Modeling Supply Response in a Multiproduct Framework Revisited: The Nexus of Empirics and Economics

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1 Modelng Supply Response n a Multproduct Framework Revsted: The Nexus of Emprcs and Economcs V. Eldon Ball Charles B. Moss Kenneth W. Erckson and Rchard Nehrng* Paper prepared for presentaton at the Amercan Agrcultural Economcs Assocaton Annual Meetng, Montreal, Canada, July 27-30, 2003 * Charles Moss s a professor n the Department of Food and Resource Economcs at the Unversty of Florda. Eldon Ball, Kenneth Erckson, and Rchard Nehrng are economsts wth the Unted States Department of Agrculture, Economc Research Servce. The vews presented n ths manuscrpt are those of the authors and do not necessarly represent those of the USDA.

2 Modelng Supply Response n a Multproduct Framework Revsted: The Nexus of Emprcs and Economcs Introducton Consderable effort has been devoted to the estmaton of aggregate agrcultural supply response models. Analysts and polcy makers all seek to estmate the mpacts of changes n government programs, exchange rates, commodty prces, and trade polces on relatve prces of nputs and outputs, the dstrbuton of economc rents, and on the output response of producers and nput supply response of supplers. Supply elastctes ndcate the speed and magntude of output adjustments n response to changes n product prce. The elastcty parameter for aggregate farm output s especally mportant for publc polcy snce t measures the ablty of the farm sector to adjust producton to changng economc condtons. Tweeten and Quance (1969) note that publc polces concerned wth the earnngs of all farm resources and total farm ncome must consder the aggregate response of farm output n a dynamc economy. The aggregate response of output to prce depends on total resource adjustments n agrculture. Agrcultural supply response models nclude both commodty supply response models (Weaver; Whttaker and Bancroft; Whpple and Menkhaus; Chavas and Holt; Ln et al.; Schmtz) and regonal supply response models (Fawson and Shumway; Shumway and Alexander; Melke). Marc Nerlove s adaptve model and partal adjustment model guded much of the emprcal analyss of dynamc agrcultural supply response over the last decades. Berndt notes that Nerlove s 1961 research on Returns to Scale was the frst emprcal applcaton of dualty of producton and cost. Askar and Askar and Cummngs surveyed the econometrc evdence of the effects of prces on farm supply. 1

3 The Nerlovan model of agrcultural supply response was qute popular n the past (Nerlove, 1956; Hossen and Cummngs, 1976). However, more recently, econometrc research has focused on supply analyss where agrculture s vewed as a mult-nput, mult-product ndustry (e.g., Antle (1984, 1999), Shumway (1984, 1988, 1995), Chambers and Just (1989), S. Ray, Fulgnt and Perrn, Ball (1988), Bnswanger et al., Mundlak). Ths research has produced useful estmates of agrcultural output supply and nput demand functons. Most of ths analyss of aggregate supply response has reled almost exclusvely on standard econometrc methods. One seeks flexble functonal forms that do not mpose a pror restrctons on supply-demand elastctes. These nclude the quadratc (Shumway, Vasavada and Chambers, Huffman and Evenson), the generalzed Leontef form (Lopez), and the translog (Antle; S. Ray; Ball,1988, 1997; Fulgnt; Kumbhakar). Choosng among the many possble functonal forms s dffcult because dfferent functonal forms can sometmes ft the data relatvely well whle generatng dfferent supply-demand elastctes (Debold and Lamb). Most models of supply response n agrculture focus on aggregate (across commodtes) supply or on own-prce response for a sngle commodty. Moreover, most models that do recognze multple outputs typcally specfy transformaton functons whch mpose severe a pror restrctons on the structure of producton (Ball, 1988). The objectve of ths paper s to model supply response n agrculture usng dsaggregated output data and to test statstcally key assumptons tradtonally mantaned n agrcultural supply studes. Followng Vasavada and Chambers (1986), Shumway (1984, 1988), and Ball (1988), we use U.S.-level data, to estmate a multproduct supply response model for U.S. agrculture, and report our prelmnary results. In subsequent analyss we wll mpose the a pror assumpton that the technology s weakly separable n major categores of outputs. Wth 2

4 ths restrcton, we propose to derve the dsaggregated supply response functons (Ball, 1988; Ball et al., 1997). In developng ths model, we dscuss conceptual ssues regardng the estmaton of restrcted proft functons that form the nexus between emprcs and economcs. These nclude ncorporatng restrctons from economc theory (curvature), and complcatons caused by the presence of multple quas-fxed factors and contegrated tme seres data. The artcle s organzed as follows. The next secton dscusses the restrcted proft functon model (Ball, 1988) and presents restrctons from economc theory. These restrctons on the emprcal model are mposed as part of the mantaned hypothess. Separable and nonjont producton structures are among the features to be tested n further development of the model. In subsequent sectons, the estmaton procedure and data are descrbed and emprcal results are presented. We also dscuss how measurement ssues complcate the estmaton of supply response. These ssues nclude both dffcultes n estmatng varables at the state level, and also problems assocated wth estmatng a resdual return to fxed factors n the presence of one or more quas-fxed factors. Some concludng comments are offered n the fnal secton. The Restrcted Proft Functon Estmatng the model requres that we specfy a functonal form for the restrcted proft functon and consder the queston of aggregaton (Shumway and Davs, 2001). Chambers states functonal form and aggregaton le at the heart of appled producton economcs. He notes that the form should be as general as possble and should restrct the ultmate outcome as lttle as possble. Also, choosng a functonal form lmts the range that the analyss can take. Once a general model s specfed, classcal statstcal tests can only be conducted under the presumpton that the general model s vald (Chambers, pp ). 3

5 We choose the transcendental logarthmc (translog) form (Bandt and Chrstensen, 1973; Al, 1976) because ths class of flexble functonal forms can model very general producton structures. However, ther applcaton to the many output case s hampered by the fact that the estmatng equatons are smple monotonc transformatons of prces or quanttes whch are often hghly correlated. We reduce the multcollnearty problem at the cost of mposng a pror the restrcton that the producton structure s weakly separable n major categores of outputs. Imposton of the separablty restrcton (Pope and Hallam; Blackorby et al.; Moschn, 1992) yelds two key smplfyng results. Frst, weak separablty ensures consstent aggregaton. Second, the exstence of an aggregate that s homogeneous n ts components mples a two-stage optmzaton procedure. Stage one, choose the optmal mx of commodtes wthn the aggregate (ths justfes the specfcaton of a model n the components alone). Stage two, choose the level of the aggregate (ths justfed the specfcaton of a model n the aggregates alone). In subsequent analyss we wll estmate the dsaggregated supply response functons for lvestock, crops, and secondary outputs, as descrbed n Ball (1988, 1997, p. 280). The restrcted proft functon s approxmated by the transcendental logarthmc (translog) functon wth arguments, P, X, and t, where tme t ndexes the level of technology. P s a vector of output and nput prces, whle X s a vector of output and nput quanttes. The subscrpt M refers to varable nputs and outputs, whle N refers to the fxed nput, land. Subscrpts of the parameters of the translog functon:, j = 1,,l, (outputs) k = l +1,, M (nputs), and s = 1,,N (proft share equatons). 4

6 M (1) ln = ln P M M + / j1 N N + 1/2 M + 1 j1 M j1 k1 N N j1 t jt j ln Pt N + j jk j1 ln j X j ln P ln P ln X ln P ln X ln X j j ln X j t ( t) 1/ 2 t j t k tt 2. The translog functon s vewed as a second-order Taylor s expanson about the unt pont. The followng symmetry restrctons are mposed by the equalty of cross-partal dervatves n a quadratc expanson (2). j j, jk kj Homogenety of degree one n prces requres M (3) 1, 1 M 1 j M 1 j M 1 t 0. Followng Ball (1988) let T be the set of all feasble nput and output combnatons. T s assumed to be a nonempty, compact, and convex set. In addton, the technology s assumed to exhbt constant returns to scale. Under the assumptons made on T, the restrcted proft functon s homogeneous of degree one n fxed nputs (Ball, 1988, pp ). In ths case we assume one fxed nput, land. Ths requres N (4) j 1 1 (one fxed nput), j1 N j1 jk 5

7 N j1 j N j1 jt 0. Usng Hotellng s lemma, ln ln P PY S, whch appled to (1) yelds the proft share equatons M (5) S ln P ln X t, j1 j j N j1 j j t = 1,, M. Because the translog functon s an approxmaton about a pont, the hypothess tests wll requre that the hypothess holds only at the pont of approxmaton. Approxmate weak separablty mposes the restrctons (6),, jk j k js js j s,j = 1,,l, k = l+1,,m, s = 1,,N, on the parameters of the translog functon. Lnear homogenety of the aggregator functon h() n output prces mples that the ratos of output supply functons are homogeneous of degree zero n output prces. Wrtng ths condton usng Euler s theorem yelds l k 1 P k / P / Pj P k 0, j,,j = 1,,l, whch mposes the further restrctons l (7) 0, = 1,,l. k 1 k 6

8 Fnally, nonjontness n nputs requres that the parameters of the translog approxmaton satsfy (8), j,, j = 1,,l. j j The Emprcal Model The emprcal model dentfes three output categores (crops, lvestock, and secondary outputs). Secondary outputs are those secondary actvtes whose costs cannot be separately observed from those of the prmary agrcultural actvty. Examples nclude the provson of machne servces, contract feedng of lvestock, recreatonal actvtes, and other actvtes nvolvng the use of the land and the means of agrcultural producton (Ball, 2002). There are three varable nputs (materals or purchased nputs, labor, and non-land captal), and a tmetrend ndex. Land s treated as a fxed nput. We mpose homogenety and symmetry of cross-prce dervatves restrctons. We shall test the cost functon for homothetcty snce rejecton of ths property would ndcate that aggregaton of agrcultural producton s nvald. We also examne whether there s jontness n the producton of crops and lvestock n the sense that the margnal cost of one s nfluenced by the output of the other. Data We analyze the structure of agrcultural producton usng the translog approxmaton to the cost functon usng neoclasscal dualty results. Data are from the USDA, Economc Research Servce. We use U.S.-level ( ) estmates of the varable purchased nputs, labor, and captal nputs, plus the fxed nput, land. We dvded all prces by the prce of materals (P6) so that the translog second order approxmaton holds n the neghborhood of log(1) = 0. 7

9 Gross Output The measure of output uses dsaggregated data for physcal quanttes and market prces of crops and lvestock. The Economc Research Servce (ERS) compled these data. The quantty data exclude producton that s used on the farm as nput. Prces correspondng to each dsaggregated output reflect the value of that output to the producer; that s, subsdes are added and ndrect taxes are subtracted from market values. Prces receved by farmers, as reported n Agrcultural Prces, nclude an allowance for net Commodty Credt Corporaton loans and purchases by the government valued at the average loan rate. However, drect payments under federal commodty programs are not reflected n the data. Intermedate Inputs One of the components of ntermedate nputs s feed, seed, and lvestock purchases. Intermedate goods produced wthn the farm sector are ncluded n ntermedate nput only f they also have been ncluded n output. Another component s agrcultural chemcals. To account properly for changes n nput characterstcs or qualty, we construct prce ndexes of fertlzers and pestcdes usng the hedonc regresson technque. The basc premse underlyng ths approach s that prce dfferences across goods are due manly to qualty dfferences that can be measured n terms of common attrbutes. The fnal components of Intermedate Inputs are petroleum fuels, natural gas, and electrcty; and other purchased nputs. Labor Input The ndexes of labor nput ncorporate data from both establshment and household surveys. Estmates of employment, hours worked, and labor compensaton are controlled to ndustry totals based on establshment surveys that underle the U.S. natonal ncome and product accounts. 8

10 These totals are allocated among categores of the work force cross-classfed by the characterstcs of ndvdual workers on the bass of household surveys. The resultng estmates of hours worked and average compensaton per hour are used to construct the ndexes of labor nput. Captal Input Estmates of the captal stock were constructed for each asset type. For deprecable assets, we employ the perpetual nventory method to estmate captal stocks from data on nvestment. Estmates of the stocks of land and nventores are mplct quanttes based on balance sheet data. We constructed estmates of rental prces for each type of asset. We derve mplct rental prces based on the correspondence between the purchase prce of an asset and the dscounted value of future servce flows derved from that asset. Deprecable captal assets nclude nonresdental structures, motor vehcles, farm tractors and other equpment. Data on nvestment are obtaned from the U.S. Department of Commerce s Bureau of Economc Analyss s (BEA) Fxed Reproducble Tangble Wealth n the Unted States. Land Land stocks are measured as mplct quanttes derved from balance sheet data (USDA- NASS and ERS). To obtan a constant qualty land stock we compute translog prce and quantty ndexes of land n farms. Aggregaton s at the county level (Ball, 2002). Land s treated as a fxed nput n our model. Emprcal Results Equatons for proft shares were estmated usng nequalty-constraned maxmum lkelhood methods. The parameter estmates for the most general model are reported n table 1 together 9

11 wth ther estmated standard errors. The hypothess of weak separablty n output prces and the hypothess that the technology s nonjont n nputs are also tested. The rejecton of ths hypothess s consstent wth the observaton of multproduct farms. Ths suggests that polces whch may be drected at a sngle output may be expected to affect all producton decsons, not smply those made wth respect to the partcular commodty for whch the polcy s targeted. US-level Supply Response Marshallan gross elastctes of supply and demand are estmated for the mantaned model, wthout mposng curvature restrctons (table 2). In subsequent analyss we wll mpose theoretcal curvature restrctons usng the Cholesky factorzaton. Ths ensures postve(negatve) own-elastctes of supply (demand). These results are extremely mportant because they mply a set of nequalty restrctons on the entre matrx of gross elastctes. All elastcty estmates satsfy the normal case restrctons when evaluated at the pont of approxmaton. Whereas n Ball (1988), the own-elastctes of supply were generally less than unty (table 3), these results are qute dfferent. Most sgnfcantly, the output supply elastcty wth respect to secondary output does not have the expected sgn. Elastctes wth respect to secondary output are more dffcult to nterpret. Also, whereas Ball (1988) found the nput demand functons were generally prce elastc, several of the nput demand functons do not have the expected sgns (postve) suggestng that an ncrease n the prce of the factor ncreases the demand for the factor. Ths clearly contradcts economc theory. One reason for these contrary results s that whereas Ball mposed curvature restrctons on the model n 1988, we have not yet mposed them n ths model. 10

12 Concluson Restrcted Proft Functon U.S. level equatons for proft shares were estmated usng nequalty-constraned maxmum lkelhood methods. We estmated the output supply and nput demand elastctes usng the parameter estmates from the restrcted proft functon and revenue and cost shares model wthout mposng curvature restrctons on the model. Accordngly, the sgns and szes of these estmates dd not agree wth those expected from economc theory. Therefore, n subsequent analyss, we shall mpose curvature restrctons on the model. As we estmate the translog supply response model, several ssues emerge. These nclude: mposng curvature restrctons, estmatng regonal supply response models, possble measurement errors when there are more than one quas-fxed factors, and contegraton of the tme seres data. Curvature --- Cholesky Factorzaton The estmates presented do not reflect the mposton of curvature restrctons (followng economc theory) for a well-behaved producton functon (McFadden). Ball et al. (1997) note that n order for the normalzed restrcted proft functon whch s lnearly homogeneous n prces to be a convex functon, t s necessary and suffcent that the matrx A be postve defnte. Lau (1978) has shown that every postve defnte matrx has a Cholesky factorzaton. The matrx A can thus be wrtten n terms of the Cholesky decomposton as A = LDL where L s a unt lower trangular matrx and D s a dagonal matrx wth typcal element value. Lau demonstrates that f the matrx s to be postve defnte then postvty on the D referred to as a Cholesky D >0. Thus mposng D s suffcent to mpose convexty on the restrcted proft functon. 11

13 Mundlak (2000, p. 327) notes that those studes where convexty s not confrmed should go no further because the remanng results have no theoretcal support. Ths s exactly what our prelmnary results have shown. They do not support the assumpton of convexty n the proft functon. However, we plan to re-estmate the model wth curvature mposed (Ball, 1988; Fernandez-Cornejo; Pars and Howtt; Pars and Caputo; Barnett and Lee; Shumway, 1995). Shumway s (1995) examnaton of the statc, appled, dual, agrcultural-producton economcs lterature yelds these conclusons wth respect to testng for monotoncty, curvature, symmetry, and homogenety. Frst, many of the test rejectons are not based on statstcal tests but rather on falure of the unconstraned estmates to satsfy the hypothess. The rejecton may not be sgnfcant n ether a statstcal or an economc sense. Second, there s no reason to expect that all (or perhaps any) of the four tested mplcatons would hold for an aggregate of frms even f they held perfectly for each frm. Shumway notes that to conduct a crtcal test of the theory requres mcro-level data, data that are even more detaled than that used n most frm-level analyses. Nevertheless, one mght well be cautous about mposng curvature on flexble functonal forms. The curvature condtons arse from theoretcal assumptons that characterze specal operatng envronment (e.g, free dsposablty of nputs and outputs). Many frms do not operate n ths envronment. Even f the operatng envronment satsfed the theoretcal assumptons, the fact that one needs to mpose curvature that s not supported by emprcal data should be a sgn that somethng s ms-specfed (Wennger). Ball mposed convexty on the translog specfcaton locally, as a pont of approxmaton, by the Cholesky LDL method (Ball, p. 815). Unfortunately, however, wth the translog, the curvature condton may be volated at other ponts of the regressor space (Wolff). Ths s why some mpose the curvature globally. In the case of the translog, global mposton can smply be 12

14 mplemented wth the Jorgenson and Fraumen method (Dewert and Wales, p. 48). We are wrtng software code to mplement the Cholesky factorzaton n SAS, followng some recent work by Moschn (1998a, 1999). Fnally, Barnett (2002) notes that although econometrcans often mpose curvature globally, they typcally mpose monotoncty locally f at all. But wthout satsfacton of both curvature and monotoncty, the second-order condtons for optmzng behavor fal, and supply functons become nvald (Barnett, p. 199). Extenson of the Restrcted Proft Functon Model to U.S. Farm Producton Regons The estmaton of the restrcted proft functon and of the factor demand and output supply response equatons s based on the assumpton that n long-run equlbrum and perfect competton, factor returns are exhausted (Euler s Theorem). Usng U.S.-level tme seres data, the resdual return to land (the fxed factor) s always postve. However, usng state-level data, ths s often not true. Ths s partly because measurement of farm ncome and expenses and ther proper allocaton across states s more complcated than t s at the U.S. level. Ths clearly complcates the development and estmaton of a regonal supply response model. Therefore, we wll postpone development of regonal supply response models untl we have successfully completed development and estmaton of the U.S.-level model. Quas-fxed Factors Furthermore, the presence of multple quas-fxed factors affects the measurement of resdual rents. The estmaton of the restrcted proft functon and of the shadow prces of varable and quas-fxed nputs s dependent on the proper classfcaton of varable and quas-fxed nputs. If a quas-fxed nput s treated as varable by usng a market prce n place of a shadow value, then the mputed value of the nput n queston s ms-stated (Mshra, Moss, and Erckson). Therefore, 13

15 the presence of one or more quas-fxed factors can cause ms-estmaton of resdual returns, and thus of supply response. Contegraton of Tme Seres Data and Its Effects on Parameter Estmates Lm and Shumway (1997, 1999) note that t s common practce to mplctly assume that all data are generated from statonary processes and to nclude tme n the regresson as a proxy for techncal change and/or omtted varables. However, Engle and Granger et al. have shown that many economc tme seres have characterstcs of a random walk. That s, they are not statonary around a functon of tme but are statonary n dfferences. If the data are nonstatonary, then regressng one nonstatonary seres on others and/or on a tme trend could yeld spurous results and adversely nfluence hypothess test conclusons because tests based on standard asymptotc results wll have the wrong sze. Therefore, we propose to perform unt root and contegraton tests to nvestgate whether economc tme seres are statonary and whether ther lnear combnatons representng the cost share equatons are contegrated. Based on these test results, a new dual model mght be specfed and estmated to reflect these contegratng relatonshps. 14

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21 Table 1. Iteratve Seemngly Unrelated Regresson: R-Square, Adjusted R=Square, Durbn-Watson, and Parameter Estmates Equaton R-Square Durbn-Waston LnProft RS RS RS CS CS Restrcted proft functon: LPROFIT=A0+A1*LOG(P1/P6)+A2*LOG(P2/P6)+A3*LOG(P3/P6) +A4*LOG(P4/P6)+A5*LOG(P5/P6) +LOG(P1/P6)*(0.5*A11*LOG(P1/P6)+A12*LOG(P2/P6)+A13*LOG(P3/P6) +A14*LOG(P4/P6)+A15*LOG(P5/P6)) +LOG(P2/P6)*(0.5*A22*LOG(P2/P6)+A23*LOG(P3/P6)*A24*LOG(P4/P6) +A25*LOG(P5/P6)) +LOG(P3/P6)*(0.5*A33*LOG(P3/P6)+A34*LOG(P4/P6)+A35*LOG(P5/P6)) +LOG(P4/P6)*(0.5*A44*LOG(P4/P6)+A45*LOG(P5/P6)) +LOG(P5/P6)*(0.5*A55*LOG(P5/P6)) +B15*LOG(P5/P6) +T*(C11*LOG(P1/P6)+C12*LOG(P2/P6)+C13*LOG(P3/P6)+C14*LOG(P4/P6) +C15*LOG(P5/P6)) +D1*T+0.5*D11*T**2; Revenue share equatons: RS1=A1+A11*LOG(P1/P6)*A12*LOG(P2/P6)+A13*LOG(P3/P6)+A14*LOG(P4/P6) +A15*LOG(P5/P6)+B11*LOG(A)+C11*T; RS2=A2+A12*LOG(P1/P6)*A22*LOG(P2/P6)+A23*LOG(P3/P6)+A24*LOG(P4/P6) +A25*LOG(P5/P6)+B12*LOG(A)+C12*T; RS3=A3+A13*LOG(P1/P6)*A23*LOG(P2/P6)+A33*LOG(P3/P6)+A34*LOG(P4/P6) +A35*LOG(P5/P6)+B13*LOG(A)+C13*T; Cost share equatons: CS1=A4+A14*LOG(P1/P6)+A24*LOG(P2/P6)+A34*LOG(P3/P6)+A44*LOG(P4/P6) +A45*LOG(P5/P6)+B14*LOG(A)+C14*T; CS2=A5+A15*LOG(P1/P6)+A25*LOG(P2/P6)+A35*LOG(P3/P6)+A45*LOG(P4/P6) +A55*LOG(P5/P6)+B15*LOG(A)+C15*T; 20

22 Table 1. Iteratve Seemngly Unrelated Regresson: R-Square, Adjusted R=Square, Durbn-Watson, and Parameter Estmates (contnued) Parameter Estmate Std Err t Value Pr > t A [.256] A [.013] A [.324] A [.000] A [.000] A [.001] A [.196] A [.024] A [.000] A [.000] A [.015] A [.459] A [.000] A [.000] A [.002] A [.003] A [.805] A [.005] A [.226] A [.692] A [.000] B [.412] B [.005] B [.926] B [.000] B [.000] C [.776] C [.555] C [.006] C [.000] C [.000] D [.722] D [.002] Note: 1 s lvestock, 2 s crops, 3 s secondary outputs, 4 s labor, 5 s non-land captal, and 6 s materals. 21

23 Table 2. Output Supply and Demand Elastctes 1/ Elastcty wth Respect to Prce of Secondary Commodty Lvestock Crops output Labor Captal Materals Lvestock Crops Secondary output Labor Captal Materals / Uses USDA-ERS data, Curvature restrctons are not mposed on the model. 22

24 Table 3. Output Supply and Input Demand Elastctes 1/ Elastcty wth Respect to Prce of Other Durable Farm- Pur- Lve- Flud Ol- Other Equp- Real Produced Hred chased Commodty stock Mlk Grans seeds Crops ment Estate Durables Labor Energy Inputs Lvestock Flud mlk Grans Olseeds Other crops Durable equpment Real estate Farm produced durables Hred labor Energy Other purchased nputs / V. Eldon Ball, Table 6, p. 823, Modelng Supply Response n a Multproduct Framework, Amercan Journal of Agrcultural Economcs, November, Curvature restrctons were mposed on the model usng the Cholesky decomposton (pp ). 23