Linearity, Slutsky symmetry, and a conjecture of Hicks. Abstract

Transcription

1 Learty Slutsy symmetry ad a coecture of Hcs Chrsta Weber Seattle Uversty Abstract Hcs (956) coectured that Slutsy symmetry should hold for dscrete as well as ftesmal rce chages f demad fuctos are globally lear. Ths aer roves ths coecture usg the LES utlty fucto ad the Slutsy comesato for rce chages. More mortatly shar cotrast to revous doubts exressed by Hcs Samuelso ad others ths aer rovdes the frst formal demostrato that comesated cross rce effects ca deed be symmetrc for dscrete chages rces. Ctato: Weber Chrsta (00) "Learty Slutsy symmetry ad a coecture of Hcs." Ecoomcs Bullet ol. 4 No.. 5 Submtted: July 00. Acceted: July URL: htt:// 0D0008A.df

2 . troducto t has log bee uderstood that the symmetry of the comesated cross rce effects frst demostrated by Slutsy (95) wll geeral hold exactly oly for ftesmal chages rces. Samuelso (947a) aears to have bee oe of the frst to argue that Slutsy s symmetry result seems to aly oly for dfferetal sze chages rces. He oted that the symmetry codto reflects dfferetal roertes of our demad fuctos whch are hard to vsualze ad hard to refute. (Samuelso 947a. 07) He oted that he had tred but thus far wthout success to deduce mlcatos of our tegrablty codtos whch ca be exressed fte form (Samuelso 947a. 07) Followg Samuelso s tal cotrbuto Samuelso (953) Hcs (956) ad Heady (986) all argued essece that for fte or dscrete chages rces Slutsy symmetry s a aroxmato whch should hold wth greater accuracy as chages rces become smaller. However hs dscusso of the Slutsy symmetry codto Hcs (956) wet a bt further by clamg wthout rovdg a roof that symmetry wll ecessarly hold eve for arbtrary dscrete chages rces as log as comesated demad curves are lear rces. Accordg to Hcs coecture f comesated demad curves are globally lear the symmetry should hold exactly ot merely as a aroxmato for arbtrarly large dscrete chages rces as well as for ftesmal chages rces. Ths aer uses the lear exedture system (LES) utlty fucto frst studed by Kle ad Rub (947) Samuelso (947b) Geary (950) ad Stoe (954) to rove Hcs coecture. As Samuelso (947b) ad Geary (950) showed f demad curves are to be lear come ad the rces of other goods the the utlty fucto must tae the secfc fuctoal form cosdered here. Thus we ca use ths fuctoal form to vestgate the coecto betwee learty of demad curves rces ad Slutsy symmetry for dscrete rce chages. Beyod rovg Hcs coecture ths aer s ma cotrbuto s to rovde the frst demostrato that comesated cross rce effects ca be symmetrc eve for o-ftesmal chages rces.. Dscrete Prce Chages LES Utlty Ths secto dscusses comesated demad curves derved for the lear exedture system utlty fucto. Thus recall that the LES utlty fucto defed over maret goods x x s gve by: U x x x lx ; 0 0 ; () where the s ad s are arametrc costats. Sce ths fuctoal form reduces to the Cobb- Douglas fuctoal form the secal case where = 0 for all the results derved here ecessarly hold for comesated demad fuctos derved from the Cobb-Douglas utlty fucto. Sce we wll be cocered here wth dscrete chages rces t s mortat to remember that whe rce chages are assumed to be dscrete there are two dfferet tyes of comesated demad curves. Alog the Hcsa demad curve the household s comesated for chages rces so as to hold utlty costat as rces chage. cotrast alog the Slutsy demad curve the household s comesated for chages rces so as to ermt the household to cotue urchasg the tal budle of goods as rces chage. Mosa (94) frst showed that these two

3 comesated demad curves are oly locally equvalet that s equvalet oly for ftesmal chages rces. t s straghtforward f slghtly tedous to show that whle the LES utlty fucto yelds Hcsa demad fuctos whch are o-lear fuctos of all rces t yelds a Slutsy demad fucto for good whch s o-lear oly. The Slutsy demad fucto s lear the rces of all other goods. (The Aedx rovdes a roof that the Hcsa demad fuctos are o-lear all rces.) That s for a gve ow rce the Slutsy demad fucto for good x s a lear fucto of for. To derve the Slutsy demad fuctos t s smlest to beg wth the Marshalla demad fuctos. The Marshalla demad fucto for good shows the quatty of good demaded as a fucto of come whch s assumed to be strctly ostve ad the rces of the goods whch are assumed to be strctly o-egatve. t s straghtforward to show that for the utlty fucto () the Marshalla demad curve for good s gve by: x. () Slberberg ad Sue (00) derve equato (). We ca covert ths Marshalla demad fucto to a Slutsy demad fucto whch holds total exedture o all goods costat as rces chage as follows: Assume that stead of holdg come costat as rces chage we adust come resose to a chage ay rce so as to ermt the household to cosume the gve budle [x * x * x *] at all sets of rces. That s we adust come as rces chage so that at all rce combatos = f( ) = x * wth x * treated as beg arametrcally gve for all. Substtutg ths come fucto to () yelds the Slutsy demad fucto: x x * x * x *. (3) Sce secto of (3) reveals that ths demad fucto s lear all of the rces excet the ow rce we ca easly calculate the mact of dscrete chages rces other tha o the demad for good. For examle for a dscrete chage the fuctoal form (3) mles that we have: x x * or : x x *. (4) Smlarly the effect of a dscrete chage o the demad for good wth s gve by: x x * or : x x *. (5)

4 To rove symmetry we must show that f the cosumto budle [x * x * x *] was otmal at the tal set of rces the the exressos (4) ad (5) must be equal. To see that ths s deed the case recall that sce the LES utlty fucto s quascocave ad twce dfferetable the x s at the tal set of rces the frst order codtos for maxmzg the utlty fucto () subect to the usual budget costrat clude the budget costrat ad: U (6) x x * where s the margal utlty of come that s the Lagrage multler for the come costrat the household s costraed utlty maxmzato roblem. At the tal rce vector the Lagrage multler ad the x * s solve the budget costrat ad the versos of equato (6). Fally observe that the f.o.c. s (6) mly:. (7) x * ) ( x * ) ( Multlyg equato (7) through by (x * - )(x * - ) we see mmedately that at the otmum the exressos for the Slutsy cross rce effects o the rghthad sdes of (4) ad (5) must both equal (x * - )(x * - ) ad hece that they must equal each other. Sce ths shows that the cross rce effects (4) ad (5) are equal eve for dscrete chages rces we have verfed Hcs coecture: f the (Slutsy comesated) demad fucto for each good s lear the rces of all other goods the comesated cross rce effects must be equal for dscrete as well as ftesmal chages rces. As was metoed the troducto ths seems to be the frst formal demostrato that for some refereces the Slutsy symmetry codto holds eve for dscrete chages rces. 3. Cocluso the rocess of rovg Hcs coecture we have also show that hstorcal essmsm cocerg the ossblty that cross rce effects mght be symmetrc for arbtrarly large dscrete chages rces (see e.g. Samuelso (947a 953) Hcs (956) ad Heady (986)) has bee at least artally mslaced. That s for at least oe class of refereces comesated cross rce effects wll deed be symmetrc eve for arbtrary o-ftesmal chages rces f the comesato follows Slutsy rather tha Hcs. t s already uderstood that the ow substtuto effect s egatve for dscrete as well as ftesmal rce chages. Smlarly whe the come comesato taes the Slutsy (95) rather tha the Hcs (946) form the Slutsy equato ad the mlcato that the rce weghted sum of the comesated cross rce effects s zero all stll hold for dscrete rce chages. Ths aer has added a fourth comaratve statcs mlcato of the utlty maxmzato hyothess (uder a sutable restrcto o refereces) whch cotues to hold for dscrete chages rces. t remas a mortat toc for future research to determe codtos uder whch other comaratve statcs results orgally derved oly for ftesmal chages arameters cotue to hold for dscrete chages as well. 3

5 APPENDX Ths aedx demostrates that Hcsa demad fuctos derved from the LES utlty fucto wll be o-lear fuctos of all rces. To see ths smly substtute the Marshalla demad fucto equato () bac to the utlty fucto () to derve the drect utlty fucto ( ): l l l ) ( where the secod equalty maes use of the ormalzato =. Settg utlty equal to the costat 0 exoetatg ad rearragg slghtly the yelds the followg equvalet reresetato of the household s drect refereces *( ):. ) *( 0 e The vert the drect utlty fucto to obta the exedture fucto:. ) ( 0 0 e E By vrtue of Hotellg s lemma the Hcsa comesated demad fucto for good s the dervatve of the exedture fucto wth resect to. Sce oe of the s equals the exedture fucto s ot quadratc ay rce. Ths tur mles that the Hcsa comesated demad fuctos must be o-lear fuctos of all rces. Hece we caot test Hcs coecture o the Hcsa demad fuctos but oly o Slutsy demad fuctos. Slberberg ad Sue (00) rovde a smlar dervato of the Hcsa demad for the LES utlty fucto. 4

6 REFERENCES Geary R.C. (950) A Note o A Costat-utlty dex of the Cost of Lvg Revew of Ecoomc Studes Heady C. (986) A New Loo at the Symmetry of the Slutsy Matrx Uublshed worg aer Uversty College Lodo. Hcs J.R. (946) alue ad Catal d ed. Oxford Uversty Press: Oxford. Hcs J.R. (956) A Revso of Demad Theory Oxford Uversty Press: Oxford. Kle L.R. ad H. Rub (947) A Costat-utlty dex of the Cost of Lvg Revew of Ecoomc Studes Mosa J. (94) O the terretato of the Fudametal Equato of alue Theory Studes Mathematcal Ecoomcs ad Ecoometrcs Memory of Hery Schultz by O. Lage F. Mctyre ad T. Ytema Eds. Uversty of Chcago Press: Chcago Samuelso P.A. (947a) Foudatos of Ecoomc Aalyss Harvard Uversty Press: Cambrdge. Samuelso P.A. (947b) Some mlcatos of Learty Revew of Ecoomc Studes Samuelso P.A. (953) Cosumto Theorems Terms of Overcomesato Rather tha dfferece Comarsos Ecoomca 0-9. Slberberg E. ad W. Sue (00) The Structure of Ecoomcs: A Mathematcal Aalyss rw McGraw-Hll: New Yor. Slutsy E.E. (95) Sulla teora del blaco del cosumatore Gorale degl Ecoomst 5-6; rerted as O the Theory of the Budget of the Cosumer A.E.A. Readgs Prce Theory by J. Stgler ad K.E. Bouldg Eds. (95) Rchard D. rw c. Chcago. Stoe R. (954) Lear Exedture Systems ad Demad Aalyss: A Alcato to the Patter of Brtsh Demad Ecoomc Joural