Table 4.5 summarizes the simulated and observed characteristics of PSBRc72H and PSBRC82. The calculated nRMSE for grain yield were 9% and 4% for PSBRc72H and PSBRc82.
These values were better than most of the nRMSE for grain yield reported in various studies reviewed by Timsina and Humphreys (2006), which ranged from 3% to 32%. The nRMSE for anthesis and maturity that I found were similar to the reported values in that review. This implies good predictive capacity of the model considering that only 5 out 18 possible characteristics were used in the calibration process. This suggests that the model would have even better predictive capacity if more information were available.
4.2.5. Model Application
Using the calibrated genetic coefficients, the average weather data from 2005 to 2007, and the soil characteristics of the project site, I performed several computer
experiments to determine the hybrid and inbred rice yield responses to varying amounts of
99 water and nitrogen. Later, I included potassium among the variable inputs and also
examined its effect on yield. I performed the computer experiments for the dry season to capture better the response of rice yield to irrigation. The choice of water, nitrogen, and potassium as variable inputs was due to their perceived importance to the growth of rice plants.30
De Datta (1981, pp.297-300) described rice as a semi-aquatic plant, indicating that it grows better and produces higher yields when grown in flooded soil. He stressed that water influences the physical characteristics of rice (i.e. plant height, tiller number and culm
strength), and acts as a solvent to increase availability of nutrients. Flushing rice field with water also reduces soil toxicity, which can retard root development, inhibit absorption of nutrients, and cause root rotting. The presence of standing water also serves as a method of weed control particularly in the early vegetative stage of the rice plant.
De Datta (1981, pp.350-351) also expressed the importance of major nutrients such as nitrogen, and potassium in the growth of rice plants. Nitrogen increases height, promotes production of tillers, and increases the sizes of leaves and grains. Nitrogen absorption can also lead to a greater number of spikelets per panicle, a higher percentage of filled spikelets in panicles, and in an increased protein content of grains. Potassium increases the size and weight of the grains. It also plays an important role in physiological processes of rice, including opening and closing of stomata, and improves tolerance to unfavorable weather conditions.
30 Phosphorus is another major nutrient that affects the growth of rice plants. It stimulates root development, and promotes active tillering, which enables rice plants to recover faster after being subjected in an
unfavorable condition. Phosphorus is also good for grain development. However, the DSSAT software was not configured to assess the phosphorus balance for rice, although it can in other crops. Due to this limitation, I was not able to include phosphorus among the variable inputs.
100 For the two-input case, I combined thirty nitrogen and five water application rates to form a hundred and fifty treatments.31 To constitute the three-input case, I replicated the two-input treatments for five application rates of potassium resulting in a total of 750 computer simulations for each variety. Figure 4.3 shows the DSSAT-generated yields of hybrid and inbred varieties at various levels of nitrogen and water. Theoretically, generating enough yield data through DSSAT would allow us to create a nonparametric representation of the technology frontier for a particular rice variety. However, it may not be easy to identify the profit-maximizing levels of inputs from this nonparametric form without intensive calculation techniques.
For the purposes of demonstrating the DSSAT model, I estimated a parametric form of the hybrid and inbred yield responses to simplify the optimization process and the derivation of the supply curve. Using the DSSAT-generated yield data, I estimated a
quadratic yield response function to simplify the calculation of the analytical solution for the profit maximization. The yield response function to be estimated is written as
(4-3)
0 3 3 3
In equation (4-3), ykis the hybrid and inbred rice production per hectare, X is a vector of variable inputs per hectare (nitrogen, potassium, and water), tk is the vector of genetic coefficients for hybrid and inbred cultivars, c is a vector of spatial characteristics (i.e. soil, topography, initial field conditions), z is a vector of weather variables (i.e. maximum and minimum temperatures, rainfall, evaporation rate, solar radiation), and mis a vector of
31 I considered different application rates from 5 to 150 kg/ha/season for nitrogen, from 20 to 60 kg/ha/season for potassium, and from 200 to 1000 mm/ha/season for water.
101 management variables (i.e. method of planting, age of seedlings, plant population, timing of fertilizer and water applications). Since yields were generated using the same weather conditions, soil properties, and crop management practices, the OLS estimates of the coefficients of yield response functions should not be subject to the endogeneity bias.32
Using the estimated yield response functions, the profit maximization problem of a firm can be written as
(4-4)
X p W t c z m maximizes the profit if it satisfies the following first order conditions,
(4-5)
; , , ,
i,This implies that the value of the marginal product of each input is equal to their respective prices if the level of the profit-maximizing input is positive. The convexity of the production set guarantees that the first order conditions are not only necessary but also sufficient conditions for the existence of a solution to the profit-maximization problem (McFadden 1978; Mas-Collel, Whinston and Greene 1995)33. However, a non-convex production set is also possible, leading to a corner solution. Substituting the profit-maximizing input levels
32 This problem in estimation of a production function was usually encountered when some unobserved variables (i.e. i.e. weather, soil, and management) are omitted from the analysis. Griliches and Mairesse (1998) provided a good discussion.
33The production set would be convex if the estimated yield response functions were concave. The estimated hybrid and inbred yield response functions would be concave if their respective Hessian matrices (H) were negative-semidefinite, or mathematically v Hv' 0 for any positive column vector v.
102 back to the original production function (equation 4-4) would lead to the recovery of the output supply function
S . Formally, this could be written as(4-6) S p W
,
f X
*
p W t c z m t c z m, ; , , ,
; , , ,
.By comparing the derived supply functions for hybrid and inbred rice, I would be able to examine the nature of the supply shift induced by adopting hybrid rice technology.
4.3. Results and Discussion