SUPPLY RESPONSE WITHIN THE FARMING SYSTEM CONTEXT ESTIMATING AGGREGATE SUPPLY RESPONSE FROM PRODUCTION AND PROFIT FUNCTIONS

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1 SUPPLY RESPONSE WITHIN THE FARMING SYSTEM CONTEXT WEEK 3: DAY 1 ESTIMATING AGGREGATE SUPPLY RESPONSE FROM PRODUCTION AND PROFIT FUNCTIONS by Coln Thrtle and Yog Khatr, Unversty of Readng and London School of Economcs CONTENTS 1. DIRECT ESTIMATION OF SUPPLY RESPONSE FROM THE PRODUCTION FUNCTION 1.1. The Determnants of Aggregate Supply 1.2. Estmaton of Supply Parameters from the Producton Functon 2. THEORETICAL ADVANCES: FLEXIBLE FUNCTIONAL FORMS AND DUALITY 2.1. Introducton 2.2. Flexble Functonal Forms 2.3. Dualty: Producton, Cost and Proft Functons 3. A PROFIT FUNCTION APPROACH TO LAND REFORM IN ZIMBABWE 3.1. Overvew 3.2. Introducton 3.3. The Dual Proft Functon Approach 3.4. Data 3.5. Elastcty Results and Interpretaton 3.6. Returns to Research 3.7. Shadow Prces 3.8. Conclusons REFERENCES LIST OF TABLES Table 1. Calculatng Contrbutons to Supply Response Table 2. Estmated Elastctes Table 3. Shadow Prces of Captal and Buldngs Table 4. Shadow Prces for the Fxed and Condtonng Factors LIST OF FIGURES Fgure 1. The Structure of Dualty Relatonshps 1

2 1. DIRECT ESTIMATION OF SUPPLY RESPONSE FROM THE PRODUCTION FUNCTION 1.1. The Determnants of Aggregate Supply The dscusson n ths chapter concentrates on the supply functon because of ts obvous relevance to agrcultural prce polcy. However, n the overall context of producton, the response of output to changes n the output prce s only one of many relatonshps that may be of nterest. We noted that ths ssue was addressed by Bnswanger (1990), who argued that although ndvdual crops respond strongly to prce factors, ths s at the expense of alternatve outputs. The short run aggregate response s very low, because the man nputs, land, labour and captal, are fxed. To get a good aggregate response n the long run requres more resources and/or better technology and nfrastructure nvestments n roads, markets, rrgaton, educaton and health. We now proceed to ncorporate some of these varables n our model Estmaton of Supply Parameters from the Producton Functon Supply response to prce s determned by a combnaton of (1) the physcal response of output to a change n the level of nput use and (2) the behavoural response of farmers n changng the level of nput use, n response to changes n the prce of outputs (or nputs) 1. The producton functon (Fgure 1, week 2, day 5) measures the physcal response of output to nputs. In the Fgure, we consdered only one varable nput n order to be able to draw a two dmensonal dagram. Now, the producton functon can be stated wth one output as a functon of several nputs, whch may be varable or fxed: Y f ( X 1, X 2, X 3,..., X n) 1 where Y s output and X 1, X 2,..., X n are nputs such as land, labour, anmal and mechancal power, mplements, fertlser and other agrcultural chemcals. The elastcty of output, Y, wth respect to an nput, X, s defned as: ε y x δ δy/y X / X δy/ δ X δlny Y/X δ LnX 2 whch measures the physcal response. The behavoural response of farmers to changes n the output prce s defned as: ε x p y δ δ X / X P y / P y δ X / δ P y δ X / P y LnX δ LnP y 3 whch s the elastcty of nput demand for X wth respect to the output prce, P y. 1 Some care s needed here. The elastcty of output wth respect to the nput prce was defned n the last chapter n equaton (11). The concept beng ntroduced here s the elastcty of nput use wth respect to the output prce. If we are lookng at the aggregate nput, then the two wll be the same, but wth opposte sgns, provded that the supply functon s homogenous of degree zero. 2

3 Under farly general assumptons these two elastctes can be combned to provde a measure of supply response that s an alternatve to the drect estmaton of the supply functon: y ε s p εyx εx p Σ 4 Summng the product of the output elastcty and the nput demand elastcty over all nputs, X, gves the supply elastcty of good Y, for a change n the prce of the output; the contrbuton of an ndvdual nput, X, s just (ε yx )(ε xpy ). It s possble to estmate the producton functon wth data only on outputs and nputs, but a functonal form must be chosen. It s convenent to start wth a functon that s lnear n logarthms: LnY Ln ß 0 + ß 1LnX 1 + ß 2 LnX , + ß n LnX n 5 Ths form s known as the Cobb Douglas. One attractve feature s that lttle data s requred, and another s ease of nterpretaton. It should not come as a surprse that the coeffcents, ß, are the output elastctes. Ths has to be so, snce from equaton (5): ε yx δlny δ δ δ LnX Y/ X Y/X ß 6.e. the coeffcent s just the slope of the functon, wth both varables defned n logarthms. Ths s convenent, but we can go further. Even f the functon cannot be estmated, we can calculate output elastctes, provded we are prepared to assume that the Cobb Douglas s approprate and that the system s n equlbrum. In equlbrum, the value margnal product of an nput wll equal ts prce (secton 1.1, yesterday), whch we rearranged to gve: MPP x δy δ X P P x y 7 and for the Cobb Douglas, manpulaton gves the MPP of X : MPP x δy δ X ß Y X 8 f the MPP s equal to the rato of prces, as n (7), then multplyng both sdes by X /Y gves the output elastcty: ε yx δy δ X X Y P X x P yy 9 The rght-hand term s the share of factor n total output (or total cost), meanng that the output elastctes for a (constant returns) Cobb Douglas are smply the factor shares, whch can usually be calculated drectly from the data. 3

4 The analyss so far covers only the estmaton of the physcal parameters of the producton functon, that s, the output elastctes wth respect to the nputs. To allow estmaton of the elastcty of output wth respect to the output prce, equaton (4) ndcates that we need also to estmate the elastcty of nput demand wth respect to the output prce (equaton (3)). Ths requres estmaton of ether the supply functon or the nput demand functon. The lnk between the two s made explct by Tweeten (1989, Ch.5). Suppose, the nput demand and output supply functons are assumed to be lnear n logarthms, and we stck to the case of one aggregate nput. We can specfy an nput demand functon much lke the output supply equaton, (equaton (14), yesterday), but we gnore competng products for smplcty and drop the technology term because t should not be mportant n nput demand. Then the nput demand functon for the aggregate nput could be wrtten as: where X s the aggregate nput, P y s the output prce, P x s the nput prce ndex, and I s a vector of relatvely fxed nfrastructure varables. Then, suppose technology s measured by an ndex of output quantty to nput quanttes, T Y/X. Multply both sdes of the nput demand functon n equaton (10) by ths ndex to get: whch s the output supply functon, provded that ß 3 s approxmately unty, whch t should be. Takng logarthms of equatons (10) and (11) gves almost dentcal equatons for nput demand (equaton (12)) and output supply (equaton (13)): and X ß X Y X Y ß P y 0 P x 0 1 P y ß2 I 10 Px Note that ß 1 appears twce n both equatons. It can be nterpreted as the short run supply response 2 to the output prce. But, the output supply elastcty and the nput demand elastcty wth respect to the output prce are both equal to ß 1 : ß ß 1 ß2 ß3 I T 11 Ln X Ln ß + ß Ln P - ß Ln P + ß Ln I 0 1 y 1 x 2 12 LnY Ln ß + ß Ln P - ß Ln P + ß Ln I + ß Ln T 0 1 y 1 x ε s p δlny δ LnP y ε xp y δlnx δ LnP y ß 1 14 and the elastcty of output wth respect to the nput prce s equal to the elastcty of nput demand wth respect to the nput prce, where these elastctes are just the same as those above, but wth the sgn changed: ε δlny 2 yp x xp x 1 In the last chapter, we consdered LnP δlnx ε x x the partal adjustment LnP -ß 15 δ δ model, used to estmate short and long-run response. 4

5 Thus, ß 1 can be estmated from the nput demand functon, whch s the statstcally preferred method. Or, t may be estmated from the output supply functon, wth a technology ndex. If ths s not avalable, t must be estmated from the output supply functon, wth a tme trend as a proxy for technology, but ths can ntroduce consderable specfcaton error. So, there s a varety of alternatve means of generatng a supply elastcty. Before leavng the subject, note that we have not yet used equaton (4) to explot any output elastctes that may have been calculated from the producton functon estmaton. Ths has happened because we have so far looked only at a sngle aggregate nput. The result s that Σ ε yx n equaton (4) s approxmately equal to unty 3, leavng; ε s p δlny δ LnP y ε xp y δlnx δ LnP y ß 1 16 However, f output elastctes for the separate nputs were estmated from equaton (5) and elastctes of nput demand wth respect to ther own prces had been estmated, usng the logarthmc verson of last chapter s equaton (13), the contrbutons of dfferent nputs to the supply response can be calculated. When the nputs are fertlser, land, labour and rrgaton, Table 1 gves some hypothetcal results and calculatons. The table shows that because the contrbuton to supply response s the product of the two elastctes, the greatest effect wll be for prce changes n an nput where both are reasonably large. Reducng land prces wll be neffectve, because although the physcal relatonshp to output s mportant, there s no behavoural response of land use to prce changes. Machnery prces are an effectve polcy tool, because the output elastcty and the nput demand elastcty are relatvely large, gvng ths varable the largest share n the total supply response. Table 1: Calculatng Contrbutons to Supply Response Input Output Elastcty Elastcty of Input Demand Fertlser Land Labour Machnery Irrgaton Total Contrbuton to Supply Response By now the reader should be convnced that some knowledge of theory s an asset n the estmaton of supply elastctes. We leave ths topc wth two further examples of the uses of theory to overcome data nadequaces. Frstly, there s a result assocated wth work by Mundlak and others, that shows how reasonable estmates of short run supply responses at the ndustry level can be obtaned from the output elastctes alone. Suppose that the output elastctes were 3 A value of unty would ndcate constant returns to scale.

6 as n Table 1 and that n the perod beng consdered, land and labour should be vewed as fxed, whle fertlser, machnery and rrgaton as varable. The extent to whch the short run supply can change n response to prce s shown to be the sum of the varable elastctes dvded by the sum of those that are fxed. So, n ths hypothetcal example, ε p 0.4/ The other example s attrbutable to Tweeten (1989, Ch.5) who shows that all the output prce elastctes n a complete system can be calculated from the nput elastctes. Alternatvely, all of the cross-prce elastctes for outputs and the nput elastctes can be calculated from the ownprce output elastctes alone. A more complete treatment of ths area requres dualty theory. Ths s a lterature whch has developed rapdly n the last two decades. Tweeten (1989, Ch.5) provdes an ntroducton and some references to dualty theory whch wll be addressed n the next secton. 2. THEORETICAL ADVANCES: FLEXIBLE FUNCTIONAL FORMS AND DUALITY 2.1. Introducton Three basc methodologes are used to tackle the measurement and explanaton of agrcultural effcency and productvty. They are, econometrc estmaton of the producton, cost or proft functon; the accountng approach, usng ndex number theory and non-parametrc programmng technques, sometmes called data envelopment analyss 4. For the purposes of analysng supply response, we have so far used the producton functon and wll now concentrate on the dual proft functon, snce t provdes a unfyng framework, from whch the output supply and nput demand functons can be derved. All the approaches and representatons start form the basc noton of a relatonshp between outputs, Y and nputs, X j. The smplest form s the sngle output producton functon, Y F(X j ), whch was the startng pont for week 2, day 5 s analyss. Ths s a purely techncal relatonshp and economcs s only ntroduced when an economc problem s stated, such as maxmsng profts wth the producton functon as the technologcal constrant. The three major recent advances n ths area are the development of flexble functonal forms, dualty theory, and the applcaton of tme seres technques assocated wth the concept of contegraton. We covered contegraton yesterday and wll now ntroduce new functonal forms and dualty Flexble Functonal Forms The Cobb Douglas producton functon s unduly restrctve. In the 1970's and 1980's, flexble functonal forms were developed, such as the translog. These functons dffer from old fashoned functonal forms by ncorporatng enough estmated parameters to take account of the nteractons between varables and to allow for non-lnearty n the parameters. Thus, a typcal specfcaton s the translog: 4 1 LnY Ln α 0 + α LnX + β j LnX LnX j 17 2 The nput-output approach, whch s partcularly useful for examnng sectoral nteractons, s not dscussed here. 6

7 where Y s aggregate output, the X s are nputs and all the α s and ß j s are coeffcents. Wthout the last term, ths s the famlar Cobb Douglas that was dscussed n week 2, day 5. Ths last term allows for nteractons between nputs, when j. These terms allow the elastctes of substtuton between each par of nputs to be estmated from the data, whereas the Cobb Douglas mposes substtuton elastctes of unty for all pars of nputs. 5 Thus, f two nputs, such as R&D and extenson expendtures, are complements rather than substtutes, ths wll be taken nto account. When j, the last term adds squared terms for each nput, whch allows for non-lnearty and estmaton of a system of smultaneous equatons consstng of the producton functon tself and all but one of the nput share equatons. The nternal rate of return to R&D can be estmated n ths framework, snce the coeffcent(s) of the R&D term relate changes n R&D expendtures to changes n output Dualty: Producton, Cost and Proft Functons The second major advance s the use of dual forms, nstead of drect estmaton of the producton functon. Dualty s not a new concept and now even undergraduate textbooks pont out that the average and margnal cost curves are smply the average and margnal product curves nverted. Ths s because they are derved from the famlar S shaped total product and total cost curves, whch are also the nverse of each other. Alternatvely, n the programmng lterature, reversng the objectve functon and the constrant s known to gve the same soluton. Thus, maxmsng output subject to a cost constrant (movng along a gven socost lne untl the tangency wth the hghest possble soquant s found) s the same as mnmsng costs subject to an output constrant (movng along a gven soquant untl the tangency wth the hghest possble socost lne s found). Fgure 1 shows how modern dualty theory has generalsed these farly obvous concepts. The most ntutvely appealng progresson s the dervaton of the dual cost functon on the left hand sde of the fgure. We begn by statng the problem, (1), whch s to mnmse costs, subject to the producton functon constrant, wth a partcular level of output, Q *. The X j s are varable nputs, the Z k s are fxed nputs, the Rs are nput prces, C s total cost and F represents some functonal form. Formally, the problem s solved by settng up the Lagrangan and takng dervatves wth respect to all the varables. These are the frst order condtons, (2), whch are not shown, wth each dervatve set equal to zero, to fnd maxmum or mnmum values. The system s solved wth the varable nputs as the dependent varables. Ths gves the nput demand functons (3), for all the X j s as a functon of the nput prces, the levels of the fxed nputs and the level of output. The last term ndcates that these are the output constraned nput demand functons, whch are the producton equvalent of the Hcksan compensated demand functons n consumer theory. Substtutng the expressons for the X j s nto the objectve functon (4) gves the dual cost functon, (5), whch expresses costs, C, as a functon of the nput prces, the levels of the fxed nputs and the level of output. [Fgure 1: The Structure of Dualty Relatonshps] 5 Ths follows from the fact that the Cobb Douglas output elastctes are also the factor shares (when constant returns s mposed) and they reman constant. The only way ths can happen s f nput prce changes are exactly compensated by quantty changes, whch requres untary elastctes of substtuton. 7

8 The pont of all ths s not the dervaton tself, but ts mplcatons. The dualty relatonshp ndcated between the constraned cost mnmsaton problem and the dual cost functon means that the dual cost functon contans all the nformaton n the constraned mnmsaton problem. Thus, we no longer need to solve these complcated problems, but can start nstead by defnng a functonal form for the dual cost functon. Then, Shephard's Lemma proves that all we need to do to retreve the nput demand functons s to take the dervatves of the cost functon wth respect to the nput prces. Ths enormously smplfes the process of generatng the estmatng equatons needed to model agrcultural producton wth the behavoural assumpton of cost mnmsaton ncorporated. The lmtaton of the cost functon s that lke the producton functon, t s restrcted to a sngle output. 6 Ths s unfortunate, snce most farms are multple output enterprses. The transton to multple outputs s made by statng the more general problem of proft maxmsaton, shown n (8), on the rght hand sde of the fgure. The notaton s the same as for the cost functon, wth the addton of P s, whch are output prces. Now, the output mx can be vared, so proft s maxmsed as the sum of value of all the outputs, (P Q ), mnus the costs of the varable and fxed nputs, subject to the constrant of the transformaton functon whch s ncluded to allow for crop swtchng. Solvng the frst order condtons (9) gves both the nput demand functons and the output supply functons (10), wth output prces, nput prces and the levels of the fxed factors as the ndependent varables. Rather than constructng ad hoc supply functons, as n the analyss presented n week 2, day 5, we now have a complete system of supply functons and wll generate a complete set of own and cross prce elastctes. Furthermore, estmaton of the system of output supply and nput demand functons also produces shadow values for the fxed nputs. These shadow prces are the margnal revenue products of these nputs and can be used to calculate rates of return to the nvestments n captal tems, nfrastructure and technology varables. Thus, the proft functon provdes a sound theoretcal bass for the study of long run aggregate supply response, whch takes technologcal change and nfrastructure nto account. Agan, the dervaton s only useful, because t shows the relatonshps nvolved. The pont s that substtutng the output supply and nput demand functons nto the objectve functon, gves the dual proft functon (12). Hotellng's Lemma (13) shows that takng dervatves wth respect to output prces and nput prces drectly gves a system of output supply and nput demand functons (14), whch can be estmated. Thus, we have no further need of the constraned maxmsaton problem and can begn our analyss by specfyng a flexble functonal form for the dual proft functon. Ths we do n the next secton, whch s an applcaton of the dual proft functon to commercal agrculture n Zmbabwe. 3. A PROFIT FUNCTION APPROACH TO LAND REFORM IN ZIMBABWE 3.1. Overvew 6 In fact, t s possble to generalse the cost functon to the multple output case, but for our purposes the proft functon s preferable. 8

9 The purchase of commercal farm land n Zmbabwe for resettlement has been a major factor n government polcy snce ndependence n 1980, but from 1980 to 1989 only 52,000 famles were relocated. The Land Acquston Bll of 1992 made compulsory purchase easer, and at present the government has announced ts ntenton to consderably ncrease the rate of resettlement. But, Zmbabwe has a serous food securty problem and the output effects of land redstrbuton are a matter of dspute. The World Bank estmate that 3,000,000 hectares of commercal farmland are under-utlsed s contested by the Commercal Farmer's Unon. Fttng a normalsed resdual proft functon to the data for the commercal sector, allows estmaton of the shadow prce of commercal farm land. We fnd that the model suggests that the World Bank s correct, n that the margnal value product of land s negatve, meanng that there s underutlsaton. However, negatve values of captal assets are common when real nterest rates are negatve, so the result should be treated wth some cauton. Also, the problem of dentfyng the under-utlsed land s not trval and redstrbutng ntra-margnal land would have output effects Introducton "Zmbabwe's one mllon communal farm households are restrcted to half the total area suted for agrcultural producton. The other half s occuped by 4,500 large-scale commercal farmers, most of whom are whte. To compound ths nequalty, the communal lands have a much lower agrcultural potental; 74% of the communal lands s n natural regons IV and V, and 51% of the commercal farmng area s n natural regons I-III (CSO, 1989). Ths grossly unequal land dstrbuton s the most fundamental and least tractable of all Zmbabwe's problems. It s also a sgnfcant cause of food nsecurty n the rural areas." (Chrstensen and Stack, 1992). There s also, n theory, an effcency argument for land redstrbuton, snce n any dual economy, output can be ncreased by redstrbutng resources untl ther margnal products are equal n the two sectors. But, t s wdely accepted that the communal farmers cannot produce at the same level as the commercal farmers wthout consderable support, and the government s already under extreme pressure to cut expendtures. Wthout consderable nvestment 7, the expectaton s that food producton would decrease, exacerbatng the food securty problem. Chrstensen and Stack (1992) estmate that 420,000 rural and 125,000 urban households are sufferng from chronc food nsecurty. In ths respect, the land reform ssue n Zmbabwe s qute dfferent from the South Afrcan stuaton, where output exceeds consumpton by a wde margn 8 and food grans are exported at below cost. Thus, South Afrca can afford to redstrbute land, even f the result s a substantal declne n output, but Zmbabwe cannot gnore the possblty that land reform could result n even greater food securty problems. Whereas many of the arguments over land reform are complex, the value of margnal land n the commercal sector can be estmated qute smply. One legacy of the colonal past s that Zmbabwe has a statstcal system not much dfferent from the UK, whch has collected agrcultural statstcs for the natonal ncome accounts that can be used for the estmaton of 7 The cost of resettlng 52,000 famles, from has been about US $112 mllon (Bratton, 1991) 8 Self-suffcency ndces for South Afrca, wth 100 meanng suffcency, show gran producton at 150, hortcultural products at 132 and lvestock producton at 98 (van Zyl et al (1993). 9

10 producton relatonshps. The data for the commercal sector s qualtatvely not much dfferent from the nformaton avalable n European countres (ndeed, better than some). These data were used for the Total Factor Productvty estmates n Thrtle et al (1993), but drect comparson of the two sectors was delberately avoded, on the grounds that they are too dssmlar. However, by fttng producton, cost, or proft functons to the two sectors separately, estmates of varables such as margnal products and shadow prces of nputs can be derved. These ndcate relatve factor scarctes, allowng quantfcaton of the costs and benefts of reallocatng resources between the two sectors. The next secton brefly explans the proft functon approach used n ths study, whch was more fully explaned n secton 2. Ths s followed by the elastcty results, shadow prces and the calculaton of the returns to research. The fnal secton concludes by consderng the polcy mplcatons The Dual Proft Functon Approach The proft functon provdes estmates of a full range of economc varables, whereas the producton functon and the TFP ndex concentrate only on the physcal relatonshps between nputs and outputs. The commercal sector s treated as sngle producton unts to whch the restrcted or varable proft functon (Lau 1972, 1976) s appled. Consder a multple output technology producng outputs y, ( 1,..., m), wth the respectve expected output prces p, usng n varable nputs x j, (j 1,..., n) wth prces w j. We then defne varable expected profts as: π p y - w x whch s smply the value of output mnus varable costs. Normalsng the proft functon wth respect to an output or nput prce has the practcal advantages of ensurng that the homogenety requrement s met and reduces the number of parameters to be estmated. Also let p represent nput prces as well as output prces, to keep the notaton compact. Thus, f: j j j 18 * p0 p p 19 p * s a vector of normalsed output and nput prces and the functonal form for the generalsed quadratc proft functon s defned as: π * π * * * * α 0 + α p β j p p j + γ k p z k p 0, j, k where β j β j 20 where π * s normalsed profts and z k s a vector of fxed or quas-fxed nputs and condtonng factors, such as R&D and patents. The vector α and matrces β, γ contan the parameter 10

11 coeffcents to be estmated. Applyng Hotellng's lemma, we derve the optmal levels of output supply and nput demand respectvely: * y + j p j + j w j + k z k j j k α β β γ 21 * - x α + β p + β w j + γ z k 22 j where we have agan dstngushed between output and nput prces, so that (21) are the output supply functons and (22) are the nput demand functons. Then, the prce elastctes of outputs and nputs for the non-numerare cases are: j j j j k k η η j j - p βj y * j w βj x * j 23 Convexty of the proft functon mples that the own-prce elastctes should be postve for an output and negatve for an nput. The cross-prce elastctes for pars of nputs are negatve for complementary nputs and postve for substtutes. For pars of outputs, postve cross-prce elastctes mply complementarty n supply and output substtutes are ndcated by negatve cross-products. If the elements of z are treated as short-run constrants on producton, we can derve the effects of relaxng the z varable constrants on the output and varable nput levels. We can derve these effects n elastcty form by logarthmc dfferentaton of (21) and (22) wth respect to the elements of z: ε εk jk k z γ y - z x k j k γ jk 24 Shadow prces for the varables n the z vector can be derved as partal dervatves of the proft functon (Dewert, 1974, Huffman,1987) wth respect to the z varables. The derved shadow values can be nterpreted equvalently as () the margnal change n profts for an ncrement n a partcular element of z, () as the mputed rental value for an addtonal unt of that factor or () the effects on expected proft of relaxng the partcular constrant represented by each z varable. The shadow value of land (treated as fxed) provdes the mplct value n producton as opposed to the market prce. The dfference between the market prce and shadow value ndcates 11

12 whether land s over, under or optmally utlsed. The shadow prces of the other condtonng factors (such as R&D) can be used to assess ther effectveness Data The data was compled by Thrtle et al (1992) and a detaled descrpton of the basc data can be found theren. Here we defne the complaton of the quas-fxed nputs and condtonng factors and dscuss the aggregaton of the output and varable nput groups. The outputs are Dvsa aggregated nto three groups: food crops (Y1), ndustral crops (Y2) and lvestock and lvestock products (Y3). The frst group contans largely maze and other grans. The second group contans largely tobacco, coffee and other export crops. The thrd aggregate s of anmals and anmal related outputs. The varable nputs are Dvsa aggregated nto four groups. These are, hred labour (XL), lvestock related nputs (feed, veternary costs, purchases from the communal sector, etc) (XV), chemcal/crop related nputs (fertlser, other chemcals and packng) (XC) and runnng costs (vehcle mantenance, transport, sundres, servces and lcences, etc) (XO). Vehcles and fxed captal n the form of buldngs and other fxed mprovements are assumed to be quas-fxed. Foregn exchange constrants support the fxty of farm captal. Two captal categores were constructed. These are farm vehcles (CAP) and buldngs (BLD). The number of tractors and combne harvesters was used together wth ther purchase prces to construct a value of agrcultural machnery stock (at purchase prces). Assumng, on average, that each vehcle s worth half of ts purchase prce, we dvded the value derved by two. Deflatng by the FAO machnery prce ndex for Zmbabwe, we derve a stock of machnery. No detals on gross fxed captal formaton could be found for the perod, so to construct a seres for buldngs, we assumed that the buldng mantenance and repars expendture was equal to the deprecaton on buldngs and that the average lfe-span of a buldng or fxed mprovement was 30 years. Thus, we can construct an mplct stock of buldngs. Subtractng net own account captal formaton 9 and deflatng by an ndex of buldng materal prces, we derve the stock of buldng and fxed mprovements. Total area of land (LAND) n the commercal sector s ncluded as a fxed nput. Other fxed, exogenous or condtonng factors ncluded are, Research and Extenson (RES), Ranfall (RAIN) and world patents pertanng to agrcultural machnery and chemcals (PAT). The research and extenson expendtures were aggregated nto an agrcultural knowledge stock. Patents are used to capture transferable technology (and thus nternatonal spllovers). The patents are smply the total number of agrculture-related patents regstered n the US by all countres. Ths number s the straght aggregate of the number of mechancal and chemcal patents. A stock varable of nternatonally produced and avalable knowledge was constructed from the patent numbers. 9 Own account captal formaton s both an output and an nput. Thus adoptng the net output approach (USDA, 1980), we subtract fxed captal formaton net of materals (e value added) from the output and nput sde of the account. 12

13 3.5. Elastcty Results and Interpretaton The system of output and varable nput equatons were estmated usng an teratve Zellner procedure, whch provdes maxmum lkelhood parameter estmates on convergence of the weghted error-covarance matrx. Convergence was acheved for the normalsed system wth symmetry mposed. The symmetry restrctons could not be tested due to a lmted number of degrees of freedom, but were mposed to ensure consstency wth the contnuty property of the proft functon. The system provdes a majorty of sgnfcant parameter estmates (at the 1% level) and the 'goodness of ft' measure represented by R 2 s of the estmated supply and demand system equatons vary between 0.77 and 0.99, whch s hgh, even for a tme seres model. The parameter estmates themselves have lmted economc nterpretaton, and are thus relegated to the workshop presentaton. The parameters are used to construct measures of elastctes and shadow prces for the quas-fxed and fxed nputs. 13

14 Table 2: Estmated Elastctes * Dependant Varable Regressors Y1 Y2 Y3 XL XV XC XO ** P1 0.8 (4.4) (-.99) (-1.2) (3.8) (-4.4) 0.36 (2.92) 0.33 P (-0.98) (-1.6) 0.47 (2.8) 0.33 (4.1) 0.68 (3.6) (-1.8) 0.37 P (-1.2) 0.28 (2.8) 0.83 (3.4) (-3.9) (-0.74) (-3.9) 0.49 WL (-3.8) 0.22 (4.1) -0.5 (-3.9) (-1.4) 0.33 (1.95) 0.27 (2.97) 0.06 WV (-4.4) 0.22 (3.6) 0.11 (0.74) 0.16 (1.95) 0.33 (0.97) 0.3 (2.76) WC 0.34 (2.9) (-1.8) (-3.9) 0.16 (2.97) 0.36 (2.76) -0.4 (-3.8) 0.32 WO CAP (-1.8) 0.33 (.78) 1.4 (3.6) 0.34 (1.8) 2.1 (4.5) (-0.9) 12.1 BLD (-0.2) (-1.8) (-4.4).07 (0.37) (-1.54) 0.99 (2.0) -6.3 LAND (-0.49) (-2.0) 1.0 (3.4) 1.1 (7.8) 0.6 (1.7) 0.14 (0.4) 4.9 RES 0.9 (2.1) (-0.5) (-0.64) (-0.39) 0.5 (1.8) 0.96 (3.2) -1.6 PAT 0.58 (1.7) (-0.4) (-1.4) (-3.4) 0.06 (0.3) 0.48 (2.1) -2.7 * t-values are n parentheses; the crtcal value s taken to be 2.26 ** t-values are not computed for numerare nput elastctes as the numerare nput and the derved elastctes are ganed resdually from (6). Table 2 summarses the short-run elastctes of supply and varable nput demand wth respect to prces, quas-fxed nputs and condtonng factors at the varable means. The sgnfcant own-prce supply and demand elastctes (on the dagonals of the upper shaded blocks) 10 have the expected sgn, and are of plausble magntudes. Note that at ths level of aggregaton, we get supply elastctes for food crops and lvestock of about 0.8, whereas the hghest elastcty for aggregate output n Table 3 n the last chapter was The own-prce elastcty of the ndustral crop aggregate has the wrong sgn, but the t-statstc ndcates that the elastcty s not sgnfcantly dfferent from zero. The nput demand elastctes form the other upper shaded block. Apart from lvestock related nputs (not sgnfcant), the varable nput own-prce elastctes have the expected sgns. However, the hred labour elastcty s of low sgnfcance and for runnng costs, whch was the numerare, we get no t-statstc. Thus, 10 The elastctes that have meanngful economc nterpretatons are n the shaded areas of table 2; the others are not dscussed. 14

15 only the crop nput own-prce elastcty s sgnfcantly dfferent from zero at hgh confdence levels. Indeed, all the own-prce output supply and nput demands are nelastc. For the outputs, complementarty (substtutablty) s ndcated by a postve (negatve) crossprce elastcty. Thus, ndustral crops and lvestock are complements and food crops are not related to ndustral crops or lvestock, due to the low t values. Input complementarty (substtutablty) s ndcated by a negatve (postve) cross-prce elastcty. Thus, lvestock nputs may be complementary to runnng costs (no t statstc for the numerare) and labour, lvestock nputs and crop nputs are all substtutes for one another. If we consder the quas-fxed, fxed and condtonng factors as constrants n producton, the long-run output and varable nput elastctes wth respect to these factors can be regarded as the responses to relaxng these constrants. The quas-fxed nputs are stock varables that are endogenous n the long-run, but changng ther levels requres nvestment. Thus, n the short run, the costs of adjustng these stock levels may be consdered n terms of forgone producton. The levels of the condtonng varables are assumed to be beyond the control of farmers and the costs of adjustment are not consdered to be ncurred by farmers. Thus, snce the reported elastctes are for the short-run, we mght predct net negatve output elastctes wth respect to fxed and quas-fxed factors and postve output elastctes wth respect to the condtonng factors representng technology. However, the effect on ndvdual outputs cannot easly be predcted, as changng captal stock levels, or technology levels may favour certan outputs and also affect the varable nput levels, whch n turn affects output. These elastctes are reported n the shaded lower part of the table. The food crop output elastctes follow the predcted pattern; wth respect to machnery, buldngs and land the elastctes are negatve (but nsgnfcant) and postve and sgnfcant wth respect to research and nternatonal technology spllovers. All the elastctes for the ndustral crops are nsgnfcant and for lvestock, machnery and land appear to ncrease output, even n the short run. The effects of changes n the fxed nputs and technology varables on the varable nputs are mostly nsgnfcant, but ncreasng machnery ncreases lvestock nputs and runnng costs. Increasng buldngs reduces runnng costs and ncreasng land rases both labour nputs and runnng costs. For the technology varables, R&D ncreases crop nputs (whch s reasonable, snce mproved varetes use more fertlser and pestcde), but technology spllovers reduce both labour nputs and runnng costs. Ths s entrely sensble, snce the majorty of patents are for machnery Shadow Prces The shadow prces for the quas-fxed, fxed and condtonng factors provde measures of the mplct value n producton of addtonal unts of the factors. In equlbrum, the shadow prces of a quas-fxed factor should equal ts opportunty cost, or rental value. Excess capacty or under-utlsaton of a quas-fxed nput would be ndcated by an estmated shadow prces less than the opportunty cost. Smlarly, under-nvestment s ndcated by a shadow value greater than the opportunty cost, ndcatng that revenue can potentally be ncreased by ncreasng the stock of the quas-fxed factor untl the shadow prce equals the opportunty cost (Berndt and Fuss, 1986; Morrson, 1986). 15

16 The economc reasonng behnd these propostons s sound enough, but does not take good account of economes wth persstent hgh nflaton and negatve nterest rates. In such crcumstances, the opportunty cost of captal nvestment s negatve, but rental rates are not. For Zmbabwe, f the opportunty cost s taken to be the real return on bank deposts, the rate has been negatve snce For long term nvestments, such as 25 year government stock, the average real rate of nterest has been negatve over the same perod. Ths mples that a ratonal farmer should nvest n captal up to the pont where the return s negatve. The Zmbabwe case s further complcated by ratonng and allocaton of farm machnery, whch would suggest that the supply s nadequate (at these prces). These factors should be taken nto account n nterpretng Table 3, below, whch reports the values of the estmated shadow prces of captal, buldngs, land and the technology varables, and the opportunty costs. The opportunty cost of machnery captal s taken to be the real rate of return on bank deposts and that for buldngs the real rate of nterest on 25 year stock (Bouchet, 1987). Consderng frstly farm captal (whch we took to be harvesters and tractors), we see that there exsted an excess capacty untl the md-1970's. Ths appears consstent wth the general encouragement of the government then, to mantan growth n agrcultural output. Towards ndependence, nflaton grew and real nterest rates became ncreasngly negatve. The over captalsaton, thus became under captalsaton by the opportunty cost crtera. That s, fnancal captal nvested n machnery deprecated n real terms less quckly than the alternatve of bank deposts. The assumpton that bank deposts are the man alternatve to captal nvestment s strengthened by the exstence of foregn exchange controls and restrctons on foregn nvestment. As we see from the table, the shadow value of captal n the post ndependence perod tends toward the opportunty cost of captal. Thus, although the mplcaton of a negatve shadow prce, s a negatve margnal product (mplyng an economcally neffcent allocaton of that factor), the polcy restrctons on alternatve forms of nvestment mply that, post ndependence nvestment n machnery captal stock s consstent wth the opportunty cost crtera. Investment n buldngs and other fxed captal s a longer-term nvestment decson, and s therefore, subject to a greater degree of uncertanty. The opportunty cost of such long-term nvestments would be related to longer term fnancal assets, and thus the yeld on long bonds for example, mght be used as a measure of the opportunty cost of nvestment n buldngs and fxed captal. We use the yeld on 25 year government stock together wth the CPI to derve the opportunty cost of nvestment n buldngs n a smlar manner to that derved for captal. Before suggestng the mplcatons of comparng the shadow prce to the opportunty cost of buldng stocks, we note the complaton of the buldngs stock s far from deal (smply a multple of the costs of mantenance and repars) and the summary measure (at varable means) s not sgnfcantly dfferent from zero, and therefore, subject to assumpton that the measure of buldng stocks s relable. Table 3: Shadow Prces of Captal and Buldngs Year Shadow prce of Captal Opportunty Cost of Captal Shadow Prce of Buldngs Opportunty Cost of Buldngs 16

17 At Var means * (-3.36) (0.143) 1.02 * t-values n parentheses The fgures n the table mply that there was under-nvestment n buldngs and fxed captal before ndependence and a movement toward the negatve rate after ndependence. The possble under-nvestment n the pre-ndependence perod mght be attrbutable to the expectaton of ndependence and the related uncertanty. In the post ndependence perod, the shadow prce of buldngs s stll largely less than, but possbly movng towards the trend n, the opportunty cost. Ths stll ndcates under-nvestment, and as the nvestment decsons relatng to buldng are very long-term, ths may ndcate contnued uncertanty of the largely whte commercal farmers wth respect to ther future postons, n the lght of past and current land reform polcy. We now move on to consder the fxed factors n the model, shown n Table 4. Table 4: Shadow Prces for the Fxed and Condtonng Factors Year Shadow Prce of Land Shadow Prce of Research Shadow Prce of Patents

18 At Var. Means * (-4.6) (.43).109 (2.05) * t-values n parentheses The table ndcates a negatve shadow prce for land n the commercal sector for the perod and shadow prces evaluated at the varable means s negatve and hghly sgnfcant. A negatve shadow value for land mples that land area s not an effectve constrant to producton n the commercal sector. The shadow values become a larger negatve over the perod, even after the polcy of land redstrbuton from the commercal sector to the communal areas. Possble reasons for ths nclude, the adopton of new chemcal and bologcal technologes that are effectvely land substtutes (less land area requred to produce a gven quantty of a gven crop). Ths does seem to be supported wth respect to the food crop and ndustral crop outputs by the negatve elastctes of these outputs wth respect to land area. It s also possble that the land redstrbuton has largely been restrcted to underutlsed or low qualty land n the commercal sector. Furthermore, even wth the 15% or so of land that has already been redstrbuted, there s stll suffcent land n the commercal sector such that t stll does not represent an effectve constrant to producton. Ths result appears to support the land reforms programme n Zmbabwe. The shadow prces of the research knowledge stock (RKS) and the patent knowledge stock (PKS) ndcate the addton to proft for a unt ncrease n the stock varables. The shadow prce of research peaks n the early 1980's and dmnshes thereafter, becomng negatve n the last four years. Ths mples the counter-ntutve result that addtons to the RKS n these years reduced proftablty. Ths s possble f funds for research compete wth other technology or nfrastructure projects, thus research funded at the cost of other potental 18

19 projects mght show a net negatve margnal product. We note, however, that the shadow value of RKS evaluated at the varable means s nsgnfcant. The shadow value of the PKS at the varable means s large relatve to that of the RKS but they are not strctly comparable as the RKS relates to a stock value and the PKS relates to patent numbers. The shadow value at the varable means s sgnfcant and represents the addton to profts attrbutable largely to the research systems of other countres (.e., spllover effects). Some developmental research expendture may be requred to customse the avalable stock of nternatonal technology for local use. Thus, the PKS may be takng some of the credt for developmental research, resultng n a declnng or even negatve shadow prce for the RKS Returns to Research We found above that research augmented food crop producton and that there s some a pror expectaton as well as emprcal evdence that research s neutral wth respect to other outputs. Assumng that research relates only to food crops, we can derve the margnal product of the RKS from the supply equaton of food crops as the partal dervatve of food crops wth respect to the RKS. The margnal product of the RKS n food crop producton s found to be 2.18 (sgnfcant at the 95% level). We can use the formula n Ito (1991, p.7) to estmate an nternal rate of return (IRR), gven the complaton of the RKS and ts margnal product, as: where r s the IRR, L s the dffuson lag and the partal dervatve gves the margnal product of the RKS. Assumng a dffuson lag of 5 years (Thrtle et al, 1993) we derve an estmated IRR to publc sector research of 36%. Thrtle et al (1993) derve an IRR of 43% usng the same RKS n a prmal (translog producton functon) model. However, no account was taken of nternatonal spllovers n that model, mplyng that some upward bas exsted n the estmated IRR due to the omsson of a varable representng nternatonally avalable technology. Lastly, the shadow prce of patents s postve and sgnfcant, ndcatng that nternatonal spllovers are mportant. Ths cannot be easly quantfed because the seres s the number of patents regstered, whch has no obvous connotatons, n terms of fnancal magntudes Conclusons exp(r, L) 0 Y exp(-rt)dt 25 (RKS) The model generates supply elastctes of 0.8 for two of the three enterprse groups, whch s consderably hgher than the results reported n the prevous chapter. The model suggests that the World Bank s correct, n that the margnal value product of land s negatve, meanng that there s under-utlsaton. However, negatve values of captal assets are common when real nterest rates are negatve, so the result should be treated wth some cauton. The extent of the dstortons of macroeconomc varables, such as the nterest rate, must have a consderable effect on the effcency of resource allocaton n the agrcultural sector. The combnaton of the over-valued exchange rate and negatve real nterest rates would lead to undue substtuton of captal for labour. Wth unemployment estmated at about one mllon, 19

20 mnmsng employment n agrculture makes no sense at all, and has only been restrcted by the shortage of foregn exchange. Lastly, the returns to research appear to have fallen at the end of the perod and Zmbabwe s now more dependent on nternatonal spllovers of agrcultural technology. 20

21 REFERENCES Berndt, E. and Fuss, M. (1986). Productvty measurement wth adjustments for varatons n capacty utlsaton and other forms of temporary equlbrum. Journal of Econometrcs (33): Bnswanger, H. (1990). The polcy response of agrculture. World Bank Economc Revew. (4): Bouchet, F. (1987). An Analyss of the Sources of Productvty Growth n French Agrculture, Unpublshed Ph.D. Thess. Vrgna Polytechnc Insttute and State Unversty. Bratton, M. (1991). Ten Years After: Land Redstrbuton n Zmbabwe, In Prosterman, Temple and Hanstad (eds.), Agraran Reform and Grassroots Development: Ten Case Studes. Curry Foundaton. Chrstensen, G. and Stack, J. (1992). The Dmensons of Household Food Insecurty n Zmbabwe, Workng Paper No.5, Food Studes Group, Oxford. Central Statstcal Offce, Zmbabwe Government, (1989). Statstcal Yearbook, Harare. Dewert, W. (1974). Applcatons of Dualty Theory. In M.D. Intrlgator and D.A. Kendrck (eds.), Fronters of Quanttatve Economcs: Volume 2. Amsterdam: North-Holland. Huffman, W. E. (1987). Research Bas Effects for Input-Output Decsons: An Applcaton to U.S. Cash-Gran Farms. Proceedngs of a Symposum, Atlanta, Georga, Jan.29-30, 1987, Mscellaneous Publcaton , Mnnesota. Agrcultural Experment Staton, Unversty of Mnnesota, St.Paul, MN. Ito, J. (1991). Assessng the Returns of R&D Expendtures on Post-War Japanese Agrcultural Producton. Research Paper No. 7, Natonal Research Insttute of Agrcultural Economcs, Mnstry of Agrculture, Forestry and Fsheres, Tokyo. Revew of Economcs and Statstcs, (54): Lau, L. J. (1972). Proft Functons of Technologes wth Multple Inputs and Outputs. Lau, L. J. (1976). A Charactersaton of the Normalsed Restrcted Proft Functon. Journal of Economc Theory, (12) 1: Morrson, C. J. (1985). On the Economc Interpretaton and Measurement of Optmal Capacty Utlsaton wth Antcpatory Expectatons. Revew of Economc Studes (52): Thrtle, C., Atkns, J., Bottomley, P., Gonese, N., Govereh, J. (1992). The Effcency of the Commercal Agrcultural Sector n Zmbabwe, Hull Papers n Developng Area Studes, No.6. Thrtle, C., Atkns, J., Bottomley, P., Gonese, N., Govereh, J. and Khatr,Y. (1993). Agrcultural Productvty n Zmbabwe, Economc Journal (103): Tweeten, L. (1989). Agrcultural Polcy Analyss: Tools for Economc Development. Boulder, Colorado: Westvew Press. Unted States Department of Agrculture (1980). Economcs, Statstcs and Cooperatves Servce, Measurement of US Agrcultural Research Productvty: a Revew of Current Statstcs and Proposals for Change. Techncal Bulletn No.1614, Washngton, D.C. van Zyl, J., van Rooyen, C. J., Krsten, J. and van Schalkwyk, H. (1993). Land Reform n South Afrca: Optons to Consder for the Future. Paper presented at the ESRC Development Economcs Study Group, London. by Coln Thrtle and Yog Khatr 21

22 Unversty of Readng Department of Agrcultural Economcs and Management 4 Earley Gate, Whteknghts Road P.O. Box 237 Readng, Berks., RG6 2AR Unted Kngdom 22