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 V₂ 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, V₂ is known and
the farmer will undertake VRT if RVRT^* > V₁, because V₂ is a sunk cost in making the VRT
versus URT decision. If, on the other hand, V₂ 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^* - V₁, 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 inλ_i . 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 λ₁ 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 , λ_m-1 , and λ_m for given levels of λ₁, λ₂,..., λ_m-3, P_Y, P_j, and V1 such

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