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

Forcella; Roberts, English, and Mahajanashetti). Spatial and yield response variability, along
with the crop price, the input prices, and the additional cost of using VRT versus URT, in concert
with farmer and farm characteristics, factor into the decision to adopt VRT (Roberts et al., 2004).
In the end, no general formula exists for determining whether VRT or URT should be used on a
particular field because each field presents a different case (Roberts et al., 2002).

The objectives of this paper are to 1) present an analytical framework for the VRT versus
URT decision for applying multiple inputs in fields with multiple management zones and 2)
illustrate the decision-making framework for irrigated cotton fields with nitrogen and water
applied to three management zones.

Analytical Framework

Assume farmers are profit maximizers who can classify their fields into m management
zones and have knowledge of the management-zone-specific yield response functions for a given
crop and set of n inputs. Suppose further that yield responses can be represented by concave
functions and fields can include any of these m management zones in any proportions. Let the
response functions be represented by equations (1).

(1)    Yi = Yi ‰...,X_ta)                     i = 1,2,...,m

where Y_i is crop yield/acre for management zone i and X_ij is the amount of input j (j=1,...,n)
applied per acre to management zone i.

Economically optimal quantities of the n inputs are determined for a particular
management zone by equating the marginal physical products of the yield response function for
that management zone with the input-to-crop price ratios and solving these equations
simultaneously for input quantities. These n equations are the first order conditions for profit
maximization for that management zone. Optimal quantities of inputs are different for each

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