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

management zone. Optimal return above input costs per acre for the field under VRT (R^*_VRT ) is
then calculated from the following profit function (Nicholson):
mn

(2)     RVrt = ∑ λi [Pγ Yi (Xiι ',..., Xin ^,) - ∑PjXij ’]

i=1                                                        j=1

= RVRT(λ1,λ2,...,λm-1, PY, P1,...,Pn)

where PY is the crop price; Pj is the price of input j (j=1,.. .,n); X_ij * is the optimal input
(j=1,.. .n) application rate for the i^th management zone; π* is optimal net return above input costs
for the i^th management zone; and λ_i is the proportion of the field in the i^th management zone

m

such that ∑λ_i = 1. Thus, R^*_VRT is the weighted average over λ_i of the optimal returns above
i=1

input costs per acre obtained for each management zone. The proportion of the field in
management zone m (λ_m ) is not included as an argument in the R^*_VRT function because λ_m = 1
m-1

- ∑λi.
i=1

Numerous decision rules could be assumed for URT application of the inputs (English,
Roberts, Majajanashetti). In this paper, farmers are assumed to base URT decisions on the
profit-maximizing input levels obtained from a field-average yield response function, with the
proportions of the field in each management zone (λ_is ) serving as weights. Determining the
optimal uniform rate based on the weighted average response function is analogous to some
methods used to develop fertilizer recommendations. For example, receiving a recommendation
from a soil-test laboratory based on a soil sample that mixes soil cores drawn at random across a
field (VanEck and Collier) is similar to weighting the recommendations for the management
zones by the proportions of the field in each management zone. In addition, soil-test laboratories

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