Models Based on Positive Mathematical Programming: State of the ...

Models Based on Positive Mathematical State of the Art and Futher

Programming:

Extensions

Thomas Heckeiei, Wolfgang Britz

Unn'ersiw ,-rfBonn, Insnrute for Agicultural Policy, Xlarket Researchand Hconomic Sociologl.,Bonn, Germanv

Absttact Nlathematicalprograrnming models have received renewed interest in the area of agricultural and asri-environmentalpolict' an:rl)'sis.Theit abiliry to explicidv representphysical constraints mzrkesthem specificailysuited for connecting economic and bio-physical aspectsof agricultural systems.Furthermore, they allou'for a direct representationof manv current agriculturalpolicv measuresrelated to production activiw levels.The introduction oi Positive X{athematicalProeramming (PX'IP) addressedproblems rvith plausibiLityof simulated behaviour and lack of enipirical validation of these models. This paper reviews the development of PMP in its various forms and takes a look at approachesbeyond PN{P contributing to the quesr for empiricalli' specifiedprogramming models.

1. Inttoduction During the 1960's.lrd 70's, aggregateLinearprogramming models were en vogue in agriculturai economics. Large scale systems,dis-aggregatedby region and farm ry'pes,covering regional ttansPort floivs or including recursive-dynamicfearureswere der.elopedboth in the US and Europe. However, iew of the svstems urre kept alive, and, gradually, nerv model wpes like multi-commodirv- models, computable general equilibrium models and dualiw based farm/farm-household models dominated the scene.Part of the developmenr$/asprobably due 'business to cycles' in the researchcommunitl'. But equallv likely, as the large-scalemodels based on mathemaricalprogramming were marketed as policl'decision supporr systems,their clrarvbacksbecame obr.ious. Besides tremendous data and computing needs, thelr struggled with poor tracking records of obsen-edbehaviour and jumpy responsesin simulation exercises. The introduction oi Positive NlathematicalProgramming (PN{P,Howitt 1995a)made the supplv response oi agriculrural programming models more realistic b1' allowing perfect calibration tt.r observed base vear behariour and bv avoiding providing a more smooth simulation re-

48

1. N{odelling Ägricultural Policies

sponse compared to linear programming models. Since its introduction, PNIP and related methods have been applied to a rapidly grorving number oi models at the farm, regional, and sectorallevell. Coincidently, the increasing use of PMP-type approachesduring the last decade was accompenied by at least three other developments supportiflg the renerved interest in programming models: Firsdy, the EU's switch from price support to other policy instruments tied to farms and production activitiesmade econometric approachesbased on behavioural funcuons more complicated or not sufficiendy direct in representingpolicy instruments.Since the mid of eighties the CÄP introduced - among others - dairy quotas, set-asideobligauons in conjunction rvith crop specific pavments per ha, voluntary set-aside,stocking densiry resrrictions,ierm specific ceilings and the nitrate directive. Most of these policv instruments are much easierto model in the context of a programming model. Secondly,the increasinS5 interest in joint production ofagriculrural outputs and environmentai goods (and'bads) is elegandyhandled bv an activitv based approach which allows for a straightforward link between economic and biophysical models. At the same time, activiry based approachesease communication and exchange in multi-discipiinary research projects. Thirdl.v, restrictions such as the land balance or requirement constraints ior animal feed enhancecredibiliw of simularion exercisesby preventing implausible results. Ä fearure getting more relevant as quantitative economic models are increasingly adopted as a tool rn the agriculrural poücv decision process br. many European public administrations. These der-elopments are [0]

rvhere: x', = (NiX 1) vector of observedactivity levels r Schlble (199f and (2000) uses a ven,similar approach, but does not reflr at all to rhe PMP literarure.

52

l. Nlodelling Agriculrural Policies

t = (Nx1) vector of a small positive numbers .r'ariablesassociatedwith the calibration consüainrs P = dual The addition of the calibration constraints will force the optimal solurion of the linear programming model (2) to exacdy reproduce the observed base year acrivity levels x", given that the specified resource constraints ailow for this solurion (which they should if thi data are consistent, see Hazell and Norton 1986: 26611.'Exacdy' is accurately understood to mean within the range of the positive perturbations of the calibration constraints, e, which are included to guaranteethat all binding resourceconsüaints of model (1) remain binding here and thus avoid a degenerate dual solution. We can partition the vector x into rwo subsets,an ((lrJ-N.!x1)vector of .preferable'activi_ ties, xr, which are bounded by the calibration constrainrs,and a O{x1) t'ector oi'marginal' activities,x-, which are constrained solely by the resource constraints.To simplify notation, without loss of generality, ,l'e assume that all elements in x', are nonzero and ail resource con* straints are binding. Then, the Kuhn-Tucker conditions implv that

p, = po _cp _ AP'.1

(3)

p' = [0]

(1)

1 = ( A^ ' , \ ' ( p ' - r ^ )

/5)

where the superscriptsp and m indicate subsetsof original vecrors and matricescorresponding to preferable and marginal acdr,-ities, respectivelv.Whereas the dual values of the calibration constraints are zeto for marginal actirities (p*) as shown in (1.a), rhev are equal to the difference of pnce and marginal cost for preierable activities (pr) as seen in (1.5), latter being the sum of variabie cost per acdvitt' urut (c) and the marginal cost of using fixed resources(Ar'1,). It should be noted here, that the dua.lvalues of the resource constraints (l) o"ly depencl on obiective function entries and coetEcientsof marginal activities. ln Pbase2 of the procedure, the dual values of the calibrauon constraints pn are emplor.ed to speci$r a non-linear objective function such that the margnal cost of the preferable actvities are equal to their respectir-eprices at the base vear activirJ-levels x". Given that the implied variabie cost function has the right curvature properties (convex in activitv levels) the solrrtio., to the re sulting programming problem vill be a "boundarv point, which is the combination of bindine constraintsand first order conditions" (Howin 1995a:330). Hou''itt (1'995a)and Paris and Howitt (1998) interptet the dual variable vector p associated with the calibration constraints as capruring any rype of model mis-specification,data errors, aggregation biasa,risk behavtour and price expectations. r To deal with aggregation errors in regional or sector modelllng, Önal md \IcCarl (1991) proritle the theoretical basis oi an exact aggregation ptocedure based on extreme point ,.pr.r.rr"t or rnder the assumption of full information on cvery tärm and suggest empincal approximation procedures using the available agqregate information on all farms.

53

I

li

\toclelüng Ägriculturai Policies: State oi the Art and New Challenges

In principle, anr q'pe of non-lineat function with the required properries qualifies for Phase2. For reasonsof computational simpliciw and lacking strong arguments for other q-pe of functions, a quadratic cost function is often emploved (exceptions:Paris and Howitt 1998 and 2000). The generalversion of this veriablecosr function to be specifiedis then I

(6)

C'=d'x+;x'Qx 1.

with: d = Q

(\ x 1) vector oi paralneters associated .r,-ith the linear term and OlxN) symmettic, positive (semi-) definite matrix of. parametets associatedrvith the cluadraticterm.

The parametersare then specified such that the linear 'marginalr'ariablecosC (MC\) functions iulfil

a c _ ! K '=) d + e x " =c+p MC' =

(.7)

r'iote, horvever, that the derivaLives(7) of this t'ariab/ecost tunction do not incorporate thc opportuniq' cost oi fired resources(Ar'1";u'hich remain captured in the ultimate moclel bv the clualvaluesof the resourceconsrrelnrs. Given that we have a set of p2rrameterssatisf ing (7), rve obtzrinthe frnal non-lrnc:rr proeramming problem that reproducesobsen'ed acrivitl'levels as

I

m ? x Z= p ' x - d ' x - - x ' Q x snbject to

A.r< 0 x>0

(8)

lf)

It should be noted at this point that the dual r-aluesof the resourceconstrainrsin model (8) at x') do not differ from the one in model (2). They are still determrnedbv the margnal protrtability of the marginal activities at their obsen-ed levels x,,-, (Am)-t 1On- (d'" + *,,-q"'11,u.hich remains equal to (Am)-1bm-cm]in the speciflcadonstep, becauseof (4) encl (7). Consequentlv, the r-'.rlue oieqr,ration (5) remainsunchanged.

54

f . i\lodelling r\gricultural Policies

7-be Paran efer SPeci|icdtiln Problem Calibration of an agriculturel farm, regional or sectoralprogramming model to obsen ed quantities is not the distinctive properqvof the PN{P-approach.This can be achievedby appropriate constraints- see model (2) - as well. N{ore interesdng is, w-hethera PN{P calibrated programming model is able to capture the behar,'ioura.l responseo[ thrmers to changing economic conclitions, so that it is capable oi evaluating impacts of political-, market-, or technjcal developments on agriculture.This responsedepends on the interplav between the constraints and the norv non-linear objective function. However, the flrst and second order conditions - (7) and the positive (semi) definitcnessof Q* sull allou' tbr almost any magnitude of responsebehaviour of the resultins model. The problem oi condition (7) is that it implies an underdetermined specificationproblem as long :rswe consider a tlexible functionai tbrm. In the caseof the second order tlexible quadradc function u'e have N+N(I\I+1)/2 paramerersrvl-richrve rry to speci$' on the basis of N pieces of information (the marginal variable cost equations (r)). There are an intlnite number of parameter sets rvhich srtisfi' these conditions, i.e. Iead to a perfectly caLibratrngmodel, but cach set impJiesa diiterent responsebehaviour to cl-rangingeconomic incentrves. In order to see this, r'"-ederive the supplv tunctrons inplied bv the PÄ.{Pcalibrateclmodel 11.8).If we start from the I-agrangiantbrmularion

L ( x ) = p ' r - r l ' . r- 0 . 5 , r ' Q x +A [ b - . a r ]

(e)

rncl continue

to issllme that all optim,il ectrr-i6. levels are positive rve obtain the t-lrst order conclitions in grrdjent frrrmrt as

;x Y l = p _ d_ o r - A ' )= 0 ^

(10)

and oL

e =b-A.r=0

(11)

Solvinq i l0) for x resultsin .r.=e'(tt-,i-A',t)

1l

and substrtuongthe right hand side of (12) into (11) allows to solve for ,.-t,

) = e O ' + ' )( s e - ' ( p - a ) - u )

,1r)

The vector of optimal activity levels as a function of exoge nous model parameter c:rn then be expressed as

* =e - '( p - d )- e ''A '(l q -'a ')'(n e -'(p -a )-r) 55

(11)

Nftdelling AgricuJtural Policies: State of the Art and Nel. Challenge -.

The gradient of (14) with respect to the price vector is proporrional ro the mar€linal supply response in this case (since product supplv is consranr per activitv unit) and given by

ax= Q - ' - Q - ' A ' ( A Q - ' A ' ) - ' AQ_' ap

(1s)

u'htch finally reveals that the full Q-matrix is relevant for the supply response of each single product. Thjs is even true when Q is di4gonal (and consequendv e-r as well), becausethe fixid allocable inputs (resource constraints) still link all production activiries with each other. The second summand in (14) rvhich is -Q't4' times the gradient of 2vwith respecr ro p ensuresthar all elements of Q 1 enter each element of the supply gradient. The different methods developed to choose among the infinite number of calibrating parameter sets increasingiy recognised the need to introduce addrtional information in order tt> avoid arbitran' simulation behaviour. We give here a short oyefview on the pfinciples employed without an extensirc discussion(seealso Umstätter 1999:30ff . or for a detailed er-aluation Röhm 2001:8ff . with respectro som€ of the approachesmentioned bero*).

E ar! Specif cati on Ra /es In the 'earl1" days of PMP the specification problem with respect to the quadraric cosr function was simplv solved bv letting d = c and setting all off-diagonal elements of Q to 0 (e.g. Hou,itt and Mean 1983, tsauer and Kasnakoglu 1990, Schmitz 1994; Arfirl and paris 1995). The N diagonal eiements crf Q, qti, v/er€ then calculated as

q,,=+ for alli= 1,...,N xi

(16)

It is easilyverified that the resulting vanable cost functi()n satisfiescondition (7). This specification rule leads to a cost function which is linear in 'marginal'acuvitl' levels,becausethe elements of P- = 0. This in turn implies that l, remains coristänr,becauseit is determined bv the profitabilifl- of the marginai acrivitiesalone which is constanr per acdvirv unit. Consequ"r,iy, price increase for products of the preferzble production acdvities leads to a substitution of" marginal activities,but leavesthe other preferable activitv levelsunchanqed until the frrst marginal activiry is driven out of the basis. This specificationis purel1'motivated bi' computational simplicin in the absenceof aclditional information. Its repeated use can onlr- be explained bl a focus on the calibration propert)' in hope that a rich technological specification in terms of constraints u'ould provide- a realis.tic simulation response. In hindsight, it is easl to arguc that technological cänstraints which are not even closell' capable of reproducing base vear obsen-ationsare not rn an1'u'a1. more likell' to capture behaviouralresponseto clianging economic .incentives.Ex,post simulations performed bv Ci'pris (2000) t'ith the (ierman regionalisedsector model RAÜMIS shou. that this approachresultsin a lcn po()r resp()nsebchärjrur crf thc rrsulüng modcl ch:Lracrcr. ised by strong overreactionsto changesin economic incenur.,es(i.e.high implieci elasticities).

56

l. lIoclelling .{.criculturaiPolicles

Paris (1988) used an alternadvespecificationrulc u'here the linear cost function ijaralr-letets d are set to zero in addititain

q ,= l 4 o c , , = 1 4 v i= i , . . , N Qii xi

€,, x,

Qt)

as the appropriate value for a given fu. In orcler to satisfythe calibrarioncondrtion (7) the linear parametersof the variablecost funcrion are then determined as d, =ci+pi-qiix;

Vi=l,...,N

(22)

Becauseof the ignored effect on shadow prices of limitecl resources,the actual elasricitiesof the resulting model will deviate from €,,, and are generallr lov,er. The exact calibranon to exogenousou'n-price elasticitiesis generallypossiblebut cannot alv'avsbe obtained as a closed form solution5.

5 See Hcckelei 2002 for an extended discussion on this sublect

58

1. Nfodelirng Agticultural Policies

C a/i bra ti on ai t b A,La.rim unt E ntropy Cil ei ort Paris and Hovitt (1998) proposed to overcome the problem of somewhat arbitran parameter specificationsthrougl-rthe use of an econometric criterion, specificallvby an applicariooof the GeneralizedMaxin.rum Entropy (GN{FI)eslnator (Golan et al. 19965.This path breaking contribution allov'ed rc elegandl,addless the underdetermined specrficationproblem using pnor information and provided a full specificauonof the Q-marix. Nlore importandl', it defined a methodological frames'ork u'hich principallr allows for the incorporation of more than one calibration and estimation exercises. obsen'ation on activi6,'ler-elsand bridged the gap betrr,'een However, their example still used one observation onJ.y,demonstrat-ing an alternative ll'al to introduce a priori information in the calibration process.The paper characterisedthe process as an estimator using an uninfcrrmauvc prior. That characterisatjonma1' have been misinterpreted bt' the reader as the a-priorr intcrrmrdon together with the entropv criterion allowed to identifi' the estimatedparamerersdespite the underdeterminednarure of their problern. The generalidea u'as apphed to an e\ample u.ith multiple obsen'adonsin a cross-secdonal anall'sisby Heckelei & Rlitz 2lr()()s'hch borrou'ed the idea of elasocin'priors for tlie Q-matrix On the one hand, pnors u'ere defined direcdy on Q, and proposed some furrher n'rodrfrcations. and not as in Paris and Hou'itt (1998) on the elementsof a LDL' decomposition of Q. On the other hand, a size deper-rdenr scal.ingwas proposed to define region specific Q-matrices in the cstrmruon :tcp. Thc resulting regional models proved to be superior in an out-of-samPlevaüdatron erercise con-rparedwith models specifiedbl the averagecost approach.Both the examp1e applicat.ionin Paris and Howitt (1998) and the cross-sectionalestimation in Heckelei and Brttz 2L100recovered cost fuflctions sarisivrng condiuon (7) u'hich n-rakesthese approaches subject to the generalcriricism put fonvard in section 3. Paris (2001) and Paris and Horvitt (21)it1)erpand the iramework oi the GNItr-based PMP methodologl' to overcome some of tl-recriticisms that have been rarsedallalnst the use of a linear technologr'' in limring re sources and the zero-marginal product for one oi the calibrating constraints presenr in thc original PIIP version. The f-irst step of this neu' strucrure is now expressedas an equilibrium problen-rconsisting of svmmetric primal and dua.lconstraintsancl the third step as an equilibriLrmproblem betu'een demand and supplv funcdons of inputs, on the one hand, and marginal cosr änd marginal revenue of the output activines,on the other hand. This nert'irameu,ork has inspired the authors to name it a Svmmetr.icPositive Equilib rium Problern (SPEP). The kev contribution of SPEP to the PN{P literature is rendering the availabiLt'n' of limiting inputs responsiveto output levels and input price changes. Äs in standardPMP, the first step of SPE,Pdeterminesthe levels of the marginal costs of ourputs and the shadou' prices of l-imiting inputs. Instead of the traditional dual pair of LP models, Paris and Howitt (2001) specify an equilibriunr problem rvith the follorving structure and their associatedduals using in addition the ar-aüableinformation on market rental prices to avoid degeneracl. in the duals of the limiung input constraints. The,v propose the fcrllor.ving dual constraintsas a starting point:

A'A+p+>pLx>0

(23)

2>r

()4\

I

Ax-b20 59

Modelling Agricultural Policies: State of the Ar and Nev. Challenges

where r is a yector of observedrental Drices. The additional constraints introduce iower limits on the dual r.'alues of bindins consrraints for which external renting prices r are available. Indeed, the seccrndconsrrainrs 124; states that farmers ate able to sell resources at prices t, but once the endowment b is completely used for production, the marginal values l, may exceed r. After determining the dua.lvalues for both calibration and resource constraints, Paris and Hou'in (2001) nov- propose to use a Generalized l-rontief specification for the limiting inputs and a quadratic specificarion for the ouput vector x in step 2 and 3 as follows:

C(x,lv)= 1'1,(d'x)+ i'1, (*'Q *) /Z + J7 ' S .,4

Q5)

which then sen'es simulatiofl purposes after some modifrcadons. Paris and Hou-itt finaliy claim that because the final SPEP specifrcation neither imphes nor excludes optimizing behavior, it removes the last remnant of a normadve behavior from tl-re origina.l PMP approach. In a critique of the SPEP approach.Britz et al. (2003)point out r-harthe emplovedcäst f,lncdon and the ultimate simulation model obtained in phase 3 of the approach .rnnoi b. derived from any optimizing behaviour and_question their general interpretabilirv. As the argumenrs brought forward bv Britz et al. had been published as a direct commenr to Paris (20ö1;, but .rot b."n commented upon bv the author in return, we refratn here from a conclusive evaluation of the SPEP framework. The most rec€nt farm modei applicarion of the SPEP framework is Arfini et al. (2005\. Sp ecif cati on B ased on D ere asing M a rgina/ Y i eld s All the PMP specificarions mendoned above specilied a non-linear cottfuncflcn. By assump tion, tJrey attribute the marginal mis-specification of the original iine ar model to rhe inpr-rr s,ä. of tl'reproduction problem. It is probably obvious, hov'ever, that a misrepresentalionof horv revenue depends on activitl' ler.els would have the same effect. In fact, the dual values on the ca.libration consLrainrscan just as well be explained bt' decreasing marginal ),ields uith increas, ing activiq' levels, which is not reflected b,v the constant 1,ieldassumption of model (2). Horvitt (19_95a)trsrs this interpretation and introduces non-linear terms into the objecrive funcrion t1r reflect differences between margrnai and observed aver€e crop yields caused bv changing iand qualiq'. The theoretical as well as empirical validi6' of the pure cost funcrion and the pure i,ield function approach is somewhat guestionable. Any reasonable technologica.l and behavioural assumption in agriculrural production would make it highll. unlike\. ttrai input applicarion is changed but;'ieid remains constant or vice versa. Röhm (2001, 51ff.) acknou'leiges this b1, combining the decreasing f.ield with the increasing cosr assumplion. Hou,er.er, his Jpproach is also not based on a ciear technologicalhypothesis,i.e. a u,eli ripresented relarionshipiberween inputs and outputs, and again does not provide a strong empiricai base fcrr the specification of parameters and the implied simulation behaviour of the resulting model. Therefore. rve dci not further elaborat c'n the details of the yield function ^ppro^.I t,r combined r.i.id/.ost upDroaches.

60

i. N'Iodelling Agncultural Policies

Suntruai{ng

evaludtiln af PLIP related caliltratizn apPrnaches

Before turning to approachesthat do not require a phase 1 with calibration constraints, rve \\'ant to close this section u'ith a short summar\-of the merits and problems associatedu'rth the main bodv of PN{P applicationsin the contert of a5liculruralsector models. 1.

2.

3.

4.

The PN{P approacl.rprovides an elegantu'a1,t6."1.1bt"te programming models to ob suPPlf responserelasen'ed behar-iourand renders a more realistic smooth agpiregate tive ro a linear ptogramming model. These merits lead to a widespreadapplication of PMP approaches in the context of aggregateat'ricultural programming models. The dual r.aluesassociatedrvith the calibration constraintsin phase 1 of PN{P potcnrialll caprurc an):r)?e of marginal model mis-specificationoi technolop', data errors, agp;regarion bias, representationoi risk behaviour, price expectations,non-linear cot.r straints etc. For an intelligent model specificationand appropriate interpretatiotl oi resu.lts,expLicitassumptionsused in the model specificationare desirable. C)ne obsen-ationon base year allocationsalone does not contain any informarion on horv the marginal incentives change if one mor''es arvay from the obsened allocation. The infinite number of calibrating sets of parametersgenerallr'implies a difftrent simulation response of the calibrated model. Extremelr' unreasonable suPpl.Yresponsesha','ebeen laefleratedin the past rvith oversimplifred P\tP specifications. The PA{P literature pror.ided a muldrude of approachesto obtain sensibleand interpretable approacl'resro the parameter specification problen-r.Those ccimprised the emplol'ment oi the relationshipbenveen averageand margtnalcosts, exPlicit assump dons on decreasingrield in land allocation, and prior inforn-rationin terms oi eroge nous elasuciries.\{'ith the introduction of the Maxmum Entropl cnterion, a tool has been suggestedto fleiblv incorporate general prior tnfcrrmatron.It also allo*'s in principle to use more than one obsen'ation u.hich - in combrnadon u-ith structural th,e assumptionson the objective function and constraints- s'ouid en:1bleto estitttale oi producers. response behaviour model parametersunderlt'ing the obsen'ed

Follorving up on rhe last point, the subsequent section u'ill fcrcus no\\: ()n the general problenr of the PlrfP-approachrelated to phase 1 which renders its use for Parameterestimation based on multiple obsen'ations problernatic. After this demonstration rve have a look at approaches for caübrating and estimating programming models u.ithout a phase 1, j.e. r',ithout the use of calbration constraints.Since v.e consider the phase 1 as a defining character of PX{P, wc do not talk about "PN{P-merhods" an\1nore, but räther of alternative approachesto the calibration and estimation of programming models.

and estimation 3. Calibntion cali bration consttaints

of optimization

models

withoat

dual

values

of

The phase 1 of the PMP approach can be looked upon as a mean of providing a ceftain shadorvprice value for limiting resourcesin the absenceof other information. It v'as pointed

61

Modelling Ägricultural policies; State of the Art and Neu, Challenge s

out above in the context of equation (5) that the inclusion of calibration constraints makes shadou'prices of resourceconstraintsdependent on the profitabiliq'of margrnalactivitiesonly. And those shadorv prices are transferred to the calibrating solution of the quaclrao. prugr^Äming model (8). In realiry- and in rhe context of model (8) as indicated by equadon (t:; - ,1",. shadow prices u'ill rather depend on more complex interacdonsdescribedby the level technology of production activities.ln fact, the determination of shadou-prices bv phase 1 of p1{p cre^te ^ fundamental inconsistencv between pz;r^meterspecification and the ..iulring quadratic optimisation model rvhich renders an\. estimation of parameters using multiple ob...r-utions inconsistent, as pr-rinted out in Heckelei Q002) and Heckelei and \\Iolff (2003). U'e uill norv explain this inconsrstencybased on thes€ two referencesin morc detail and then look at alternatrve suggested approaches of estimating and calibrating programming models avoiding phase 1 of PMP.

Inconsistenry o;fPAIP aben asittg nahiple ob.reruations The last section pointed out the danger of specifiing models based on PMP that imply arbitrary simulation behaviour.One problem is the thin informadon baseprovidecl bv just än. yea, of observations on activiry levels. In fact, the data in this case do not provide anv information on secondorder properdes(Hessianmatrix; of the objecdi'etuncdon.If a changein economic incentir-esand the resulting behar.iour is not observed, then the informauon for prrameter specification must come from other sources.Even if one u.ould be able to specifi. the ,true' model with respect to behavioural assumptionsand functronal form, the parameters are sull not identified. The only conr.'incinguse of pMp *.'ith just one obsen-ationls the u,seas a cajibration method in combination with eiasticitiesor othet exogenousinfonnation on technol,g., or behavioural resporisewith respect to changes in activiq, levels. The main focus of this section, however, shall be the inclusion of additional data looking for the bridge to q'pical economettic modeis. Pans and Hou-itt (1998) alreadypointecl at thä potential of inffoducing more than one obsen'ation.Howe','er,the question u,e need to address first is, whether the PMP procedure itself is designed to make b.st ore of additional data infotmation. We shou' that the margrnal conditions derived from the first phase of pN,Ip are inappropriate. They represent a mis-specified model in the sensethat the:rnclusion of acldi, tional obsen'ations q'ill never allou, to recover the underlf ing model v,hich is assumed to have generated the data. In order to see this, we will use some of the elements alreadyintroduced in the pre1ious sections,but look at the methodologl from an econometrician'spoint of vieu'. This includes the assumption that the ultimate model to be specifiedis the 'true'' model structure, or ar least one that is believed to be a good approximalion of the true model: Apparentlv, mau, pl\.fp modellers thought that the final model with a non-iinear objective f.rn.tton to-be oprimised under linear resourceconstraintsis a reasonabletepresentationof the behaviour of agricultural producers, otherwise it w'ould not have made an1.senseto use this stfucrure as the uitimate specificat-ron'The PNfP procedure, however, enforces shadow prices and marginal cosr values that differ from the ones implied b1,the nonJinear model. the quadratic model (8) is the tn:e data generating process. The derivations (9) to - luppose (13) have shou'n that the shadou,'pricesof the resourceconstraints under the ässumptlon that

62

1. Modelling Agticultural Policies

all

acti\-iq,

levels

^r.e

l=@O'A\'(Ae.b-rt)-b).

posirive

^t

the

optimum

can

be

calculated

as

This is ciearlydifferent from the dual r'aluesof the resource

on quanticonstrainrsobtained in the flrst phase of P1!{P(seeequatic,n(5)) u,hich c-rnl1'depend ties related to the marginal activitiesand u'ere given bv 2 = (A' I (p' - . ) The secondphase of PN{P then uses these dual values at the obsen-ed activiq' levels through enfolcement of the 'mafginal cost' equations (7), therebv implicid,v imposing \\'fong \-a.luesfot the mafgin'ältariGiven this discrepanq', it is impossrblc t() recover the true non-linear objecable iost as r,",e11. tive iunction lto matter hou' manr, obsen'ations on actiyin levels are used. The use of the exercrseu'ith tnulbiasedmxginal cost equationsas estimating equationsin sc,meect.rltometr.ic is fundanrenapproach The PN{P estitnatcs. inconsistent tiple observaticlns€ienerallvleads to tally fiarveclrn the sensethat ir imposes first order conditions s.hich are incorrrpariblewith the nonlinear model it ultimatelv tries to recover. constraintshave been recIn principle, the problems rvith the dual values of the resc-,urce (20tt0) and Röhm (2tXl1) Clptis (1999), Chantreuil and authors. Gohin ognised by several including obscn.ed land rents into the specification step to ensure a ln()rc rcasonlllle sugXsest yalue. This can either be done by introducing a land rent actir.iw at this price into the original linear model or bv adjusting the conditions for specificadon.\\'ild (2000) emplovs simulaoon exercisesto sho\\' the impossibilitlt to recover a true quadratic model u'ith more than one obse1l'adonbased on phase 1 of PI\{P. He also shou,san alternativecalibration approacl.rspecificall1'designedfor the quadratic specificationu,hich does tect.rvertl-retrue model bv simultaneouslr''calibratingparametersand shadorvprice of land. Gohin and Chantreuil and Wild alreadr,observ'ethat in the caseof a simPle model struc ture with just one resource constraint, phase 1 of Pl{P ls not needed as the impiied value of the shadou' plce 9f land can be deduced directlt'. Belor.vrverevieu'a generalalternativeintroduced bl,Heckelei 2002 and Heckelei and Wolff (2003) lr'hich does not reqr-Lircphase 1 of PN{P to calibrateor estimateanl programming n-rodeleven for ntore cotnplex c.)nstralntstructures. A direct use of the first order conditior-rsof the assumedbehavioural optirnisation model prices and theretrv the use oi the PN1P-approachaltogether makes the use of distorted shadc'ru' obsolete.

A Genera/,4ltematirc to PAIP 'gelreral' alternative to PN{P u.ith respect to calibrating ot' estimaung a The Programrullg model is nothing but a simple methodological principle: al'''a1's16 directlv use the first order conditions of the r.en' optimisation model that is assumedto represent or apptoümate pr,, ducer behar-iollrand is suitablefor the simulation needsof the analvsts.No first phasecalculating dual valuesof calibrarionconstraintsbased on a different model is necessar\'.\\'e can avc-ricl the irnplied methodological inconsistencl.altogether and generalll.estimate shadow prices of simultaneouslvu'ith the other parametersof the model. resourcecc,,nstraints The basic principle can be illustrated by rvriting a generalprosamming model u'ith an ob= ]ectir.efunctio" h(y | ) to be optimised subject to a constfaint lectof g(y | ) l) in Lagyang'ren fbrn-r:

63

Modelling Agricultural Policies; State of the Art and Neu' Challenges

(y, la,F)= hSlo)+ ?,'ic$lp)l

Q6)

where y, L, a, and p represent column vectors of endogenous variables, unknown dual values, parameters of the objective function, and parameters of the constraints, respectivell.. The appropriate first order optimalitl conditions äre the gradients u'ith respect to y and ], set to zero:

al _ an(JIa)*^,ae(_v IF)_ o

(27)

AL = e ( v l F ) = o AT

(28)

d)'

dy

dy

For the case of inequalitv constraint. g$lp) ( 0 u'e need to substitute the gradient uith respect to l, bv the complementary slacknessrepresentarion('

& o s0p)=or ;d = s(yl|)o lI ti k, Witterschlick/Bonn, M. Iflehle. Umstätter J' (.1999): Calibmting RegionalProdnt:tionModels nsing Pa.ritiueA.[athenatital Prograntring. :ln -1,9m-ennronntental Pokg,Ana/1:i: in Southwest Gennanl,Shaker Verlag, Äachen. \{'ild L. (2000): SdtäQang tnt Kostenfttnkionenin Ra/tnen der Pasitiu Mat/tematischenPropramnierung Diplom Thesis, Univetsiq'of Bonn.

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