The Variable-Rate Decision for Multiple Inputs with Multiple Management Zones

zones. These equations were estimated by Hexem and Heady in the mid 1970’s using field data.
They were estimated as quadratic yield response functions similar to those in Arce-Diaz et al.,
Agrawal and Heady, Mjelde et al., Vanotti and Bundy, and Schlegel and Havlin. These
functions would be somewhat different if estimated with current data. Nevertheless, they are
plausible irrigated cotton yield response functions, chosen for illustrative purposes to serve as
examples in this paper. Their use facilitates exposition of the aforementioned concepts because
they are continuous and concave. The response functions are portrayed graphically in Figure 1.

An average cotton lint price received by farmers (PY=$0.52/lb) and an average nitrogen
price (P_n =$0.26/lb) over the 2000-2003 period and an irrigation water price of $4.00/acre-inch
were used in the analysis. Optimal yields, input application rates, and net returns above input
costs were determined for each management zone (Table 1). R^*_VRT was determined as a
weighted average of the last column in Table 1, given the assumptions about the λ_is . R^*_URT was
calculated using the field-average yield response function to determine optimal field-average
input application rates, corresponding yields, and net returns above input costs for each
management zone, weighted by the assumed λ_is . In this example, RVRT* was evaluated for
hypothetical cotton fields for all combinations of the λ_is when each λ varied between 0.0 and
0.9 in increments of 0.1 (eg., λ₁ = 0.0, λ₂ = 0.4, and λ₃= 0.6 or λ₁ = 0.2, λ₂ = 0.5, and λ₃= 0.3).

For illustrative purposes, Table 2 presents average RVRT^*s for all combinations of two
λ_is assuming the λ for one management zone is fixed at the level in the first column. For
example, if the proportion of the field in management zone 1 is fixed at λ₁ = 0.0, the average
RVRT^* is $44.41/acre for fields with all combinations of λ₂ and λ₃between 0.0 and 0.9.

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