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 λ_m-2 ,
λ_m-1 , and λ_m .

(6) RVRT = RVRT (λm-1,λm-2 I λ 1,λ2,...,λm_₃, Pγ, Pι,...,Pn)= V1

where λ₁,λ₂,...,λ_m-2, P_y, P_j (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)     Y}3^=a3^+b3^X31^+c3^X231^+d3^X32^+e3^X3²2^+f3^X31^X32

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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