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); Xij * is the optimal input
(j=1,.. .n) application rate for the ith management zone; π* is optimal net return above input costs
for the ith management zone; and λi is the proportion of the field in the ith management zone
m
such that ∑λi = 1. Thus, R*VRT is the weighted average over λi 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 (λm ) is not included as an argument in the R*VRT function because λm = 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 (λis ) 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
3