View of Long-term fertilizer field trials: comparison of three mathematical response models

Vol.6(1997): 151-160.

Long-term ^{fertilizer field trials:} comparison of three mathematical response models

StefanT.Bäckman

DepartmentofEconomics and Management, P.O. Box27,FIN-00014 UniversityofHelsinki, Finland, e-mail:Stefan,^backman@helsinki.fi

S.Vermeulen,V.-M. Taavitsainen

KemiraAgro,Luoteisrinne2,FIN-02771 Espoo,Finland

Accession to theEuropean Union causedadropofnearly 60per cent from1994to1995 in pricesof wheat,barleyand oats inFinland. The economicuseof fertilizer therefore decreasedaccordingly.To

calculatetheeffect of theprice changes on the economic optima, thephysical production function mustbe known. Threephysical production functions, thequadratic,the linearresponse andplateau (LRP) and the exponential function were estimated for this purpose. The models differed little in respectof^theR_>djvalue (0.82-0.90) but the calculatedoptimum varied, depending on^theproduction function. Dataon along-term field trial (21 years) wereanalysed. The field trial wasestablishedin 1973todemonstrate the effect of mineral fertilizerincropproduction. Thecropsgrown in the trial werebarley, wheat andoats.Different varietieswereincluded inthe models.

Keywords: economicoptimum, nitrogen, wheat, barley, oats, Finland

ntroduction

Fertilizer field trials yield avast amountof data that^cannotbe analysed without theuseofaprop- ermathematical model. Fertilizer recommenda- tions and political decisions onfertilizerusede- pend directly on the parametersestimated from the data.However,in agronomy there isnostand- ardsystemfor choosing between modelstorep- resent the relationship between fertilizer input and yield formation ^(Cerrato and Blackmer

1990).

All the models chosen, the quadratic model, the exponential model usually known as the Mitscherlichmodel, anda modified form of the plateau model, are commonly used in cropre- sponse analyses(Bockand Sikora 1990, Cerra- toand Blackmer 1990, Frank etal. 1990, Paris

1992, Sumelius 1993).Even though the models give comparable

R 2

^values, they may give dif- ferent optimal fertilizerrates. Optima based on the quadratic functionor onthe exponential func- tion have been criticised for giving excessively high optimal fertilizerrates. A model basedon a linear response and plateau(LRP)function gives

©Agricultural and Food ScienceinFinland Manuscriptreceived June1996

Bäckman^,S. T. etal. Fertilizer trials: mathematical response models

Table I. Potential benefits and drawbacks of the^LRP,quadraticandexponentialmodels.

LRP Quadratic Exponential

Fit withtheory vonLiebig law ofdiminishingreturn Mitscherlich Fit withexistingrecommendations lowerorequal higher higher

Computing feasibility verycomplex easy complex

Diminishing phase ^no yes no

Symmetric no yes no

Usedbyextension service seldom often seldom

lower optima (Paris 1992).Frank etal. (1990) found that thecosts of wrong specification of optimal input amounts are substantial. They claimed that neither polynomial nor plateau growth should be assumedapriori onagronom- ic grounds. Paris ⁽¹⁹⁹²⁾ pointed ^outthat ^most people interpret the_vonLiebig hypothesis _{as a} linear relationto limitingnutrient, and that this interpretation is notconsistent withvonLiebig’s concept.AccordingtoParis, thevonLiebig hy- pothesis claims that there isa direct,butnot nec- essarily linear,relation to the limiting nutrient.

Incorrect specification of optimal input has not only economic but also environmental im- pacts. Excessive application rates lead tonutri- entlossestothe environment by leachingorvol- atilization. Miettinen (1993) studied the effect of fertilizer policy and leaching and found that imposition of input quotas caused farmers the lowest additionalcostsin barley production. The loss of profitswas greater with other measures and the decrease in nitrogen leachingwassmall- er.Evaluation of these inputquotas (ornutrient quotas) again requires the use of mathematical models.

We here compare three models (quadratic, LRP, exponential) in aneffort toestablish their benefits and drawbacks and tocalculate optima accordingtothem. Some of the potential bene- fits and drawbacksareassumed in advance (Ta- ble 1). The use of these models in practice and

the theory behind them are also discussed. To testthe modelsweused dataon along-term field trial (21 years) in which compound NPK ferti- lizers were used. We found that nitrogen ex- plained mostof the yield response; fertilization

with phosphorus did not increase the yield, though atthe lowest levels the yield wasdeter- mined by both nitrogen and phosphorus (Yli- Halla 1991).

Material and methods

The fertilizer field trial waslocatedonthe same plot, with five fertilizer levels being applied each year. The trialwas established in 1973 and the data used _arefrom the period 1973-1993. New varieties replaced old ones but the cultivation techniques remained the same throughout the period. The field trial was conducted atKemira Agro's research farm at Kotkaniemi, 40 km northeast of Helsinki. The change in varieties was gradual. Barley (29 varieties), wheat (21 varieties) and oats (16 varieties) were cultivat- ed. The five fertilizer levelsarereferred toas A, B, C,D and E in increasing order of fertilizer application, with level Aas zero. The primary purpose of the trialwas demonstrative, and the treatmentswere notrandomized.

The amountsof fertilizerapplied ^{inthe trial} werereduced from 1986 onwards(Table 2), ex- ceptat levelA,atwhich itwas zerothroughout.

The NPK fertilizer applied was the compound fertilizermost commonly used in Finland. The composition and amounts of this fertilizer are listed in Table2.

The nutrientsN, P and K are consideredas linear combinations and N is assumedtobe the main limiting nutrient. Partial correlation anal-

Vol.6(1997): 151-160.

Table^2,Applied NPKcompound fertilizer (kg/ha)indifferent periods.

Year 1973-1979 1980-1985 1986-1988 1989 1990-1993

Fertilizer 15-9-12 16-7-13 16-7-13 17-6-12 20-4-8

LevelA,kg/ha 0 0 0 0 0

LevelB,kg/ha 300 300 300 300 200

Level 6OO 600 500 500 400

Level^D,kg/ha 900 900 700 700 600

LevelE, kg/ha 1200 1200 900 900 800

yses also show_ahigher value for the N fertilizer than for the compound fertilizeras a whole(Ta- ble 3). The N fertilizer applied was chosen _as the independent variable in the model, but note thatP and K were also applied as macronutri- entsin the field trial. Differences in thecompo-

sition of the compound fertilizer had little effect on output(Fig. 1).Annual effectswereanalysed with F tests. Inclusion of annual effects allowed for annual physical production functions. The effect of residual Non levelsB, C,D and Ewas assumed tobe nil from yeartoyear. The effect of residual Pwas assumed tobe close to nil at level C. At levelA, yield decreased from 1973

onwards.

The functions arequadratic, LRP and expo- nential. The LRP and exponential functionswere chosen because they reflect alevelat which no further increase in yield response is achieved.

The LRP function showeda“breakpoint" accord- ing tovon Liebig’s Law of the Minimum. The exponential function increases asymptoticallyto amaximal yield level.

Polynomial (quadratic) function

y.. ⁼a ⁺ b x..⁺c..x^2.+e

JU U U >J u u u (1)

The indices i and j denote variety and year variationsrespectively, y stands for the yieldre- sponsein kg/ha and x for the N fertilizer applied in kg/ha.a, b and c are parameters ande are er- ror terms that areassumed ^to be normally dis- tributed and with zero mean. This model, in which all coefficients vary both among varieties and in time, is easy to calculate and is theone mostcommonly used in fertilizer trial analyses.

Table3.Partial correlations between fertilizer components and wheat yield.

Controllingfor Yield P

Nfertilizer 0.3525 0.000 Pfertilizer Pfertilizer 0.0272 0.356 N fertilizer NPKfertilizer -0.0386 0,190 Nfertilizer andP fertilizer

The parameter c is expected to give anegative estimate in ordertoreflect diminishingreturns.

There are separate calculations for every crop.

All indices^i.,.nand j...konevery coefficientare tested separately with Ftestsand refertoeffects in this text. Only effects significant according tothe F testare included in the final results.

Linearresponse and plateau ^(LRP)function

{a.

^u+yJ^b.,max’^ux..,,^rxij^O><X...^x.bij^u<xub'J

^.

^JIj ⁺

^e«

⁽²⁾ The indices i and j denote variety and year variations respectively, ^ystands for the yieldre-

Fig. 1.^Averagewheat yields at five fertilizer levels.

Bäckman,S. T. etal.Fertilizertrials: mathematical response models sponsein kg/ha and x for N fertilizer applied in

kg/ha. The index b on x stands for the level of fertilizer that achieves the breakpoint in the yield response toadditional fertilizer nitrogen, y is the yield response level achieved inagiven year for a specific variety. The function imposes a maximal yield levelatwhich further application ofa specific input will _not give any further in- crease in the yield.

Exponential function (Mitscherlich)

yu^{. =} au(1-buec^»x"001₎+ eu (3) The indices i and j denote variety and year variations, respectively, y stands for the yield response in kg/ha and x for the N fertilizer ap- plied in kg/ha. A factor, 0.01,was included in the model because ofa problem with decimals when printing the results, and this factor was thereafter considered in the results. The func- tion givesanasymptotic maximal yield level. The growth to the maximal level diminishes expo- nentially, whichmeans that theparameter c. is negative.

The quadratic function was computed with IMSLsubroutines; RGLM for fitting the multi- variate linear regression model and RSTAT for computing and printing statistics. These subrou- tineswereretrieved from the MATLAB program.

The quadratic function could thereafter be esti- matedsimply by changing the dummy variable effects. For estimation of the LRP and the expo- nential functions _we used the subroutine DUNLSF. This IMSL command solves a non- linear least squares problem using a modified Levenberg-Marquardt algorithm andafinite-dif- ference Jacobian. The Levenberg-Marquardt method is a modification of the Gauss-Newton algorithm for solving non-linear least squares

problems (Appendix 1).From one ^currentpoint another point is calculated by the trustregion approach(Dennisand Schnabel 1983).This pro- cedure is repeated until stopping criteriaaresat- isfied. Each separate estimationwas made by a different FORTRAN program. All effects were

analysed with Ftests.

For estimation of the LRP model, aprogram

loop was used torestrict the ^movementof val- uesof theparameterxb|^, with values outside the domain of the field trial(0-192) givingapenal- ty to the minimization of the non-linear least squares. The first restricting penalty loop (algo- rithm x_b ⁼ (200-x_b)2₎ we tried did^not perform effectively enough on all data. The secondre- stricting penalty loop,^x_h⁼ _xb (Fig. 2),was moreeffective and x._bij^finallyJ varied between -4 and 4, whichmeant optimal solutions between 0 and200 kg of N per hectare. These penalty loops served ^tobring the routinestoaclear stop.The first algorithm probably did not work well because the sample range was too small _{to re-} flect the decreasing response phase in all cases.

The barley data were nevertheless calculated with the first algorithm, and so barley reflects a higher plateau response than wheat or ^oats (Fig. 3).

We assumed that all species and varieties in the trial would respond differently to fertilizer, and that all species and varieties in the trial would give different yields without fertilizer. We also assumed that every year would give differ- ent yield responses. F ^tests were performed ^to test ourassumptions.

The R_a²_d value will give the estimated func- tion response ^to the distributionpattern, is the R_a²_d value corrected for the number of degrees of freedom and is therefore better suited forcom- paring different models with different numbers

Fig. 2. Restricting penalty algorithmfor theLRPmodel.

Vol.6(1997): 151-160.

ofparameters. The number ofparametersis also acriterion in which the number ofparameters as wellas the flexibility of the function should be related_to the information they give. Preference should be given to function estimates that are easy bothtoexplain andto calculate.

Table4.Optimal inputfor the maximal economic return.

Quadratic^{model LRPmodel} Exponentialmodel

f

^{-b X=X" i„(}

^_E*E_)

X=X^{= *}

2c c

Economic analysis

Making marginal cost equal to marginal reve- nuegivesusthe maximal economicreturn.This occurs,accordingtothe production theory, when the marginal physical product ^(MPP)is equalto the priceratio, w/p, where w is the unit price of fertilizer and p the unit selling product price. Due

tothe short growing season inFinland, annual variations in the corresponding yield responses arehigh. Knowledge of the physical production function is therefore valuable. Another factor adding^tothe interest of the physical production function interesting is the change in price rela- tions in Finland caused by accession to the Eu- ropean Unionin 1995. The ensuing switch from anagricultural policy favouring price _supportto onefavoring _more direct _support meant a drop of about60 percent in the price of cereals.

The maximal yield is the pointat which the slope of the production function equalszero.The fertilizer input for maximal economic output, whencosts are included, is lower than that for maximal yield. The maximal economic return calculated with the LRP function is approximat- edto equal zero, or breakpoint. The first-order derivatives of the exponential function show four solutions, depending on the signs selected for the parameters. We chose the solution that is mostlikelytooccur onthe basis of diminishing returns (Table4). The optimal economicoutput is calculated from themeanLSestimate, which givesalower result than the average of the sum of the optima calculated with the included ef- fects.

The prices used for calculating maximaleco- nomicoutput are w ⁼FIM 9.43/kg, oatsFIM

1.54/kg, barley FIM 1.63/kg and wheat FIM 2.19/kg in 1993and w ⁼FIM 6.41 /kg,^oatsFIM

0.70/kg barley FIM 0.73/kg and wheatFIM 0.87/

kg in 1995.

Results

The values estimated for the quadratic function were checked with F tests. All the effects were significant _except for the variations in there- sponse of the ^oats variety toN. The second de- greevariety variations were not significant ei- ther. Estimation of the quadratic function suc- ceeded without major drawbacks but resulted in manyoptima thatwereexplicit(outside thesam- ple range).

The parameters estimated(Table 5) are the values thatcan be fitted tothe modelspresent-

ed. Note that the intercepts(parameter a, LRP and quadratic ^model) for oats and barley are higher than those for wheat. Wheat also responds lesstoN fertilizer than doesoats orbarley (Fig.

3). The restricting penalty algorithm used for barley did not allow_us toinclude _as many ef- fects _as did the algorithm used for wheat and oats. This is probably why the breakpoint for barley is higher than that foroatsand wheat. The

parameters a for the exponential model show (Table 5)the asymptotic yield level. The stand- ard_errorin the_parameterof the asymptotic lev- el for wheat is remarkably high, possibly imply- ing that the annual and variety variations donot explain the entire yield variation for wheat at high fertilizer levels.

A comparison of the distributionpatternand the LRP model estimated for oats shows that _a

Bäckman,S. T. ^etal. Fertilizer trials: mathematical response models

reduction in the yield response for oats at high fertilizer levelscannot be reflected by the LRP model. Oats shows_a clear decrease in yieldre- sponseathigh fertilizerrates,but the LRP mod- el does not have a decreasing phase at which MPPcO. The residuals _{now cross} the _zero line twice and the optimum should therefore be found between the crossings. The yearly variation in the breakpoint ^atthe x-axis_was not significant for the LRP model, but the resulting yield pla- teau or N response had significant variations.

Thereweretoofew replications for each variety

in the dataon the barley experiments. The LRP and the exponential results for barleywerethere- fore calculated without considering variations dueto variety.

Estimation of the exponential function suc- ceeded well. The

R 2 values

were higher than those for the quadratic function. The exponen- tial function does^nothaveaphase foradecreas- ing yield response and sois it less suitable for yield response analysis than the quadratic func- tion. Only a few of the optima obtained were implicit. Moreover, the sample ^range was too Table 5.Estimated parameters and standarderrorsforquadratic, LRPand exponential models.

a SK b SE c SE

Quadratic

Oats 1414 ^(64.5) 51.5 (1.6) -0.204 (0.01)

Barley 1010 (73.3) 52.9 (1.3) -0.173 (0.01)

Wheat 1274 ^(72.0) 35.8 (1.2) -0.094 (0.06)

LRP

Oats 1379 (114) 43.17 (3.1) 0.27 (0.08)

Barley 1051 (42.5) 43.13 (1.2) 9,16 (0.09)

Wheat 1336 (81.3) 28.56 (1.5) 0.27 (0.07)

Exponential

Oats 5339 (341) -0.71 (0.02) -1.94 (0.17)

Barley 6044 ⁽³³²⁾ -0.81 (0.01) -2.13 ^(0.66)

Wheat 7493 (1953) -0.79 (0.02) -1.23 (0.15)

Fig. 3.Oats, barleyand wheatyieldresponses toapplied nitrogen(21 years) with theLRPmodel.

Vol.6(1997): 151-160.

Table6.Statisticalresults of estimation of the three models.

R 2 Hst. SSres SStot Number of Calculationof number

error parameters* of parameters

Quadraticfunction

Oats 85.2637.5 3.910E+08 1.637E+10 82 =3+3*21+16

87.8664.0 6.088E+08 2.501E+10 124 =3+3*21+2*29

88.3526.6 2.922E+08 1.393E+10 106 =3+3*21+2*20

Barley Wheat LRPfunction

Oats 82.7688.8 4.516E+08 1.637E+10 88 =3+2*2l-2+3*16-3

86.2705.4 7.229E+08 2.501E+10 63 =3+3*2l-3

87.1552.3 _2.914E+08 1.393E+10 100 =3+2*2l-2+3*20-3 Barley

Wheat

Exponentialfunction

Oats 83.3675.6 4.322E+08 1.637E+10 93 =3+3*2l-3+2*16-2

86.0711.0 7.264E+08 2.501E+10 63 =3+3*2l-3

90.2479.8 2.382E+08 1.393E+10 120 =3+3*2l-3+3*20-3 Barley

Wheat

*The numbers ofparameterswith included effects. Effect includedonthe basis of F-tests. There are16varietes for oats,29 forbarley and20 for wheat. The number of years are21. Allyear variation effects included except those for theLRP functiononoatsand wheat.Allthe year effectsinthe exp. model couldonlybe included for wheat.In theLRPmodel all wheat and oatsyeareffects could be included.

small when the exponential function was used

onthe data for wheat.

The number of_parameters varies according

tothe number of effects included(Table 6).When onlyR ^* wascompared, the quadratic function performed better than the exponential _or LRP function on oats and barley. The exponential function performed beston wheat,but therewas

aproblem with the sample range, which mayex- plain the higherror^termfor the coefficientafor wheat. The LRP function performed only slightly worse than the others. R_adj^{2 , ,} however, is not a very good criterion for selecting a model, as shown by, among others, Kvålseth (1985), McGuirk and Driscoll (1995). It would be more appropriateto testthe models against each oth- er.This would be aninteresting topic for afol- low-up study. We didnotdosohere becauseour main interestwasin finding ^out which effects^to include, and also because we lacked reliable methods for testing the models against each other.

It is strikingthat, accordingtothe quadratic and exponential ^models, a decrease in optimal

fertilizeruse was aresult oflower prices in 1995 (Table 7).According tothe LRP ^model, howev- er,changes in price do_notaffect economic opti- mal fertilizer use. This is because the MPP is zeroafter the breakpoint, which tellsusthat any additional fertilizer input gives no further in- crease in yield. Optimal fertilizer_useaccording to the LRP modelis, however, still lower than according _tothe quadraticor exponential mod- el. Newer varieties show anincrease in optima.

Discussion

Recommendations based_{on an}“incorrect" mod- el have an impact on both profitability and on the environment. Preference has been given to functions that enable_ustorelate theparameters achieved _to biological and physical processes.

Biological theories and economic theories _are not, however,always comparable. In agronomy, there isnosatisfactory relation between growth

Bäckman,S. T. etal. Fertilizer trials: mathematical response models

Table 7.Economic fertilizer ^useandcorresponding yield with threeprice relations and three different regression models, kg/ha.

Crop Quadratic LRP Exponential

Nitrogen Yield Nitrogen Yield Nitrogen Yield 1993Prices

Oats 111 4619 77 4720 128 5024

Barley 136 5005 116** 6054 136 5772

Wheat 168 4633 113 4575 230* 7143

1995Prices

Oats 104 4563 77 4720 108 4870

Barley 127 4942 116** 6054 116 5630

Wheat 151 4540 113 4575 187 6897

*Valuebeyond sample range and therefore not reliable.

**Calculatedwithadifferentrestricting loopfrom oats and wheat.

inputs and yield formation. Attempts have been madeto construct deterministicmodels, but as van Keulen and Stol (1991) remarked: "These modelscannot be used in practice. They are an aidto structure thinking about the system." Sto- chastic modelsare moreoftenused,butthey may

lack any idea with which the results ^can be ex- plained.

The optimization of fertilizerusedependsonthe aim ofaparticular optimum. Possible optima are:

a) biological (includesquality and quantity) b) economic (includes micro- and macroeco-

nomic optima) c) environmental.

The biological optimum dependsonthe quan- titymeasuresand will give the biologically max- imal yield. The biologically optimal input of fertilizers may be different if quality is taken into

account.Economic optimaarecalculatedtogive the maximal economicreturn for the individual farmeratthe microlevel. The optimal macroin- putof fertilizer is the optimal input foracertain number of farmers with different production options(functions).The results of these experi- mental data^cannotbe used assuchat themac- rolevel. Herewehave concentrated onderiving the economic and biological quantity optima for this experiment, and considered only the^varia- ble fertilizer^costs.

The quadratic function has been criticized for giving excessively high estimates (Ackello-Ogu- tu etal. 1985, Paris 1992).We, however, found that it was easy toboth calculate and use for optima. The quadratic function has a diminish-

ing phase that was accurateif comparisons are made with the distributionpatternof the dataon oats. The high estimates are partly due to the inflexibility of the quadratic function but partly to the size of the domain.

The lack of the diminishing phase in theex- ponential function is adisadvantage. A dimin- ishing phase would, however,require inclusion ofafourth parameter in the model. The advan-

tage of afour-parameter exponential function over the quadratic function would be that the former could includea steeper increasing phase and aflatter diminishing phase. The disadvan- tage ofafourthparameter is the decrease in the degree of freedom. The quadratic function is symmetric in the increasing and diminishing phases. The distributionpattern appears^not to be symmetric. The diminishing phase is inter- esting only ifit affects the earlier (increasing and rational) phases; otherwise it hasnosignificance for the optima. A non-linearfunction with three parametersthat could include _{a steeper} increas- ing phase and aflatter diminishing phase would probably perform well. These non-linear func- tions _are available but they do not necessarily have any relevance_tobiological theories. These

Vol. 6(1997): 151-160.

functions might beaninteresting topic forafol- low-up study.

The LRP model gives optimal input levels thatareclosertothe recommendations used to-

day than the optimal input levels given by the exponential _or quadratic models. According ^to the exponentialfunction,the optimaluseof fer- tilizer is highly dependentonthe unit price rela- tion between input and output. This is not so according to the LRP function. The LRP func- tionis, however,quite cumbersometocalculate and therefore probably notthat useful.

If the error term were brought further into the calculation of the optimal solutions in the form ofaneconomicrisk,the optimizations ob- tained would be more accurate.This was done for the quadratic function by Carmer et al.

(1991). It is, however, a very time-consuming

task and the results should be considered in the perspective of variability in actual growingcon- ditions. Optimising under varying conditionscan giveushigher optima because of the price rela- tion and attitudes torisk. The average curve is still,however,the expectationcurve.

In conclusion, we stress that none of these three models, quadratic, LRP or exponential, performed satisfactorily in all respects. The quadratic model performed best foroatsbecause of the diminishing phase. The LRP performed best in view of therecommendations, and the exponential function performed best in respect of thesteep increasing phase. As atopic for fur- ther research new models should be tried^outor new methods introduced that include other fac- torsaffecting economic output.

References

Ackello-Ogutu, C,Paris,Q.^&Williams,W.A. 1985.Test- inga vonLiebig crop response function against pol- ynomial specifications.American Journal of Agricul- tural Economics 67: 873-880.

Bock, B.R. ^& Sikora, ^F.J. 1990. Modified-quadratic/pla- teaumodelfor describing plant^responsestofertiliz- er.Soil Science SocietyAmerican Journal54:1784- 1789.

Carmer, S.G.,Walker, W.M. ^& Swanson, E.R. 1991. A risk assessement-risk management approach to selecting^fertilizerapplication rates. Journal of Pro- duction Agriculture4:67-73.

Cerrato, M.E. ^& Blackmer, A.M. 1990.^Comparisonof models fordescribingcorn yieldresponseto nitro- gen fertilizer. Agronomy^Journal82: 138-143.

Dennis,J.E. jr.^&Schnabel, R.B. 1983.Numerical^meth- ods for unconstrained optimization and nonlinear equations.Prentice-Hall,lnc., New Jersey.378p.

Frank, M.D., Beattie,B.R.^&Embleton,M.E. 1990. A com- parisonof alternative crop response models. Amer- ican Journal of AgriculturalEconomics 72: 597-603.

Keulen, H.^van& Stol,W. 1991.Quantitative aspectsof nitrogennutrition in crops. Fertilizer Research 27:

151-160,

Kvålseth,T.0.1985.Cautionary noteaboutR 2.^{The Amer-}

ican Statistican39: 279-285.

McGuirk, A.M.^& Driscoll, P. 1995.The hot airin R²and consistent measures of explained^variation.Ameri- canJournal of AgriculturalEconomics 77: 319-328.

Miettinen, A. 1993. The effectiveness and feasibility^of economic incentives of input control inthe mitiga- tion of agricultural water pollution. Agricultural Sci- ence inFinland2: 453-463.

Paris, Q. 1992. The von Liebig hypothesis.American Journal of Agricultural Economics 74: 1019-1028.

Sumelius, J. 1993. Aresponse analysis of wheat and barley to nitrogen in Finland.Agricultural Sciencein Finland2: 465-479.

Yli-Halla,M. 1991.Phosphorus supplying capacityof soils previouslyfertilized with different rates of P.Journal of Agricultural ScienceinFinland63: 75-83.

Bäckman, S. T.etal. Fertilizer trials: mathematicalresponsemodels

SELOSTUS

Lannoituksen pitkäaikaiset kenttäkokeet: kolmen matemaattisen mallin vertailu

Stefan T.Bäckman,S. Vermeulen ja V.-M. Taavitsainen Helsingin yliopisto jaKemiraAgro Oy

Vehnän, ohran jakauran hinta laski Suomessa noin 60 prosenttiavuodesta 1994vuoteen1995 Euroopan unioniin liittymisen myötä. Myös lannoitteidenhin- talaski, muttahuomattavastivähemmän. Tämä suh- teellistenhintojenmuutosmerkitsisitä, ettätaloudel- lisestioptimaalinenlannoitteidenkäyttömäärä piene- ni.Lannoitteiden taloudellisestioptimaalisen käytön määrittäminen edellyttää fyysisten tuotantofunktioi-

den tuntemista. Tutkimuksessa estimoitiin kolme fyy-

sistä tuotantofunktiota; toisen asteen polynomi-, LRP-ja eksponenttifunktio.Tutkimusaineistona käy- tettiin 21 vuoden kenttäkokeita, joissaoli tutkittu mineraalilannoitteiden vaikutusta viljanviljelyyn.

Kokeissa viljeltyjä kasvejaolivatohra,kaura javeh- nä. Lajike-erot otettiin malleissa huomioon. Mallit

eivätpoikenneet suuresti toisistaan selitysasteenpe-

rusteella⁽0,82-0,90), muttamalleista lasketut talou-

dellisetoptimit poikkesivat toisistaan.

Appendix 1

The DUNLSF^algorithm:

1 1 ^m

min-F(x)^T

F(x)--gf,(x)

²

where min,F:R" ^-»R^m, and is the ith component function of F(x) ^• From a currentpoint, the algorithm usesthe thrustregion approach:

™?||f(Xc)+

^j(Xc)(x„

-Xc)||

₂

subject^to

||x„ -Xc||

2 sö_c

to geta newpoint x„ ^{which is}computed^as

X„ ^-Xc^~(J(Xc)Tj(Xc)⁺ J(Xc)Tp (Xc)

where |».-0 if

6. »|[J(x.)

^{TJ(x.))_II(X.)T}

F(xj|

_l ^and n^{c >O}

F(Xc)^and J(Xc) ^{are tne}function values and the Jacobian evaluated at the currentpoint

x

^c>

respectively. This procedure is repeated until the stopping criteria _are satisfied (IMSL MATH/LIBRARY).