Generalised Translation of Indirect Utility Functions

Gene alised T ansla ion o Indi ec U ili y Func ions DENIS CONNIFFE Na ional Uni e si y o I eland Maynoo h Abs ac This pape conside s he de i a ion o new demand sys ems om exis ing ones h ough eplacing an indi ec u ili y unc ion by , whe e p is a ec o o p ices and y is income. This is a gene alisa ion o Go man ansla ion ),( yU p})/(,{ j ypyyU jj   p ,( yU ) jj p    p and will be shown o be e ec i e in e ms o p oducing new demand sys ems wi h bo h good egula i y and lexibili y p ope ies. JEL Classi ica ion: D 11 Keywo ds: T ansla ion, indi ec u ili y unc ions, demand equa ions. Add ess o co espondence: Denis Conni e, NIRSA, Economics Depa men , Na ional Uni e si y o I eland, Maynoo h, Co. Kilda e, I eland. Tel: 353 1 7086299; Fax: 353 1 7083934; e-mail: [email p o ec ed] 2 I INTRODUCTION Go man (1975) in oduced he “ ansla ion” de ice o inco po a e ex a pa ame e s in o u ili y unc ions and demand equa ions. I is he o iginal indi ec u ili y unc ion, whe e p is a ),( yU p ec o o p ices and y is income, he ansla ed u ili y unc ion is ),( jj pyU   p, (1) whe e he j  a e he subsis ence quan i ies, y is assumed jj p   and summa ion is o e n commodi ies. Go man showed ha i we e he o iginal demand equa ions, he ansla ed ),( yqip equa ions a e ijji pyq      ),(p. (2) Fo example, i is he simple homo he ic u ili y unc ion y/P, whe e P is a weigh ed ),( yU p geome ic mean o p ices, )log(log jj pP   , he demand equa ions iii pyq /   a e ansla ed o he amous S one-Gea y linea expendi u e sys em (LES) )( jj i i ii py p q     . The idea o his pape is o eplace (1) by he mo e gene al ansla ion o o ),( yU p                   j y p yyU j j   ,p. (3) This educes o (1) i all 1 j  . The condi ion jj py    is eplaced by . (4)  j jypy jj    1 which will hold o posi i e j  i y is no oo small. I a i  is nega i e, i  mus also be nega i e. This pape is pa icula ly conce ned wi h how gene alised ansla ion can p oduce demand sys ems wi h bo h good egula i y and lexibili y p ope ies. Regula i y means ha , gi en app op ia e anges o he pa ame e s, he indi ec u ili y unc ion complies wi h he cons ain s implied by a ional economic beha iou .1 Ideally, his should be possible o all p ices and incomes (global egula i y), bu 1 Tha is, a con aximises di ec u ili y unde a budge cons ain . This implies he indi e u ili y unc ion ),( yU p should be homogeneous o deg ee ze o in income y and p ices p, non- sume m c 3 should a leas hold o all alues o hese a iables ele an o he si ua ion unde s udy. Flexibili y is also equi ed in ha he co esponding demand sys em, while sa is ying egula i y, should be able o model a easonably comp ehensi e spec um o consume beha iou - so he possible alues o income and p ice elas ici ies, which a e unc ions o pa ame e alues, should no be se iously es ic ed2. The in luen ial ‘ lexible unc ional o ms’ app oach, employing Taylo se ies app oxima ions o gene al u ili i y (o cos ) unc ions, sough sys ems embodying lexibili y and hoped o egula i y, bu i seems ha o en hei lexibili y depends on hei pa ame e s being allowed o ake alues ha con adic egula i y3. Such models gene ally canno es i obse ed consump ion pa e ns do o do no acco d wi h economic heo y. The app oach in his pape will be o s a om globally egula sys ems and o imp o e hei lexibili y by gene alised ansla ion4. P ope ies o he gene al ansla ed u ili y (3) a e examined in sec ion 2 and he co esponding demand sys em de i ed. Income and p ice elas ici ies a e ob ained and p esen ed in e ms o he elas ici ies o he pa en sys em and he pa ame e s o he gene alised ansla ion. Sec ion 3 illus a es hese esul s by conside ing a pa icula case, ai ly pa simonious in pa ame e s, ha is in e es ing in i s own igh . Sec ion 4 applies gene alised ansla ion o mo e pa ame e ich, hough s ill globally egula , u ili y unc ions. In sec ion 5 he possible applica ion o gene alised ansla ion o non- globally egula u ili ies is discussed. Finally, in sec ion 6, connec ions be ween gene alised ansla ion and Hou hakke ’s (1960) indi ec addilog sys em a e explo ed5. II. TRANSLATED DEMAND EQUATIONS AND ELASTICITIES Le dec easing in y, non-inc easing in p, and con ex o quasi-con ex in p. These cons ain s imply co esponding condi ions (agg ega ion, homogenei y, Slu sky symme y and nega i i y) on he demand equa ions. 2 Mo e o mal de ini ions o lexibili y exis , di e ing in de ail. See, o example, Diewe (1974). 3 Ca es and Ch is iansen (1980), Ba ne and Lee (1985) and Coope and McLa en (1992) ha e discussed he di icul y o econciling lexibili y and egula i y o such sys ems. 4 O cou se, he e a e o he app oaches o imp o ing egula i y p ope ies. Ba ne (1983), Ba ne and Lee (1985) and Chal an (1986) epo ed wide egula i y egions esul ing om app oxima ions based on Lau en , Mun z-Sa z and Fou ie expansions a he han Taylo se ies. Lewbel (1987) and Coope and McLa en (1996) ha e also p oposed sys ems. 5 Lewbel (1985) has al eady used he e m gene alised ansla ion in a pape on inco po a ing demog aphic e ec s in o demand equa ions, which was a heme o conside able in e es o Go man. Bu he gene alised ansla ion o his pape is so ob iously a gene alisa ion o Go man’s ha i seems inapp op ia e o call i any hing else. 4 j y p y   , yz j j         ha he ansla ed u ili y is I s homogeneous o deg ee ze o in income and as i ough o be i a alid u ili y unc ion, i is clea ha is also. ),( zU p. ),( yU piso p ices, ),( zU p y z z zU y zU       ),(),( pp                 yyz j jj j j  1 (5)              jj pp zU  ),(p ssuming he o iginal u ili y is non-dec easing in income, he de i a i e o wi h espec o z non nega i e. Taking he e ms wi hin he chain b acke s, equa ion (4) implies he i s and second ms combined a e posi i e while he hi d is also posi i e because, as p e iously men ioned, ),( zU pA is i  e and i  a e p esum o hed o ha e he same sign. S e ansla ed u ili y is non-dec easing in income. y z z U p U pi zU i        ),(p                   p UU i ii  (6)     1 i yzpi  u ili ies. show ha abo e so s will be gula . This co esponds o he si ua ion wi h Go man ansla ion. Fo example, he simple omo he ic u ili y unc ion y/P,  So assuming he o iginal u ili y is non-inc easing in p ices, he ansla ed u ili y is also. P oo s o he con exi y o ansla ed u ili ies a e mo e complica ed and depend on he o ms o he o iginal They a e p o ided in Appendix 1 and me income le el he ansla ed sys em e )log(log jj pP    h , is globally con ex in p ices, while ypy jj /)(   , which gi es he LES, is con ex i . (7) s nds o inc ease wi h ime. The e m ‘e ec i e global egula i y’ has been applied o egula i y 222 2)( ppy   iijji So al hough Go man ansla ion inc eases lexibili y by in oducing ex a pa ame e s, i may in alida e egula i y a e y low incomes. This has no been seen as a di icul y, because, a leas o analyse wi h ime se ies da a, in e es ocuses on in e ences alid o ecen o cu en ime pe iods and income e 5 e e ywhe e excep a low incomes6. Gene alisa ions o (7) o gene alised ansla ion can be deduced a e sec ions. F om Roy’s lemma he demand equa ions a e om he esul s o Appendix 1 and will appea in l    y jij )1(1                      j i p y p z zU p zU yq j i ii i i     ),( / ),( ),( 1 pp p, e deno es he demand equa ion om he ansla ed u ili y. Also by Roy’s lemma, i q whe                        j y p yyqzq z zU p zU j jioio i   ,),( ),( / ),( pp pp , whe e eno es he demand unc ion de i ed om he o iginal u ili y unc ion. So io qd                                          j i j y p y p y p yyq yq j jij i ii j jio i      )1(1 , ),( 1 p p , o , mo e idily,                   1 , 1i y p zi ii   p, o , in e ms o budge sha es   ),( q V yq ioi p                     i y p y z zw V yw i iiioi   , 1 ),( pp , (8) whe e j y Vjj     )1(1 pj        c oss-p i ms o and , he income and p ice elas ici ies o he o iginal sys em. They a e: and ),( ywi pis he budge sha e o he ansla ed sys em as a unc ion o p and y, and ),( zwio pis he budge sha e o he o iginal sys em as a unc ion o p and z. The income, own-p ice and ce ))( zp, io E),( zeiko p elas ici ies o he ansla ed sys em a e de i ed in Appendix 2, in e 6 The e m seems due o Coope and McLa en (1996). 6        jjoi i io jj iio i io i zw yw zw yV z ywzE yw zw E  ),()1( ),( ),( 1),(),( ),( ),( p p p pp p p           yV z yw zw zEzwzeywzee i io ioioiioi iiioii ),( ),( 1),(),(),(1),()1(),( p p ppppp                ywzee k kikoik ),()1(),( pp  yV z yw zw yw yw zEzwze k ko i k ioioiko ),( ),( 1 ),( ),( ),(),(),( p p p p ppp he xample o he nex sec ion illus a es his. D TRAN E en i he elas ici ies o he o iginal sys em a e e y es ic ed, hese elas ici ies a e much less so. T E III GENERALISE SLATION OF THE CONSTANT BUDGET SHARE MODEL Re u ning o PyU /, whe e )log(log jj pP    , he o iginal demand equa ions, in budge sha e o m, a e iio w  , o cons an budge sha es. Income and own p ice elas ici y a e uni y and c oss-p ice elas ici ies a e ze o. F om (8) he ansla ed demand equa ions a e                                         j j y p y p y p w j jj i i ii j ji i      11 1 , idily,. o                    i  i iiii y p y z V w  1 (9) is shown in Appendix 1 ha his sys em7 will sa is y e ec i e global egula i y i I 22 222 1 22)1(           ii pp ii  (10)           y pz y pz iiiiiiii  , which holds i   iiji     ,1, and .1    I could e en hold o an 0 i  i  10 i  , al hough ha may be pu ely academic. Ob iously, (10) educes o (7) i all j  = 1. I (9) is o ha e much p ac ical alue, i should be mo e lexible han he LES, which i becomes 7 This sys em has been examined in g ea e de ail in Conni e (2002a), bu was no hen unde s ood as a case o gene alised ansla ion. 7 when all j  = 1. The limi a ions o he LES include he linea i y o all i s Engel cu es8, i s inabili y e inc pa a io o income elas ici ies ansla ed income elas ici y, d opping, o con enience, ip s indica ing ansla ed and o iginal, o ca e o in e io goods and he ying o p ice e c s o ome aspec s o p ice changes. In icula , he a io o he c oss-p ice elas ici ies ik e and jk e equals he i Eand j E, and he possibili y o complemen a y goods is excluded. F om he o mula o he p e ious sec ion o he subsc     i jjjji i iwyVw     iii z wE )1( 1    . y ne anges d i Eis no gene ally a simple mono onic unc ion o income. A commodi y could be a luxu , o a cessi y a di e en o income, and could e en be an in e io goo ( o a posi i e i  and small i  ). Howe e , as y he budge sha es (9) end o cons ancy and i E o uni y, which is no un easonable and is a p ope y o many o he demand sys ems. Tha Engel cu es can ake a la ge he demand equa ion and he income elas ici y, al hough as F om he o mulae o he p e ious sec ion, he own-p ice and c oss-p ice elas ici ies a ie y o shapes is clea om y ey will app oach linea i y. h a e          wwe iiii  )1(1 yV z wi i ii   1 and             Kk i kik yV w w e  1,   iz  s o c oss-p ice elas ici ies need no equal a ios o income elas ici ies. The compensa ed p ice so a io e ec y q q p qi k k i      is 8xample, Lau, 1986) is Eons an p ices, a e no One o he ew gene ally ag eed indings om empi ical s udies (see, o e ha ngel cu es, he ela ionships be ween expendi u e and income a c n-linea o a leas some goods. 8 )}1()1({)}1(1{ i ikki ppppyVyVy ikkikiki qqy zz qq  kjjjjki w        and his can gene ally be nega i e o posi i e, allowing o complemen s as well as subs i u es. So he ex a n pa ame e s con ained in (9), ela i e o he LES, do subs an ially inc ease lexibili y9. O ou se, any demand sys em wi h jus 3n-1 pa ame e s, will all sho o ull lexibili y i he e a e mo e lly i da a e sca ce, o obse ed p ice anges do no embody su icien independen a ia ion o es ima e a e y e ailed model o in e ac ing p ice e ec s. NIOUS UTILITIES igina . Ini ially, a u il o so10, pa s om wo u ili on ame e s, a e onca e in p ices. Then (Conni e, 2002b) he sum, p oduc and he ecip ocal o ecip ocals (ha monic mean) a e also globally egula u ili y unc ions. Fo example, wi h and c han a ew commodi ies. Some imes, howe e , pa simony o pa ame e s is desi able, especia a d IV GENERALISED TRANSLATION OF LESS PARSIMO ea e lexibili y o p ice e ec s can be modelled by gene alised ansla ion o a less pa simonious, G bu s ill globally egula , o l u ili y i y o he o m Py/will be conside ed, whe e is a unc ion o mo e han n pa ame e s, bu is conca e in p ices. P I is easy o gene a e a globally egula homo he ic u ili y unc ion wi h 2n, ame e y unc i s /y and 2 /Py , whe e 1 P and 2 P, bo h unc ions o n pa 1 P c j j pP  = 1 jj pP  = 2, wi h i  and i  posi i e, he ha monic mean o he u ili y unc ions is    jjj pp y Uj   , which is globally egula wi h 2n – 1 pa ame e s. I gi es he demand sys em jjj iiji ij j wp+p p+p =      , which s ill, o cou se, gi es a uni a y income elas ici y, bu p ice elas ici ies o ollak (1972) desc ibed a class o demand equa ions o he o m ),,/( Wyp q iii  9 P whe e W is a homogenous unc ion o all p ices and income, as exhibi ing “gene all alised sepa abili y”. Because p ices, excep , ake e ec h ough W, he e a e implied symme ies in how commodi y demands a e a ec ed by p ices o o he goods. The LES and he Indi ec Addilog Sys em (IAD) o Hou hakke (1960) a e o his class. Bu (9) is no , p o ided he i p j  a e non ze o. 10 Cons ain s such as 0  j  educe pa ame e s by one. 9 )( jjji iiiii ppw j    )1( j we   pj   and j pj   )( jjji kiikik ppw j    we   . sys em wi h 4n pa ame e s, gi en by equa ion (8), wi h elas ici ies gi en by subs i u ing he abo e in o he o mulae o sec ion II. Many o he sys ems a e o p c ion Combining simple u ili y unc ions is no he only way ex a pa ame e s could ha e been ob ained. iginal homo he ic u ili y unc ions wi h mo e han 2n pa ame e s a e qui e possible. Fo example, Gene alised ansla ion hen p oduces a -1 ob ainable by o he combina ions ai s o u ili y un s. O   2 1 2 1 kjjk pp y U  , could ha e been aken whe e he jk  a e posi i e and jk  = kj  11. This has n(n+1)/2 pa ame e s and gi es he demand sys em 2 1 2 1 2 1 2 1 2kjjk jiji i pp pp w     , wi h p ice elas ici ies 2 2 1 2 1 )(4 2 1 kjjki iii iii ppw p we     and 2 2 1 2 1 2 1 2 1 )(4 kjjki ikki kik ppw pp we     . spi e o a ela i e p o usion o pa ame e s his sys em s ill embodies es ic ions on p ice ec s as In e y q p i k i   = qq k  2 2 1 2 1 ki 22 )(4 kj ik pp pp    si i e i 11  jk is po ik  is, so uling ou complimen a i y. So gene alised ansla ion, which gi es a 11 This is ac ually he homo he ic case o he gene alised Leon ie u ili y unc ion. 16 iii p P Pp P P pz U z U Pz U              log11 - and 0, 1 2 2 2 2 . u he mo e F                                           ji 22 2 i 2 2 2 2 p Plog p Plog P z log p Plog P z log jiji ii pp P P z pp U and p P P z p U o (A1) and (A2) become and s                                                2 2 ii 2 i 2 2 p z p Plog 2 p Plog log1 ii p z z p P z P and   P                           ijjiji 2 p z p Plog p z p Plog p Plog p Plog log1 z pp P z ji . be ew i en as These can                                          2 2 2 2 2 211loglog1 i i ii ip z z p z p z z p P z p P z P and                                   jijjiiji ppzp z pp z p z pp z             zzzP z zPP P 11log1loglog . a i he Hessian ma ix o wi h espec o p ices, which is posi i e e ini e17, plus which, is posi i e semi-de ini e, whe e G is he ec o wi h i h e m 2 1 o he Hessian m x (wi h espec o p ices ) o he ansla ed u ili y is (apa om he S PlogP/1 mul iplie ) z by minus 'GG , d ii p z z p P z    1log , lus a diagonal ma ix wi h i h e m p 1 2 i  2 )1(             y p p p zi i iii i  minus , whe e H is he ec o wi h i h e m zHH /' 17 I P is conca e, is conca e. Plog 17 1           i y p p zi ii i   . While 'H H  is nega i e semi-de ini e, he addi ion o i o he diagonal ma ix wi h i h e m 2 22 2   i p z  22  i gi es a nonnega i e de ini e ma ix (diagonals posi i e and p incipal mino s o highe o de ze o). ed u ili y is con ex wi h espec o p ices i he n ul iplied by z minus he diagonal ma ix wi h elemen s 2          yp ii i  So he ansla ega i e o he Hessian o log P m z y p p yiiiii    pii i i i 22 22 1 2)1(                  (A3) a i e de ini e. on exi y o he cons an budge sha e model is nonneg C )log(log jj pP   Now le . The Hessian o log P wi h espec o p ices is So U is on ex wi h espec o p ices i diagonal. c z y p p y p p z ii i ii i i iii i i 22 22 1 2 2)1(                        is posi i e, o 22 2222 2)1(        ii p p p zi iii i iiii  , (A4) 1            y z y pi  hich is equa ion (10) o sec ion 3. I all h ej  a e posi i e, he le hand side is o o de , while e igh hand side e ms a e o o de and , so since 2 y w i y  2i y  22 i  his posi i e he condi ion holds o ided y (and hence z) is no small. The i s igh hand side e m is nega i e i 0 < i  p < 1, so ha a e o i  migh e en be compa ible wi h con exi y in ha si ua ion. Fo a leas one o he j  znega i e, be he smalles (mos nega i e). The g ea es powe o y in z is hen wi h coe icien s y  1 le s  18 s ss s p   (  is nega i e i s  is). So he le hand side de , while he igh hand ms a e o o de and and i is clea he c i ical e m is i= s. Compa ing coe icien s , he condi ion is s y  22 is o o si y  2i y  22 e  22g ea e han . This will be ue i 0 > 2 s  s  s yss    > -1, e en i s  ois small (o e en ze o). So e ec i e global egula i y holds18 i     iiji     ,1, and  .1  ain con ex wi h espec o p ices i he nega i e o he Hessian o log P ul iplied by z minus (A3) is nonnega i e de ini e. Le he smalles eigen alue o minus he Hessian a ix o log P, ha is, he ma ix wi h i,k h e m Con exi y wi h Pconca e in p ices The ansla ed u ili y is ag m m ki pp P   log 2  . I log P is s ic ly conca e minus he Hessian is posi i e de ini e and can be w i en equal o be Q+  I, wh de e I is he iden i y ma ix, Q is posi i e semi-de ini e an  is posi i e. Then he condi ion o egula i y is 22 222 )1               p z p zi ii iii    . 1 2 (  ii yypi i    (A5) ms he same as (A4) wi h eplacing i  /2 i p  This see , bu i is ac ually a much s ic e condi ion. I s equi alen , whe e s  deno es he smalles i  would ha e been 2 2 22 1 22)1(            ip pz p pz ii   2          i yy iiiiiiis   , all i and no jus i=s. So (A5) p obably exagge a es he income equi ed o e ec i e global gula i y, al hough as he le hand side is o o de (assuming o 2 y)  i  e , while he igh hand side ms a e o o de and , he e is no doub i will hold as y inc eases. Less demanding ed o sp i ic ases o P. Again, i i y  2i y  22 e  condi ions could be de i ec c =0 o some obse a ions (log P no 18 The condi ions on he j  and j  co espond o hose o he egula i y o he u ili y unc ion o he indi ec addilog sys em. The alidi y condi ions o he IAD ha e been deba ed in he li e a u e mo e han once, as he exchanges be ween Gamela os (1973, 1974) and Some maye (1974), and be ween Akin and S ewa (1979) and Mu y (1982) es i y. 19 i  s ic ly conca e) e ec i e global egula i y could s ill hold i 0 < < 1, al hough he case is e  p obably no o p ac ical in e es . Fo nega i he same a gumen as be o e gi es he condi ion nd again his will be ue i 0 > ss p   2>2 s  as  > -1. ice index. Hessian A(3) o he He Con exi y wi h Pa anslog p As be o e, con exi y equi es he nega i e o he o log P mul iplied by z minus be nonnega i e de ini e. The nega i e o ssian o Log P is    LL , i p/1 ,  is he ma ix o coe icien s jk  whe e L is he diagonal ma ix wi h i h elemen and is he diagonal ma ix wi h i h elemen I , ollowing es ima ion o he  ./ 2 ii pS jk  ,  is ound o be conca e he con exi y equi emen becomes 22 22 1 22)1(            ii z p pzS i   . 2       y p p y i iiiiiiii  his gi es he condi ion o sec ion V. The condi ion is he same as (A4) wi h eplacing i S i  T and e same a gumen s apply as ega ds he equi ed anges o i  and i  . Bu while he i  hwe e ons an s, he a e unc ion o p ices and he alidi y o he a gumen depends on hei being exi y wi h a Gene alised Leon ie u ili y e u n o (A1) and (A2) o ake accoun o e ms ha no longe anish. Howe e , uch o he p e ious app oach will s ill apply. The ( ansla ed) u ili y unc ion is i Sc osi i e. p Con I is now necessa y o m 1 2 1 2 1 2 1 2                   U                 z p z p z pj i ij j j  nd could be w i en in he o m , whe e */ Pz a 2 1 2 1 2 1 2 1 2* jiijiij pppzP   , 20 al hough is now a unc ion o income, as well as p ices. The con exi y o he o iginal u ili y wi h espec o p ices mus imply ha ha is conca e in p ices and he e o e is also. *P *P *logP 2 i * *2 2 *2 2 p logz *log                P Pp P P z p U ii nd a  iii pz PzPPU                *logloglogz 2**2 P P p P P pz           pz *log1 * i 2 * * i i C p P P    *log1 * , say. (A6) Al hough he i s e m o (A6), which is easily shown o be 2 1  i  2* )(2        i p z P z/1 beis o o de , he e m , which may be shown o i C 2 1 2 1 2 jiji pp pz       2 1 3* 1 )( jiij i pP . This will be impo an la e . Re u ning o (A1), wo o i s e ms z/1 is o o de ii ip z pz U p U       2 2 22 now gi e i i i ippzP PpP          * *2* i z C z p z z P z                 2 11 p log1 *22 i * . he es o (A1) is Pz log 2 T 2 2 2 2 2 i ip z z U p z z U                 . (A7) a ly, e ms om (A2) Simil ijjiji p z pz U p z pz U pp U            222 gi e 21 jijiji p z p z zPp z z P z p z z P z Pz      log1 *log ** P pp P                        * 11 p log1 p **2 us  ji pl i ji p C p C j zz     nd, om he emaining e m o (A2), a ji p zzU  2 2 . (A8) p z  ome e ms a e almos iden ical o hose occu ing in he case, wi h Pz/* P Sins ead o , and, llowing om m o a posi i de ini e ma ix (minu y he Hessian o ) plus wo posi i e semi-de ini e ma ices minus wo diagonal ma ices i h diagonal e ms P as be o e, can be exp essed as o he su e s * /Pz * logPb w 2  2 * 2       i p z zP and 2 i p z z U     . espec i ely o o de i  22/1  and i   2/1 I is wo h no ing ha hese e ms a e in income. , hose in and can be seen as ollowing om i Cj C Tu ning o he ex a e ms in ol ed ' * *1 ''                  pp zz zCPLL , (A9)  zP C whe e Lis he ec o wi h elemen s i ip z CzP zP    *1, * n s . The i s ma ix in (A9) is posi i e semi-de ini e. The second l ma ix wi h diagonal e ms e diagonal ma ix wi h diagonal e ms i C and Cis he ec o wi h eleme is nega i e semi-de ini e, bu becomes posi i e semi-de ini e i he diagona 2* 2i zCP is added o i . The same applies o he hi d e m i h 2 * 2          i p z zP is added. These diagonal ma ices ha e o be sub ac ed again, o cou se. The i s e m in (A7) and he (A8) e m a ise om he ma ix 22 ' 2 2                       pp zz z U and since         2 1 2 1 2 1 2 1 2 2 3 jiijjj pp z p P p U   (A10) is nega i e he ma ix is nega i e semi-de ini e. Adding he diagonal ma ix wi h diagonal e ms     3 * 2 jj z 2 2 2      zU 2   i p z p oduces a posi i e semi-de ini e ma ix. No e ha (A10) is o o de 2 3  z. o he inal sub ac ed diagonal ma ix is So he i h elemen 2 2 2 * 4    p z                i z U zP 2* 2 22i i zCP p z z U      , (A11) whe e, o cou se, 1           i y p p zi ii i   and 1 2 2)1(             i y p p p zi i iii i   . ei  F om ea lie commen s i is clea he e ms o (A11), assuming h posi i e, a e espec i ely o de o i y  2 1 2, i y   1 2 1  y. 2 and As al eady said, minus he Hessian ma ix o , mul iplied by , is posi i e de ini e in ices and i is easily seen o be o o de * logP * /Pz z o y. p So he same a gumen s as p e iously will o income and own p ice elas ici ies o ansla ed sys ems n enience o n , he dependence o quan i ies on p ices and income will no be indica ed xplici ly excep o z, he ansla ed income. The income elas ici y is emons a e e ec i e global egula i y. d Appendix 2: De i a ion Fo co o a ion e 23 =yq y i                               1 1i y p zq V i iiio   . y q q y Ei i i   = V 1                         1 )1( i y p yy z z zq iiiiio   y V V qi   . y qi   ha No ing and j y p yy Vj jjj             )1( 1 V y z   his is  ii y p yV q y p yVz zq  i ijji iiiiio                         )1( 1 )1( 1 . o S  i E  ii pq iiiiio     1 )1( 1 y p VyVqz z q yi ijj i i                   )1( . Using  )(( 1 zw y z VwzqVq y p y p y p y i ii       p ioi ioi ii ii i ii                , e he and e he budge sha es, and ha i w)(zwio awhe z zq zq z zEio )( io io  )( )( becomes i E        jjoi i io jj iio i io zw yw zw yV z wzE w zw  )()1( )( )( 1)( )( . The o icwn-p e elas ici y is = ii i pq p                  i i i i ii p q q p   e              1 1i y p zq V i iiio   . V 1                           2 )1()( i y p yp z z zq p zq iiii i io i io   i i p V V q   . i i p q  = No ing ha 1 )1( 1           i i iii iy p yp V   , 1          pi i y p zi ii   and 24 his is  yV zq qq V zq q pV zq q z zq p zq V io i ii io i i iio i io i io } )( ){1( } )( { )1( } )( { )( 1                . o equals  ii e S               ze zqio )( )( V zq q y p q zE zq zqp Vq io i ii i i io i ioi iio i )()1()1( )( )(  , o          yV z w zw zEzwzewzee i io ioioiioi iiioii )( 1)()()()1)(1()(  . 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