Incorporating demand shifters in the Almost Ideal demand system

Transcription

1 Ecoomcs Letters 70 (2001) locate/ ecobase Icorporatg demad shfters the Almost Ideal demad system Jula M. Alsto, James A. Chalfat *, Ncholas E. Pggott a,1 1 a, b a Departmet of Agrcultural ad Resource Ecoomcs, Uversty of Calfora, Davs, CA 95616, USA b Departmet of Agrcultural ad Resource Ecoomcs, North Carola State Uversty, Ralegh, NC, USA Receved 13 July 1998; accepted 14 December 1999 Abstract Itercepts of share equatos geerally clude demad shft varables. I the Almost Ideal demad system ad related models, ths results estmates that deped o uts of measuremet. Solutos to ths problem are detfed ad dscussed Elsever Scece B.V. All rghts reserved. Keywords: Almost Ideal demad system; Geeralzed Almost Ideal demad system; Demad shfters; Scalg JEL classfcato: D12; Q11 1. Itroducto I cosumer demad aalyss, varables other tha prces ad come are ofte cluded demad equatos. For stace, studes usg cross-sectoal data, demographc varables mght be cluded, ad studes usg tme seres data, treds ad seasoal varables mght be cluded. Researchers must be careful to corporate these demad shft varables a maer that matas the theoretcal propertes of the model. Importatly, the estmated demad system must be Closed Uder Ut Scalg (CUUS), a property that esures that estmated ecoomc effects are varat to the scalg of the data (Pollak ad Wales, 1992, p. 70). I applcatos of the very popular Almost Ideal demad system, the tercepts of the expedture share equatos are commoly expressed as lear fuctos of other explaatory varables, as orgally suggested by Deato ad Muellbauer (1980, p. 320). Below we show that the resultg model s o loger cosstet wth CUUS. We explore two *Correspodg author. Tel.: ; fax: E-mal address: jm@prmal.ucdavs.edu (J.A. Chalfat ). 1 Member of the Ga Foudato of Agrcultural Ecoomcs / 01/ $ see frot matter 2001 Elsever Scece B.V. All rghts reserved. PII: S (00)

2 74 J.M. Alsto et al. / Ecoomcs Letters 70 (2001) specfc alteratve approaches for corporatg demad shfters ad preservg CUUS whle retag the other desrable features of the Almost Ideal model. 2. The problem The Almost Ideal model s derved from the expedture fucto where ad E( p,u) 5 exp[a( p) 1 ub( p)], (1) ˆ ˆ ˆ a( p) 5 l P 5 d 1 a9p 1 (1/2)p9Gp (2) b j j b( p) 5Pp. (3) The correspodg vector of share equatos s ˆ w 5 a 1 Gp 1 b l(m/p), (4) ˆ where w deotes the vector of budget shares, p deotes the vector of prces, M s total expedture, p deotes the vector of logarthms of p, ad the parameters to be estmated are d, the vectors a ad b, ad the matrx G. Let deote a ut vector. To satsfy homogeety, addg-up, ad symmetry, the followg restrctos must hold: 9a 5 1, 9G 5 0, 9b 5 0, G 95G. Whe the tercepts (a) of the share equatos are expressed as lear fuctos of some demad shft varable Z, a s redefed as a 5 a0 1 l? Z, ad the restrctos 9a0 5 1 ad 9l 5 0 are requred for addg-up. The effects of chages Z are thus see as chages the parameters of the expedture fucto, ad tegrablty s preserved the share equatos by lettg each elemet of a vary wth Z, whle retag the addg-up restrcto. The share equatos become w 5 a01 l? Z 1 Gpˆ 1 b l M 2 b[d 1 (a01 l? Z)9pˆ 1 (1/2)pGp]. ˆ ˆ (5) Ufortuately, augmetg a ths fasho results estmated ecoomc effects that are ot varat to uts of measuremet of the prces ad quattes of the goods the model, a shortcomg that has ot bee documeted prevously. Specfcally, a dsproportoate chage the uts of measuremet for ay or all of the goods chages the estmates of G ad b so that elastctes ad cosumer welfare measures also chage. Cosder chagg the uts of measuremet for quattes, usg q 5 q p, ad p 5 p /p (p. 0). I the model wthout Z, whe we replace pˆ wth pˆ 2 p, ˆ where ˆ p 5 l p, the share equatos (4) become

3 J.M. Alsto et al. / Ecoomcs Letters 70 (2001) w 5 a 1 G(p ˆ 2 ˆ p ) 1 b l(m/p ) 5 a 1 G(p ˆ 2 ˆ p ) 1 b l M 2 b[d 1 a9(p ˆ 2 ˆ p ) 1 (1/2)(pˆ 2 ˆ p )9G(p ˆ 2 ˆ p )] 5 a 2 Gpˆ 1 Gp ˆ 1 b l M 2 b[d 2 a9pˆ 1 (1/2)p9Gp ˆ ˆ 1 a9p ˆ 1 (1/2)p9Gp ˆ ˆ 2 p9gp] ˆ ˆ (6) 5 a 2 Gpˆ 1 Gp ˆ 1 b l M 2 b[d 2 a9pˆ 1 (1/2)p9Gp ˆ ˆ 1 (a 2 Gp ˆ )9pˆ 1 (1/2)p9Gp] ˆ ˆ 5 a 1 Gpˆ 1 b l M 2 bd1 f a9pˆ 1 (1/2)p9Gp ˆ ˆg, provded that a 5 a 2 Gpˆ ad d 5 d 2 a9pˆ 1 (1/2)p9Gp. ˆ ˆ Hece, rescalg leaves G 5 G ad b 5 b, ad a ad d adjust to chages uts of measuremet, a famlar result whe costats are added to regressors. Predcted shares, elastctes, ad welfare measures are thus varat to such scalg choces. The same varace s ot acheved whe a smlar rescalg s doe the augmeted demad system (5), whch becomes w 5 a 1 l? Z 1 G(p ˆ 2 ˆ p ) 1 b l M 0 2 b[d 1 a 9(pˆ 2 ˆ p ) 1 Z? l9(p ˆ 2 ˆ p ) 1 (1/2)(pˆ 2 ˆ p )9G (pˆ 2 ˆ p )]. (7) 0 Collectg the terms that volve p, ˆ as (6), the share equatos become ] ] w 5a 1 l? Z 1 Gp ˆ 1 b l M 2 b[d 1 (a] 1 l? Z)9pˆ 1 (1/2)p9Gp], ˆ ˆ (8) 0 0 ] ] ] where a ˆ 05 a02 Gp ad d 5 d 2 a90ˆ p 1 (1/2)p9Gp ˆ ˆ 2 Zl9p. ˆ Notce that d s ot costat, because the term Zl9p ˆ vares across the data, ad so ths model wth the prces rescaled s ot equvalet to the orgal model. As a result, the other parameters (.e., G, b, ad l) are affected by the scalg, ad hece the prce ad come elastctes ad cosumer welfare measures ow deped o the uts 2 chose for measurg quattes. Ths meas that ay emprcal results from a Almost Ideal model augmeted ths way are codtoed o some choce of quatty uts. 3. Possble solutos Closure Uder Ut Scalg mposes the codto that f some parameters deped o scalg, the others may eed to as well. Specfcally, whe we make a a lear fucto of demad shfters, CUUS requres that we also make d a lear fucto of the same demad shfters. That s, f oe adopts 3 a 5 a 1 l? Z, the oe must also adopt d 5 d 1? Z. The augmeted share equatos the become 0 0 ˆ ˆ ˆ ˆ w 5 a 1 l? Z 1 Gp 1 b l M 2 b[d 1? Z 1 (a 1 l? Z)9p 1 (1/2)pGp]. (9) After rescalg the quatty uts, as above, the share equato s 2 The two versos of the model are equvalet oly f the scale factor p does ot vary across the goods, so that p5 p ;, whch mples that l9pˆ 5 0 (sce 9l 5 0 for addg-up). 3 We are grateful to a aoymous referee for potg out ths alteratve.

4 76 J.M. Alsto et al. / Ecoomcs Letters 70 (2001) w 5 a 1 l? Z 1 G (pˆ 2 ˆ p ) 1 b l M 0 2 b[d 1? Z 1 a 9(pˆ 2 ˆ p ) 1 Z? l9(p ˆ 2 ˆ p ) 1 (1/2)(pˆ 2 ˆ p )9G (pˆ 2 ˆ p )]. (10) 0 0 Collectg the terms that volve ˆ p results w 5 a 1 l? Z 1 Gpˆ 1 b l M 2 b[d 1? Z 1 (a 1 l? Z)9pˆ 1 (1/2)p9Gp], ˆ ˆ (11) where a ˆ 9 ˆ ˆ ˆ 05 a02 Gp, d 5 d 2 a0p 1 (1/2)p9Gp, ad 5 2 l9p. ˆ Notce that a 0, d, ad are all costat ad are merely rescaled, ad G, b, ad l are uchaged (.e., G 5 G, b 5 b, ad l 5 l). The desred varace property s acheved. Deato ad Muellbauer (1980) poted out that t may be dffcult to estmate d, ad subsequet experece has bore out that predcto. Such dffcultes would be compouded f we were to make d tself a fucto of other varables. Hece, may staces, t wll ot be feasble to estmate a model whch demad shfters augmet both a ad d, ad a alteratve soluto must be foud. Geeral methods for corporatg demad shfters ad preservg CUUS demad systems clude scalg, traslatg, ad other modfyg fuctos, as descrbed by Lewbel (1985). Amog these theoretcal alteratves, we would lke to detfy a practcable alteratve that wll preserve the other desrable features of the Almost Ideal model, ad wll also allow demad shfters to be corporated parsmoously, ad flexbly. Oe such alteratve s to adopt a geeralzed verso of the Almost Ideal model, the Geeralzed Almost Ideal (GAI) model, frst derved by Bollo (1987). The expedture fucto used to characterze the GAI model allows for a porto of total expedtures to be allocated to pre-commtted quattes, whch are uobservable ad must be estmated alog wth the other parameters the demad system. The GAI expedture fucto ca be expressed as E( p,u) 5 p9c 1 E *( p,u), (12) where c deotes the vector of pre-commtted quattes ad E *( p,u) s the expedture fucto for superumerary (beyod pre-commtted) expedtures, defed to be of the Almost Ideal form, as Eqs. (1) (3). I the GAI model, the share equato for the budget share of good s S D pc M* w5]] 1]] a1ogj l pj1 b l(m * /P), (13) M M * where M 5 M 2 p9c ad l P s defed as (2). The GAI model ca, of course, be vewed as a geeralzato of the Lear Expedture System (LES), whch the margal budget shares are o loger costat, but are stead of the Almost Ideal form. The pre-commtted quattes are depedet of prces ad expedture, suggestg that augmetg the c vector by makg each pre-commtted quatty a fucto of Z could avod the scalg problems descrbed earler. Such a approach was referred to as demographc traslatg by Pollak ad Wales (1992, p. 12), who showed how to use t to corporate demographc varables the LES. The same approach ca be appled the GAI model to corporate ay type of demad shfter, producg estmates that are varat to chages quatty uts. To see ths, replace each pre-commtted quatty c wth a lear fucto of Z:

5 J.M. Alsto et al. / Ecoomcs Letters 70 (2001) c 5 c 1 l Z. (14) 0 The the share equatos wll take the form S p [c0 1 lz] M* w5]]]] 1]] a1ogj l pj1 b l(m * /P). (15) M M Whe the quatty uts are redefed so that prces are rescaled, oce aga by dvdg by p, the demad system becomes S ( p /p )[c 0 1 lz] M* w ]]]]]] ]] 5 1 a1ogj l( p j/p j) 1 b l(m * /P). (16) M M The margal budget share paretheses s the rght-had sde of the share equato of the Almost Ideal demad system; we showed above that ths term s varat to chages quatty uts. I addto, M s fxed ad M* chages oly f the pre-commtted expedtures, p9c, chage. To see that pre-commtted expedtures do ot chage whe quatty uts chage, cosder the frst term: p [c 1 l Z] (p /p )[p? (c 1 l Z)] ( p /p )[c lz] ]]]] 5]]]]]]] 5 ]]]]]]. (17) M M M Hece, the ew c s smply c p ad the ew 0 0 l s smply pl. Thus, chagg the uts of measuremet the GAI demad model s of o cosequece. Augmetg the pre-commtted quattes ths fasho does ot appear to mply ay restrctos o how Z affects the demad for ay partcular good. D D 4. Cocluso I oe of the most popular models used demad aalyss, the Almost Ideal demad system, t s commo to make the tercepts of the share equatos lear fuctos of demad shft varables, a procedure that meas results are ot varat to dsproportoate chages uts of measuremet. We have descrbed two specfc alteratve approaches for corporatg demad shfters the Almost Ideal model that overcome ths varace problem. The more appealg approach, from a emprcal stadpot, volves adoptg the Geeralzed Almost Ideal model, ad allowg the pre-commtted quattes to be lear fuctos of demad shft varables. Ths approach allows the demad shfters to be cluded a fasho that s flexble, parsmoous, ad matas the model s varace to chages uts of measuremet. It s easly see that smlar results wll be foud wth related models of the PIGLOG class. For stace, Lewbel s (1989) model that ests the Almost Ideal ad Traslog models wll exhbt the same depedecy o uts as the Almost Ideal model. Oe soluto s to clude demad shft varables as modfcatos of the pre-commtted quattes, Bollo ad Vol s (1990) geeralzato of Lewbel s model.

6 78 J.M. Alsto et al. / Ecoomcs Letters 70 (2001) Refereces Bollo, C.A., GAIDS: a geeralzed verso of the Almost Ideal demad system. Ecoomcs Letters 23, Bollo, C.A., Vol, R., GAITL: a geeralzed verso of the Almost Ideal ad traslog demad systems. Ecoomcs Letters 34, Deato, A.S., Muellbauer, J., A Almost Ideal demad system. Amerca Ecoomc Revew 70, Lewbel, A., Nestg the AIDS ad traslog demad systems. Iteratoal Ecoomc Revew 30, Lewbel, A., A ufed approach to corporatg demographc or other effects to demad systems. Revew of Ecoomc Studes, Pollak, R.A., Wales, T.J., Demad System Specfcato ad Estmato. Oxford Uversty Press.