Testing Separability in a Generalized Ordinary Differential Demand System: The Case of Nigerian Demand for Meat

Transcription

1 Testng Separablty n a Generalzed Ordnary Dfferental Demand System: The Case of Ngeran Demand for Meat Ebenezer O. Ogunynka Graduate Student Department of Agrcultural Economcs Kansas State Unversty, 342 Waters Hall, Manhattan 66506, Kansas emal: eogun@agecon.ksu.edu Thomas L. Marsh Assstant Professor Department of Agrcultural Economcs Kansas State Unversty, 342 Waters Hall, Manhattan 66506, Kansas emal: tlmarsh@agecon.ksu.edu Selected Paper prepared for presentaton at the Southern Agrcultural Economcs Assocaton Annual Meetng, Moble, Alabama, February 1-5, 2003 Copyrght 2002 by Ebenezer Ogunynka and Thomas Marsh. All rghts reserved. Readers may make verbatm copes of ths document for non-commercal purposes by any means, provded that ths copyrght notce appears on all such copes.

2 Abstract Ths study nvestgates consumer demand for meat products n Ngera and tests f any of the meat products are separable from one another. Estmatng a generalzed nested demand system, the Rotterdam model was selected usng an adjusted lkelhood rato test. Results ndcate that mutton and pork are more elastc than beef and poultry. Only pork was found to be separable from other meat types. 1. Introducton To date there has been very few emprcal studes focusng on demand analyss for Ngera. The majorty of studes have focused on producton ssues because of lmted data avalablty for consumers and other constrants. An exhaustve search on consumer demand for meat n Ngera usng varous natonal and nternatonal archves (ncludng that of Agrcola) has not been frutful. Pursung an analyss of consumer demand for meat types n Ngera wll provde ntal nsght nto consumer responsveness to prce changes and allow more precse welfare calculatons that are necessary for polcy analyss. The study s objectves are: () to nvestgate the approprate demand model n the analyss of the consumer demand for meat usng the Generalzed Ordnary Dfferental Demand System; () to test the separablty structure between the meat products n order to gan a better understandng of Ngeran consumer choces n proten (meat) demand; and () to suggest the polcy as well as the dscplnary mplcatons of the fndngs. The paper s organzed as follows. Frst, some background nformaton on Ngera s provded. Then, Secton 2 presents the separablty concepts and the revew of related studes. The dscusson of the data and the estmaton procedure are gven n 1

3 Secton 3. Secton 4 contans results and dscusson. Secton 5 ncludes the summary and the concluson. 1.1 Background Informaton Domestc consumpton of Ngeran meat products: beef, mutton, goat meat, pork and poultry have been met largely by domestc producton over the years. Avalable nformaton ndcates that mutton and goat meat have never been mported nto Ngera (table 1). Sheep and goat have been rased almost by every household especally n the rural communtes, where agrculture s stll the major occupaton and ther meat are manly consumed n local restaurants and rural markets. The values shown n the beef column show that domestc beef producton has not been adequate to meat ts demand, although substantal part of these values could be sad to represent mport of veal ncluded n beef nformaton and whch were normally consumed by foregners manly from the west. The northern normads are the predomnant rearers of cattle and asde from the fact that they need the female calves for local mlk producton, the male calves are usually kept ether for an ndcaton of wealth or reared for future sales. Slaughterng calves for market s thus never a usual nor an acceptable practce for the normads. Beef consumpton, on the whole, has been most erratc of all meat type (fgure 1). The fact that about half of Ngeran populaton are Muslms has had an adverse effect on producton, mportaton and consumpton of pork. The average of 1.7 percent and 8.8 percent of pork mport n the later years mght have been due to reducton n local producton that necesstates ts mportaton or foregners demand for mport. 2

4 Poultry has been ncreasngly popular among Ngerans over the years and local producton has been mmense due to huge government s nvestment n day-old chcks producton. Its mportaton was however noted durng the late 70s tll early 80s. Ths concded wth the tme of meat shortage followng the great draught of early 70s. Although, there has been generally hgh dependence on mport n order to meet natonal food requrement, there has not been major mport of meat products n Ngera. Ther demand has, however, been growng due to the nutrtonal need of the ever ncreasng populaton, whch the World Bank (1996) estmated to be growng at 2.9 percent per annum. Importaton, generally, has reduced n real term. Ths mght be due to the declnng ncome (fgure 2), whch presently stood below US$400 per capta. The declne could have a detrmental effect on household food consumpton n general snce Ngeran households devote about 75 % of ther total expendture to food (fgure 3). 2. Conceptual Framework and Lterature Revew 2.1 Concept of Separablty The foundaton work on separablty was lad by Leontef (1947). Three concepts of separablty n demand analyss have been summarzed by Goldman and Uzawa (1964). These are based on the earler works of Leontef (1947), Sono (1961), Strotz (1957, 1959), Gorma (1959), Frsch (1959), Houthakker (1960) and Pearce (1961). In the sense of Blackorby, Prmont, and Rusell (1978) as well as Goldman and Uzawa (1964), these three concepts have been found to be extremely useful and effectve for economc analyss of the structure of consumers preference because of ts wde rangng 3

5 mplcatons for the exstence of consstent aggregates and the decentralzaton of optmzaton decsons. In the sense of Goldman and Uzawa (1964), denotng a set of all n fnte commodtes by N, where N = {1,.,n}, and whch can be parttoned nto a class of mutually exclusve and exhaustve subsets, {N 1,.,N s }; namely, N = N 1 N s, N s N t s empty for s t. A commodty bundle, x = (x 1,., x n ), may be correspondngly parttoned nto (x (1),.,x (s) ), where, for each s, the subvector x (s) s composed of x, ε N s. Gven that {N 1,.,N s } s a partton or subset of the set N, let u(x) be a utlty functon that states the necessary and the suffcent condtons of a preference relaton. Let u (x) and u j (x) be partal dervatves of u(x) wth respect to x and x j, respectvely, there are several concepts of separablty that can be stated as follows. A utlty functon u(x) s weakly separable wth respect to the partton {N 1,.,N s } f the margnal rate of substtuton u (x)/u j (x) between two commodtes and j from subsets N s does not depend upon the quanttes of commodtes outsde of N s, namely, u( x) / uj( x) = x k 0, for all, j ε N s and k N s (2.1) A utlty functon u(x) s strongly separable wth respect to the partton {N 1,.,N s } f the margnal rate of substtuton u (x)/u j (x) between two commodtes and j from dfferent subsets N s and N t, respectvely, does not depend upon the quanttes of commodtes outsde of N s and N t ; namely, 4

6 u( x) / uj( x) x = k 0, for all ε N s. j ε N t, and k N s N t (s t) (2.2) Goldman and Uzawa (1964) defned Pearce separablty as the thrd concept, whch demands that the margnal rate of substtuton u (x)/u j (x) between two commodtes and j from the same subset N s be ndependent upon the quanttes of all other commodtes, that s, u( x) / uj( x) = x k 0, for all, j ε N s and k, j. (3) Another concept of separablty that has been used n the lterature s addtve separablty. Ths s the case n whch each good s ts own group and ts partal dervatve does not depend upon any other goods. All the separablty concepts stated above n terms of utlty functons can be characterzed by the Slutsky terms of the correspondng demand functon. 2.2 Lterature Revew Prevous studes have nvestgated separablty usng alternatve approaches. These are the demand system approach (ordnary and nverse form) as well as dfferental demand system approach. Moschn, Moro, and Green (1994) tested for the structure of preferences n three dfferent demand systems, based on the work of Balckorby, Davdson, and Schworm (1991). Eales and Wessells (1999) argued that the easest demand system wthn whch such test can be conducted s the Rotterdam model, the system used n Capps et al (1994), who examned the demand for meat products n the 5

7 Pacfc Rm regon. The Rotterdam model does not assume a specfc functonal form and dfferent versons were developed by Barten (1964) and Thel (1965). One unque feature n a Rotterdam system s that the restrctons used to test for separablty (ether asymmetrc or symmetrc) depend only upon coeffcents and not on any varable and the results of such test are global. If the Rotterdam system s not consstent wth a partcular data set, then such tests are unavalable (Eales and Wessells, 1999). Accordng to Eales, Durham, and Wessells (1997), a number of studes have examned the plausblty of theoretcally consstent nverse demand systems, e.g., Barten and Bettendorf (1989), Moschn and Vssa (1993), Eales and Unnevehr (1993, 1994), and Brown, Lee and Seale (1995). Barten and Bettendorf develop dfferental nverse demands for applcaton to monthly demand for fsh n Belgum. Specfcally, they develop nverses of the Rotterdam, Central Bureau of Statstcs (CBS; Keller and van Drel (1985), Latnen and Thel (1979)), and dfferental Almost Ideal Demand System (AIDS) models as well as the nterpretatons of the coeffcents. Just as the ordnary CBS demand model s a hybrd of the ordnary Rotterdam and AIDS models, the nverse CBS (orgnally developed by Latnen and Thel) s a combnaton of nverse AIDS scale effects. The Natonal Bureau of Research demand model (NBR; Neves (1987)) s also a hybrd. It has an nverse analog, the nverse NBR, whch combnes nverse AIDS quantty effects wth an nverse Rotterdam scale effect. As far as Generalzed demand models are concerned, Barten (1993) developed a synthetc dfferental demand model, whch nests such dfferental demand systems as Rotterdam (Barten 1964, Thel 1965), the dfferental AIDS, and the two hybrd demand systems: CBS and NBR. Lee, Brown, and Seale (1994) appled ths system to examne 6

8 Tawanese demands. Ths should, however, not be confused wth the generalzed nverse demand system developed by Brown, Lee and Seale (1995). An alternatve parameterzaton to the Barten s synthess above, known as Generalzed Ordnary Dfferental Demand System (GODDS), was developed later by Eales, Durham and Wessells (1997) n a study of Japanese demand for fsh. Ths system nests the Rotterdam demand system and the dfferental form of the AIDS as well as the two hybrd models: NBR and CBS models. 3. Data and Analytcal Procedures 3.1 Data Collecton and Dervaton Tme-seres annual data ( ) on consumpton of meat products as well as ther nomnal retal prces were collected from the Food Balance Sheet secton of the Food and Agrculture Organzaton (FAO) as well as the Ngera s Federal Offce of Statstcs, Lagos. The consumpton data per 1000 metrc tonnes are presented n aggregate form and these were converted to per capta level per klogram, usng the populaton data also avalable n FAO as well as the metrc converson. The total expendture were derved by addng up ndvdual expendture, whch s a product of the observed quantty that s consumed and ts correspondng nomnal prce, whle expendture share of each of the meat products was derved from a dvson of the ndvdual expendture by the total expendture. 3.2 Demand Modelng The Generalzed Ordnary Dfferental Demand System (GODDS) dscussed n subsecton 2.2 s used n ths study. The content (equatons 3.1 to 3.10 below) as well as 7

9 method of testng for separablty are defned Eales and Wessells (1999) and others as follows: dw N ( γ + θ w ( δ w )) d ln( p ), = ( β + θ1w ) d ln( Q) + k 2 k k k (3.1) k = 1 where: d ln( Q) = w β γ k 1 = θ and θ δ k p q / x and w 2 j w d ln( q ) j j = 0.5 ( w + lag( w )) ( Dvsa volume ndex), ( budget shares), (exp endture coeffcents), ( prce coeffcents), ( nestng parameters), ( Kronec ker' s delta). (3.2) The outcome of dfferent restrctons on the nestng parameters s as shown n table 2. The coeffcent γˆ j wll be Almost Ideal prce effects f ˆ 2 = 0 = ˆθ and Rotterdam prce θ 1 effects f ˆ = 1, θ ˆ =-1. For other values of θˆ, the prce effect s nether that of the θ 2 1 Almost Ideal nor the Rotterdam systems. Further dscusson of the relatonshp between the models can be found n Neves (1994). In the analyss below, the functonal forms nested n GODDS were tested wth homogenety and symmetrc restrctons n place to nvestgate ther consstency wth the data n order determne ther approprateness for further analyss. Important economc effects can be derved usng the approprate functonal form. The compensated, cross-prce elastctes for the GODDS model are: 2 e * k γ k = + ( θ 2 1)( δ k w w k ), (3.3) whle expendture elastctes are: β e = + θ1 +1 w (3.4) 8

10 The restrctons mpled by asymmetrc separablty for the GODDS model are of the form (for, j ε A,whch s asymmetrcally separable from k ε B): γ k ( β + w ( θ + 1) ) = θ ( ) ( ) 1 γ ( 1) ( 1) ( 1) jk w j wk θ 2 + w wk 2 β + w θ + j j 1 (3.5) These restrctons are mposed at the mean shares unless the resultng model s Rotterdam. Consequently, the test of separablty n GODDS are local (unless the restrctons that result n the Rotterdam model are mposed. To test for separablty between the meat products, numbers can be gven to them so that 1 4 s for beef and veal; mutton and goat meat; pork as well as poultry, respectvely. The nonredundant restrctons that are necessary and suffcent for the asymmetrc separablty are thus (e.g., separablty of beef/veal from mutton/goat, pork, and poultry): σ σ e e σ =, = (3.6) σ e e where σ j s the Allen-Uzawa elastcty of substtuton gven as: γ k + ( θ 2 1) w ( δ k wk ) σ k =. (3.7) w w k Usng the above framework, there are two steps of analyss. The frst one nvolves the model that was restrcted by homogenety and symmetry usng teratve Seemngly Unrelated Regresson (ITSUR), whle the second estmaton ncorporates the restrctons mpled by the separablty of meat products, usng the approprate model n a nonlnear framework. All restrctons n equatons ( ) are tested by comparng estmates from the above two estmatons usng a lkelhood-rato test. 9

11 4. Analyss and Results 4.1 Test of Models and Separablty The nested model (GODDS) s restrcted for homogenety and symmetry and estmated wth a system of equatons by ITSUR n SHAZAM. Conductng an adjusted lkelhood rato test yelded the Rotterdam as the approprate model (Table 3). Gven the restrctons of separablty are easly done and the results of ts tests are global n Rotterdam, further analyss was therefore done usng the Rotterdam model. The results for the asymmetrc separablty tests are presented n Table 4. A detaled descrpton of the method for mposng separablty n the Rotterdam model can be found n the artcles of Moschn, Moro and Green (1994) and Capps et al. (1994). The results ndcate that only pork s statstcally separable from other meat types at the 5% level. The result s not surprsng snce as ponted out n secton 1, the composton of the Ngeran populaton has a tremendous effect on pork consumpton. 4.2 Elastctes of Demand Table 5 reports the compensated elastcty estmates from the Rotterdam model. Curvature was mposed usng Cholesky decomposton for consstency wth the economc theory. Ths s because ntal results volated curvature condtons. All of the own-prce elastctes are nelastc, whch ndcates that f an own-prce ncreases by 1%, consumpton decreases by less than 1% n all meat type. Moreover, beef and poultry are less responsve to own-prces than are mutton or pork. In regards to cross-prce effects, the meat products are more often substtutes than complements. Pork and poultry are ndcated to be complments, whle almost all others are substtutes. 10

12 The expendture elastcty estmates, are also provded wthn Table 5. Beef, pork, and poultry have postve expendture elastctes and are thus normal goods. Importantly, as ncome ncreases, consumers tend to purchase more beef than other meat products. Ths s consstent wth expectatons as beef s relatvely more expensve than other meat n Ngera and s consumed by those wth hgher ncome. Mutton and goat meat appear to be the nferor good, snce ther combned consumpton s negatvely responsve to ncome change. Ths s understandable snce Ngerans spend about 70% of ther dsposable ncome on food generally and there s tendency for consumpton of meat products to shft from the low patronzed products to the expensve ones as ncome ncreases. 5. Summary and Concluson The study estmated the demand for meat products n Ngera. To do ths, t s approprate to dentfy the model provdng the most robust estmates that could be adopted for polcy actons. The GODDS that nests four dfferental models was used and tested, yeldng the selecton of the Rotterdam model, whch was not only used to mpose and test for separablty but also to estmate the prce elastctes of consumer meat demand. As would be expected, pork s the only meat type that s separable from others and ths should be noted when modelng demand for meat n Ngera. The own prce elastctes are all nelastc, ndcatng that an ncrease n own prces by 1 percent wll not change the demand for meat products much. 11

13 The areas for further research are as follows: () the ncorporaton of addtonal meat substtutes such as fsh, demographc varables as well as food safety varables (f any) that can have mpact on meat demand, () analyzng the demand for meat wth more data than used here could generate better estmates, and () nvestgatng further the effect of separablty restrctons on the elastcty estmates and other economc parameters. 12

14 References Barten, A. P. Consumer Allocaton Models: Choce of Functonal Forms. Emprcal Econ. 18 (1993): Blackorby, C., R. Davdson, and W. Schworm. Implct Separablty: Characterzaton and Implcatons for Consumer Demands. J. Econ. Theory 55 (Dec. 1991): Capps, O., Jr., R. Tsa, R. Krby, and G.W Wllams. A Comparson of Demands for Meat Products n the Pacfc Rm Regon. J. Agr. And Resour. Econ. 19,1 (1994): Eales, J., C. Durham, and C. Wessells. Generalzed Models of Japanese Demand for Fsh. Amer. J. Agr. Econ. 79 (November 1997): Frsch, R. A Complete Scheme for Computng All Drect and Cross Demand Elastctes n a Model wth Many Sectors. Econometrca, Vol. 27 (1959): Goldman, S. M. and H. Uzawa. A Note on Separablty n Demand Analyss. Econometrca, Vol 32, Issue 3 (Jul.,1964): Gorman, W. M. Separable Utlty and Aggregaton. Econometrca, Vol. 27 (1959): Hauthakker, H. S. Addtve Preferences. Econometrca, Vol. 28 (1960): Lee, J. Y., M.G. Brown, and J. L. Seale, Jr. Model Choce n Consumer Analyss: Tawan, Amer. J. Agr. Econ. 76 (August 1994): Leontef, W. W. A Note on the Interrelaton of Subsets of Independent Varables of a Contnuous Functon wth Contnuous Frst Dervatves. Bulletn of the Amer. Math. Socety, Vol. 53 (1947): Leontef, W. W. Introducton to the Theory of the Internal Structure of Functonal Relatonshps. Econometrca, Vol. 15 (1947): Moschn, G., D. Moro, and R. D. Green. Mantanng and Testng Separablty n Demand Systems. Amer. J. Agr. Econ. 76 (February 1994): Neves, P. D. A Class of Dfferental Demand Systems. Econ. Letters 44 (1994): Pearce, I. F. An Exact Method of Consumer Demand Analyss. Econometrca, Vol. 29 (1961):

15 Sono, M. The Effect of Prce Changes on the Demand and Supply of Separable Goods. Internatonal Econ. Revew, Vol. 2 (1961): Strotz, R. H. The Emprcal Implcaton of Utlty Tree. Econometrca, Vol. 25 (1957): Strotz, R. H. The Utlty Tree-A Correcton and Further Apprasal. Econometrca, Vol. 27 (1959): World Bank Development Reports, Issues. Oxford Unv. Press. 14

16 Table 1: Import Quantty Percentage Share of Ngeran Domestc Consumpton (5-year averages) 1 Percentage Share Year Span Beef a Mutton b Pork Poultry c Source: Computed from FAO Food Balance Sheet, a = ncludng Veal; b = ncludng goat meat; c = only 4-year average. 1 ndcates that per capta values are used to derve the percent share. Table 2. Restrctons on the Generalzed Models Whch Yeld Alternatve Functonal Forms Restrctons Model AIDS 0 0 Rotterdam -1 1 CBS 0 1 NBR -1 0 Source: Testng Separablty of Japanese Demand for Meat and Fsh Wthn Dfferental Demand Systems, Eales and Wessells (1999), J. of Agrc. & Res. Econs., p117. Note: Notaton follows that of equatons (3.1) and (3.2) n the text. 15

17 Table 3: Tests of Models Nested wthn GODDS Log- Lkelhood Adj LR Statstc GODDS AIDS CBS NBR Rotterdam Ch-square crtcal value at 5% wth 2 df = 7.38 Table 4. Separablty Test Results Separablty # Restrctons Log- Adj LR Structure Lkelhood Statstc Rotterdam Unrestrcted a Beef vs. rest of meat Mutton vs. rest of meat Pork vs. rest of meat Ch-square crtcal value at 5% wth 2 df = 7.38 Table 5. Compensated Prce Elastcty Estmates Beef a Mutton b Pork Poultry Income Beef a Mutton b Pork Poultry a beef ncludes veal and b mutton ncludes goat. 16

18 7 Quantty Consumed(kg) Year Beef & Veal Mutton & Goat Pork Poultry (Chcken) Source: Drawn from the data n FAO Food Balance Sheet, Fg 1. Per Capta Meat Consumpton n Ngera ( ) 17

19 Income (n U.S. Dollars) Source: World Bank, Fg 2: Per Capta Income n Ngera, % Expendture on Food South East South West North West North East Note: Food Share = Annual Total HH Food Expendture / Annual Total HH Expendture. Source: Partcpatory Infomaton Collecton Study, Obafem Aw olow o Unv., Ile-Ife, Fg. 3: Percentage of Household Expendture on Food (Food Share) by Regon: