assessing spatial break-even variability in fields with two or more

. l o i tn~c~/ of A,yricrrlrrtrcrl tir~cl Applied Ecorzonlic..~. 33,3(Dccember 200 1 ):55 1-565 0 2001 Southern Agricultural Economics Association

Assessing Spatial Break-even Variability in Fields with Two or More Management Zones

Burton C. English, S.B. Mahajanashetti, and Roland K. Roberts

ABSTRACT

Farmers are interested in knowing whether applying inputs at variable rates across a field is economically viable. The answer depends on the crop, the input. their prices, the cost of variable rate technology (VRT) versus uniform rate technology (URT). and the spatial and yield response variability within each tield. Methods were investigated for determining the range of spatial variability over which the rctum t o VRT covers its additional cost compared with URT in fields with rnultiple management zones. Models developed in this article, or variants thereof, could be uscd to hclp farmers make the VRT adoption decision.

Key Words: nrrrncigrincnt :one.\. riitrogctz. prc~c.isioi1 , f i r rn~in~ , site,-.s/):l,c~cijic. mcrrzagerncJiz/, .spc~/iul hrc~cik-ei~cv~ c.crriahility propol.rior~.s, spatir/l vcrrinhility, vcrrinhlp rci/c7 teclznology. vic~l(l rrsponsc ~~c~ricrhiliry.

Agricultural fields consist of numerous areas that differ from one another with respect to the factors that condition crop growth (Can- ' t (11.; Hannah, Harlan, and Lewis; Hibbard c,t a/.; Mal- zer ct ~11. ; Sawyer; Spratt and Mclver). Precision farming uses a set of technologics to gather information about the heterogeneous makeup of a farm field and uses that information to make management decisions that address site-specific crop nceds within the field (Swinton and Low- enber-DeBoer). Its component technologies en-

Burton C. English is a professor, S.B. Mah:~jjanashetti is a fnrmer graduate research assistant and Roland K. Roberts is a profes\or, Department of Agricultural Eco- non~ics, The Llniversity of Tennesscc. Knoxvillc, TN. This article contributes to ~ h c objectives of Southern Regional Rexarch Project S-283 and wa\ supported by the University of Tennessee Agricultural Euperi- ment Station. The authors acknowledyc the contribu- tions of the anonymous reviewers and Stevc Zidcs of the Llniversity of Tennessee Mathematics Department. Authorship is shtu-cd cqually.

able farmers to understand the changing plant- growth environment across a field, estimate input requirements for relatively homogeneous smaller-than-field-size units, and apply inputs on a site-specific basis. Two important benefits of precision farming are claimed to be increased profits to farmers and I-educed environmental harm res~~l t ing fiom more precise placement of inputs (Kitchen et al.; Koo and Williams; Saw- yer: Watkins, Lu. and Huang). The key. how- evel; to the acceptance of site-specific farming is the profitability of using these technologies (Daberkow; Reetz and Fixen; Sawyer).

Lowenberg-DeBoer and Swinton reviewed 17 precision farming studies conducted before 1998 and found inconclusive evidence about the profitability of site-specific management in field crops. Of the studies reviewed. 12 used

empirical yiclds and five used simulated yields to determine proiitability. At least nine additional studies have been conducted since Low-

enberg-DeBoer and Swinton's review. one of which used empirical yields (Lowenberg- DeBoer and Aghib), while eight used simulated or hypothetical yields (Babcock and Pautsch; Bongiovanni and Lowenberg-DeBoer. 1998; Bullock et al.; English, Roberts. and Mahaja- nashetti; Lowenberg-DeBoer; Roberts. English, and Mahajanashetti; Thrikawala et al.; W~itkins. Lu, and Huang). With these additional studies the profitability of site-specific input management is still inconclusive. The disparity in results stems from differences in assumptions about costs, yield response, and the value of the crop (Lowenberg-DeBoer and Swinton).

Another reason for finding different profitability results across fields is differences in spatial variability, where .sl~trtial ~.(1ri(1bilitj. is detined as the distribution across a field of management Lanes with different crop yield responses to an input (Roberts, English, and Ma- hajanashetti). Within-field variability in soil physical and chemical characteristics is a nec- essary condition for the economic viability of using variable rate technology (VRT) (English, Roberts, and Mahajanashetti: Forcella; Hayes. Overton, and Price; Roberts, English, and Ma- ha.janashetti: Snyder). Relationships among crop yields, the level of input applied. and soil characteristics determine spatial variability within 3 field. These relationships also determine yield response variability, where yiel~I re- .sponsr vuricrhility is defined as the differences in magnitudes of yield response among management zones (English. Roberts, and Maha- .janashetti; Forcella; Roberts, English, and Ma- hajanashetti). Spatial and yield response variability, along with the crop price, the input price, and the additional cost oT using VRT versus uniform rate technology (URT) factor into the economic decision to adopt VRT.

Roberts, English, and Mahajanashetti developed a theoretical model for evaluating the economic viability of V R I for fields with two management zones. Frequently. however, a decision-maker is faced with more than two management zones within a given tield. The research presented in this article extends their model to multiple management zones. The ob- jective of this research was to investigate methods for determining the range of spatial

variability over whicli the return to VRT covers its additional cost compared with URT in fields with two or more management zones. The methods are presented in theoretical form and evaluated with sensitivity analyses using hypothetical examples.

Theoretical Model

Assume farmers are protit maximizers who can classify their fields into ni ~iianagement zones and have knowledge of the management-zone-specific yield response functions for a given crop and input. Suppose further that yield responses can be represented by concave functions (diminishing marginal physical product) and that fields can include any of these In management zones in any proportions. Assuliie the cost of obtaining knowledge about the management zones and their yield response functions is a sunk cost with regard to the decision of whether to use VRT instead of URT. Let the response functions be represented by equations ( 1 ).

where Y , is crop yield per acre for the it" management Lone and X, is the amount of input applied per acre to the it'' management zone.

A farmer using VRT on a particular tield determines the optimal application rates for the In management zones by equating the marginal physical products of the respective response functions with the input-to-crop price ratio. Op- timal return above input cost per acre for the field under VRT (R;!,,.,) is then calculated from the following profit function (Nicholson ).

= C A,(7i ; ) I I

R?l<T(Al. A ? , . . . , A,,, 1 . Py, Px)

where P, is the crop price: P, is the input price; XT is the optimal input application rate for the i"' management Lone; .rrT is optimal return above input cost for the i'll rnanage~nent

zone; and A , is the proportion of the lield in the it'> management zone such that Cyl, A, = I. Thus. R:;,,,. is the weighted average over A, of the optimal returns above input costs per acre obtained from each management zone. The proportion of the tield in management zone m (X,,,) is not included as an argument in the R:,: ,,, ,- function because A,,, = I - X:"I' A,.

Numerous decision rules could be assumed for URT application of thc input, two of which are explored as exa~nples below. The first rule assumes farmers base URT decisions on the profit-maximizing input level obtained from a weighted average yield response function, with the propo~.tions of the tield in each management zone serving as weights. The second rule assumes farmers determine the uniform rate for the entire field as the profit-maximizing level of input obtained from one rnanage- ment zone (eg., the highest or rnetiium response management zone). These two examples are presented to demonstrate that the return to variable rate technology (RVRT) is a nonlinear or linear function of A, depending on the decision rule used for URT rather than depending on the shape of the response functions assumed in equations ( I ).

approximately weighted by the proportions of the field in each management zone.

Assume the farmer determines the optimal uniform application rate based on the field average respon\e tunction expressed as

where Y,,(X,,) is the weighted average crop yield response function and X,, is the uniform input application rate. The optimal return above input cost per acre for URT (R?,,) is calculated from the following profit function:

where Xlj: is the optimal unit'or~n application rate obtained by equating the marginal physical product of the average yield response function in equation (3) with the input-to-crop price ratio. Again A,,, is excluded as an argument because 2:" , A, ecl~~als I.

The difference between R:,,, and R;,.,., which is the optimal return to VRT (RVRT*), can be specified as a profit function:

( 5 ) RVRT* = R": - R:* VKT 11I<1

Determining the optimal uniform rate based on the weighted average response function is analogous to some niethocls used to develop fertilizer reco~nmendations. For example, re- ceiving a recommendation from a soil test lab- oratory based on a soil sample that mixes soil cores drawn at random across a lield (VanEck and Collier) is si~nilar to weighting the recommendations for the management zones by the proportions of the tielcl in each management zone. In addition. soil-test laboratories and the Extension Service often base their fertilizer recommendations on yield goals developed by farrners (Savoy and Joines). These yield goals can be formed in a variety of ways (O'Neal et at.). If the farmer forms the field yield goal by i~nplicitly averaging yield goals across management zones, the tield yield goal and the fertilizer recommendation would be

= RVRT"(A,, A,, . . . . A ,,,- ,, P,. P,)

where all variables are defined earlier-. Equation (5) is concave in A,. Its concavity

can easily be understood by considering fields with only two management zones-Manage- tnent Zones I and 2. For fields that are uni- formly Management Zone 1 (A, = 1 and A, =

0), RVRT* -- O because the weighted average response function and the response function for Management Zone 1 are the same. Fields with a positive A, ( A , (- I ) have both rnanage- ment 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 average response function, RVRTZk now becomes positive and continues to increase to a maximum as A? increases (A, decreases) over some range.

Eventually, RVRT:'; begins to decline until it reaches zero for fields with only Management Zone 2 ( A , = 0 and A, = 1 ). At this point, the average response function and the response function for Management Zone 2 are the same. The above d i sc~~ss ion can be generalized to m management zones for all concave functional forms, including the linear-plus-plateau function. which is not strictly concave.

Spatial Break-even Variability Proportions (SBVPs) (English, Roberts, and Mahajanash- etti; Mahajanashetti; Roberts, English, and Mahajanashetti) for two particular management zones. say Management Zones m-1 and m, are defined as the lower and upper limits of A,,, , and A,,, thr given levels of' A , , AL, . . . . A ,,,- ~ ? , P,. Px. and V such that RVRT'* = V, where V is the :idditional cost of using VRT compared to URT. The SBVPs for A,,, vary in- versely with the SBVPs for k , , , ~ - , because Z::, A, equals I. Mathematically. equation ( 5 ) can be modified as follows and used to locate the SBVPs for A , , , , and A ,,,.

where h , . h 2 , . . . , h,,, 2 , P,, P,, and are given levels of the respective variables and A,,, =

y,,, 2 1 - A,,, I - -, I .

Solving equation (6) for A,,, , provides the SBVPs for A ,,,-, and A,,, that bound the range over which RVRT'" 2 V. However, for certain - - A , , A,, . . . . h ,,,-,, P, and P,, RVRT*: !nay be less than V for all possible levels of A,,,-, . implying that SBVPs do not exist and that economic losses froni using VRT would occur at all levels of A,,, ,. In some cases, RVRT* may be greater than V for all possible levels of A , , , , . implying that SBVPs do not exist and that economic gains would occur from using VRT regardless of the level of A,, ,_ , . Finally, in the remaining cases, only an upper or a lower SBVP exists, but rlot both. If RVRT'!' > V for A,,, , = 0, and RVRT:* = V for 0 < A,,, , 5 ( 1 - Z:'ll2 i , ) , only an upper SRVP exists. In this case, the maximum this upper SBVP can be is i - C;?'' X, when A,,, = 0. However, if RVRT*

> V for A,,,-, - 1 - C:"' hi and RVRT'* = V for 0 5 A,,, , < ! I - ki7, ' A , ) , - only a lower SBVP exists. In this case, the minimurn this SBVP can be is 0 when A,,, = I - CyL;' h,,

As a more specific example using a concave functional form, assume three management zones and express ecluations ( I ) as quadratic yield response f~~nc t ions . Given these assumptions, the functional forms of equations ( 2 ) . (4), ( 5 ) . and (6) can he determined and the SBVPs can be identified. Let the respective management-zone proportions be A , . A?, and A,, and let equations ( 1 ) be represented by equations (7 ) , (8), and (9).

where Y, and X , are as defined in equations ( I ) for Management Zones 1 . 2, and 3.

For VRT, set the first derivative of each function equal to P,/P, and solve for X r . XT, and X f . Substitute these optimal input rates into equation ( 2 ) to get equation ( l o ) , which is the profit function for VRT.

For UR1: substitute equations (7) , (8), and (9) into equation (3) and set X I = X, = X I =

Xu. Set the first derivative of the resulting tield average yield response function equal to P,/ P, and solve for Xrj:. Substitute this opt i~nal uniform input application rate into equation ( 3 ) to get equation ( I I ) , which is the protit function For UR1:

English. Muhtijunu.shelti, und Robevt,~: Asse.c.ting S/~utiul Brruk-even Vtrl-itrbilitj 555

Table 1. Maximum Return to Variable Rate Technology and Spatial Break-even Variability

Proportions for Hypothetical Corn Fields with Three Management Zones with the Proportion

of the Field in the High-Yield Response Management Zone Held Constant, Weighted Average Re\yon\e Function

RVRT:@.J Maximum SBVPs,' for SBVPs" for

RVRT*~' for Percentage of Percentage

Maxirni~ing Field in Low-

Percentage of Percentage of Percentage of of Field in Response Ficld in High- Field in Low- Field in Low- Medium-Response

Management Response Management Response Response Management Management Management Zone (i,) Zone (A2)

Zone ( A , ) Zone ( A , ) Zone ( A , ) Lower Upper Lower Upper

%# $lac 70 0 58 1.95 b b b 11

20 79 5.22 22 80' 0' 58 4 0 60 7.03 8 60' (Y 5 2 60 40 6.38 7 40' CY 3 3 8 0 20 3.89 12 20' 0 8

. 'RVRT" i > the return-to-variable-rate technology defined in equation (12) and the SBVPs are sp;ltial brcnh-cven variability PI-oportions Sound by solving cquation (13).

Bccauw the maximum RVRT* attainable by varying A , is less than the additional custom charge for V K T o f $3.001 ac. brcak-even values for A , and A? d o not exist. ' This nurnbcr is the rnaxirnurn or minimum Ihr A, or A?, respectively. Upper or lower SBVPs do not rxi\t hecauw KVRT' is greater than the ;~dJitional custorn charge of $3.00/ac when A , or A? arc at their constrained n i a x i ~ n u n ~ o r minimum.

function in A,, which can be solved using the quadratic formula for the lower and upper SBVPs for A? if they exist:

I

The optimal return to variable rate technology is given by

+ 4nTT::X1[c3 + (c, - c3)X,]

+ 2Px(Px/P,) - 2P,[b, + (b, - b,)Xll

(12) RVRT'" = R:,, - R;,, + 2P,lb,(b2 - b,)

+ (b, - b,)(b, - b,)X,l JA, Setting A,, P,, and P, equal to X,, P,, and P,, + [4.rr:(cz - c,) - 4nf(c2 - c,) setting equation (12) equal to V, and consol- idating terms gives the following quadratic + P, (b, - b,)']Xt

556 . /OILYIZN/ of A S Y ~ ~ ~ U ~ I L I ~ I I I und Applied E~.onomics, Decernh~r 2001

The SBVPs for A, are found from the restric- 111

(14) RVRT* = i , [aT (XF) - .rr,(X;;,)] tion A, = 1 - A , - A:. Equation (13) dem- I = I

onstrates that equation (12) is concave (quadratic in this case) in i ? . This concavity results fi-om assuming the farmer uses the weighted- average yield response function to choose the uniform rate. More specifically it results because the average response function approaches the response function for Management Zone i as A, approaches 1 and diverges from that response function as A, approaches 0.

The RVRT* maximizing h2 is found by setting the partial derivative o f equation ( 12) with respect to A2 equal to zero and solving for X 1 (given X,). The resulting A2 is substituted into equation ( 1 2 ) to find the maxirnum RVRT*'. If this maximum RVRT''' is less than V, SBVPs for A, and A, do not exist and a farmer would have no economic incentive to use V R T on the field in question, given XI.

where .rrT(X:) is optimal return above input cost per acre for Management Zone i and .rr,(X;t) is return above input cost per acre for Management Zone i when Xz is applied to it. The expression in brackets is zero for i = m beca~~se applying the input to Management Zone rn at its optimal rate gives the same return above input cost under V R T and URT.

For given crop and input prices, the ex- pressions in brackets are constants for each management zone; therefore, RVRT* is linear in A,. When all management-zone proportions except two are fixed, only one SBVP can exist for A, . I f the expression in brackets is greater (less) than V regardless o f the level o f A, ( for i f m ) , no SBVP exists for A, and VR'T is more (less) profitable than URT. Also, because the exprewion in brackets is Lero for Manage-

Respolzse Functiotl for O?zr M~inag~rricnt ment Zone m. the larger (smaller) A,,, the

Zone srnaller (larger) RVRT'I' and an SBVP will ex-

Using the response function for one management zone to determine the uniform input application rate is a less appealing criterion than the aforementioned criterion, but anecdotal evidence suggests that some f i~r~ners make uniform-rate decisions based on this criterion. For example, some farmers who use URT for fertilizer application may fertil i~e the entire field based on the yield goal for the "best land." Obviously, a farmer would use this method only i f a considerable proportion o f the field were in the targeted management zone. Nev- ertheless. for illustrative purposes, the example presented below explores the entire range o f possible proportions o f the field in the targeted management Lone.

Assume the farmer determines the optimal

ist for A,,, only i f the expression in brackets is greater than V for Management Zone i f m . The SBVPs for any pair o f A,s can be f o ~ ~ n d by setting equation (14) equal to V . holding prices and all other X,s constant. and solving for A,. Finally, as a more specific example for concave functions. the parameters o f the quadratic yield response functions in equations (7) through ( 9 ) can be substituted into equation (14) as in the previous case and solved for the SBVP i f it exists.

Equation ( 1 3 ) is linear in A, because the uniform rate i s constant and independent of A,. Even i f the uniform rate were chosen as a constant, R, determined by family tradition, for example, equation (14 ) would still be linear in A, after substituting R for X:.

uniform input application rate based on the response function for a single management zone, Illustrative

say Management Zone m . The uniform application rate is now determined as X: using To illustrate the concepts presented above, as- Y,,(X,,) = Y,,,(X,,,) instead o f equation ( 3 ) . Sub- sume hypothetical fields s~lited to corn pro- stitute X;: for Xz in equation ( 4 ) to get the cii~ction can be classified into three manage- new profit function for URT. Subtract the new ment zones and that the following quadratic R&, from equation ( 2 ) to get the new RVRT" functions represent corn yield response to fer- function in equation (14) . tilizer nitrogen for the management zones:

(15) Y , = 120 + 1.1 I N , - 0.0013Ny

(16) Y2 = I00 + 1 .0SN2 - 0.0026Ni

(17) Y, = 75 + O.SN, - 0.0014NZ

where Y, , Y,. and Y, are corn yields (bulac) and N , , N2, and N, are nitrogen application rates (Iblac) for high-, medium-, ancl low-response management zones. respectively.

Equations ( IS)-( 17) are p l a ~ ~ s i b l e corn yield response functions chosen for illustrative purposes. They were not e\timated from site- specltic field data. hut were assumed for ease of exposition and because similar ones have been used historically to represent corn yield response to nitrogen (eg., Arce-Diaz et al.; Agrawal and Heady: Mjelde et al.; Vanotti and Bundy; Schlegel and Havlin). Their use facil- itates exposition of the aforementioned concepts because they are continuous and exhibit diminishing marginal physical productivity throughout, and because a mathematical so- lution to equation (6) exists as expressed in equation (13). The latter cannot be said when equations (1) are expressed in semi-log form (also concave), for example. Even when they are expressed as quadratic-plus-plateau or Mitscherlich-Baule functions (Bullock and Bullock; Cerrato and Blackmer; Frank, Beat- tie, and Embleton; Llewelyn and Featherstone; Stecker et al.), which were shown for those cases to more accurately represent corn yield response. mathematical solutions would be difficult. Also. if quadratic response functions overstate nitrogen use at the economic optima (Cerrato and Blacknier; Llewelyn and Feath- erstone) for all management zones, the effects o n RVRT* may be mitigated somewhat. Con- sequently. the less complicated quadratic functional form was used in this article. Even when mathematical solutions do not exist for other functional forms, the concepts presented above still hold and iterative procedures can be used to find approximate solutions for the SBVPs by adjusting A , , , , (A2 for this specific example) until the left-hand side of equation (6) equals V .

After defining A , , A,, and A, as the proportions of the field in high-. medium-, and low- yield response management zones, spatial

break-even analyses were conducted. The average Tennessee corn price received by farmers (P,- = $2.79/bu) and the average nitrogen price (P, = $0.26llb) over the 1993-1 997 pe- riod (Tennessee Department of Agriculture) were used in the analysis.

The additional custom charge for variable rate nitrogen application compared to uniform rate application was assumed to be ?' = $3.001 ac. This additional charge was close to the mean of $3.08/ac (range $ l .SO to $5.50/ac) obtained from personal telephone interviews with firms providing precision farming services to Tennessee farmers in 1999 (Roberts, En- glish, and Sleigh). Responding firms indicated that the additional charge would include the difference in application costs for VRT versus URT and a charge to create a nitrogen application map based on soil survey maps in con- junction with the consultant's knowledge about corn response on various soils, a visit to the field to observe conditions, and an inter- view with the farmer about historical yields.

Sensitivity analyses examined the effects on the SBVPs of 10-percent increases and decreases in P,., P,. and the linear (b,) and squared (c,) ternis of equation (9) as found in equation ( 17) (low-response management zone). Sensitivity of the SBVPs to changes in V was examined by decreasing V by $I.SO/ac and increasing V by $2.50/ac, which is the range in cost differences found by Roberts, English, and Sleigh. These analyses were conducted for the weighted-average-response- function case and for the case where the uni- for111 rate is determined as the optimal rate for the high-response management zone.

The maxin~um RVRT* for example fields with n o land in the high-response management zone (XI = 0 percent) was $l.C)S/ac (Table 1 ) . This maximum RVRT" occurred in fields with 58 percent of their area in the low-response management zone and 42 percent i n the medium-response management zone. Thus, a farmer with a field containing only low- and metiium-response management zones would not be able to cover the additional custom

Table 2. Maximum Return to Variable Rate Technology and Spatial Break-even Variability Proportions for Hypothetical Corn Fields with Three Management Zones with the Proportion of the Field in the Low-Yield Response Management Zone Held Constant, Weighted Average Response Function

RVKT*.' Maximum SBVPsa for SBVPs" for

RVRT*" for Percentage of Percentage of Maximizing Field in Medium- Percentage of Percerltage of Percentage or

Field in High- Response Response Field in Low- Field in Medium- Field in Medium-

Management Management Response Response Response Management Management Management Zone (A2) Zone ( A , )

Zone (X,) Zone (A,) Zone (A2) Lower Upper Lower Upper

,' RVRT" is the return-to-variable-rate technology detined in equation (12) and the SBVPs are spatial break-even variability proportions found by solving equation (13).

Because the maximum RVRT" attainable by varying A 2 is less than the additional custoni charge for VRT of $3.00/ ac, break-even values for A, and A , d o not exist. 'This number is the tnaximum or minimum ('or A? or A , , respectively. Upper or lowcr SBVPs do not exist because RVRT" is greater than the additional custom charge of $3.00/ac when A? or A , are at their constrained m;rximum or ~ninimuni.

charge of $3.00/ac, implying that the adoption of VRT would lead to economic losses on that tield. The maximum RVRT* ($2.33/ac) for example fields having only medium- and high- response management zones ();, = 0 percent) also was less than the additional custom charge (Table 9), suggesting that adoption of VRT would not be profitable. For tields with only low- and high-yield response management zones (X2 = 0 percent), SBVPs were clearly identified at 15 and 90 percent of the field in the low-response management zone, with the tnaximum RVRT* ($7.07/ac) occur- ring at 56 percent in the low-response management zone (Table 3). Thus, for fields with only high- and low-response management zones, farmers would have an economic incentive to adopt VRT on those fields with between 15 and 90 percent of their area in the low-response tnanagetnent zone or between 85 and 10 percent in the high-response tnanage- ment zone.

When the percentage of a field in the high- response management zone (A,) was specified at 20, 40, 60, or 80 percent, economically vi-

able ranges of spatial variability in the low- and medium-response management zones were identified (Table 1 ). These ranges. however, had only lower SBVPs for the low-response management zone and upper SBVPs for the mediurn-response tnanagenlent zone. No upper or low SBVPs existed for these management zones because RVRT"' was greater than $3.00/ac when A, reached its maximum and A, reached its minimum. A similar kind of result occurred when the percentage of a field in the low-response management zone (Xj) was set at 20, 40, 60, and 80 percent (Ta- ble 2).

When the share of an example tield in the medium-response management zone was specified at 60 or 80 percent ();, = 60 or 80 percent), no economically viable mix of Manage- ment Zones 1 and 3 could be found (Table 3). However, given X, = 20 or 40 percent, VRT could be employed more protitably than URT on fields provided they had land in all three management zones. For example, for A, = 20 percent, tields with between 9 and 73 percent of their area in the low-response management

Table 3. Maximum Return to Variable Rate Technology and Spatial Break-even Variability Proportions for Hypothetical Corn Field\ with Three Management Zones with the Proportion of the Field in the Medium-Yield Response Management Zone Held Constant, Weighted Av- erage Re5ponse Function

RVRT“:.~ Maximum Mauimi~ing RVRT"-.I for

Percentage of Percentage of Percentage of Field in Medium- Field in Low- Field in Low-

Response Response Response Management Management Management

SBVPs' tor

Percentage of F~eld In Low-

Re\pon\e M,~nagenient

Zone ( A , )

SBVP\,' for Percentage of Field in High-

Re\ponse Management

Zone ( A , )

Zone ( A 2 ) Zone ( A , ) Zone (A,) Lower Upper Lower Upper

% $lac 76 0 5 6 7.07 15 90 10 85

20 43 5.68 9 7 3 7 7 1 40 3 1 4.28 7 5 3 7 5 3 60 18 2.89 13 h 11 17

80 5 1 .SO 11 h 11 h

-' RVR'C':: is the ret~~m-to-variable-rrite technology dctincd in equation (12) ~u id t h e SBVPs are spatial break-even ~ariabi l i ty proportion\ Ibund by solving equation ( 13). " Bt.cau\e the maximum RVRT':' attainable b! \ n ry inp h , is less than the additional custoni charge f o r VR'T of $3.001 ac, breah-even value\ for A, and A , do not exist.

zone (A,) and between 7 and 7 1 percent in the high-response management Lone (A,) would be considered for VRT instead of URT.

Illustrative sensitivity-analysis results are presented in Table 4 for example fields with 20 percent of their area in the tneclium-response managenlent Lone (x,). As the difference increases between the upper and lower SBVPs with changes in a parameter, a particular field would be more likely to have RVRTZk 2 V , increasing the economic incentive for the farmer to use VRT on that field. Ten-percent increases in prices result in only slightly wider ranges of spatial break-even variability, implying for this example that economic incentives to use VRT are relatively insensitive to price changes.

The model seems quite sensitive to changes in response function pararneterx. As the yield response functions for high- and low-response management zones become more similar in slope (b, or c, increases). spatial break-even variability decreases, decreasing the econoniic incentive to use VRT. Sensitivity to changes in these parameters suggests that accurate estimation of the management-zone yield response functions is critical to obtaining accurate estimates of RVRT''' and the SBVPs.

For fields with );, = 30 percent. a decrease in the cost difference between VRT and URT (V) widens the range of spatial break-even variability and an increase in v narrows it (Ta- ble 4). At the lower end of the range in V ($1 .Solac) found by Roberts, English, and Sleigh, a field would need to have between 0 and 79.8 percent of its area in the low-response management zone (A,) for VRT to at least break even with URT. The range of SBVPs for A , nan-ows to between 35 and 5 1.6 percent at the upper end of the range in V ($S.SO/~C).

Ferrili7e fbr tlze Highest Responye Mc~nagenzent Zone

Table 5 presents the SBVPs for URT f. - '11 mers who are assumed to fertilize the entire field at the optimal nitrogen rate for the high-response management zone. Farmers with fields having high percentages of their areas in low- and medium-response management zones have economic incentive to use VRT. In general, VRT has its greatest economic advantage over URT in fields with smaller proportions of land in the high-response management zone because more can be gained from adopting VRT.

Table 4. Impacts of Changes in Nitrogen and Corn Prices, Response Function Parameters, and the Additional Cost of VRT on Spatial Break-even Variability Proportions for Hypothetical Corn Fields with Three Management Zones with 20 Percent of the Field in the Medium- Response Management Zone, Weighted Average Response Function

SBVPs,' for Percentage SBVPs,' for Percentage o f of Field in Low-Response Field in High-Response

Management Zone (A,) Management Zone ( A , )

Parameters that Change Lower Upper L*owel- Upper

Price5 of nitrogen and corn % Mean prices 9. I 73.2 6.S 70.9 Increase P, by 10% 8.8 74.0 6.0 7 I .2

Increase P, by 10% 7.1 73.9 6.1 72.6

Low response Function Original parameter values 9.1 73.2 6.8 70.9 Decrease h, by 10% 5.4 80.0h O.Oh 71.6 Increase b,; by 10% 20.0 56.7 23.3 60.0 Decrease c , by 10'% 5.5 79.3 0.8 74.5 Increase c, by IO'h 70.7 57.5 22.3 59.3

Additional cost of VRT Decrease v by $1 .SO 0.0 79.8 0.2 80.0 Original V = S3.001ac 9.1 73.2 6.8 70.9 Increase V by $2.50 35.0 51.6 28.1 45.0

,' SRVPs are spatial bre;lh-evell variability proportions found by solving ecluation (13). "This nirrnber is the niuxirnum or minimurn l i ~ r A, or A , , respectively. Upper 01- lower- S B V P do not csist because KVRT" is greatcr than the additional custom chnrpc ( v ) when A , or A , at-c at their constr:~ined rnasimuni o r minimum.

On these fields, URT greatly over f e r t i l i~es the low- and medium-response management zones, while VRT provides each management zone with its optimal level of nitrogen. Also. for a fixed proportion of a field in the high- response Iilanagement Lone, the larger the proportion of the field in the low-re\pon\e management tone and the smaller the proportion in the medium-response management zone, the more profitable VRT is relative to UKT. For example, ti)r VRT to be profitable when 80 percent of the field is in the high-response management zone ();, = 80 percent), at least 8 percent of the tield must be in the low-response management zone (8 5 A3 5 20) and at most 12 percent can be in the ~nedium-response management zone (0 5 A? 5 12).

Table 6 shows sensitivity-analysis results for prices. low response function parameters, and changes in the additional cost of VRT versus URT. A 10-percent change in the nitrogen

price (P,) has imperceptible effects on the SBVPs and n 10-percent change in the corn price (P, ) has only slightly larger impacts. The SBVPs also seem insensitive t o changes in the low-response function parameters. Nev- ertheless, the SBVP for the high-response management zone ( A , ) increases slightly and the SBVP for low-response management zone (A,) decreases slightly when the low-response function parameters decrease by 10 percent. Thus, us the marginal physical product of the low-response function diverges from the marginal physical products of the other two response functions, more of the field can be in the high-response management zone for VRT to break even with URT. Alternatively, as the cost of VRT compared to URT (v) changes over the range found by Roberts, English, and Sleigh. the minimum proportion of the tield that must be in the low-response management zone (A,) increases froin 0 to 15.7 percent,

Orglislz, Mtrhrrjutztrslzc~tti, and Roberts: A.s.vessi/~g Slxitiril Bre~rk-e~.c.tr \/trr.iuhilit~ 56 1

Table 5. Spatial Break-even Variability Proportions with Farmers Fertilizing for the High-

Response Management Zone for Hypothetical Corn Fields with 'Three Management Zones

Spatial Break-even Spatial Break-even Percentage of Fieltl in High V. '11 .' lability Proportions Variability Proportions

( A , ) , Medium ();,), ancl (SBVPs) for Low ( A , ) and (SBVPs) for Medium ( A Z ) Low ( A , ) Response Medium (AL) Response and High ( A , ) Response Management Zones Management Zones Management Zones

C/r % '%, High Response ( X I ) Low Response (A,) Medium Response ( A l ) 0 0.' 1 OOh

20 (P 80h 40 0.' 60" 60 0.' 40" 80 8 12 Mediuni Response ( A Z ) Low Response (A,) High Response ( A , ) 0 13 87

20 5 75 40 0.' 6( )I' 60 0,' 40t' 80 0,' 20t' Low Response ();,) Mediulri Response (A:) High Response ( A , )

(1 30 70 20 0.' 80" 40 0.' 60,' 60 0.' 40h 80 0.' 20"

, ' A n SHVPs doe\ not cxi\t hccaicse RVRTQs greater- than the additional custom charge for VRT n l X3.00/ac when A , or A, are at their con\lrained ~niliirnum o f Lero. I' An SBVP doe\ not exis1 h e c a u e KVKT1- i \ ptcatel- than the atltlitional custom charge fix VKT of $3.OO/nc when A , or A , are at their Ina\ltiiuni 01 I - x,.

while the maximum proportion allowed in the high-response management zone (A,) decreases from 80 to 63.3 percent.

Discussion

This hypothetical example emphas i~es that obtaining inforriiation about a management zone's yield response potential is more iiiiportant than obtaining information about its yield potential ( m a x i r n ~ ~ m yield). This point can be generalize to all concave functional forms and is illustrat- ed for the q~~udrat ic case by the absence of the intercept terti~s (a , , a2, and a,) in equations (13) and ( 14). Even for linear-plus-plateau response functions, which do not exhibit diminishing marginal physical productivity (not strictly concave), RVRT'I' is cieterniined by the yield re-

sponses for the management zones that are not at their respective yield pl;iteaus when the ~uni-

form input rate is applied, rather than by the maximum yields themselves.

If a farmer can gain knowledge of the tield- specific management /ones for a particular crop and input and the parameters of the cor- responding yield-response funct ions , the methods discussed above could be used in de- ciding whether to use VRT or URT on a tield. Unfort~rnately, this knowledge is difficult to obtain with certainty, but farmers are currently using other precision farming technologies (eg., yield monitors, grid soil sampling, field mapping) that can be used to identify management zones and their yield-response potentials (English, Roberts, and Sleigh). Yield- monitol- and grid-soil-sampling data can provide information about yield-response potential, especially when a historical database of those data is available. The uncertainty about yield-response potential can be further

Table 6. Impacts of Changes in Nitrogen and Corn Prices, Response Function Para~neters, and the Additional Cost of VRT on Spatial Break-even Variability Proportions for Hypothetical Corn Fields with Three Management Zones with 20 Percent of the Field in the Medium- Response Management Zone, Farmers Fertilizing for the High-Response M~anagernent Zone

SBVPs,' for Percentage of SBVPs,' for Percentage of Fielci in Low-Response Field in High-Response

Parameters that Change Management Zone ( h , ) Management Zone ( A , )

Prices o f corn and nitrogen Mean prices Increase P, by IOVr Decrease P, by I OIF Increasc P, by 10'k Decrease P, hy 1 0 %

Low response function Original parameter values Decrease b; by 10% Increase b, by 10% Decrease c, by 10% Increase c, by 10'%1

Additional cost of VRT Decrease v by $1 .SO Original v = $3.00/nc Increase V by $2.50

SBVPs are spatial break-even variability proportions

reduced when data collected through precision technologies are combined with expert per-cep- tions or knowledge, such as 1 ) the farmer's historical perceptions about yield response in different parts of the field and 2) recommendations from experts-such as soil-test laboratories. crop consultants, input suppliers. or extension personnel-who may implicitly or explicitly assume yield-response functions based on their knowledge when making recommendations about input application.

Researchers are exploring inexpensive methods for estimating munagement-zone-specific yield-response functions from yield mon- itor data (Bongiov~lnni and Lowenberg De- Boer, 2000). Other researchers are developing methods for estimating managen~ent-zone-specitic meta-response models from crop-growth simulation models (Peeters and Booltink). As these estimation methods become Inore re- fined. the methods presented in this article will become increasingly important in the VRT- versus-URT decision.

Actual fields within a geographic area can

contain a wide variety of soil types suited to producing several major crops. Over the years :I limited number of field experiments have allowed estimation of a patchwork of yield-response functions for some geographic areas. The demand for VRT will probably increase in the future, requiring estimates of yield-response functions for a growing number of farmers. A concerted effort to estimate and document yield response for a variety of crops, soil series, and weather conditions would be beneficial to agribusiness firms who are interested in providing VRT services to farmers and to farmers who are contemplating adopting VRT. Estimation of metn-response functions for major crops and soil series within a particular geographic area could be used with the methods in this article until methods for estimating management-zone-specific response functions are improved and become less expensive for on-farm use. These meta- response functions could be made available to agribusiness firms and farmers in a user- friendly modeling fralnrwork that would al-

low them to evaluate the VRT-versus-URT decision for a specific field.

Conclusions

Adoption of VRT depends to a large extent on

the expected net economic benefits received by potential adopters. Fields generally exhibit

yield variability; however, not all fields war- rant VRT from an econorllic standpoint. Farm- ers are interested in knowing whether VRT is economically viable on their fields. The answer to this question varies from field to field depending on spatial variability as well as yield-response variability among management zones. The answer also varies with the crop, the input, prices, and the cost of using VRT relative to URT. In the end, n o general formula exists for determining whether VRT or URT should be used on a particular field because each field presents a different case. What researchers can do, however, is provide agribusiness firms, extension personnel, and farmers with a consistent means for evaluating this decision based on the economic models Dre-

of Agricrrlturul cind Re.rource Economir.~ 23(1998):385-400.

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sented above, or variants ~ ~ ~ ~ i ~ ~ d Carr, P.M., G.R. Carlson, J.S. Jacobson, G.A. Niel- sen, and E.O. Skogley. "Farming Soils. Not model inputs would include estimates of yield- Fields: A Strategy for Increasing Fertilizer Prof-

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field in each management zone, expected crop 4( 99 ):57-61. and input prices, and the additional cost of c,,,.,~~, M.E. and A.M. ~ l ~ ~ k ~ ~ ~ . c '~omparison of VRT versus URT. Even with educated guesses Models for Describing Corn Yield Response to for the model inputs, model outputs in the form of a field RVRTes or SBVPs could be used as additional pieces of information in helping farmers make the VRT-versus-URT decision for a particular field.

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assessing spatial break-even variability in fields with two or more

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