphate fertilizer in production of irrigated wheat in Iran was estimated as 97.70 kg/ha, while the current average consumption of phosphate fertilizer in production of irrigated wheat is equal to 123.06.02 kg/ha. In other words, during the years 2007 to 2017, the amount of phosphate fertilizer used in the production of irrigated wheat in Iran was 25.362 kg per hectare (equivalent to 20.609%) more than the optimal level. Also, the potential yield of irrigated wheat due to phosphate fertilizer input, about 2904.54 kg/ha has been obtained in Bayesian method. the average optimal consumption of potash fertilizer in the production of irrigated wheat and irrigated barley in Iran was estimated 39.68 and 81.81 kg/ha, respectively, while the current average consumption of potash fertilizer in the production of irrigated wheat and barley is 50.64 and 134.18 kg/ha, respectively. In other words, during the years 2007 to 2017, the amount of potash fertilizer used in the production of irrigated wheat was 10.96 kg/ha (equivalent to 21.65%) and in the production of irrigated barley was 52.37 kg/ha (equivalent to 39.02%) more than the optimal level.

According to the results of present study, farmers in the production of wheat and barley use chemical fertilizers (nitrogen, phosphate and potash) more than the optimal amount, so that the average optimal use of chemical fertilizers of nitrogen, phosphate and potash in the production of irrigated wheat, respectively 28.52, 20.59 and 78.36, and in the production of irrigated barley, the average optimal use of nitrogen and potash chemical fertilizers, respectively 74.84 and 39.03% per hectare, are less than the current amount of chemical fertilizer use in the country. According to the results of the study, in order to more efficiently use of chemical fertilizers and to reduce environmental pollution caused by their use in agricultural production, the government should reduce the direct payment of chemical fertilizer subsidies. Regarding the elimination of subsidies and pricing of chemical fertilizers (nitrogen, phosphate and potash), the importance of the type of fertilizer in crop production, input production elasticity and input demand elasticity should be considered.