The objective was to compare optimal long-term farm level investments in conservation tillage systems and annual nitrogen use in wheat production from a private and social perspective. The performance of four tillage systems was simulated on a 243 hectare (600-acre) Oklahoma wheat farm with six soil types. A decomposition method was used to find the optimal tillage system. It was found that private producers would select a disk chisel system while the less erosive sweep system was optimal from a social perspective.Crop Production/Industries,

Bocoum, Kolado

Stoecker, Arthur L.

English

Research Papers in Economics

Optimal Rate of Applied Nitrogen and Choice of Reduced Tillage System
By
Kolado Bocoum and Arthur Stoecker
a
Paper presented at Western Agricultural Economics Association Annual Meetings,
Vancouver, British Columbia, June 29-July 1, 2000
Copyright 2000 by Kolado Bocoum and Arthur Stoecker. All rights reserved.  Readers
may make verbatim copies of this document for non-commercial purposes by any means,
provided that this copyright notice appears on all such reports.
                                                          
a  Respectively, former Research Assistant, now with Planning and Statistical Unit of the Ministry of
Agriculture, Republic of Mali, and Associate Professor, Department of Agricultural Economics,
Oklahoma State University.  Agricultural Economics paper no. AEP0002.1
Abstract:  The objective was to compare optimal long-term farm level investments in
conservation tillage systems and annual nitrogen use in wheat production from a private
and social perspective.  The performance of four tillage systems was simulated on a 243
hectare (600-acre) Oklahoma wheat farm with six soil types.  A decomposition method
was used to find the optimal tillage system.  It was found that private producers would
select a disk chisel system while the less erosive sweep system was optimal from a social
perspective.
Introduction:  Soil erosion constitutes a major problem facing farmers in Oklahoma.
More than seven million acres are classified by the USDA as highly erodible.  Yet many
incentives for compliance with soil conservation practices within the framework of the
federal commodity program will be removed in the future.  The majority of compliance
plans for wheat production land include the use of conservation tillage systems designed
to retain plant residue on the soil surface throughout the year.  Studies by Epplin et al
(1994), Aw Hassan (1992), and Klemme (1983) have compared the profitability of
conservation tillage systems.  However, these studies have not fully considered the
interaction between nutrient management, tillage system choice, and externality costs at
the farm level.
The objective of this research is to compare the optimal social and private use of
conservation techniques and commercial fertilizer to sustain productivity of soils in
Garfield County, Oklahoma subject to limitations of offsite damages from soil erosion
and fertilizer losses.2
Problem and Model:  The main objective of the study is to determine the most profitable
long-term use of tillage systems and inorganic nitrogen when there is a concern about off-
site damages from soil erosion and nitrogen loss.  A representative 243 hectare (600 acre)
wheat farm with six soil types was defined.  The tillage types (in order of decreasing
erosion and increasing herbicide use) to be considered were the plow (PL), disk-chisel
(DC), sweep twice (S2), and sweep once (S1) systems.  Non-zero externality charges are
used in the social analysis.  Assume a total horizon of 100 years (T) and that during this
100 year period the tillage machinery is replaced every 10 (mt) years.  Therefore, the
problem is to choose a tillage system every mt years and choose the wheat area and
nitrogen use on wheat each year t on soil i to maximize the Net Present Value (NPV)
over a T year period.
( ) ( ) [
][ ] ()
() (1) r 1 TC M
r 1
E t Nloss t R AP E t Nloss t
M , N VC M , TN03 , N , D Y P Aw
NPV Max
mt
i i mt,
T
1 mt
t
pt e pt n pt jt ijt e jt n
mt jt t mt jt jt jt ijt w jt T
1 i
T
1 j
T
1 t M N,
/
/
+ −
+
 



 



− − + − −
−
=
∑
∑ ∑ ∑
=
= = =
Subject to:
) , ( 1 it jt ijt jt jt M D E D D − = + (2)
0 0 j j D D
−
= (3)
∑ =
=
I
i
it t all for M
1
1( 4 )
) 3 , , ( jt jt jt jt TNO N D f Nloss = (5)
) 3 , , ( 3 1 jt jt jt jt TNO N D g TNO = + (6)
0 0 3 3 j j TNO TNO
−
= (7)3
A A A jpt jwt jt + = (8)
0 3 , , , > TNO N M D
where
t =1,2,…,T represents the annual planning horizon
mt = 1,11,21,…T is the machinery or tillage investment planning horizon
i= 1,2,…,I is the number of alternative tillage systems
j =1,2,…,J is the number of soil types on the farm
Yijt is the yield of wheat in soil type j, using tillage i at time t, in
metric ton per hectare
Pw is the price of wheat in dollars per metric ton
Rp is rental rate per one hectare of pasture
Ajwt is the hectares of wheat on soil type j in year t
Ajpt is the area of pasture on soil type j in year t
Djt is the depth of soil j at time t in meters
VCt is the variable cost at time t in dollars per hectare
TCi is the Present Cost of Tillage System i at the time of replacement.
Njt  is the amount of nitrogen applied to wheat on soil type j at time t
in kilograms per hectare
Mitm is a zero-one integer variable of the type i of tillage used in time t
tn is the off-site cost in dollars per kg of nitrogen loss
Nlossjt is the amount of nitrogen loss in soil type j at time t in kilograms
per hectare
te is the cost in dollars per metric ton of soil lost
Eijt is the amount of eroded in depth in meters from soil type j when
tillage system i is used at time t
Sjt is the index for soil type j in the farm at time t
r is the discount rate
The objective function equation (1) is the present value of net returns per hectare
above the cost of nitrogen, machinery, and off-site damages aggregated across all tillage
systems and soil type of the farm for the entire planning period.  The constraints are4
given by equation (2) through equation (8).  Equation (2) is a soil depth transition
equation for soil type Sj.  Equation (3) sets the initial soil depth at a given level.  Equation
(4) ensures that no more than one tillage system is chosen for a given year.  Equation (5)
allows determining the amount of nitrogen lost each year per soil type from surface
runoff and leaching below the soil surface.  Equation (6) determines nitrogen carryover.
Methods:  The proposed model is of the discrete non-linear form.  The size and
complexity of the problem increases with the length of the planning horizon and the
number of non-linear constraints.  We were unable to obtain an optimal solution with the
GAMS-MINOS  software.  However a decomposition method which used a spreadsheet
solver was successful.
The study area was Garfield County, Oklahoma.  Farming is assumed to take
place in the Renfrow-Vernon-Kirland association, which contains six soil types: Kirland
53%, Vernon 21%, Renfrow 19%.  Norge, Miller, and Zane soils account for 7% of the
area.
Data required for estimating the functions for crop yield, soil nitrogen, soil
erosion, and nitrogen loss for each tillage system were obtained by using the Erosion
Productivity Impact Calculator (EPIC) (William et al., 1983).  Simulation runs for this
study were made over a 100 year-period with four level runs of applied nitrogen
application (16, 50, 100, and 150 kg per hectare) in combination with each of the four
tillage systems and six soil types.  Thus, 1600 observations were generated.
The Oklahoma State University Enterprise Budget generator was used to estimate
a budget for each of the four tillage systems.  Variable costs include the costs of wheat
seed, phosphorus, harvesting, pesticide, annual operating capital, machinery labor, fuel5
and repairs.  Machinery ownership costs are the sum of depreciation, interest, and taxes.
The budget summaries are shown in Table 1.  The study assumes a base line wheat price
of $110 per metric ton, while the price of nitrogen is  $0.55 per kilogram.
Table 1. Annual and Net Purchase Cost for Alternative Wheat Tillage Systems for
the Representative Farm
Type of Cost Disk Chisel Plow Sweep Twice Sweep Once
Operating Costs
a ($/ha) $138.20 $142.58 $174.34 $197.27
Fixed Costs ($)
b $84,864 $157,346 $67,231 $67,231
a Source: OSU Enterprise Budget
b  Total cost to buy one machinery complement every 10 years less discounted salvage
value for the 243 hectare farm.
The statistical estimation was made by considering each treatment or level of
fertilizer application over a 100 year-period as one of 96 cross sectional units (4 fertilizer
levels X 4 tillage systems X 6 soil types).  Data from the four treatments for each soil
type were pooled and arranged so that all observations appear together by tillage system
within soil type.  It is assumed that tillage system and treatment (applied nitrogen) have
fixed effects.  There are 100 replications corresponding to 100 years of simulation with a
random effect assumed to be distributed as N(0,
2
δ σ ).
The statistical model for each soil type can be represented as:
100 ,..., 1 , 4 .,..., 1 , 4 .,..., 1 = = =
+ + + + =
t k i
Y ikt t k i ikt ε δ β α µ
where
Yikt  is the observation when tillage system i is used and k'th  level of nitrogen
applied at time t.
µ is the overall population mean6
αi the tillage effect
βk is the treatment level effect
δt is the year effect assumed to be iid N(0, 
2
δ σ )
εikt is the experimental error associated with Yikt, assumed to be iid N(0, 
2
ε σ )
A modified version of the Mitscherlich-Spillman (M-S) function (Taylor 1982,
Young et al. 1985) was used for the yield response function.  The estimated yield
function was expressed as:
Where
Yt is the yield in metric tons per hectare
Ym is the maximum attainable yield
1/Dt is the inverse of topsoil depth at time t (m)
1/Nt is the inverse of applied nitrogen at time t (kg/ha)
1/TNO3t is the inverse of residual nitrogen in soil at time t
α, β1, β2, β3   are parameters
All models were estimated using the SAS PROC MIXED procedure.
The data to estimate the offsite damage costs in the study area are not available.  Based
on estimation by Rebaudo and associates (1990), the study assumed an offsite damage
cost of $1.5/mt.  An 8% discount rate was also assumed following Aw Hassan (1992).
The problem was too large to solve with  GAMS MINOS software.  The
empirical model specified above has more than 2400 linear and non-linear equations and
3620 variables.  A branching method, which used the EXCEL solver software, was
developed.  The method (outlined in Figure 1) consists of decomposing the optimization
problem into a large number of smaller optimization problems.  The branching method is
t t t t m t TNO N D EXP Y Y ε β β β α + + + − = ) 3 / / / ( * 3 2 17
.
                                                       .
                                                       .
                    NPV from Soil S WRT Annual N Application
Figure 1.  Outline of the Spreadsheet Algorithum.
Let T = 100 Total Years in  Planning Horizion
Let nH = 5,   Number of 10 year Machinery Planning Periods
Let nS = 6,    Number of Unique Soils on Farm
Let MaxFarmNPV = 0,     Highest Farm Level NPV Found
Let MachComp be an index variable for machinery complements
Arbitrarily Select a Machinery Compliment to use in each 10-year period
Index Cur_Mch_Horz = 1,5         (Use Disk Chisel in periods 6-10)
Let Change_Flag  = 0     Binary variable set = 1 if any change in
                          Machinery Complements increases MaxFarmNPV
For mH = 1 to nH          (machinery horizion, years 1, 11, 21, 31, 41)
For MachComp = 1 to NMch_Comp
Call Spreadsheet Solver to find Max  NPV from Soil 1
WRT Annual N Application over period T
Call Spreadsheet Solver to find Max NPV from Soil 2
WRT Annual N Application over period T
.
.
Call Spreadsheet Solver to find Max  NPV from Soil nS
WRT Annual N Application over period T
         Calculate CurFarmNPV =
Sum NPV over all soils Less Discounted Purchase Price
of all Machinery Complements in planning horizon mH
Is CurFarmNPV >   BEST NPV?
IF NO
IF YES
Set BESTNPV = CURRENT NPV
Set ChngFlag = 1
NEXT MachComp     (Increment MachComp )
Next  mH            ( Increment index for Machinery Planning horizon)
Is   Change_Flag =1? IF NO
IF YES, Repeat With mH=1 STOP: Best Sequence of
Machinery Complements Found8
similar to the methods of successive approximation algorithums in dynamic programming
reviewed by Yakowitz (1982).  First a tillage system was chosen for use in each of the
10-year machinery planning periods which begin in years 1, 11, 21, 31, and 41.  (Use of
the DC system was specified for the last 50 years). Equations (1) through (8) with their
recursive linkages are entered into a separate spreadsheet page to represent a 100 year
period for each soil. The solver was then called to determine the optimal annual level of
nitrogen for wheat on each soil.  The annual choice between allocating that soil to wheat
or pasture was then made.  The farm level NPV from the initial sequence of tillage
systems was calculated.
Then, in year 1, each of the alternative tillage systems was examined to see if
their selection at that time would increase the NPV.  The tillage system adopted in year 1
that gave the highest NPV was chosen.  Each of the alternative tillage systems was then
tested to see if its selection in year 11 would increase the NPV.  (Unfortunately, each
time a different tillage system is considered, the solver must be called to re-optimize
nitrogen applications and crop choice on each soil).
The examination of alternative tillage systems was then examined for years 21,
31, and 41.  If none of the alternative tillage systems increased the NPV during any of the
machinery planning periods then the problem was terminated.  If an increase in the NPV
was observed during any of the 5 machinery periods, then the process was repeated.  The
optimization process required from two to three hours on a 500 MHz microcomputer.
The advantage is that for decomposable problems, solutions can be obtained by
microcomputers and spreadsheet software which are generally available.9
Results:  The results of the estimation showed that all coefficients have the expected sign
and that most were significantly different from zero at the 5 percent level.  In general, the
estimated equations fit the data well though the yield and nitrogen loss predictions were
over estimated for some soils.  The estimated functions for Kirkland soil type are shown
on Table 2.
Data collected from variety trails on farmer fields in North Central Oklahoma,
were used to adjust EPIC yields to reflect the actual yield of wheat on soil types in the
study area.  During the 100 year simulation period, average estimated yields with 100 kg
of nitrogen applied per hectare varied from 3.03 MT/ha to 3.11 MT/ha for the S1 and S2
systems and from 3.09 MT/ha to 2.09 MT/ha for plow system.  Figure 2 below shows
yield changes in relation to soil depth by soil type.
Table 2. Functions for Wheat Yield, Residual Nitrogen, and Erosion for the
Kirkland Soil Derived from EPIC Simulation Data.
Functions Intercept
a D N TNO3 DC PL SW1
Erosion 7.6521 -3.6315 1.4395 9.5861 -0.0372
(m) (3.18)
c (-3.06) (9.48) (16.27) (-2.66)
Yield
b 0.4761 0.3350 20.1668 10.1141
MT/ha) (8.01) (3.37) (11.41) (12.00)
Nitrogen -48.4702 22.3470 0.2852 0.7589
Carryover (kg/ha) (-8.27) (7.71) (12.37) (44.98)
Nitrogen Loss -8.6885 46.9016 0.0209 29.7562 105.4920 -86.8852
(kg/ha) (-5.52) (6.07) (4.16) (26.79) (32.10) (-5.55)
a   For dummy variable tillage intercept represents SW2
b   For yield variables are inverse of variable shown on Table
c   Numbers in parenthesis are t-values.
The study found that the optimum levels of applied nitrogen were slightly lower
for the sweep systems, (varying from 117 kg/ha to 132 kg/ha, depending on the soil type)10
than for the disk chisel, (varying from 123 kg/ha to 135 kg /ha).  At the farm level, as
shown in Table 3, the optimum nitrogen application for the private case varied from 126
kg/ha to 134 kg/ha depending on the soil type.  For the social optimization, nitrogen
applications vary from 104 kg/ha to 130 kg/ha.  When off-site damages were not taken
into account, the net present value of the disk chisel was higher than the net present value
of all the other systems on all soils, and the net present value of the S2 system was the
lowest.  The analysis shows that if externality charges were considered producers would
adopt the less erosive sweep systems which would almost maintain current yields without
increased nitrogen as indicated in Figure 3.  In Figure 3 the nitrogen-soil depth isoquants
for the farm with the DC system after 10 and 50 years are weighted averages of the
individual soil isoquants.  The optimal path shows nitrogen application would increase by
about 3 kg per ha over the 40 year period.  The socially optimal S2 system shows a nearly
constant level of nitrogen and soil depth.
Figure 2.  Yield Response to Soil Depth by Soil Type with 100 kg N/ha Applied N.

OPTIMAL RATE OF APPLIED NITROGEN AND CHOICE OF REDUCED TILLAGE

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