Demand Systems Based on Regular Ratio Indirect Utility Functions

Transcription

1 Demand Systems Based on Regular Rato Indret Utlty Funtons by Russel J. Cooper Department of Eonoms Dvson of Eonom and Fnanal Studes Maquare Unversty NSW 2109 Australa Emal: and Keth R. MLaren Department of Eonometrs and Busness Statsts Monash Unversty Clayton, Vtora 3800 Australa Emal: June 2006

2 Abstrat Empral demand analyss s usually onduted n the ontext of the perennal tradeoff between regularty and flexblty. The lass of known globally regular demand systems s qute small and omes at the pre of nflexblty. At the other extreme are demand systems suh as Translog or the Almost Ideal Demand Systems desgnated as loally flexble, n that they do not put any pror restrtons on slopes (or elasttes, other than those mposed by the regularty ondtons, at the pont of approxmaton. The ost of ths flexblty at a pont s that these systems usually exhbt only small regons of regularty about the pont of approxmaton. A onvenent ompromse between these two extremes s the lass of effetvely globally regular demand systems, whh preserve regularty n an nterestng unbounded regon. Lewbel extended Gorman s defnton of rank to non-aggregable systems, and also showed that rank s equvalent to the mnmum number of pre ndes n the ndret utlty funton. The purpose of ths paper s to ntrodue a new demand system that s effetvely globally regular, potentally loally flexble, and of potentally arbtrary rank. 2

3 1 Introduton Empral demand analyss s usually onduted n the ontext of the perennal tradeoff between regularty and flexblty. A demand system s sad to be regular f t satsfes the restrtons mposed by the paradgm of ratonal onsumer hoe.e. f t s onsstent wth the paradgm of maxmzng utlty subjet to a budget onstrant. In the ontext of Marshallan (ordnary demand systems, ths means that the demand systems expressng quanttes demanded as funtons of expendture and pres satsfy the propertes of non-negatvty, homogenety, Engel aggregaton, Cournot aggregaton, and the symmetry and non-negatve defnteness of the Slutsky matrx. A system s sad to be globally regular f t s regular for all non-negatve values of expendture and pres. The lass of known globally regular demand systems s qute small (Cobb-Douglas, CES, Indret Addlog,? and omes at the pre of nflexblty. For example, n a Cobb-Douglas system nome, own-pre and rosspre elasttes are a pror onstraned to be +1, -1 and zero, respetvely. At the other extreme are demand systems suh as the Translog (see Chrstensen et al (1975 or the Almost Ideal Demand Systems (AIDS (see Deaton and Muellbauer (1980 desgnated as loally flexble, n that they do not put any pror restrtons on slopes (or elasttes, other than those mposed by the regularty ondtons, at the pont of approxmaton. That s, they possess just enough free parameters to approxmate any theoretally possble elastty at the gven pont. The ost of ths flexblty at a pont s that these systems usually exhbt small regons of regularty about the pont of approxmaton. A onvenent ompromse between these two extremes s the lass of effetvely globally regular demand systems, n the sense of Cooper and MLaren (1996. By effetvely globally regular s meant that there exsts a pre ndex P( p suh that the regularty propertes are satsfed for all expendture-pre ombnatons satsfyng P ( p. Thus the regularty regon s an unbounded regon n pre-expendture spae, potentally nludng all ponts n the sample, and all ponts orrespondng to hgher levels of real nome. A well-known example of suh a system s the Lnear Expendture system, whh s regular over an unbounded regon but not flexble at a pont. Suh systems an also be flexble at a pont, suh as the orgnal MAIDS of Cooper and MLaren (1992. Another nterestng haraterst of demand systems s ther rank. Gorman (1981 defned the rank of a demand system as the dmenson of the spae spanned by ts Engel urves. For example a system has rank one f and only f t s homothet, n whh ase all nome elasttes are unty. Gorman showed that exatly aggregable regular demand systems have maxmum rank of three. Lewbel (1991 extended the defnton of rank to non-aggregable systems, and also showed that rank s equvalent to the mnmum number of pre ndes n the ndret utlty funton. Thus the Cobb-Douglas system s rank one, whle the LES, AIDS and MAIDS are rank two. Nonparametr tests suh as Banks, Blundell and Lewbel (1997 suggest that rank should be at least three, and ths led them to generalse AIDS to the Quadrat Almost Ideal Demand System (QUAIDS. Lewbel (2003 extends QUAIDS to a rank four demand system. But lke AIDS, nether QUAIDS nor Lewbel s rank four system s effetvely globally regular. 3

4 The purpose of ths paper s to ntrodue a new demand system that s effetvely globally regular, potentally loally flexble, and of potentally arbtrary rank. 2 The Rank of Demand Systems Let > 0 denote a level of expendture (ost and let p be an n-vetor of non-negatve pres. Let S(, p denote an n-vetor of shares. Then followng Lewbel (2003 a demand system an be wrtten n the form m k k S(, p = A ( p f (, p k = 1 (2.1 for some and the k m n, where the f (, p are salar funtons of expendture and pres, k A ( p are n-vetors of funtons of pres. Defne the n m matrx 1 2 m Ap = A( p, A( p,, A( p. Generalzng Gorman (1981, Lewbel (1991 defnes the rank of a demand system to be the maxmum rank, over all possble pre vetors p, of the matrx Ap. Lewbel (1991 then shows that f the system (2.1 s onsstent wth utlty maxmzaton, then the rank m wll also be the mnmum number of salar funtons (pre ndes Ak ( p suh that the ndret utlty funton an be wrtten n the form [ ] U, p = U, A( p,, A ( p. (2.2 Gorman (1981 was onerned wth the ase where (2.1 an be wrtten n the restrted form 1 m m k k S, p = A ( p g (. k = 1 (2.3 He defned suh systems to be exatly aggregable, and proved that homogenety mples that suh systems have maxmum rank of three. There s no suh restrton on the rank defned aordng to (2.1. However t s the ase that a rank of m<n mposes restrtons on a demand system. 3 The Class of Regular Rato Indret Utlty Funtons A system of Marshallan demand equatons wll be regular over the regon Ω, an n+1 dmensonal subset of expendture-pre spae, f the orrespondng ndret utlty funton satsfes the followng ondtons n the regon Ω : (, p RIU:V s non-dereasng n expendture ; non-nreasng n pres p; homogeneous of degree zero n p and ; quas-onvex n p. 4

5 The lass of Regular Rato 1 Indret Utlty funtons s spefed as havng the general form of (, U p (, (, V p = (3.1 W p where the omponent funtons have the followng regularty propertes n a regon Ω of the n+1 dmensonal postve orthant: (, p RV:V s postve; non-dereasng n expendture ; non-nreasng n pres p; homogeneous of degree zero n p and ; onvex n p. RW: W (, p s postve; non-nreasng n ; non-dereasng n p; homogeneous of degree zero n p and ; onave n p. Struture (3.1 has an attratve ntutve appeal. Roy s Identty appled to any ndret utlty funton wll n general generate demand equatons or shares as rato forms (ratos of partal dervatves of U, exept n the speal ases when the denomnator ollapses to a onstant. Roy s Identty appled to (3.1 also generates demand equatons or shares as rato forms (ratos of ombnatons of partal dervatves of V and W. Our man result s the followng. Theorem 1: Provded the two omponent funtons V and W satsfy these propertes RV and RW, respetvely, then, n the regon Ω, the orrespondng ndret utlty funton U defned by (3.1 wll satsfy the regularty propertes RIU of an ndret utlty funton. In addton, U wll be postve n the regon Ω. Proof: Homogenety of degree zero follows by onstruton. The monotonty property follows by straghtforward dfferentaton and from the postvty and V, p and W, p. The urvature property follows monotonty propertes of from the fat that the rato of a postve onvex funton to a postve onave funton s quas-onvex, a result that s dsussed more fully and proved n Lemma 1 n the appendx. The power of the above onstruton follows from the followng well-known propertes: postve lnear ombnatons of postve non-nreasng onvex funtons are postve non-nreasng onvex funtons; postve lnear ombnatons of postve non-dereasng onave funtons are postve non-dereasng onave funtons. j Thus f V, p, 1,, n satsfy propertes RV, f W, p, j = 1, m satsfy RW, = 1 In earler versons, the adjetve ratonal was used n plae of rato, and had a ertan appeal. However, tehnally, a ratonal funton s a rato of polynomals, and we do not want to neessarly restrt the onsttuent funtons to be polynomals. Further, n the demand systems lterature the adjetve ratonal s also used n an alternatve sense to desrbe ratonal onsumer hoe, as above, and hene would ause unneessary onfuson. It s also worth notng that when the word ratonal s used n ths sense n the onsumer demand lterature, t often appears to mean onssteny wth the addng up and homogenety ondtons only, gnorng the other mplatons of utlty maxmzaton. 5

6 and f the onstants θ, = 1,, n, and π, j = 1,, m satsfy 0 θ 1, 0 π 1 then j n = 1 (, = U p m j= 1 θ V j π W (, p j (, p (3.2 satsfes propertes RIU. Thus the spefaton (3.1 defnes a whole lass of regular ndret utlty funtons, and the above result mples that t s possble to onstrut demand systems wth arbtrary rank, n the sense of Lewbel (1991. Agan, the ntuton s nformatve. Use of lnear ombnatons of funtons s a onvenent and smple way to aheve nreased rank. But sne quas-onvexty s not preserved when takng lnear ombnatons, t s approprate to take lnear ombnatons of omponent funtons whose (more restrtve propertes are preserved under postve lnear ombnatons. In addton, t should be relatvely easy to hoose funtonal forms n suh a way that the resultng ndret utlty funton s effetvely globally regular, n the sense of Cooper and MLaren (1996. By effetvely globally regular s meant that there exsts a pre ndex P( p suh that the regularty propertes are satsfed for all expendture-pre ombnatons satsfyng P( p. Provded ths ondton s 0 0 satsfed at some base expendture-pre ombnaton, say P( p, then the regularty ondtons wll be satsfed for the entre sample and for typal post-sample analyss whh usually orresponds to evaluaton at hgher levels of real expendture, n the above sense. Thus, by onstruton, the regular regon Ω exludes only unnterestng expendture-pre ombnatons. As an llustraton of the power of ths result, the Lnear Expendture System s an example of a demand system that s effetvely globally regular. In the usual notaton of the lnear expendture system, P( p = γp and the regular regon s the set of expendtures and pres that satsfy > γ p. A onvenent and ntutvely appealng way to onstrut funtons to use as buldng bloks n (3.2 s to buld them up from funtons of varous defntons of real nome, defned as, where the funtons P( p are defned for all nonnegatve pres, and satsfy the regularty propertes of a unt ost funton: non- P ( p negatvty; non-dereasng n pres; homogeneous of degree one n pres; onave n pres. In addton, these funtons are typally normalzed to unty at base perod pres: P (1 = 1. Ths proedure wll be llustrated n the next seton. Alternatvely, t s sometmes onvenent to explot the homogenety propertes, and defne funtons n terms of the n-dmensonal spae of normalzed pres. For example, by homogenety of degree zero, V (, p ( 1, p ˆ ˆ ( p = V V V ( r = =,say. 6

7 Usng analogous defntons, the orrespondng result s: f Vˆ ( r s postve, nonnreasng and onvex n normalzed pres r, and f Wˆ ( r s postve, non-dereasng and onave n normalzed pres r, f Uˆ r Vˆ r = (3.3 Wˆ r then Uˆ ( r s postve, non-nreasng and quas-onvex n normalzed pres r. For example, the Lnear Expendture System orresponds to an ndret utlty funton of the form κ A( p B p, κ > 0 (3.4 where A s lnear and B s Cobb-Douglas. (In the usual notaton κ = γ s a normalzng onstant that allows the pre ndex Ap to have the same base as the omponent pres. A natural queston s whether ths an be generalzed to the ase n whh A and B are arbtrary unt ost funtons. Now (3.4 s not mmedately n the form of (3.1. However, normalzng by n numerator and denomnator, then (3.4 above an be wrtten as 1 κar whh s n the form of (3.3, leadng to an affrmatve answer, provded the >κ A p, the ondton for effetve global B r numerator s postve.e. provded regularty. Another alternatve way to onstrut regular ndret utlty funtons from (two omponent funtons s the followng. Theorem 2. Let 1 (, p and V 2 (, p utlty funton onstruted as V satsfy propertes RV. Then the ndret 1 2 (, = (, (, U p V p V p (3.5 s postve and satsfes the regularty ondtons RIU. Proof: Defne W (, p Theorem 1 apples. = 1 V 2 (, p. Then by Lemma 2 W satsfes RW and 7

8 4 An Example of a Regular Rato Indret Utlty Funton Pursung the noton that a onvenent way to onstrut regular ndret utlty funtons s to buld them up from funtons of varous defntons of real nome, defned as nome deflated by pre ndes that satsfy the regularty propertes of a unt ost funton, onsder the followng onstruton: (, U p θ + (1 θ ln G( p = B( p ( Ap η (4.1 where the pre ndes, and A p B p G p are anddate pre ndes P( p. Ths system wll be of rank 3 n Lewbel s termnology. 4.1 Relaton to Prevous Spefatons The spefaton (4.1 s presented as a more regular alternatve to the Quadrat Almost Ideal Demands System (QUAIDS, a rank 3 demand system that orresponds to an Indret Utlty funton spefed as (, U p ln ( Ap + ln = B p G p A p whh s usually wrtten n Cost Funton form (on nverson n as (4.2 * B p u ln C( u, p = A ( p +. (4.3 1 G p u and where B( p and G( p are homogeneous of degree zero (and thus not monoton. Condtons for the requred urvature (and monotonty propertes of ether (4.2 or (4.3 are not obvous. The spefaton (4.1 also ontans MAIDS as a speal nested ase when θ= 0 and η= 1 (, ( = ln Ap ( ( B( p = U p B p ln A p. (4.4 Fnally, (4.1 nests regular homothet demands (suh as Cobb-Douglas or CES when η= 0 and when θ= 0 or The Share Equatons The share equatons an be derved as follows. Correspondng to (4.1, 8

9 η B θ (1 θ B B + +η θ + (1 θ ln 2 ( A G G U = 2η B η 1 (4.5 and η η 1 B θgp (1 θ A p B Bp (1 ln 2 ( A + G A +η θ + θ G U p = 2η B (4.6 where A = ln A( p ln p, B = ln B( p ln p, and G G( p Therefore by Roy s Identty the Marshallan share equatons are: = ln ln p. (1 θ A +θ G +η B +ηb(1 θ ln A S( p, = px( p, = G (1 θ +θ ( 1 +η +η(1 θ ln ( A G (4.7 gvng shares as weghted averages of two measures of real nome. 4.3 Regularty Propertes Homogenety Sne the pre ndes A( p, B( p and G( p ndret utlty funton as deflators of expendture, then the ndret utlty funton wll be homogeneous of degree zero. The HD1 propertes also mply A = 1, B = 1 and G = 1 and hene S 1. = Monotonty. Sne the pre ndes A( p, B( p and G( p are all HD1, and they enter the are all nondereasng, the numerator of (4.1 s nonnreasng. Hene a suffent ondton for the monotonty of the ndret utlty funton s that 0 1 and 0 ln A p 0. Ths θ η>, and that ( latter ondton wll be satsfed for all pre and expendture values that satsfy the ondton A p (4.8 whh s a one n pre-expendture spae. At least for these pre and expendture values the monotonty ondton together wth the homogenety ondton wll be 9

10 suffent to guarantee that the shares wll be bounded to the unt nterval 0 S p, Curvature. The pre ndes A( p, B( p and G( p are all onave. Now ln ( A( p s the negatve of a onave funton (an nreasng onave funton of a onave funton, and thus onvex. Not so well known s that the reproal of a postve, nreasng and onave funton s onvex, and ths property s addressed n Lemma 2. Thus by Lemma 1 a suffent ondton for the quas-onvexty of the ndret utlty funton s that 0 1, 0 1 and ln A p 0. Ths latter ondton θ <η ( wll be satsfed for all pre and expendture values that satsfy the ondton (4.8. Thus the suffent ondtons for urvature ft well wth the suffent ondtons for monotonty. 5 An Empral Applaton MAIDS an be generalzed by replang the natural logarthm of real nome by a Box-Cox transformaton, resultng n the Generalzed Exponental Form (GEF of Cooper and MLaren (1996, n whh ase the LES an be nluded as a nested ase. Sne the LES s the arhetypal example of an effetvely globally regular system, a smlar generalzaton wll be used for the empral applaton, to gve an effetvely globally regular system that nests both LES and MAIDS. 5.1 The Empral Spefaton Thus onsder the spefaton (, U p θ + (1 θ G( p = B p ( κa( p η µ µ 1 (5.1 subjet to the restrtons µ 1,0 θ 1 and η> 0. Spefaton (5.1 nests LES when θ= 0, η= 1, µ= 1, Ap lnear ( κ Ap = γ and B( p Cobb-Douglas ( = p β. When θ= 0, η= 1, µ= 1, and A and B are arbtrary (unt ost funtons the resultng spefaton s the Gorman Polar Form. When θ = 0, 0 η 1, µ= 0 the gener form of MAIDS results, and θ = 0, 0 η 1, µ 1 gves the gener form of GEF. Fnally, when 0 θ 1 eah of these models s generalzed to a rank 3 form. Correspondng to (5.1 p 10

11 η η 1 µ µ B θ (1 θ 1 B B + ( κ A( p +η θ + (1 θ 2 ( κa( p G A G U = 2η B ( κ η µ 1 B Gp (1 A p A η θ θ p B Bp µ + +η (1 2 2 θ + θ ( κa( p G A G U p = 2η B gvng the share equatons as S µ (1 θ A( κ A( p +θ ( G +η B +ηb(1 θ G = µ (1 θ ( κ A( p +θ ( 1 +η +η(1 θ G ( κa( p ( κa( p µ µ 1 µ. (5.2 µ Data As an llustratve example, we use aggregate data from the Australan Natonal Aounts, Prvate Fnal Consumpton Expendture. Quarterly data for the perod 1959:3 to 2005:2 was aggregated to annual (fnanal year data, to avod seasonal omplatons, and the possbly spurous alloaton of annual fgures to ther quarterly omponents. Aggregate date was onverted to per apta unts by dvdng by populaton. Pres were represented by mplt pre deflators, alulated as the rato of a nomnal seres to the real (han volume measure. Intally nne ategores were onstruted, but ths level of dsaggregaton would not allow estmaton of even the LES. Ths appeared to be due to the hgh orrelaton among the pre seres. Calulaton of parwse orrelatons revealed qute hgh orrelatons between many pars, wth one orrelaton as hgh as Usng the szes of these orrelatons as a gude, and estmaton of LES as a prerequste, the nne ategores were further amalgamated down to fve, for whh the LES estmates were well behaved. The resultng fve ategores are: Food; Cgarettes & tobao plus Alohol beverages plus Hotels, afes & restaurants; Clothng & footwear plus Furnshngs & h'hold equpment; Rent & other dwellng serves plus Eletrty, gas & other fuel; Other. 5.3 Results Wth the varous nestngs, and the varous hoes avalable for the pre ndes A, B and G, there s learly a large number of models that ould be estmated, espeally when allowng for the mposton of the suffeny ondtons for regularty. Sne the theme of the paper s regularty, only suh regular models wll be reported. As 11

12 usual, the LES provdes a remarkably good ft, and wll be taken as the benhmark effetvely globally regular model. In addton, two generalsatons wll be presented. Model 1 s LES. Model 2 allows η and µ to be freely estmated ondtonal on the LES pre ndes. Fnally, Model 3 allows θ to be freely estmated, and ntrodues a Cobb-Douglas pre ndex for G, agan ondtonal on the LES pre ndes. Log lkelhoods are summarzed n Table 1. η µ θ Log Lkelhood Model Model Model Table 1: Log-lkelhoods for 3 nested models As noted, the LES fts the data remarkably well. One mght have thought that freeng up η and µ would add sgnfant Engel flexblty, as well as (presumably removng the pror restrtons between nome elasttes and pre elasttes mposed by an addtve dret utlty funton, leadng to sgnfantly mproved explanatory power, but ths appears not to be the ase. On the other hand, the generalsaton to Model 3, whh mproves Engel flexblty by movng from a rank 2 to a rank 3 system, does appear to mprove ft sgnfantly. Some other nterestng summary statsts are presented n Table 2. Category R Model Model DW Model Model Table 2: Comparson of R 2 and Durbn-Watson statsts 6 Conluson The onept of rank s an nsghtful haraterst of demand systems. Ths paper has extended the lass of effetvely globally regular demand systems to ranks three and hgher. 12

13 7 Appendx The followng two results appear to be well-known, although t has been surprsngly dffult to fnd an aessble referene to an explt proof. The frst result relates to quas urvature propertes of ratos of onave/onvex funtons. The most aessble referene s probably Mangasaran (1969 who sets the result as a problem. Hs Problem 6.1.( states: Let Γ be onvex. Show that f=g/h (our notaton s quas-onvex on Γ f [{g s onvex on Γ, h>0 on Γ } or {g s onave on Γ, h<0 on Γ }] and [{g s lnear on g 0 on Γ }]. n R } or {h s onvex on Γ, 0 g on Γ } or {h s onave on Γ, Ths result s also quoted n Greenberg and Perskalla (1971, who attrbute t to Mangasaran and to three less aessble papers datng bak to 1964 and Greenberg and Perskalla was the referene used by Cooper and MLaren to prove the quas-onvexty of MAIDS (1992, 1994 and GEF (1996 demand systems. Avrel, Dewert, Shable and Zemba (1981 quote, wthout proof, a smlar result. For our purposes we need the followng. gx Lemma 1: Defne f( x = where gx s onvex and postve on Γ and hx s hx onave and postve on Γ. Then f ( x s quas-onvex and postve on Γ. Proof: Assume (a f( x1 α and (b f( x2 α. By postvty of hx, f x1 α g x1 α h x 1 and f x2 α g x2 α h x 2 and hene (a and (b mply Now ( λ gx + gx [ hx hx ] (1 λ α λ + (1 λ 0 λ ( λ + (1 λ λ + (1 λ α[ λ + (1 λ ] α ( λ + (1 λ g x1 x2 g x1 g x2 h x1 h x2 h x1 x2 where the frst nequalty follows by the onvexty of g, the seond nequalty follows by (, and the thrd nequalty follows by the onavty of h and postvty of f. Usng the two extreme nequaltes 13

14 ( (1 ( (1 whh mples ( (1 g λ x1+ λ x2 α h λ x1+ λ x2 f λ x1+ λ x2 α and hene provng the quas-onvexty of f f( x αand f( x α f λ x + (1 λ x α 2, The seond result relates to the reproal of a onave funton. Whle t s obvous that the reproal of a quas-onave funton s quas-onvex (and ve-versa, leadng to the ommon use of the reproal ndret utlty funton n dualty results, a orrespondng type of result for onave funtons appears not so well known. Agan, n Mangasaran Problem 6.4.( he states Hnt: Show frst that the reproal of a postve onave funton s a postve onvex funton, and that the reproal of a negatve onvex funton s a negatve onave funton, Ths result s then used by Mangasaran, together wth Lemma 1, to prove analogous results to Lemma 1 for the produts of onave/onvex funtons. Lemma 2: Let gx be a postve onave funton. Then f( x = 1 gx s a postve onvex funton. Proof: The j th element of the Hessan matrx of f, H f s f xx j 2 ggxx 2ggg j x xj = so 4 g f 1 g 2 H = H + g 2 3 x g x. g g 1 2 g xh x= xh x+ xg x sne xh x 0 and g 0. g g f g Therefore ( 2 Alternatve Proof: Sne g s onave Now g λ x1+ (1 λ x2 λ g( x1 + (1 λ g( x2. 14

15 1 f ( λ x1+ (1 λ x2 = g x x ( λ + (1 λ λ g x + λ (1 g( x λ + (1 λ gx gx 1 2 =λ f x + (1 λ f x. 1 2 The frst nequalty follows from reproatng the nequalty mpled by the onavty of g, and the seond nequalty from the fat that the reproal s a onvex funton when onstraned to the postve orthant. Referenes Avrel, M., W.E. Dewert, S. Shable and W.T. Zemba, Introduton to Conave and Generalzed Conave Funtons, n S. Shable and W. Zemba ed. Generalzed Conavty n Optmzaton and Eonoms Aadem Press, New York, 1981 Banks J., R. Blundell and A. Lewbel, Quadrat Engel Curves and Consumer Demand, The Revew of Eonoms and Statsts 79, 1997, pp Barnett, Wllam A. and Ousmane Sek, Rotterdam vs Almost Ideal Models: Wll the Best Demand Spefaton Please Stand Up?, Unversty of Kansas, February 12, Chrstensen, L.R., D. Jorgenson and L.J. Lau, Transendental Logarthm Utlty Funtons, Ameran Eonom Revew, 65, 1975, pp Cooper, Russel J. and Keth R. MLaren, An Emprally Orented Demand System wth Improved Regularty Propertes, Canadan Journal of Eonoms, XXV, No.3, 1992, pp Cooper, Russel J., Keth R. MLaren and Prya Parameswaran, A System of Demand Equatons Satsfyng Effetvely Global Curvature Condtons, Eonom Reord, Vol. 70, No. 208, Marh 1994, pp Cooper, Russel J. and Keth R. MLaren, A System of Demand Equatons Satsfyng Effetvely Global Regularty Condtons Revew of Eonoms and Statsts, Vol. 78, No 2, May 1996, pp Deaton, A.S. and J. Muellbauer, An Almost Ideal Demand System, Ameran Eonom Revew, 70, 1980, pp Gorman, W. M., Some Engel Curves, n Essays n the Theory and Measurement of Consumer Behavour n Honour of Sr Rhard Stone, ed. by Angus Deaton. Cambrdge: Cambrdge Unversty Press,

16 Greenberg, Harvey J. and Wllam P. Perskalla, A Revew of Quas-onvex Funtons, Operatons Researh 1971 pp Lewbel, Arthur, The Rank of Demand Systems: Theory and Nonparametr Estmaton, Eonometra 59, 1991, pp Lewbel, Arthur, A Ratonal Rank Four Demand System, Journal of Appled Eonometrs 18, 2003, pp Mangasaran, Olv L., Nonlnear Programmng. MGraw-Hll, New York,