Short- and Long-Run Demand andSubstitution of Agricultural Inputs

Jorge Fernandez-Cornejo

Short- and long-run Hicksirtn and Marshallian elasticities arc estimated, along with Morishimaelasticities of substitution, using a restricted profit function and a series of decompositionequations. Convexity in prices and concavity in quasi-fixed factors of the restricted profitfunction are simultaneously imposed using Bayesian techniques. The empirical model isdlsaggregated in the input side, utilizes a Fuss-quadratic flexible functional form, incorporatesthe impact of agricultural policies, and introduces a new weather index. The methodology isapplied to Illinois’s agriculture, and implications for agriculture in the Corn Belt and theNortheast are briefly dkcussed.

Measuring ease of substitutability between produc-tion factors is of practical and theoretical impor-tance in economics. Many pessimistic predictionsof natural-resource depletion proved to be grosslyincorrect because the models used failed to recog-nize important substitution relationships (Field andBemdt). Concerns over ground- and surface-waterquality, food safety, land retirement, and altern-ativeproduction systems call for quantitative as-sessments of the effect of changing market condi-tions or government policies on the demand foragricultural inputs. In U.S. agriculture, it is par-ticularly important to assess policy alternatives tar-geted to limit the use of fertilizers and pesticides(chemical inputs), which have contributed to thecontamination of water resources and to the pres-ence of toxic residues in food. Groundwater con-tamination is particularly serious in areas with highapplication rates of nitrogen or mobile pesticides,shallow water tables, and permeable, coarse-textured soils (Miller). About 10% of the nation’s3,000 counties, located chiefly in the Upper Mid-west and East, have been identified to have poten-tial for contamination of groundwater by both ni-trates and pesticides (Lee).

The failure of many econometric models to in-

Jorge Femandez-ComejoisanagriculturaleconomistwiththeResourcesand Technology Division, Economic Research Serwice(ERS), U.S, De-partmentofAgriculture.Theviewsexpressedaretheviewsof theauthoranddo not necessarily represent policies or views of the U.S. Depart-ment of Agriculture.

The author wishes to thank Charlie Haflahan from ERS for developinga computer program to impose inequalities using the Bayesirmapproachand Dick Shumway from Texas A&M University for providing a com-puter version of Remain’s procedure for calculating expected prices offarm-program commodities. Valuable comments about earlier versionsof this paper received from C.M. Gempesaw, M. Spilker, U. Vasavada,and H, Vmomen are gratefully acknowledged.

corporate substitution possibilities is due in part tolack of reliable information on price elasticities ofinput demand. Published empirical estimates of in-put demand elasticities in U.S. agriculture varywidely (Tables 1 and 2), even excluding earlierestimates based on models not derived from opti-mizing decisions of economic agents. To a largeextent, differences among elasticity estimates aredue to differences in model specification, includ-ing levels of aggregation over inputs/outputs andfirms, functional form, price expectations, and in-troduction of exogenous variables (e.g., weather,government policy). Also, there are differences inthe behavioral assumptions, that is, profit maximi-zation or cost minimization. In addition, modelsare often inconsistent with economic theory, andfrequently long-run equilibrium is implicitly as-sumed.

Economic theory requires the restricted profitfunction to be convex in prices and concave inquasi-fixed factors. Convexity in prices has beenimposed by Shumway and Alexander, Ball, andothers. However, unlike this work, previous stud-ies have not imposed convexity in prices and con-cavity in quasi-fixed factors simultaneously. Bothcurvature properties are essential, in particularwhen using decomposition methods. While con-vexity in prices ensures that short-run Marshallianelasticities are of the “correct” sign, Le Chate-lier’s principle may be violated if the restrictedprofit function is not concave in quasi-fixed inputs.For example, some long-run own-price elasticitiesmay be smaller (in absolute value) than corre-sponding short-run elasticities, and own-priceHicksian elasticities may be larger than corre-sponding Marshallian elasticities. Among the few

Fernatrdez-Cornejo Demand and Substitution of Agricultural Inputs 37

Table 1. Input Demand Elasticities-Selected Empirical Studies in Agriculture

Number ot?outputs/var. Allowance for

Inputs/Quasi-fixed Weather andAuthor, Year, Type of Functional Inputs Curvature GovernmentCountry/Region Input Demand Form (Y/x/Z) Imposed Policy

Binswanger, 74, USA Hicksian, LRa Translog 1/5/0 No NoLopez, 80, Canada Hicksian, LR G. Leontief 1/4/0 No NoRay, 82, USA Hicksian, LR Translog 1/5/0 No NoBrawn & Christensen, Hicksian, SR’, Translog 113/2 No No

82, USA LRCapalbo, 88, USA Hicksian, LR Translog 1/4/0 No NoCapalbo, 88, USA Hicksian, LR Translog 2/5/0 No NoWeaver, 83, ND Marshalliao, SR Translog 3/5/1 No YesShumway, 83, TX Marshallian, SR N. Quadratic 61312 No YesImpez, 84, Canada Marshalliarr/ G. Leontief 2/4/0 No No

Hicksian, LRAnde, 84, USA Marshallian, LR Translog 1/4/0 No NoCapaIbo, 88, USA Marshallirm, SR Translog 1/3/1 No NoShumway & Marshallian, SR N. Quadratic 51412 Yes Yes

Alexander, 88, USABail, 88, USA Marshallian, SR Translog 5/6/1 Yes NoMcIntosh & Shumway, Marshallian, SR N. Quadratic 10/4/3 Yes Yes

89, CABurrell, 89, UK Marshallian, Translo8 61313 No No

Hicksian, SR

‘LR is long-run; SR is short-run.

decomposition studies in agricultural economics,Lopez (1984) and, recently, Higgins use a long-run profit function to derive long-run Hicksianelasticities. In both cases, the convexity require-ment is violated. Hertel uses a restricted profitfunction to obtain short- and long-run Marshallianelasticities. Convexity in prices is satisfied in Her-tel’s model using pseudodata, but the issue of con-cavity in quasi-fixed factors did not arise becauseHertel considered only one quasi-fixed factor andassumed constant returns to scale.

This paper provides the methodology to deter-mine theoretically consistent Hicksian and Mar-shallian input demand functions and elasticities ofsubstitution (ES) in the short and long run. Short-run Marshallian demands are calculated directlyusing a restricted profit function. Long-run Mar-shallian and short-run Hicksian demands are de-rived via decomposition equations, and long-runHicksian demands are calculated from the corre-sponding short-run demands by a second transfor-mation. Altogether, this decomposition techniqueprovides a total of four types of demand functionsthat, if estimated directly, would require estima-tion of two cost functions and two profit functions.This technique also makes possible the calculationof short-run and long-run elasticities of substitu-tion while maintaining the assumption of profitmaximization.

The empirical model uses a Fuss-quadratic nor-

malized restricted profit function and is disaggre-gate in the input side to a larger extent than inprevious dual models. For example, feeds, seeds,fertilizer, pesticides, fuels, hired labor, and familylabor are each a separate category. In addition, themodel allows for the impact of agricultural pro-grams and policies on farmers’ price expectations,and a new weather index is introduced. The esti-mated input demand functions are used to provide

Table 2. Own-Price Elasticities of InputDemand, Selected Estimates in Agriculture

HiredAuthor/Year Labor Fertilizer Chemicats

Binswanger, 74 –0.911 –0.945Lopez, 80 – 0.897a –0.391Ray, 82 -0,839 -0.128Brown &

Christensen, 82 -0.650 -0.188Capalbo, 88 – 0.207” –0.068Capalbo, 88 – 0.492a –0.876Sbumway, 83 –0.43 –0.70Weaver, 83 – 1.016” –1.377LOPCZ, 84 –0.3771- 1.24Antle, 84 –1.311 -0.194Capalbo, 88 –o.594a –0.606McIntosh &

Shumway, 89 –0.593 -0.038Burrell, 89 – 0.42

“Includes hired and family labor.

38 April 1992 NJARE

an approximate measure of the impact on chemicalinput use of imposing ad valorem taxes on chemi-cal inputs, This paper reports empirical results forIllinois, which has a large potential for both nitrateand pesticide contamination of groundwater (Lee).However, the methodology developed in this papermay be used on similarly exposed areas in the CornBelt and the Northeast.

Hicksian and Marshallian Elasticities

Duality theory allows the determination of supplyand demand functions without explicit solution ofthe optimization problem, making possible the useof flexible functional forms with weaker main-tained hypotheses than in traditional primal meth-ods, increasing the enerality of the inference

F(Gallant and Golub). Dual models require somebehavioral assumptions about the firm and themarket where it operates. In agriculture, it is usualto assume that markets are competitive and that theobjective of firms is either cost minimization orprofit maximization. The type of dual model spec-ified (e. g., cost or profit function) has an importantimpact on the input demand and output supplyelasticities directly derived from the model. If acost-function approach is used, input demands ob-tained from the application of Shephard’s lemmaare characterized as conditional on output level.These Hicksian (or compensated) input demandsreflect movements along an isoquant for a givenoutput level (Sakai; Lopez 1984) and are used tocalculate elasticities of substitution. When a profitfunction model is specified, unconditional inputdemands obtained from application of Hotelling’slemma are known as Marshallian (or uncompen-sated) demands and include substitution effectsalong the old isoquant and expansion effects alongthe expansion path (to the new isoquant). Thesigns of the Hicksian cross-price input demandelasticities of input pairs are often used to classifyinputs into net substitutes (positive) in productionor net complements (negative), The correspondingcases for Marshallian elasticities are referred to asgross substitutes/complements.

A version of Le Chatelier’s principle requiresown-price Marshallian elasticities to be larger inabsolute value than the corresponding Hicksianelasticities because the latter hold output constant,while Marshallian elasticities allow both inputsand outputs to adjust to their new equilibrium lev-

1In addkion, multicollinearity is likely to be less severe among (fac-tor) prices required in the dual appruach than among factor quantitiesused in primal metbuds, and the erogeneity of prices is more likely tohold.

els. A procedure similar to the Slutsky decompo-sition may be applied to obtain a relationship be-tween Marshallian and Hicksian elasticities (Sakai;Lopez 1984; Higgins).

Measures of Substitutability

The degree of substitutability between productionfactors is measured by the elasticity of substitution(ES). For a production process with two inputs, iandj, the ES (mu)is defined by the elasticity of theinput quantity ratio with respect to the marginalrate of substitution (MRS) between inputs; u in-creases as substitution between inputs becomeseasier. Under profit maximization, the MRS isequal to the price ratio. For a two-variable inputcase there is no ambiguity in the meaning of pricechange and one elasticity measure suffices since au= Uji (Kang and Brown).

When more than two variable inputs are in-volved in production, there are as many possibledefinitions of ES as there are possible combina-tions of elements of the underlying Hessian matrix(Mundlak). Several definitions are used in the lit-erature. The direct elasticity of substitution (DES)is an extension of the ES for two inputs with thecondition that output and all the other inputs areheld constant. Because of this inflexibility, DES isnot commonly used. The Allen-Uzawa partial ES(AUES) can be expressed as the cross-price elas-ticity of the Hicksian input demand (:ij) divided bythe respective cost share. The AUES 1san exampleof the one-factor one-price ES. Its popularity maybe due to its appealing symmetry, although theeconomic interpretation of cross-price elasticitiesis more direct (Field and Berndt).

The Morishima elasticity of substitution (MES),proposed by Robinson and Morishima, is classi-fied as a two-factor, one-price elasticity of substi-tution and may be interpreted as the cross-priceelasticity of relative (Hicksian) demand because itmeasures the relative adjustment of factor quanti-ties when a single factor price changes. Intuitively,if inputs i and j are net complements (negativecross-price Hicksian elasticity), an increase in theprice, Pj$ will lead to a decrease in the quantityemployed, Xi. However, since the decrease in Pjalso decreases Xj, the own-price effect must besubtracted to obtain the net effect. This is whatMES represents.2

2 Koizumi (1976) first suggested this interpretation, He noted that“the Morishima elasticity of substitution of Xj for Xl measures tbe per-centage change in employment in Xf caused by 1% change in the priceP, of Xj after the percentage change in Xj due to the pure demand ef-fecthas been partialed out.” Note that the MES can be expressed as

(MES),j = abr(x~x,yaltij = a(lfi, – hx,yahwj = ●O– ●J.

Fernandez-Cornejo Demand and Substitution of Agricultural Inputs 39

Kang and Brown recommend the use of theMES because it has the desirable property of beinginvariant to the separability assumption usuallymade. They show that the MES is independentfrom the “unestimated characteristics of the func-tion, ” while a partial measure, such as the AUES,does not have this property. Thus, it is possible, asBerndt and Wood find empirically, that the AUESSin a three-input model may yield different valuesthan the AUESS for a four-input model, even if thefourth input is separable. Kang and Brown showthat the calculation of MES does not even requiredata for omitted inputs and that values of two dif-ferent studies are directly comparable.

Moreover, Blackorby and Russell show that theAUES is not a measure of the “ease” of substi-tutability or curvature of the isoquant; it is mean-ingless as a quantitative measure; it provides noadditional information to that contained in theHicksian cross-price elasticity, and it cannot beinterpreted as a logarithmic derivative of an inputquantity ratio with respect to a price ratio (orMRS). The MES, on the other hand, provides anexact measure of the curvature along an isoquantand may be interpreted as a logarithmic derivativeof an input quantity ratio with respect to an inputprice ratio. Consequently, MES is a more appro-priate measure of input substitution. Blackorbyand Russell also note that the asymmetry of theMES is natural because, while in two-dimensionalinput space the curvature of an isoquant at a pointis an unambiguous idea, in more than two dimen-sions, curvature may be measured in many direc-tions. For example, the same change in the priceratio, Pi/Pj, may be obtained when Pj changes andPi is held constant or vice versa. However, eachcase leads to a different change in the quantity ratio(Xi/Xj).

The Restricted Profit Function

Early empirical work based on the dual frameworkimplicitly assumes that firms are in static (long-run) equilibrium. Recognition of the short-run fix-ity of some production inputs makes estimation ofa full equilibrium profit (or cost) function inappro-priate. As Brown and Christensen observe, inmany cases the assumption of full static equilib-rium “is suspect and so are the empirical results. ”In order to relax the assumption of static equilib-rium, two basic approaches are available. The firstuses full dynamic models within the costs of ad-justment framework. By combining techniques ofdynamic optimization and the notion of adjustment

costs, this approach not only provides estimates ofshort- and long-run demand functions, but also de-scribes the nature of the adjustment and the timerequired for the adjustment of the quasi-fixed fac-tors. However, estimation of these dynamic mod-els at the level of disaggregation required to exam-ine input-substitution issues is often not feasiblewith available data. The second method is basedon the use of restricted profit or cost functions. Thefirm is assumed to be in static (short-run) equilib-rium only with variable factors, conditional on lev-els of the other factors. When the nature of thestock-adjustment process is not the focus of anal-ysis, but rather the characterization of both short-and long-run production structure, models basedon the restricted profit (or cost) function are theproper choice (Hazilla and Kopp 1986). The the-ory of the restricted profit function is well devel-oped (Diewert; Lau 1976). Its framework is gen-eral enough to accommodate as special cases costand revenue functions and all possible intermediatecases (by allowing a subset of inputs and outputs tobe variable).

This study assumes profit-maximizing produc-ers operating in competitive markets, and the re-stricted profit function is used to capture the infor-mation about the production structure in both theshort and long run.3 Consider n + m + s ‘‘com-modities” including n variable net inputs/outputs(netputs), m fixed inputs/outputs, ands exogenousvariables such as time or weather. Let X = (Xl. . . Xn)’denote the vector of variable netputs withthe sign convention Xi > 0(<0) if the ith variablenetput is an output (input); Z = (Zl . . . ZJ’ is thevector of non-negative quasi-fixed netputs; R =(R, . . . RJ’ is the vector of exogenous factors; P=(P, . . . PJ’ is the price vector of variablenetputs; and IV = (WI . . . WJ’ is the price vectorof quasi-fixed netputs. The restricted profit func-tion is defined by

(1) m(P,Z,R) = MAXx[P’X:XETl .

The production possibilities set T is assumed tobe nonempty, closed, bounded, and convex. Inaddition, if Z includes only inputs, T is assumed tobe a cone (Diewert; Ball). Under the above as-sumptions on the technology, the restricted profitfunction is well defined and satisfies the usual reg-

3 Recently, Lim and Shumway (1989a)performed nonparametric testsfor each of the 48 contiguous states using agricultural production data forthe period 1956-82. They found “little departore” of the data from thejoint hypothesis of profit maximizationand a convex technology in all 48states. In addition, they found that for about 9090of the states (includlngIllinois), the data were consistent with constant returns to scale.

40 April 1992

ularity conditions (Diewert). In particular, withonly the inputs fixed, ITis homogeneous of degreeone in variable netput prices (P) and quasi-fixednetput quantities (Z). In addition, IT must satisfysymmetry (the Hessian matrix must be symmet-ric), monotonicity, and curvature conditions. Cur-vature conditions require IT to be convex in P forevery Z and R, and concave in Z for every P and R.That is, m is a saddle function in (P,Z) for all R.No curvature assumptions are made about R. Theunrestricted profit function may be expressed as

(2) T(P, W,R) = MAXZ[T(P,Z,R) – W’Zl,

where m(P,Z,R) is defined by equation (1).

Theoretical Consistency

One of the penalties of using flexible functionalforms in dual models is that the estimated func-tions may not be theoretically consistent becausethe number of parameters is sufficient to allow theelasticity matrix to have any value at any point inthe data space (Gallant and Golub). The restrictedprofit function IT is theoretically consistent if itsatisfies assumed homogeneity, symmetry, mono-tonicity, and curvature conditions. Symmetry andhomogeneity are usually easier to impose becausethey translate into equality restrictions on the pa-rameters, which reduces the number of free param-eters (the dimensionality of the parameter space).Monotonicity and curvature require inequality re-strictions on the parameters, which are more dif-ficult to impose because they reduce the parameterspace but not its dimensionality.

Dual methods require more strict curvature con-ditions than primal methods. As Lau (1978) notes,the production function may not be convex, but theprofit function must always be convex in priceswhen output and input markets are competitive andfirms are profit-maximizers. Therefore, a noncon-vex profit function is inconsistent with the behav-ioral assumption of profit maximization. Empiri-cally, consequences of the violation of convexityare that signs of output supply and input demandelasticities are inconsistent with economic theory,and tests of functional structure (e.g., separability)are meaningless because duality theorems do notapply (Hazilla and Kopp 1985; Ball), Impositionof curvature avoids these adverse consequencesand provides a gain in statistical efficiency by us-ing a priori information (Gallant and Golub).

Imposition of curvature frequently uses theproperty of a twice continuously differentiable

function that is convex (concave) with respect to asubset of its arguments if and only if its Hessianmatrix (H) is positive (negative) semidefinite. Anecessary and sufficient condition for the Hessianto be positive semidefinite is that all the eigenval-ues are non-negative. Alternatively, the Choleskyvalues must be non-negative. The last condition al-lows the transformation of the restrictions of pos-itive semidetlniteness of the Hessian matrix intosimple inequality restrictions on the parameters,and it is used in the nonlinear programming/maximum-likelihood (NLP) approach to imposecurvature. In agricultural economics, Shumwayand Alexander, Ball, and others impose concavityon a profit function using this approach. Some ofthe weaknesses of the approach stem from diffi-culties in the statistical interpretation of the resultsand inapplicability of the likelihood-ratio test(Chalfant and White).

An alternative Bayesian approach was reexam-ined recently by Kloek and van Dijk, Geweke, andChalfant and White, No prior information is re-quired beyond the inequality restrictions, whichare treated as prior beliefs about the model. TheBayesian approach consists of estimating the pa-rameters without imposing the restrictions. If therestrictions are violated, they are imposed follow-ing Geweke. For example, to impose convexity,the prior distribution is the indicator function:

{

lif6GDP(o) = O otherwise ‘

where 6 is the parameter vector, p((3) is its priordensity function containing all information about 6before the data are examined, and the set D isdefined by D = {6 c 9tq I eigenvalues of H = O},where q is the number of free parameters. Themean of the posterior distributionfltl Iy) is a Bayesestimator4 that minimizes expected loss for a qua-dratic loss function. In practice, Monte Carlo in-tegration (Kloek and van Dijk; van Dijk andKloek; Geweke) is used to calculate E(6), sinceanalytical procedures are not available and nu-merical procedures become too complicated be-yond three to four dimensions.

Assuming, for example, that the parameter vec-tor follows the multivariate normal distributionwith a mean vector 6 and known variance-covariance 2, then the posterior distribution willbe a truncated (multivariate) normal such that Ohas

4 A Bayes estimator can be shown to be consistent aod a Best As-ymptotic Normal (BAN) estimator under quite general conditions(Mood, Graybill, and Bees).

Fernandez-Cornejo

non-zero values only in D. The proc~dure involvesfirst estimating the unconstrained 6 and its vari-ante-covariance matrix by the usual procedures(e.g., iterative seemingly unrelated regression,ITSUR), Then, random samples are drawn fromthe multivariate normal distribution and the eigen-values of H arecalculated to verify if the particularvalues of O lie in D. All draws that yield O @ D(i.e., a Hessian with some negative eigenvalues)are excluded. E(6) is calculated from the mean ofall values of Othat are in D.

Since Z is usually unknown, the posterior is nolonger multivariate normal. It is necessary to con-sider a joint prior P(6, X), and a procedure called“importance sampling” is used (Kloek and vanDijk; Geweke). The procedure indicated above ismodified by sampling from a multivariate t ratherthan from a multivariate normal distribution (seedetails in Geweke and in Chalfant, Gray, andWhite). In addition to parameters, it is possible tocalculate their numerical standard errors that are“analogous to the usual standard error of the esti-mate of a population mean” (Chalfant, Gray, andWhite).

Imposition of curvature conditions (convexity orconcavity) on the approximating function may becarried out either globally or locally (e .g,, at thepoint of expansion). If convexity conditions areimposed locally, there is no guarantee that the ap-proximating function will be globally convex (animportant exception noted by Lau (1978) is thequadratic function). Global convexity/concavityoften requires such severe restrictions on parameters

Demand and Substitution of Agricultural Inputs 41

Decomposition Analysis

This section presents a series of decompositionequations required to retrieve all demand functionsusing the restricted profit function as a startingpoint. It is based on properties of the Hessian ma-trices examined first by Lau (1976) and used byLopez (1984) and by Hertel. Assuming sufficientdifferentiability of the profit function and using theenvelope theorem, the long-run (Marshallian) netsupply function is

(3) [hl(P,w,R)/lNqnx, = X(P,W,R).The first-order condition for long-run (full-equilibrium) profit maximization from (2) is

(4) [ihr(P,Z,R)/8Zl~x , = W,

which states that the shadow-price vector is equalto the corresponding vector of rental prices W.From (4), the optimum value for Z, Z* =Z(P, W,R), is obtained. From (2), using again theenvelope theorem,

(s) [hl(P,w,R)mqmx,= –Z(P,W,R).

The decomposition equations of the Hessians ofthe restricted and unrestricted profit functions areobtained by differentiation of the above expres-sions with respect to P. After some algebra, theHessian of the (unrestricted) profit function is ex-pressed as a function of the Hessian of the re-stricted profit function in the neighborhood of thelong-run equilibrium as follows:

‘6) rf’~)lnxn=rp’z:’w’R)’R)lnxn-r’H(::)’R)lnxJ’2H(:~”)’R

of the approximating function that the flexibility The Hessian of the restricted profit function mayof the functional forms is often destroyed, leading be expressed in terms of the unrestricted profitto upward bias in the degree of input substitut- function:

ability (Lau 1978). Thus, the only traditional flex- It is also useful to express the restricted profitible functional form that satisfies global curvature function in terms of a less restricted profit func-conditions is the normalized quadratic. tion. The vector of netputs held fixed in the re-

42 April 1992 NJARE

stricted profit function is Z. Denote the fixed net- positive semidefinite, then [tlXi/aPi]H s aXi/aPi]M,puts in the less restricted profit function by Zf with which is another manifestation of Le Chatelier’sprices Wf, and the netputs that become variable by principle. Finally, an expression similar to (10) isZ“ with prices W“: used to express the long-run Hicksian demand, and

[ 1[r3m(P,zf,z”,f?) = m(P,w’’,zf,R) 1[ 1[ Hd2TI(P, W“,Zf,R) d211(P, W“,Zf,R) - * a211(P,Wv,Zf,R)(8) ap2 ap2 – apawv a(w”)2 1aw”ap “

The long-run Marshallian elasticities are ex- the corresponding long-run Marshallian demand ispressed in terms of the corresponding short-run obtained using an expression analogous to (9).elasticities using equation (6), noting that in theshort run only some of the inputs (K) belong to the The Empirical Model

fixed netput category (Z). Using Hoteiling’s The empirical model uses the Fuss-quadratic nor-lemma on (6) yields (in derivative form) realized restricted profit function (Fuss; Diewert

‘9) R~’R)I=nf’R)lHwrp’K:’w’R)’R)l-’FlFinally, to calculate the derivatives of the quasi- and Ostensoe). This flexible functional form is ca-

fixed factors with respect to price in the long run, pable of satisfying curvature globally. Imposingit is simpler to obtain first Z* = (P, W,R), as symmetry and linear homogeneity in P and Z, itshown in the next section. may be expressed as

If the Hessian d2r(.)/r3Z2 is negative semidefi-nite, it follows that [aXi/aPi] ‘R 2 a[xijapi]s:,(11) ii(~,~,R) =

which illustrates that Le Chatelier’s principle wdl

[.

-.II(P,Z,R)

P

be satisfied if the restricted profit function is con-Plzl =

uo + (a’b’c’) ~vex in P and concave in Z. In terms of elasticities, Rthe above show that long-run own-price elasticitiesare larger (in absolute value) than the correspond-

[

BE

ing short-n.melasticities. The Hicksian short-run in- + Y2(P7?’R’) E’ C

put demand elasticities are obtained from (8), notingthat in this case the fixed netputs vector Z includesall outputs (Y) and some inputs (K). Thus, Z = where P = (P2/P1 . . . PNJP1)’, Z = (Z2/Z1 . . .[Y’ ,K’]’. In this case, the restricted profit function Z~/Z1)’,R = (Rl...RJ’;a. is a scalar parameter;becomes m(P,Z,R) = – Co,st(P,Y,K,R), From (8) and a, b, and czarevectors of constants of the sameand (3), the matrix of derivatives of the short-run dimension as P, Z, and R, respectively. B, C, andHicksian input demand with respect to price is D are symmetric matrices of parameters of the ap-

In this expression, the first term of the right- propriate dimensions, e.g., B is (n – 1) x (n –

hand side is the familiar substitution effect and the 1). Similarly E, F, and G are matrices of unknownsecond the expansion effect. Since 32T(.)/a(Wy)2 is parameters, Because no interaction is expected be-

tween exogenous factors and quasi-fixed factors,

5Similarly, a long-inn Hicksian demand may be obtained from the G is a null matrix and D is diagonal. Using theunrestrictedprofitfunction(Lopez 1984). envelope theorem, the vector of short-run net sup-

Fernandez-Cornejo Demaud and Substitution of Agricultural Inputs 43

ply functions divided by Z1, i.e., X = (XJZI . . .XJZl)’, is

(12) X(P,Z,R) = VPfr(P,Z,R)=a+B’P+EZ+FR,

which provides n – 1 equations. The numeraireequation is obtained from

(13) ; = ti(~,2,R)– P’X

T(P ,Z,R)

= P,z,- ~ ~iXi; and

,G2

x*(14)

~=ao+b’Z+c’R–~P’B~

+ ~ z’c.Z+ ~R’D R.

Short-run Marshallian elasticities are obtained di-rectly by calculating first the derivatives of the~’s. Other elasticities are derived from the decom-position equations. Long-run elasticities requirethe optimum Z vector, which is obtained by solv-ing for 2 in the expression

(15) W = WIP1 = V2ii(P,Z,R)=b+E’P+CZ.

The derivatives of Z with respect to W and P areobtained from (15), resulting in dZ/dP = – C- 1Eand dZ/dW = C-1. This allows the calculation oflong-run elasticities for the quasi-fixed factors.Long-run elasticities for variable factors are ob-tained from the decomposition equations,

Linear homogeneity is imposed by normalizat-ion and symmetry by sharing of parameters.Monotonicity is verified when the predicted X’shave the correct sign (i.e., negative for variableinputs and positive for outputs). In order to satisfythe curvature conditions, the restricted profit func-tion must be a saddle function, convex in pricesand concave in the quasi-fixed inputs. The Fuss-quadratic normalized profit function % is globallyconvex in prices if and only if (iff) the HessianVPP2f@,2,R) = B is positive semidefinite, and%is globally concave in Z iff C is negative semidef-inite. Curvature conditions are imposed followingthe Bayesian (statistical) approach. The initial (un-constrained) parameter estimates are obtained bythe iterative seemingly unrelated regression(ITSUR) technique, which is asymptoticallyequivalent to maximum-likelihood estimation.Given that homogeneity and symmetry are main-tained and monotonicity is verified, the mean ofthe posterior density, that is, the mean vector ofthose replications that satisfy the curvature condi-

tions (non-negative eigenvalues for B, nonpositivefor C), provides the estimate of the parameter vec-tor. A total of 14,000 replications are carried outfor the estimation. After the parameters are ob-tained, the model may be used to carry out policysimulations, such as examining the effect of im-posing taxes (on fertilizer and pesticides) on inputuse, output supply, and farm income.

The desire to specify a highly disaggregatedmodel in terms of outputs and/or inputs is oftenhampered by multicollinearity and degrees of free-dom limitations. Thus, separability and nonjoint-ness assumptions are usually maintained. In somecases separability assumptions are not tested. Moreoften, researchers maintain some separability as-sumptions in their models and they test (and usu-ally reject) separability at a higher level of aggre-gation than that of their models. For this study, wedraw on recent empirical evidence (Lim andShumway 1989b) that finds consistent aggregationof all outputs in a single category is justified in 11of the 48 contiguous states, including Illinois. Themodel is specified as part of a two-stage optimiza-tion (Fuss). The output submodel is a revenuefunction that includes five output categories. Theinput submodel includes aggregate output and nineinput categories: hired labor (numeraire), feed,seeds, fertilizer, pesticides, fuels, capital and re-lated services, operator/family labor, and land (in-cluding buildings). Operator/family labor and landare considered as quasi-fixed inputs. In addition,the model includes time as a proxy for disembod-ied technical change, a weather index discussed inthe next section, and a government policy variableto account for diversion payments.

Previous studies (e.g., Shumway and Alex-ander; Huy, Elterich, and Gempesaw) have docu-mented considerable regional differences in the ag-ricultural production structure. Recent empiricalstudies (Poison; Lim and Shumway 1989b) havefound that the production structure, in particularseparability, varies even among states in the sameproduction region. As a result, aggregate studies ofU.S. agriculture, and even regional studies, maysuffer from unknown specification bias. Conse-quently, current research is focusing on states (orcounties) as data become available. The estimatedmodel consists of eight equations (12) and (14)with additive disturbances appended to reflect er-rors in optimization. After imposing symmetry andlinear homogeneity, 61 parameters are estimated(that is, 6 is a 61-dimensional vector). After theparameters are estimated, short-run Marshallianelasticities are calculated and the decompositionequations are used to obtain Marshallian long-runand Hicksian short-run/long-run elasticities. Fi-nally, MES is obtained by using (A4ES)0 = ~ti – ~ti.

44 April 1992

Data

The model is estimated using annual data for thestate of Illinois for the period 1950-86. The dataset used was compiled by Evenson and updated byMcIntosh using various U.S. Department of Agri-culture publications. For this study, a fuel dataseries is added and other minor changes intro-duced. Fuel expenditures are obtained from Eco-nomic Indicators of the Farm Sector (EIFIS) andprices are from Agricultural Prices. For thoseyears that state-level fuel prices are not available,regional (Corn Belt) or U.S. prices are used asproxies. All aggregation is made using Tornqvistindices.

Producers are assumed to make their productionplans based on subjective evaluations of future out-put prices and government programs. Lagged out-put prices are used as proxies based on results byMcIntosh and Shumway, which show that one-period lags of output prices are better predictors ofoutput price than other ARIMA models or futures-based models. The concepts of effective supportprice (ESP) and effective diversion payments(EDP) (Houck et al.; Ryan and Abel) are usedfollowing McIntosh and Shumway. Since an-nounced government programs may affect farm-ers’ decisions even when the ESP is below theexpected output price (Shumway, McIntosh, andPoison), a weighted average of expected marketprices and ESP is calculated using Remain’s tech-nique, in which the weights depend on the relativemagnitudes of ESP, expected market prices, andloan rates.

While weather has been recognized as a veryimportant factor in the supply of agricultural com-modities, few empirical studies using the dualframework have incorporated weather into theirmodels. Shumway used the Stallings index, whichis the ratio of actual to calculated yields based ona linear trend. A drawback of the Stallings index isthat it is not directly related to weather variables.More recently, Shumway and Alexander, andMcIntosh and Shumway have incorporatedweather variables (such as rainfall and temperaturein critical planting and growing months) in theirdual models. However, they report statistically in-significant weather coefficients. One difficultywith this direct approach is that many weather vari-ables would need to be introduced to the model tocapture the effect of weather on agricultural out-put, consuming scarce degrees of freedom. Thisstudy introduces into the model a weather index,R2, that synthesizes the weather information rele-vant to each specific crop. The index is defined asthe ratio of actual to normal yields and is calcu-

lated for each majorweather variables Vi:

Yactua,(16) R2 = p- =

normal

NJARE

crop as a function of the

The coefficients P, are calculated using Thomp-son’s multiple-regression technique (Thompson1970, 1986) to capture the effect of weather onyields using state-level yield (USDA) and weather(Teigen and Singer) data. Following Thompson,relevant weather variables for the Corn Belt arepreseason precipitation; June, July, and Augustrainfall; June, July, and August temperature (all indeviation form); and their squares.

Empirical Results

Table 3 compares the short-run Marshallian own-price elasticity for the variable inputs in 1986, withand without imposing curvature conditions. Ingeneral, the theoretically consistent own-priceelasticities are larger in absolute value than thecorresponding unrestricted elasticities; differencesrange from 970 for hired labor to 64% for pesti-cides.

Table 4 and 5 present the short-run Marshallian,short-run Hicksian, and long-run Hicksian elastic-ities for the last observation (1986). As expected,own-price elasticities are negative for inputs andpositive for outputs, and Le Chatelier’s principle issatisfied in both the long-run/short-run and theMarshallian/Hicksian cases. All elasticities are inthe inelastic range except for hired labor. The re-sults of Table 4 show that in the short-run, exceptfor feeds, the difference between Marshallian andHicksian own-price elasticities is very small, dueto a small short-run expansion effect. This effect is

Table 3. Own-Price Short-Run MarshallianElnWicities, Illinois, 1986, with/without aTheoretical Consistent RestrictedProfit Function

DifferenceWith Without (%)

Hired laborFeedsSeedsFertilizerPesticidesFuelsCapitalOutOut

–2.076–0.242–0.291–0.078–0.104–0.048–0.601

0.061

– 1.884–0.134-0.255–0.065-0.037–0.039–0.487

0.032

9.244.612.416.764.418.819.047.5

Fernandez-Cornejo Demand and Substitution of Agricultural Inputs 45

Table 4. Marshallian/Hicksian Short-Run Elasticities of Input Demand, Illinois, 1986

HiredLabor Feeds Seeds Fertilizer Pesticides Fuel Capital

H. LaborMarshalliarr –2.076 –0.166 0.154 –0.091 –0.149 -0.282 2,878Hicksian – 2.035 –0.308 0.146 –0.117 –0.136 -0.291 2,741

FeedsMarshallian –0.028 –0.242 0.029 0.027 0.017 0.013 0,027Hicksian -0.052 –0.159 0.033 0.042 0.010

Seeds0.018 0.108

Marshallian 0.135 0.149 –0.291 –0.051 –0.075Hicksirm

–0.0250.128

0.1130.173 –0.290 – 0.046 –0.077 –0.024 0.136

FertilizerMarshalliarr –0.041 0.071 –0.026 – 0.078 0.072Hicksian

–0.017–0.052

– 0.0570.111 –0.024 –0.070 0.068

Pesticides–0.015 -0.018

Marshallian –0.100 0.069 –0.057 0.107 –0.104 0.111Hicksian

0,030–0.091 0,039 –0.059 0.102 –0.101 0.109

Fuels0.001

Marshallian –0.233 0.061 -0.024 –0.032 0.138 – 0.048 0.088Hicksian – 0.240 0.088 – 0.022 –0.027 0.135 – 0.047

Capital0.114

Marshallian 0.423 0,024 0.019 –0,019 0.007 0.016Hicksian

-0.6010.403 0.093 0.023 –0.006 0.000 0.020 –0,533

larger in the long run. Compared to the results ofthis paper, Lopez (1984) finds moderate long-runexpansion effects in Canadian agriculture, whileHiggins finds large expansion effects for Irish ag-riculture.

Focusing on chemical inputs (Table 6), the es-timated own-price short-run elasticities for fertil-izer lie at the lower end of the range of previouseconometric estimates using dual models (e.g.,Binswanger; Burrell), but they are higher than amore recent estimate by McIntosh and Shumwayfor California. Long-run Marshallian estimates aremore in line with previous estimates. The resultsare consistent with the estimates for nitrogen fer-tilizers for Illinois for 1980/89 by Vroomen andLarson using a direct approach and are also similarto estimates for the United Kingdom and WestGermany based on LP techniques (Buren). Thedemand for pesticides is also quite inelastic andthere is little published information on elasticitiesderived from dual methods to compare our resultswith. Earlier research (Miranowski) reports own-price elasticities of – 0.19 for herbicides and– 0.62 for insecticides used in the production ofcorn, based on 1966 cross-sectional data.

The intensification of agricultural production,made possible by fertilizers and pesticides, hasalso led to contamination of ground- and surface-water resources and contributes to the presence oftoxic residues in food. Environmental concernshave generated interest about alternatives to limitthe use of inorganic fertilizers as well as pesti-cides. The simplest alternative, and easiest to con-

trol, is the imposition of ad valorem taxes on thoseproducts.b The effectiveness of such taxes dependson the responsiveness of fertilizer and pesticidesdemand to increases in their prices. The results ofthis study show that such response is quite small.For example,’ a 10% reduction in fertilizer usewould require a 128% tax in the short run and a65% tax in the long run. To reduce fertilizer use tothe point that excess nutrient8 (available for leach-ing or runoff) is negligible would require a tax ofmore than 300% in the short run and near 200% inthe long run.9 In the case of pesticides (mainlyherbicides), a 10% reduction in use would requirea 96V0tax in the short run and a 269k0tax in thelong run. Thus, even moderate reductions in fer-tilizer or pesticide use would require substantialtaxes. On the other hand, taxes have a significant

6 Sweden, Austria, and Finland currently impose a 25% tax on rrhro-gen and phosphorous fertilizers (OECD). Iowa has a very small tax onnitrogen fertilizers for the purpose of raising revenue for research andextension activities related to gromrdwaterprotection.

7 The results presented in this paragraph are only approximate sincepoint elasticities are assumed to be approximately applicable. In addi-tion, feedback output effects from the markets of agricultural productsare beyond the scope of this paper.

s Following Huang and Lantin, excess nutrient is defined as tbe dif-ference between the amount nf nutrient applied from all sources (on anacre of crophmd) and the amount removed at the end of the growingseason in the grain and stalks. For example, they estimate the amount ofnitrogen required tn achieve zero excess in the production of com to beabout 110 lbs/a per year, The average amount of nitrogen used on comin 1986 was 155 lb/a in Illinois (Vroomen),

9 These figures can be compared to results for the European Commun-ity (EC), where the per acre application of fertilizer is more than twicethat of the U.S, According to Henrichmeyer, a 400% to 500’%tas onnitrogen would be required in the EC to achieve a noticeable effect,

46 April 1992 NJARE

Table 5. Hicksian Long-Run Elasticities of Input Demand, Illinois, 1986

With Respect to the Price of

HiredLabor Feeds Seeds Fertilizer Pesticides Fuels Capital Land

Hired laborFeedsSeedsFertilizerPesticidesFuelsCapitalLand

–11.450.0150.132

–0.028–0.191-0.218

0.497–0.044

0.090–0.191

0.1710.1000.0860.0770.0490.021

0.1500.033

–0.290–0.024–0.058–0.022

0.0220.000

– 0.0620.038

–0.047–0.072

0.108– 0.028–0.012

0.003

–0.2870.022

–0.0760.072

–0.1190.1390.017

–0.008

–0.2640.016

– 0.024–0.015

0.113–0.048

0.0170.001

3.3830.0560.133

–0.0370.0780.097

– 0.6050.034

8.4420.0110.0010.004

–0.0160.0040.015

–0.007

impact on farm income. A simple calculationshows that in the short run, the tax necessary toachieve a 10% reduction in fertilizer use wouldcause a 41?ZOdrop in farm income, while the taxrequired to decrease pesticide use by 10% wouldreduce income by 1770.

The signs of the cross-price input-demand elas-ticities are often used to classify inputs into net(gross) substitutes in production when the Hicksian(Marshallian) elasticity is positive or into net(gross) complements if negative. It is found thatfor Illinois, the classification into net and grosssubstitutes/complements coincides for all pairs. Inthe short run, all pairs are weak net and grosssubstitutes/complements except for the pair hiredlabor/capital, although this weakness is moderatedin the long run. Nearly 6070 of all pairs are short-run substitutes and about 70!Z0are long-run substi-tutes. It is interesting that in both the short run andthe long run, pesticides behave as net and grosssubstitutes for feeds, fertilizer, fuel, and capital,and fertilizer is a net and gross substitute for feedsand pesticides. The substitutabilities between pes-ticides, and fuels and capital may be related toalternative tillage practices, while substitutabilitiesbetween feeds and fertilizers, pesticides, and otherinputs may be due to the presence of purchasedfeeds.

Table 7 presents the short- and long-run Mo-rishima elasticities of substitution (MES) for 1986.For most inputs, differences in MES between theshort run and the long run are small except forsome of the input pairs that involve hired labor or

Table 6. Own-Price Elasticities of ChemicalInIW& Illinois. 1986

Short-Run Short-Run Long-Run Long-RunHicksian Marshallian Hicksian Marshallian

Fertilizer –0.070 –0.078 –0.072 –0,155Pesticides –0.101 –0.104 –0.119 -0,382

capital. The estimates also show more than 9090 ofthe input pairs exhibit short-run Morishima substi-tutability, while only 60% of the input pairs be-have as net (Allen) substitutes. This behavior issimilar to that noted by Ball and Chambers in adifferent context.

Strong Morishima substitutability is found forthe pair hired labor/capital for both the short andlong run. In addition, the large degree of asymme-try for that pair suggests that any policy that causessimilar percent decreases in the price of capital orincreases in the price of hired labor will inducevery different increases in the capital/hired-laborratio. For example, an increase of 10% in the priceof hired labor will lead to an 11Yoincrease in thelong-run capital/labor ratio. However, a 10% de-crease in the price of capital will lead to a 148%increase in the capital/labor ratio. All other pairsshow much weaker Morishima complementarilyin both the short and long run. For example, anincrease of 10% in the price of pesticides willincrease the fertilizer/pesticide ratio by onlyabout 2%.

The inherent asymmetry of the Morishima elas-ticities is very pronounced in the short and longrun, Only four input pairs exhibit a small/moderatedegree of asymmetry, They are fertilizer/feeds,pesticides/fuels, capital/seeds, and feeds/pesti-cides. The asymmetry in the pair fuels/capital,noted by Taylor and Gupta for southeastern agri-culture, is even more pronounced for Illinois.

Concluding Comments

The main contribution of this study is that it pro-vides a procedure whereby, based on estimation ofa theoretically consistent restricted profit functionand using a series of decomposition equations, alldemand functions (Hicksian and Marshrdlian in theshort run and the long run) and the elasticities ofsubstitution are readily calculated without estimat-

Fernarrdez-Cornejo Denrand and Substitution of Agricultural [nputs 47

Table 7. Short- and Long-Run Morishima Elasticities of Substitution, Illinois, 1986

HiredLabor Feeds Seeds Fertilizer Pesticides Fuels CaOital

Hired LaborSRaLR”

FeedsSRLR

SeedsSRLR

FertilizerSRLR

PesticidesSRLR

FuelsSRLR

CapitalSRLR

0.000.00

1.7211.5

2.1811.6

1.9211.4

1.9011.2

1.7411.2

4.7714.8

0.110.21

0.000.00

0.190.22

0.200.23

0.170.21

0.180.21

0.270.25

0.420.92

0,460.46

0.000.00

0.240.24

0.210.21

0.270.27

0.430.42

0.020.04

0.180.17

0.050.05

0.000.00

0.140.14

0.060.06

0.050.04

0.01-0.07

0.140.21

0.040.06

0.200.23

0.000.00

0.210.23

0.100.20

-0.19-0.17

0.130.12

0.020.03

0.020.02

0.180.19

0.000.00

0.160.14

0.941.10

0.630.65

0.560.63

0.530.59

0.530.62

0.550.62

0.000,00

aSR is short-run; LR is long-run.

ing the cost function. Unlike previous work, thisstudy imposes simultaneous convexity in pricesand concavity in quasi-fixed factors. Both curva-ture properties are essential in the use of decom-position methods to avoid violations of Le Chate-lier’s principle. In addition to the short-run andlong-run Hicksian and Marshallian elasticities,both Allen-Uzawa and Morishima ES can be cal-culated, although the theoretical evidence favorsthe use of the Morishima elasticities. Thus, theproposed methodology provides a wealth of infor-mation, facilitating analysis of the flexibility ofproduction systems, The model is disaggregated inthe input side to obtain information (e.g., pesti-cides) not readily available from this type ofmodel, and a new weather index is introduced.

More empirical work is needed using dual mod-els with a fair amount of disaggregation in theinput side. While these efforts are facilitated by theuse of two-stage modeling, some separability andhomotheticity assumptions still need to be estab-lished. The task can be simplified by drawing onnonparametric results such as Lim and Shum-way’s. Provided the estimated model is theoreti-cally consistent, all elasticities will be of the cor-rect sign and Le Chatelier’s principle will be sat-isfied. For the case of Illinois, the producers’responsiveness to price changes for fertilizer andpesticides is small, in particular in the short run,implying that the impact of taxes imposed on theseinputs of the size imposed in some countries in

Western Europe (25%) will be negligible in theshort run and small in the long run.

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