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

and the Extension Service often base their fertilizer recommendations on yield goals developed
by farmers (Savoy and Joines). These yield goals can be formed in a variety of ways (O’Neal et
al.). If the farmer forms the field yield goal by implicitly averaging yield goals across
management zones, the field yield goal and the fertilizer recommendation would be weighted by
the proportions of the field in each management zone.

Assume the farmer determines optimal uniform application rates based on the field-
average response function expressed as:

m

(3)    Yu = Yu (Xui,..., Xun) = ∑λi Yi (Xu1,..., Xun)

i=1

where Yu is the weighted average crop yield response function for the field and Xuj is the
uniform application rate for input j (j=1,...,n). The optimal return above input cost per acre for
URT (R^*_URT ) is calculated from the following profit function:

mn

(4) R*urt = Pγ ∑λ_l Yi (Xui^,,., Xun∙) — ∑ P_jX_u,∙
i=i                                                 j=i

= RURT(λi,λ2,...,λm-i, PY, Pi,...,Pn)

where Xu,^* is the optimal uniform application rate for input , obtained from the field-average
yield response function through the simultaneous solution of the n first order conditions for profit
maximization, which equate the marginal products of the inputs with their respective input-to-
crop price ratios. Again λ_m is excluded as an argument because the sum of the λ_i s equals i.

The difference between R^*_VRT and R ^*_URT , which is the optimal return to VRT (RVRT^*),
can be specified as:

(5)    RVRT^* = R^*VRT - R^*URT = RVRT^*(λi,λ2,...,λm-i, PY, Pi,...,Pn)

where all variables have been previously defined.

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