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



VRT is more profitable than URT if RVRT* - V1- V2 > 0, where V1 is the application
cost for VRT minus the application cost for URT and V
2 is the cost of gathering spatial
information and using it to identify management zones and their yield response functions. If the
management zones and their response functions have already been identified, V
2 is known and
the farmer will undertake VRT if RVRT* > V
1, because V2 is a sunk cost in making the VRT
versus URT decision. If, on the other hand, V
2 is not known, the farmer can use conservative,
educated guesses about the λ
i s, the corresponding yield response functions, and V1 to estimate
RVRT* - V
1, which can be thought of as an education guess about the maximum amount a
farmer can invest in gathering spatial information and identifying the field’s management zones
and their yield response functions.

Equation (5) is concave inλi . Its concavity can easily be understood by considering
fields with three management zones; management zones 1, 2, and 3. For fields that are all in
management zone 1 (
λ1 = 1, λ2 = 0, and λ3 = 0), RVRT* = 0 because the weighted average
response function and the response function for management zone 1 are the same. Fields with a
positive
λ2 and/or λ3 (0 <λ1 <1) have multiple management zones and farmers can consider
using VRT. Since optimization of input use with VRT is more suited to the site-specific yield
response functions than to the field-average response function, RVRT* now becomes positive
and continues to increase to a maximum as λ
1 decreases over some range.

Spatial Break-even Variability Proportions (SBVPs) (English, Roberts, and
Mahajanashetti; Mahajanashetti; Roberts, English, and Mahajanashetti) are defined as the lower
and upper limits of λ
m-2 , λm-1 , and λm for given levels of λ1, λ2,..., λm-3, PY, Pj, and V1 such



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