Estimation of Consumer Demand Functions When the Observed Prices Are the Same for All Sample Units

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Dpartmt of Agricultural ad Rsourc Ecoomics Uivrsity of Califoria, Davis Estimatio of Cosumr Dmad Fuctios Wh th Obsrvd Prics Ar th Sam for All Sampl Uits by Quirio Paris Workig Papr No.

1 Dpartmt of Agricultural ad Rsourc Ecoomics Uivrsity of Califoria, Davis Estimatio of Cosumr Dmad Fuctios Wh th Obsrvd Prics Ar th Sam for All Sampl Uits by Quirio Paris Workig Papr No Sptmbr by Quirio Paris All Rights Rsrvd. Radrs may mak vrbatim copis of this documt for o-commrcial purposs by ay mas, providd that this copyright otic appars o all such copis. Giaii Foudatio of Agricultural Ecoomics

2 Estimatio of Cosumr Dmad Fuctios Wh th Obsrvd Prics Ar th Sam for All Sampl Uits Quirio Paris Uivrsity of Califoria, Davis Sptmbr 2003 Abstract With th advt of th almost idal dmad systm (AIDS) of Dato ad Mullbaur, th stimatio of cosumr dmad fuctios rvolvs aroud spcificatios that us flxibl fuctioal forms of th idirct utility fuctio. This dual approach has put o th backburr th traditioal primal approach bcaus th dirct utility fuctio xists oly i a latt stat. Th lack of xplicit, aalytical ivrtibility of ithr systm, howvr, is a idicatio that focusig xclusivly o th dual sid of th cosumr problm is quivalt to disrgard pottially importat ad idpdt iformatio rsidig with th primal sid. This papr suggsts that fficit stimats (i th ss of usig all th availabl iformatio) of th dmad fuctios rquir th joit stimatio of all th primal ad dual rlatios. Th spcificatio of this objctiv assums that riskutral housholds maximiz thir xpctd utility subjct to thir xpctd budgt costrait. This thortical framwork lads to a two-stp procdur that producs cosistt ad fficit stimats of th modl s paramtrs. Th grality of th approach proposd hr ca hadl also th frqutly coutrd cas wh all th sampl uits fac th sam obsrvd commodity prics. Fially, w prst a gral solutio of tholiar rrors-i-variabls problm with a ovl stimatio procdur that avoids th pitfalls of th traditioal approach. EL Classificatio: D0 Kywords: Cosumr dmad fuctios, primal, dual, oliar rrors-i-variabls Quirio Paris is a mmbr of th Giaii Foudatio of Agricultural Ecoomics. 1

3 Estimatio of Cosumr Dmad Fuctios Wh th Obsrvd Prics Ar th Sam for All Sampl Uits Itroductio Statistical survys of cosumr bhavior rarly gathr pric iformatio that is spcific for ach idividual sampl uit. How ar, th, dmad fuctios stimatd i th absc of ay pric variability? Th aswr varis from th us of tim sris ad pal data to costructio of uit valu prics. Idd, this statistical rality is th cosquc of th followig coudrum of cosumr thory. Thory suggsts that obsrvd commodity prics should b idtical for housholds that mak thir cosumptio dcisios i th sam markt viromt, but coomtric stimatio of cosumr dmad fuctios rquirs a sigificat variability of prics without which th duality approach for thir stimatio is ifasibl. Aothr usatisfactory aspct of th traditioal approach to dmad fuctios stimatio cosists i th absc of ay us of th primal iformatio rprstd by th first-ordr-cssary coditios for utility maximizatio. With th advt of flxibl fuctioal forms such as th almost idal dmad systm (AIDS) of Dato ad Mullbaur, th primal rlatios cotribut idpdt iformatio which xists oly i a latt stat. Hc, fficit stimators of th dmad fuctios rquir th utilizatio of th complt systm of primal ad dual rlatios. This papr cotributs thr mai rsults. First, w propos th joit ad ovl stimatio of primal ad dual rlatios i ordr to obtai fficit stimats of systms of dmad fuctios. Scod, w prst a thortical ad mpirical framwork that admits 2

4 th stimatio of dmad fuctios also wh th obsrvd commodity prics ar th sam for all th sampl uits. Third, w offr a gral solutio of tholiar rrorsi-variabls problm that rsults from th spcificatio of th mpirical framwork outlid abov. Ths rsults ar obtaid o th basis of plausibl ad gral assumptios: Th typical houshold is risk utral ad its objctiv is th maximizatio of th xpctd utility fuctios subjct to a xpctd budgt costrait. I othr words, th houshold uit maks its cosumptio dcisios through a plaig procss that uss xpctd iformatio. Th xpctatio procss is spcific to ach houshold. Wh th coomtricia dsirs to masur commodity quatitis ad prics for th purpos of ifrrig th dmad fuctios that wr usd by th sampl housholds, masurmt rrors occur. W assum a additiv rror structur for ach pic of sampl iformatio, that is, commodity quatitis, prics ad th houshold s disposabl moy icom. Th rsultig oliar rrors-i-variabls modl will b stimatd usig a ovl approach. Traditioally, rrors-i-variabls spcificatios iclud uobsrvabl quatitis that ar rplacd by thir obsrvabl coutrparts. This rplacmt stp, howvr, iducs udsird proprtis o th stimator with icosistt stimats ad othr difficultis. I ordr to avoid ths udsirabl proprtis, w will ot rplac th latt xpctd quatitis ad prics. A two-phas procdur will first grat cosistt stimats of thos xpctd quatitis ad prics which, th, will b usd as istrumtal variabls to produc fficit stimats of th modl s paramtrs. 3

5 Th Thortical Framwork W assum that risk-utral housholds maximiz thir xpctd utility subjct to a xpctd budgt costrait. Lt p b th ( 1) vctor of xpctd prics ad y th xpctd disposabl icom availabl to th risk-utral houshold that solvs th followig problm (1) U * (p,y ) df = max{u (x) s.t. y = p x} x whr x is a ( 1) vctor of commodity quatitis. Th first-ordr-cssary coditios ar statd as (2) L x = U x(x)- lp = 0 L l = y - p x = 0 whr L is th Lagraga fuctio ad l is th Lagrag multiplir. Aftr solvig quatios (2), th vctor of commodity quatitis x will bcom th vctor of xpctd quatitis x. Similarly, th Lagrag multiplir l will bcom th xpctd Lagrag multiplir l which is quivalt to th margial utility of moy icom. W assum a itrior solutio of quatios (2) that will grat commodity dmad fuctios with valus (3) x = d (p,y ). I cas th first-ordr-cssary coditios (2) do ot admit a xplicit, aalytical solutio, th dmad fuctios (3) xist via th duality pricipl. Th idirct utility fuctio is obtaid by isrtig th optimal quatitis (3) ito th dirct utility (4) U * (p,y ) = U (x ) with th margial utility of moy icom dfid as U * y (p,y ) = l. 4

6 Th abov dvlopmt corrspods to th txtbook discussio of cosumr thory. Th coomtric spcificatio of th modl rquirs th spcificatio of th rror structur that accompais ay statistical masurmt. A Masurabl Modl of Cosumr Dmad W assum that all th commodity quatitis ad prics ad th disposabl icom ar masurd with rrors. That is, th coomtricia obsrvs x, p ad y that bar a additiv rlatio with thir xpctd coutrparts, that is x = x +, p = p + ad y = y + 0. Th vctor of rrors = (,, 0 ) is distributd as ~ N(0,S). W furthr assum that th rrors ar idpdtly distributd across sampl uits. Th masurabl coomtric modl of cosumr bhavior ca thus b assmbld usig all th primal ad dual rlatios ad th postulatd rror structur as follows: Primal rlatios (5) p = U x (x )/U * y (p,y )+ (6) y = p x + 0 Dual rlatios (7) x = d (p,y )+ Error structur (8) p = p + (9) y = y + 0 (10) x = x +. Equatios (5) rprst th latt first-ordr-cssary coditios ivolvig commodity prics ad th margial utilitis of goods. Ths rlatios ar traditioally disrgardd 5

7 durig th stimatio of dmad fuctios (7). Howvr, i th cas of modl spcificatios ivolvig flxibl fuctioal forms, rlatios (5) covy iformatio idpdt from th dmad fuctios (quatios (7)). Th pursuit of fficit stimats, thrfor, rquirs th joit stimatio of all th primal ad dual rlatios of th cosumr modl. Ecoomtric Modl W assum a sampl of N houshold with idx = 1,...,N. Th stimatio procdur proposd hr is articulatd i two-phass. Th Phas I objctiv cosists i obtaiig cosistt stimats of all th idividual housholds xpctd quatitis ad prics. Ths stimats, th, will b usd as istrumtal variabls durig Phas II to obtai fficit stimats of th modl s paramtrs. Th Phas I spcificatio assums th form of a oliar rrors-i-variabls modl to b stimatd by oliar last-squars. Spcifically, lt b = (b U,b x ) b th vctor of paramtrs that costitut th fial goal of th stimatio procss. Th, th Phas I stimatio problm is statd as follows: (11) mi subjct to b,p j,x j (12) p = U x, N Â,y, =1 (13) y = p x + 0 (x * )/U y (p,,y * )+ (14) x = d (p,y,b x )+ (15) p = p + 6

8 (16) y = y + 0 (17) x = x + (18) Â p j N =1 j (19) Â y = 0 N =1 (20) Â x j N =1 j = 0, j =1,...N = 0, j =1,...N Th sampl idx was rmovd from th obsrvd vctor of prics i ordr to idicat that all th sampl uits fac th sam commodity prics. Th solutio of th Phas I modl is ot a trivial pursuit. For xampl, with N=100 ad =5, thumbr of costraits is (2+1)(2N+1)+1=2211. I th cas of first-ordr-coditios that admit a xplicit solutio thumbr of costraits is rducd cosidrably bcaus ithr th primal or th dual costraits ar rdudat. With a utility spcificatio corrspodig to th flxibl AIDS modl, all th primal ad dual costraits ar rquird for obtaiig fficit stimats. Thumbr of xpctd quatitis ad prics, xpctd moy icom, paramtrs ad rrors is somwhat largr tha Th Phas I problm ca b solvd with commrcially availabl computr applicatios for oliar programmig such as GAMS (Brook t al.) Costraits (19)-(20) ar th orthogoality coditios for th xpctd uobsrvabls. Th stimats of ths quatitis ad prics p ˆ, y ˆ, x ˆ will b usd as istrumtal variabls i th Phas II stimatio problm. Lt ˆ S -1 b th stimatd covariac matrix computd from th stimatd rsiduals of Phas I. Th, th Phas II stimatio problm is th followig oliar smigly urlatd (NSUR) quatio modl: 7

9 (21) mi subjct to N Â b, =1 ˆ S -1 (22) p = U x (ˆ, (23) y = p ˆ x ˆ + 0 x * )/U y, (24) x = d ( ˆ p, ˆ y,b x )+. ( p ˆ, y ˆ * )+ Th Phas II problm ca b solvd with covtioal coomtric packags such as SHAZAM (Whistlr t al.). Th cosistcy of th abovoliar last-squars stimator has b discussd by Davidso ad MacKio (1993) i thir Thorm 5.1. Empirical applicatios W prst two mpirical applicatios of th mthodology dvlopd i th prvious sctios. Th first xampl dals with a Cobb-Douglas utility fuctio that admits a xplicit solutio of th first-ordr-cssary coditios. I this cas, ithr th primal or th dual st of rlatios is rdudat for th Phas I problm. It is ot rdudat, howvr, for th Phas II stimatio problm, as ach compot of th rror structur i th NSUR spcificatio covys idpdt iformatio. Th scod xampl dals with a flxibl fuctioal form of th idirct utility fuctio such as th AIDS spcificatio. I this cas, th dirct utility fuctio xists oly i a latt form ad its coomtric spcificatio will b giv as a scod-ordr Taylor sris xpasio. Thus, this primal spcificatio of th dirct utility fuctio covys iformatio that is idpdt from th dual st of rlatios ad all th primal ad dual rlatios, thrfor, ar rquird i th Phas I problm. 8

10 Cobb-Douglas Utility Fuctio Th dirct utility taks o th follow spcificatio: (25) U (x) = Â b j log(x j - g j ) j=1 whr x j > g j 0 ad th paramtr g j rprsts a subsistc lvl of th corrspodig good. Th Phas II coomtric modl taks o th followig spcificatio: subjct to Primal rlatios N b, =1 mi Â ˆ S -1 (26) p j = U x j, = b j Â k=1 b k (ˆ x j * )/U y ( ˆ, p j ( y ˆ - Â p ˆ k=1 k g k ) + ( ˆ j - g j ) x j, y ˆ * )+ j (27) y = Â ˆ Dual rlatios p j x ˆ j j (28) x j = d j ( ˆ p j = g j + b j Â k b k, y ˆ,b x j )+ j ( y ˆ - k=1 p ˆ j Â p ˆ k g k ) + j Th Cobb-Douglas modl of cosumr bhavior admits a xplicit solutio of th firstordr-cssary coditios. W r-mphasiz, howvr, that, durig th Phas II stimatio rlatios (26) ad (28) covy idpdt iformatio through diffrt rrors ad thir distributios. Thy ar both rquird, thrfor, if o wishs to obtai fficit stimats of th modl s paramtrs. 9

11 AIDS Idirct Utility Fuctio Th AIDS modl is traditioally spcifid as th followig xpditur fuctio: (29) log E (p,u ) = (1- U )log[a(p )]+U log[b(p )] = log y whr a(p ) ad b(p ) ar pric idxs dfid (by Dato ad Mullbaur, 1980) as (30) log[a(p )] = a 0 + Âa k log p k + Â Âg kj log p k log p j /2 (31) log[b(p )] = log[b(p )]+ b 0 (p k ) b k. Th AIDS idirct utility fuctio, thrfor, ca b writt as (32) U * (p,y ) = log y - log[a(p )] (p k ) b k k=1 b 0 k with th margial utility of moy icom giv by (33) U * y (p,y ) = 1/ y b 0 ( p k ) b k = l. k k=1 k j Th Phas II stimatio problm of th AIDS modl ca th b statd as follows: subjct to Primal rlatios N b, =1 mi Â ˆ S -1 (34) p j = U x j, (ˆ x j * )/U y ( ˆ, p j, y ˆ * )+ j È First drivativ of 2d ordr È = Í Î Taylor sris of U (ˆ x ) Î Í ˆ y b 0 ( p ˆ k ) b k k=1 + j (35) y = Â ˆ Dual rlatios p j x ˆ j j

12 y (36) x j = a ˆ j ˆ p j y p j + ˆ ˆ Âg kj k=1 log p ˆ k y + b ˆ j ˆ p j log[ ˆ y /a( ˆ p )]+ j. Th Taylor sris xpasio of th latt dirct utility fuctio provids th complmtary primal iformatio for obtaiig fficit stimats of th AIDS paramtrs. Coclusio W hav tackld thrs glctd aspcts of th coomtric stimatio of cosumr dmad fuctios. First, w hav proposd a thortical framwork basd upo th assumptio that housholds ar risk-utral ad maximiz thir xpctd utility subjct to a xpctd budgt costrait. W th assumd that all th sampl iformatio is masurd with a additiv rror. Scodly, w suggstd that fficit stimats of th cosumr dmad fuctios rquir th joit stimatio of primal ad dual rlatios. As a byproduct of this gral framwork w obtai th possibility of stimatig dmad fuctios also wh all th sampl housholds fac th sam obsrvd commodity prics. This statistical rality corrspods to a rathr commo vt associatd with may atioal survys ad csuss of cosumr bhavior. Third, th rsultig oliar rrors-i-variabls problm was solvd usig a twophas approach that producs cosistt ad fficit stimats of th modl s paramtrs. This ovl approach avoids th pitfalls of th traditioal mthod for hadlig rrors-i-variabls modls whr th latt, uobsrvabl iformatio is rplacd by its obsrvabl coutrpart. This rplacmt stp iducs udsirabl proprtis o th 11

13 stimator such as icosistt stimats. Th mthodology prstd i this papr is rathr gral ad ca b applid to svral othr micro-coomtric cotxts. 12

14 Rfrcs Brook, A., K., D. Kdrick ad A. Mraus. GAMS, Usr Guid. Boyd ad Frasr Publishig Compay. Davrs, MA, Davidso, R. ad. G. MacKio. Estimatio ad Ifrc i Ecoomtrics. Oxford Uivrsity Prss, Nw York, N.Y., Dato, A ad. Mullbaur. A Almost Idal Dmad Systm, Amrica Ecoomic Rviw 70(3), u 1980, Sto,. Th masurmt of Cosumrs Bhaviour i th Uitd Kigdom, Cambridg Uivrsity Prss, Cambridg, Whistlr, D., K. Whit, S.D. Wog, ad D. Bats. SHAZAM, Th Ecoomtric Computr Program, Usr s Rfrc Maual, Vrsio 9. Northwst Ecoomtrics Ltd. Vacouvr, B.C. Caada,

Chapter (8) Estimation and Confedence Intervals Examples

Chapter (8) Estimation and Confedence Intervals Examples Chaptr (8) Estimatio ad Cofdc Itrvals Exampls Typs of stimatio: i. Poit stimatio: Exampl (1): Cosidr th sampl obsrvatios, 17,3,5,1,18,6,16,10 8 X i i1 17 3 5 118 6 16 10 116 X 14.5 8 8 8 14.5 is a poit