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A Fully Calibrated Generalized Constant-Elasticity-of-Substitution Programming Model of Agricultural Supply

ABSTRACT This article provides a methodology to exactly calibrate land-constrained programming models of agricultural supply against supply elasticities using generalized constant-elasticity-of-substitution, crop-specific production functions. We
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  A F ULLY  C ALIBRATED  G ENERALIZED C ONSTANT -E LASTICITY - OF -S UBSTITUTION P ROGRAMMING  M ODEL OF  A GRICULTURAL  S UPPLY P IERRE  M ÉREL ,L EO  K. S IMON , AND  F UJIN Y I This article provides a methodology to exactly calibrate land-constrained programming models of agriculturalsupplyagainstsupplyelasticitiesusinggeneralizedconstant-elasticity-of-substitution,crop-specific production functions. We formally derive the necessary and sufficient conditions under whichthe model can replicate a reference allocation while displaying crop supply responses consistent withexogenous information on supply elasticities.When it exists,the solution to the exact calibration prob-lem is unique. Subject to a caveat,the proposed specification is shown to be more flexible with respectto calibration than the quadratic specification that has been used extensively in policy models basedon positive mathematical programming. Key words :calibration,positive mathematical programming,regional supply,supply response. Good policy models should respond to sim-ple shocks according to reasonable expecta-tions.The purpose of this article is to elucidatethe conditions under which a popular classof agricultural supply models can meet thisrequirement.Positive Mathematical Programming (PMP)models of agricultural supply have been usedextensively for agricultural policy analysis,both before and after their formalizationby Howitt (1995b). Like linear programmingmodels, they require minimal data; but, incontrast to such models, they permit theexactreplicationofobservedcroppingpatterns(without the use of flexibility constraints) andyield smooth responses to policy changes.Many countrywide or regional agriculturalsupply models are based, in part or in total,on PMP principles, for instance in the UnitedStates (the Regional Environment and Agri-culture Programming model,formerly the U.S. Pierre Mérel is an assistant professor, Department of Agriculturaland Resource Economics, University of California, Davis and amember of the Giannini Foundation of Agricultural Economics.Leo K. Simon is an adjunct professor,Department ofAgriculturaland Resource Economics,University of California,Berkeley and amember of the Giannini Foundation of Agricultural Economics.Fujin Yi is a PhD candidate, Department of Agricultural andResource Economics, University of California, Davis. The authorsthank Quirino Paris, Richard Howitt, David Hennessy, RichardPerrin,LilyanFulginitiandseminarparticipantsatIowaStateUni-versity, University of Nebraska-Lincoln and Toulouse School of Economics for helpful comments.The usual disclaimer applies. MathematicalProgrammingRegionalAgricul-tural Sector Model; along with California’sStatewide Agricultural Production model),Europe (the Common Agricultural PolicyRegionalisedImpact[CAPRI]model),Canada(the Canadian Regional Agricultural Model),Denmark (the Dutch Regionalized Agricul-tural Model), Sweden (the Swedish Agri-cultural Sector Model), Germany (the FarmModelling Information System model), andSpain (the Programación Matemática para elAnálisis de Políticas Agrarias model). Thesemodels typically involve the maximization of agricultural profits under simple crop-specificproduction technologies and resource andpolicy constraints. Though a majority of mod-els assume fixed proportions among inputs—perhaps a leftover trait of their farm-modelsrcins—more flexible models can be specifiedand are, in fact, more suitable to the represen-tationoffarmingtechnologyataregionalscale.PMP models that use constant-elasticity-of-substitution (CES) production functions werefirst introduced by Howitt (1995a). The CES- quadratic model constitutes a natural general-ization of the classic Leontief-quadratic modelthat allows for substitutability between inputswhile retaining much of the simplicity of thestandard PMP procedure.The initial purpose of PMP was to calibratemodel parameters so that the maximizationof aggregate farm returns under resource and  Amer. J.Agr. Econ.  93(4):936–948;doi:10.1093/ajae/aar029Received October 2010;accepted February 2011;published online July 28,2011©TheAuthor (2011). Published by Oxford University Press on behalf of theAgricultural andApplied EconomicsAssociation. All rights reserved. For permissions,please e-mail:  Mérel,Simon,andYi A Calibrated Programming Model ofAgricultural Supply  937 policy constraints would exactly replicate theobserved base-year allocation (Howitt 1995b).More recently, analysts have asked that theimplied supply responses of PMP-like modelsbemadeconsistentwithexogenouspriorinfor-mation (Heckelei and Britz 2005; Helming, Peeters, and Veendendaal 2001). The purposeof these requests was to avoid selecting aset of calibrating parameters that would leadto unreasonable magnitudes for the model’simplied supply elasticities.An early answer to the issue of uncon-trolled supply responses was to calibratePMP models “myopically” (Helming, Peeters,and Veendendaal 2001; Johansson, Peters,and House 2007). In myopic calibration, themodel’s supply response is calibrated ignor-ing the changes in the shadow prices of lin-ear constraints caused by a rise in the priceof any one activity. Thus, myopic calibra-tion sets model parameter values such thatthe model’s response to output price changes would  replicate the exogenous supply elastic-ities,  if the shadow prices of linear constraintswere held constant  . Of course, when the priceof any one activity rises, these shadow priceschange and therefore the  actual   implied sup-ply response of the model will be smaller thanthe prior value to which the parameters wereinitially calibrated (Heiner 1982). Myopic cal-ibration would in fact be appropriate if theprior elasticity estimates against which cali-bration is sought were obtained keeping theshadow prices (notably the land rent) con-stant. However, prior information on supplyelasticities typically comes from econometricestimates that implicitly reflect resource limi-tations faced by farmers,notably the land con-straint (Buysse, Huylenbroeck, and Lauwers2007). 1 Thus,a PMP model of agricultural sup-ply that includes these constraints should yieldsupply elasticities that are consistent with suchprior information, allowing shadow prices toadjust in response to output price changes.While the error made by calibrating modelparameters myopically need not always belarge,thereisnoguaranteethatmyopiccalibra-tion will lead to reasonable supply responses.For instance, Jansson and Heckelei (2011)report that in the context of the EuropeanCAPRI model,“the simplified calibration wassuch that the exogenous elasticities were not 1 For instance,Russo,Green,and Howitt (2008) estimate supplyelasticities for California commodities using a partial adjustmentmodel,and they do not control for the price of land.As such,theirelasticity estimates incorporate the land constraint. properly reproduced by the model becauseof failure to take into account dual values of resource constraints.”In fact,in the context of quadratic models, Mérel and Bucaram (2010)formallyshowthatifonecropis“dominant”,inthe sense that its  desired  acreage response 2 islarge relative to the sum of the desired acreageresponses of all other crops, then myopic cal-ibration will automatically lead to poor cali-bration of the supply response of that crop. Insuch cases,making the model’s implied supplyresponse consistent with prior values shouldbe addressed by solving an exact calibrationproblem, where  total   supply elasticities, allow-ing for shadow price adjustments—as opposedto  partial   elasticities—are calibrated.Yet,thedualgoalofcalibratingaPMPmodelagainst the base-year allocation while exactlyreplicating exogenous supply elasticities is notalways achievable;in some instances no choiceof parameter values can reproduce the exoge-nous supply pattern. Despite the fact, notedby Heckelei and Britz (2005), that a single- year observation on activity and input levelsdoes not provide any information on second-order properties of the objective function, 3 notall sets of supply elasticities are compatiblewiththeinformationcontainedinthereferenceallocation. While the single-year observationcannot  determine  by itself the model’s supplyresponse,it does put  restrictions  on its pattern.Mérel and Bucaram (2010) derived the nec-essary and sufficient conditions under whichquadraticmodels,includingtheCES-quadraticspecification of  Howitt (1995a), can be cal- ibrated against an exogenous set of supplyelasticities. These conditions, referred to bythese authors as the “number of crops” andthe“no dominant response”rules,respectively,ensure that the base-year data are compatiblewith the set of exogenous elasticities and pro-vide an  ex ante  criterion to determine whetherexact calibration of the model is feasible.The present article extends their analysisto a more desirable model that we refer to asthe generalized CES model. In this model,thestrict concavity in the objective function arisesfrom a production relation with decreasingreturns to scale rather than from the additionof a quadratic adjustment cost, while the 2 MérelandBucaram(2010)definethedesiredacreageresponseof a crop as the derivative of acreage with respect to output price,divided by the base-year yield. 3 Heckelei and Britz (2000) exploit cross-sectional data torecover a full matrix of quadratic cost coefficients, by imposingrestrictions on the variability of this matrix across agriculturalregions.  938  July 2011 Amer. J.Agr. Econ. possibility of substitution between farm inputsis preserved. The change has at least threeconsequences. First, the objective functionis directly interpretable as the differencebetween a production relation and a linearcost term, as required by microeconomictheory. Second, for each activity there isonly one parameter controlling the supplyelasticity; therefore, the under determinacy of the model is less severe than that which ariseswith the use of a full matrix of quadratic costcoefficients, eliminating the need for arbitraryassumptions—a popular choice is to set alloff-diagonal terms to zero—or the use of maximum entropy methods (Paris and Howitt1998). 4 Third, while the CES-quadratic modelsingles out one input—typically, land—as thesource of concavity in the net revenue of each crop, the generalized CES “spreads” thisconcavity equally between inputs.As we showhere, this modeling difference has importantconsequences for the implied input allocationresponse to policy shocks.Themethodologicalcontributionofthisarti-cle is threefold. First, we derive a closed-formexpressionfortheimpliedsupplyelasticitiesinthegeneralizedCESmodel;calibrationagainstsupplyelasticitiescanthusbeachievedthroughthe resolution of a simple system with asmany equations as activities. This techniqueconstitutes an improvement over the exist-ing method of duplicating the model’s entireset of first-order conditions for  ceteris paribus increments in the price of   each  activity,to indi-rectly recover the value of the model parame-tersconsistentwiththeexogenousinformationon supply elasticities (Heckelei 2002). Ourelasticity equations can also be incorporatedinto “well-posed” models based on more thanone observation and estimated through gen-eralized maximum entropy (GME) wheneverit is deemed appropriate to include priorinformation on supply elasticities (HeckeleiandWolff 2003).Second, the availability of a closed-formelasticity equation allows us to derive the nec-essary and sufficient conditions under whichthe model can be simultaneously calibratedagainstthereferenceallocationandtheexoge-nous set of supply elasticities (the“calibrationcriterion”). These conditions, which relate theinformation contained in the observed alloca-tion to the set of supply elasticities, implicitly 4 An attendant implication is that the generalized CES modeldoes not allow the analyst to control for the magnitude of cross-price elasticities. delineatetherangeofelasticityvectorsthatarecompatible with this allocation and with thechosen model specification. They can be rou-tinely incorporated into the PMP calibrationalgorithm. Where calibration is not feasible,theycanserveasabasisforallowingcalibrationby minimizing the departure from the initialelasticity prior. 5 Third, we compare the generalized CESmodelandtheCES-quadraticmodelof Howitt(1995a) on the basis of their flexibility withregardtocalibrationandshowthat,subjecttoacaveat,the generalized CES model can accom-modatelargersetsofsupplyelasticitiesthanitsquadratic counterpart.The article is organized as follows. First,a fixed-proportion variant of the generalizedCES model is presented. The necessary andsufficient conditions for exact calibration arederived. When they are satisfied, the solutionto the calibration problem is unique. The rela-tive parsimony of the fixed-proportion modelallows us to interpret the calibrating equationseasily and, with little notational complexity,allows for a basic understanding of the calibra-tion criterion. We then generalize the resultsto the case of variable proportions. Finally, weprovide a comparison of the generalized CESmodel and the CES-quadratic model on thebasis of their flexibility toward calibration andtheir empirical response to two simple policyexperiments. The empirical application is con-ducted on a multi-input, multi-output agricul-turalsupplymodelforYoloCounty,California.Our results are derived for the case whereone linear constraint is binding, and we willinterpret it as a land constraint. All deriva-tions and proofs are relegated to the onlinetechnical appendix. The Fixed-Proportion Case Inwhatfollows,theletter  I   denotesthenumberof non-zero activities in the reference alloca-tion. We denote by  x i  the acreage of crop  i , by  p i  the price of crop  i  per unit, and by  C  i  theper acre cost.We use  ¯  x i  to denote the observedacreage and  ¯ q i  to denote the observed out-put.The shadow value of land in the reference 5 The calibration conditions are also relevant for “well-posed”models estimated through GME that incorporate prior informa-tion on supply elasticities (Heckelei and Wolff 2003). The reason is that such models typically require the analyst to specify a set of supports for the supply elasticities, and it is important that thesesupports contain elasticity values that are compatible with the“mean allocation”at which the elasticities are to be evaluated.  Mérel,Simon,andYi A Calibrated Programming Model ofAgricultural Supply  939 allocation is denoted  ¯ λ 1 . 6 The set of exoge-nous supply elasticities is  ¯ η = ( ¯ η 1 , ... ,  ¯ η  I  ) , and ¯ η > 0.The optimization program is written as max  x i ≥ 0  I   i = 1 {  p i α i  x δ i i  − ( C  i  + λ 2 i )  x i } (1)subject to  I   i = 1  x i = ¯ L where  x  = (  x 1 , ... ,  x  I  ) denotestheacreageallo-cation and  ¯ L  the available land.In model (1), the output of activity  i  is  α i  x δ i i  .The coefficients  δ i  lie within the interval  ( 0,1 ) and are used to calibrate against the set of elasticities  ¯ η , while the crop-specific parame-ters  λ 2 i  are introduced to allow the model toexactly calibrate against the base-year alloca-tion  ( ¯ q i , ¯  x i , ¯ λ 1 ) . For a given set of parameters δ i ∈ ( 0,1 ) , calibration is achieved by imposingthat the first-order conditions to program (1)be satisfied at  ( ¯ q i , ¯  x i , ¯ λ 1 ) : ∀ i = 1, ... ,  I  (2)   p i ¯ q i δ i = ( C  i  + λ 2 i  + ¯ λ 1 ) ¯  x i α i ¯  x δ i i  = ¯ q i .The equation set (2) determines the parame-ters  α i  and  λ 2 i  as functions of the referenceallocation and the parameters  δ i .Following the procedure described in Méreland Bucaram (2010),we can derive the supplyelasticity of crop  i  implied by model (1) as 7 η i = δ i 1 − δ i  1 − ¯  x 2 i  p i ¯ q i δ i ( 1 − δ i )   I  j = 1 ¯  x 2  j  p  j ¯ q  j δ  j ( 1 − δ  j )  an expression that shows that the implied elas-ticities depend on the base-year allocation andthe parameters  δ i , but not on the parameters α i  or  λ 2 i . Calibration against the exogenoussupply elasticities may thus be conducted 6 In the srcinal PMP model of  Howitt (1995b), ¯ λ 1  was obtainedfrom a first-stage linear programming model with calibration con-straints. Britz, Heckelei, and Wolff (2003) argue that doing so canleadtoarbitraryvaluesfortheduals.Here,weassumethatareliablevalue for  ¯ λ 1  is available.As is apparent in equation (3),the choiceof   ¯ λ 1  does not affect the calibration system and is thus irrelevantto the calibration criterion. 7 See appendix for the derivation. independently of calibration against the base-year allocation. Defining  b i =  ¯  x 2 i  p i ¯ q i , a param-eter which represents the ratio of acreageto gross revenue per acre, the correspondingcalibration system can be written as ∀ i = 1, ... ,  I  (3) ¯ η i = δ i 1 − δ i  1 − b i δ i ( 1 − δ i )   I  j = 1 b  j δ  j ( 1 − δ  j )  .In equation (3), the second term in the brack-ets captures the effect of the change in theshadow value of land induced by the changein the price of crop  i .To see why,first note thatthe myopic value of parameter  δ i , that is, theone that obtains if the change in the shadowprice of land is ignored,is simply  δ myopic i  =  ¯ η i 1 +¯ η i ,a number that necessarily lies between zeroandone.Assuch,thefactor  δ i 1 − δ i inequation(3)represents the supply elasticity of crop  i , hold-ing the price of land constant  .The second termin the bracket reflects the adjustment to thisimplied elasticity necessary to take accountof the fact that the shadow price of land  λ 1 changes with  p i .We show in the appendix that the terms b  j δ  j ( 1 − δ  j )  represent the (negative of the) acreagereactivity of crop  j  to a rise in the price of land.The acreage reactivity is defined as the par-tial derivative of acreage with respect to thelandrent λ 1 ,keepingallotherprices(includingoutput price) constant. It is proportional to  b i ,the ratio of acreage to gross revenue per acre.That is,acreage reactivity is larger (in absolutevalue) the larger the acreage and the smallerthe gross revenue per acre.Theadjustmentterminequation(3)involvesthe ratio of the acreage reactivity of crop  i to the sum of the acreage reactivities of allcrops.Thattheadjustmenttermshouldbepro-portional to the acreage reactivity of crop  i  isintuitive,sincethistermadjustsforthefactthatthe myopic elasticity  δ i 1 − δ i ignores the changein  λ 1 . This acreage reactivity is deflated by thesum of all acreage reactivities, a quantity weshow in the appendix as inversely related tothe magnitude of the change in  λ 1 : d λ 1 dp i = b i ¯ q i ¯  x − 1 i 1 − δ i   jb  j δ  j ( 1 − δ  j ) .  940  July 2011 Amer. J.Agr. Econ. Therefore, the adjustment term can be inter-preted as the product of the acreage reactivityof crop  i , keeping  p i  constant, multiplied by ameasure of the change in  λ 1  arising from thechange in  p i . 8 We shall now state the first proposition of this article, which identifies the necessary andsufficient condition under which model (1) can be calibrated against the base-year allocation ( ¯ q i , ¯  x i , ¯ λ 1 )  while replicating the exogenous setof supply elasticities  ¯ η . Since equation set (2) has a solution for all values of   δ i  in  ( 0,1 ) ,calibration will be feasible whenever system(3) has an acceptable solution, that is, a solu-tion  δ = (δ 1 , ... , δ  I  )  such that  δ i ∈ ( 0,1 )  for all i = 1, ... ,  I  . Proposition 1.  Suppose that I   ≥ 2 . Then ,  thecalibration system  (3)  has a solution in theacceptable range  ( 0,1 )  I  if and only if  (4)  ∀ i = 1, ... ,  I b i ¯ η i  <   j = i b  j ¯ η  j  1 + 1 ¯ η  j  2 . When this condition is satisfied, the set of cal-ibrating parameters  δ  is unique and satisfies δ i  >δ myopici  for all i = 1, ... ,  I. Proposition 1 establishes the conditionunder which an exogenous set of elasticities  ¯ η is compatible with a given observed allocation.Condition (4) implicitly delineates a subregionof  R  I  ++  within which the vector  ¯ η  should lie forcalibration to be feasible.The intuition behindcondition (4) is that given the set of elasticities ¯ η  j ,  j = i , the elasticity  ¯ η i  cannot be arbitrar-ily large—as would be the case under myopiccalibration—due to the change in the shadowprice of land.Interestingly, the upper bound for the elas-ticity  ¯ η i  depends on the elasticities of otheractivities in a non-linear fashion. This upperbound becomes arbitrarily large as any one of the elasticities  ¯ η  j  gets arbitrarily large or arbi-trarily small. The reason is that in either casetheeffectof   p i  ontheendogenouslandrentgetsarbitrarily small, because, as noted previously,the derivative  d λ 1 dp i is inversely proportional to 8 Formally, the optimal output  q i  and the optimal acreage  x i are functions of the output price  p i , the per acre cost  C  i  andthe land rent  λ 1 , itself a function of all output prices, all peracre costs and land availability:  q i  = q i (  p i , C  i , λ 1 ( p , C ,  ¯ L ))  and  x i  =  x i (  p i , C  i , λ 1 ( p , C ,  ¯ L )) .Therefore,  dqidpi  =  ∂ qi ∂  pi  +  ∂ qi ∂λ 1 ∂λ 1 ∂  pi  fromwhichwededuce that  η i  = η myopic i  +  ∂ qi ∂λ 1 ∂λ 1 ∂  pi piqi  =  δ i 1 − δ i  + δ i ∂  xi ∂λ 1 ∂λ 1 ∂  pi pi xi  . the sum of the acreage reactivities of all crops,that is,   jb  j δ  j ( 1 − δ  j ) . As a given elasticity  ¯ η  j ,  j = i ,approaches infinity, implying constant returnsto scale in the production of activity  j , thecorresponding parameter  δ  j  approaches one,makingtheacreagereactivityofcrop  j  arbitrar-ily large. As  ¯ η  j  approaches zero, implying thatoutput is fixed,the corresponding parameter  δ  j approacheszero,makingtheacreagereactivityof crop  j  arbitrarily large as well. The result ineither case is that the effect  d λ 1 dp i becomes arbi-trarily small, so that the calibration equationfor activity  i  resembles the myopic calibrationequation,which always has a solution.The calibration region is depicted as theshaded region in figure 1, for the case  I   = 2and  b 2 b 1 = 1.Calibrationisfeasiblewhenevertheprior  ( ¯ η 1 ,  ¯ η 2 )  lies in the space delineated by theouter portions of two cones, that is, either inthe space below the bottoms of the cones orclose enough to one of the axes or in the tun-nel that separates the cones.When  b 1 = b 2 ,thecalibration region becomes asymmetric but isstill delimited by two cones.When  ¯ η liesoutsideofthecalibrationregion,the analyst may choose a different elastic-ity vector that meets the calibration criterion.The availability of an explicit criterion in factenables a rational search for a reproducibleelasticity vector. A natural choice would beto select the elasticity vector that minimizesthe distance between the initial vector and thecalibration region. Suppose, for instance, that b 2 b 1 = 8,  ¯ η 1 = 0.6 and  ¯ η 2 = 0.8. This situation isdepicted in figure 2.The prior elasticity vector, represented by the small dot, lies outside thecalibrationregion.Theelasticityvectorthatliesinthecalibrationregionwhilebeingascloseas 0 2 4 6 8 100246810 h 1      h         2 Figure 1. Calibration region for  b 2 b 1  = 1
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