Issue 2(31)/2012 ISSN: Considerations on the Hicks Effect for N Consumer s Goods
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1 EurEcmca Issue 3/0 ISSN: Csderats the Hcs Effect fr N Csumer s Gds Cătăl gel Ia Daubus Uversty f Galat Deartmet f Ecmcs catal_agel_a@uv-daubus.r bstract: The aer aalyzes the Hcsa effect f cme ad substtut fr the case f gds whe all ther rces chage. We determe hereby the lmts f varat f the tw effects ad the relatsh betwee them. Keywrds: Hcs; gds; substtut; reveue Jel Classfcat: D0 Itrduct Let csder a csumer wh has the cme ad s faced wth the chce f gds B B wth tal rces. Fllwg a relcatg f the maret we wll csder the ew rces f gds B B as: ' '. Let als be a utlty fuct U:SC R + where SC s the sace f csumt gds relatve t thse gve. Csderg the budget ze ZB{ SC csumt baset s that the utlty be mamum becmes: ma U SC } the rblem f determg the I the cdts that the fuct U s ccave ad SC s a cve set t s shw [] [3] [4] that the tmal slut f the rblem s stuated the brder area f the budget that s t satsfes: ma U SC lyg the Lagrage multler methd results the ed: wth the slut: U m U m 6 DEELOPMENT POLICIES
2 EurEcmca Issue 3/0 ISSN: called Marshall demad. f f Csderg w the same rblem the drect f the mmzat f the allcated cme t meet a gve level f utlty the rblem s: m U u SC where u s the desred utlty. Fally t s shw that the Hcs demad satsfes: wth the slut: Um U U u m ~ g u ~ g u Nw csder terms f chagg rces that frst the csumer wll chage the demad rder t reserve hs rgal utlty level. The cmesated demad Hcs tye wll satsfes the rblem: m ' U u SC 7 where u s the tal level f the utlty. Let the slut: ~ f' ' u ~ f ' ' u ad - the cme eeded t urchase the gds baset resectvely U u ' ~ ~ ~. DEELOPMENT POLICIES
3 EurEcmca Issue 3/0 ISSN: We wll call the assage frm the tal baset f gds at effect shrt - Hs ad we have: Hs ~ -. ~ ~ - Hcs substtut The secd hase arses frm the fact that f the tal cme the csumer wll chage aga the demad vectr crresdg t ts actual cme rrt t that revusly reserved ts utlty. I ths case the rblem f the ucmesated demad s: wth the slut: I ths case we have: termedate gds baset Hv ~ - ~. ma U ' SC ~ g' ' ~ g ' ' ' ~ ~ ~ ~ at ~ The ttal effect f these tw hases s: U the btaed utlty. We wll call the trast frm ~ ~ - Hcs reveue effect shrt - Hv ad we have: 8 H Hs + Hv ~ - + ~ - ~ ~ - Cdts fr the Estece f the Cbb-Duglas Utlty Fuct Let a utlty fuct f Cbb-Duglas tye: U wth >0. The cdts f estece f a utlty fuct [4] mly the C dfferetablty ad ts ccavty. Cmutg the artal dervatves f frst ad secd rder fr the fuct we bta: U U U Q j j U j j j j j The Hessa matr s: Q U DEELOPMENT POLICIES
4 EurEcmca Issue 3/0 ISSN: DEELOPMENT POLICIES 9 H C-D U The rcal dagal mrs are therefre: U U. Fr the fuct s ccavty we must have: 0 dd ad 0 eve. Therefre: 0 s:. Hw the ly cdt f ccavty f the fuct remas: >0. 3 The alyss f the Hcs Effect fr a Cbb-Duglas Utlty Let w csder a csumer wh has the cme ad s faced wth the chce f gds B B wth tal rces whch are further adjusted t: ' '. We wll te: ' - the de f the gd rce chage. Let w a Cbb-Duglas utlty fuct: U wth >0. We wll te frst:. We have bvusly:. Calculatg the margal utltes we get: U m U frm where:
5 EurEcmca Issue 3/0 ISSN: DEELOPMENT POLICIES 0 s fally: If we dete: the share f the gd rce the dssable cme we have: The utlty crresdg t ths csumt dstrbut s: U Let us w csder the chage assets fr each B the cme remag cstat. Frm the abve relats we bta fr: ' : f ad the arrate utlty: 3 U. Let us te als that: ' '. t a rce chage f B fr the same value f the utlty U we wll have:
6 EurEcmca Issue 3/0 ISSN: DEELOPMENT POLICIES δ where δ ' ' beg the ew cme whch wll esure the utlty U. We therefre have: δ r terms f cme: ' ' ' ' The ew cme wll be: ' ' Wth ths cme we have: H δ where δ ' ' '. The Hcs substtut effect s thus: H H - δ Csderg w the tal cme stead f ' we bta: H f - H δ whch meas The Hcs cme effect. Detg fr smlcty: Γ we have therefre:
7 EurEcmca Issue 3/0 ISSN: We defe the fllwg the rats: H f f f r f H H H H - Γ H f - H Γ - the art f the ttal chage csumt due t the substtut effect; - the art f the ttal chage csumt due t the cme effect; H - the rat betwee the cme ad substtut effect. We have bvusly: + ad r. Frm the abve fllws: H H + H Γ Γ - Γ r Γ Γ Γ + Γ Γ * * Let the fuct: f: R + R + R where 0 f We have:. f f + DEELOPMENT POLICIES
8 EurEcmca Issue 3/0 ISSN: If we te: λ Γ : f λ λ + λ Let als the fuct: g λ λ + λ. Frm the eress f g we get easly: lm g gλ - lm g. But sce: g' 0 λ we get that: 0 g' <0 s g s strctly decreasg. I ths case: g λ -. g' >0 s g s strctly creasg. I ths case: g λ -. If λ the g >0 therefre f >0 that s f s creasg wth resect t. Hw f lm f 0 λ fllws: 0 lm f λ lm f 3 Le a cclus the cdts that cstat we have: f 0-λ 0 f -λ If w: λ < the as gλ -<0 fllws that f wll chage the mty. Let ϕ a arbtrary rt f g 0 that s: λ ϕ λ ϕ + λ ϕ 0. It s easy t see that the equat has tw rts: ϕ 0 ad ϕ. Therefre we have: f 0ϕ g >0 >0 f s strctly creasg s λ ϕ f ϕ 0 0ϕ ϕ f ϕ ϕ g <0 <0 f s strctly decreasg s DEELOPMENT POLICIES
9 EurEcmca Issue 3/0 ISSN: f ϕ g >0 λ f ϕ ϕ ϕ λϕ ϕ ϕ >0 f s strctly creasg s ϕ ϕ λ ϕ f ϕ ϕ ϕ Wth the smlfed tats: λ λ let the equat: We have: g' ϕ λϕ g" ϕ λ ϕ g ϕ λϕ λϕ 3 ϕ ϕ + λϕ 0 0 ϕλ 0 If ϕ > > the g" ϕ <0. If ϕ < > the g" ϕ >0. fter these we have: ϕ 0 g s strctly decreasg ad cve ϕ g s strctly creasg ad cve ad ccave. We have but: lm g ϕ gλ-<0 lm g ϕ. O the ther had: ϕ 0 ϕ 4 Because: l e y g λ fllws: l lm lme. Csderg the fuct: hy y + wth y>0 we have hy< ad lm0 hy. Therefre: g < y λ-<. Fr the determat f ϕ 0 we wll aly the Newt's methd tag t accut that the terval 0 the fuct g s strctly decreasg ad cve. S we chse the t 0 suffcetly clse t 0 s that: g 0 g 0 >0. We get therefre: DEELOPMENT POLICIES
10 EurEcmca Issue 3/0 ISSN: Fally: ϕ lm. g + - g' λ λ λ + λ 0 Fr the determat f ϕ we have tw cases: If g <0 the rt belgs t the terval where g s strctly creasg ad ccave s we wll chse the t 0 because g 0 g 0 >0. If g >0 the rt belgs t the terval where g s strctly creasg ad cve s we wll chse aga the t 0 because g 0 g 0 >0. We therefre have: ad ϕ lm. g + - g' λ λ λ + λ 0 If g 0 we have bvusly that ϕ. fter the abve relatshs we get fally: the art f the ttal chage csumt due t the substtut effect belgs t: 0-λ 0 f λ ; -λ f λ ; 5 λ ϕ ϕ 0 ϕ 0ϕ f λ <; λ ϕ ϕ ϕ λϕ ϕ ϕ ϕ ϕ f λ <; λ ϕ ϕ ϕ f λ <. ϕ the art f the ttal chage csumt due t the cme effect belgs t: λ 0 f λ ; 0λ f λ ; λ ϕ ϕ 0ϕ f λ <; DEELOPMENT POLICIES
11 EurEcmca Issue 3/0 ISSN: λϕ ϕ λϕ ϕ ϕ ϕ f λ <; λ ϕ 0 ϕ ϕ f λ <. r the rat betwee the cme ad substtut effect belgs t: λ λ 0 f λ ; λ f λ ; λ λ ϕ λϕ ϕ 0ϕ f λ <; λ ϕ λϕ ϕ λ ϕ λ ϕ ϕ ϕ ϕ f λ <; λ ϕ ϕ f λ <. λϕ ϕ 6 4 Ccluss The aalyss f the effect f cme ad substtut fr the case f gds s essetal determg the effect f rce chages csumt mvemet. The reset demarche establshes the lmts f varat f the tw effects ad the relatsh betwee them whe rce chages all the gds ad t just tw as the classcal thery. 5 Refereces Chag.C Fudametal Methds f Mathematcal Ecmc. McGraw-Hll Ic. Cbb C.W. ad Duglas P.H. 98. Thery f Prduct. merca Ecmc Revew Dt.K Otmzat Ecmc Thery. New Yr: Ofrd Uversty Press. Ia C.. Ia G. 0 -Mcrecmcs. Galat: Steze. Ia C.. Ia G. 00. The Substtut ad the Reveue Effects fr a Cbb-Duglas Utlty Fuct. als f Duarea de Js Uversty f Galat Fasccle I Ecmcs ad led Ifrmatcs Years XI Mas-Cllel. Whst M.D. Gree J.R Mcrecmc Thery. New Yr: Ofrd Uversty Press. Pdyc R.S. Rubfeld D.L Mcrecmcs. Pretce-Hall Iteratal. DEELOPMENT POLICIES
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