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



that RVRT* = V1, where V1 is the additional application cost of using VRT compared to URT.
Mathematically, equation (5) can be modified as follows and used to locate the SBVPs for
λm-2 ,
λm-1 , and λm .

(6) RVRT = RVRT (λm-1m-2 I λ 12,...,λm_3, Pγ, Pι,...,Pn)= V1

where λ1,λ2,...,λm-2, Py, Pj (j=1,...,n), and V1 are given levels of the respective variables and
m-3_

λ m = 1 - λ m-2 - λ m-1 - Σ λi

i=1

As a more specific example using a concave functional form, assume three management
zones and express equations (1) as quadratic yield response functions containing two inputs with
interaction between the inputs. Given these assumptions, the functional forms of equations (2),
(4), and (5) can be determined and the SBVPs can be identified. Let the respective management-
zone proportions be
λ1 , λ2, and λ3, and let equations (1) be represented by equations (7), (8),
and (9).

(7)     Y1 =a1+b1X11+c1X121+d1X12+e1X122+f1X11X12

(8)     Y2 =a2 +b2X21 +c2X221 +d2X22 +e2X222 +f2X21X22

(9)     Y3=a3+b3X31+c3X231+d3X32+e3X322+f3X31X32

where Yi and Xij are defined in equations (1) for m = 3 management zones (i=1, 2, and 3) and n =
2 inputs (j=1 and 2).

For VRT, take the partial derivative of the yield response function for management zone I
with respect to inputs 1 and 2, set these derivatives equal to the price of input j divided by the
price of the output, and solve the two equation simultaneously (Heady and Dillon) for X*
i1 and
X*
i2 (Equations 10 and 11).



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