Random Horizon Stochastic Dynamic Slutsky. Equation under Preference Uncertainty

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1 Aled Maeacal Scences ol. 8 4 no HIKARI Ld.-a.co ://d.do.og/.988/as.4.4 Rando Hozon Socasc Dynac Slsy Eqaon nde Pefeence Unceany Dad. K. Yeng Cene of Gae eoy S Peesbg Sae Unesy Rssa and SRS Conso fo Adanced Sdy n Cooeae Dynac Gaes Se Yan Unesy Hong Kong Coyg 4 Dad.K. Yeng. s s an oen access acle dsbed nde e Ceae Coons Abon Lcense c es nesced se dsbon and eodcon n any ed oded e ognal o s oely ced. Absac s ae eends Slsy s classc o on conse eoy o a ando ozon socasc dynac faeo n c e conse as an ne-eoal lannng ozon nceanes n fe ncoes efeences and lfe-san. Uly azaon leadng o a se of odnay eal-deenden deand fncons s efoed. A dal oble s se o dee e eal coensaed deand fncons. s eesens e fs e a eal-deenden odnay deand fncons and eal coensaed deand fncons ae obaned nde ese nceanes. e coesondng Roy s deny elaonss and Slsy eqaons n a ando ozon socasc dynac faeo ncean efeences ae deed. e analyss ncooaes ealsc caacescs n conse eoy and adances e conenonal coeconoc sdy on conson o a oe ealsc oal conol faeo. Keyods: Oal conson Socasc dynac ogang Roy s deny Slsy eqaon Inodcon e gond-beang o by Slsy 95 lad e fondaon fo goos analyss of oal conson decson n coeconocs. e analyss c

2 7 Dad. K.Yeng soed a e effec of a ce cange on e deand of a good can be decoosed no sbson effecs and ncoe effec yelds sgnfcan econoc lcaons see Vaan 99. s onen conbon n conse eoy non as e Slsy eqaon as csened by Jon Hcs as e Fndaenal Eqaon of Vale eoy. I becoes an negal a of ansea econocs and conse eoy. e aes by Allen 96 and 95 Hcs and Allen 94 Sclz 95 Dooley 98 and Es 975 oagaed Slsy s classc o. Anoe lesone n conse eoy s Roy s deny 947 c odes an ofen noed aeacal esl n conse eoy. In addon e deny s also nsenal n ong e Slsy eqaon. Yeng eended Slsy s o o a socasc dynac faeo n c e conse as a -eod lfe-san fe ncoes beng ncean and deed e socasc dynac Slsy eqaons. Yeng 4 fe eended Slsy s conse oble no a dynac faeo n c e conse as a ando lfe-san and ncean fe ncoes. In s ae nceanes n e conse s fe efeences s ncooaed o Yeng s 4 eenson of e Slsy faeo o eflec an ofen obseed ealy n conse coce. Canges n condons a cold no be efecly foeseen le eal ase abs ecnology syle cle and faly cooson conbe o nceany n fe efeences. In acla oal conson coce nde a dynac faeo nceanes n e conse s lfe-san fe ncoes and fe efeences s eaned. Ine-eoal eal-deenden odnay deand fncons and eal coensaed deand fncons ae obaned. o of e os ccal fondaons n conse eoy Roy s deny and Slsy eqaon ae deed n a ando ozon socasc dynac faeo nceany n e conse s fe efeences. e ae s oganzed as follos. e fs esen a dynac odel of ly azaon by a conse an ncean lfe-san ncean fe ncoes and ncean fe efeences n Secon. In Secon a se of eal-deenden odnay deands s caacezed. e Roy s deny esl n a ando ozon socasc dynac faeo ncean fe efeences s deed n Secon 4. e dal oble s folaed n Secon 5 and e coesondng eal coensaed deand fncons ae obaned. Socasc dynac Slsy eqaons fo e conse nceanes n lfe-san fe ncoes and efeences ae folaed n Secon 6. Secon 7 concldes e ae. Dynac Uly Mazaon nde Unceanes n Incoes Pefeences and Lfe-san Consde e case of a conse ose lfe-san noles eods ee

3 Rando ozon socasc dynac Slsy eqaon 7 s a ando aable ange { } and coesondng obables { }. Condonal on e eacng of eod e obably of e conse s lfe-san old las o eods becoes esecely see Yeng and Peosyan :.. n e se o denoe e qanes of goods consed and n e coesondng ces n eod { }. e efeence o ly fncon of e conse n eod s non o be. Hs fe efeences ae no non ceany. In acla s ly fncon n eod { } s non o be obably fo { } f e ses n eod. e se ~ o denoe e ando aable ange } and coesondng obables { }. { e conse azes s eeced ne-eoal ly E c c E c. c sbec o e bdge consan caacezed by e eal dynacs n n.. ee s e nees ae s e conse s sbece one-eod dscon faco fo e daon fo eod o eod e nal eod dscon faco s e ando ncoe a e conse ll ecee n eod and fo { } s a se of sascally ndeenden ando aables and E s e eecaon oeaon

4 74 Dad. K.Yeng esec o e sascs of. e ando aable as a non-negae ange { } coesondng obables { }. e ando aable as a ale of zeo obably becase e conse ll ecee no ncoe n eod. Moeoe nde e ao of non-saaon e conse ll send all s eal n e las eod of s eeced lfe-san and eefoe. e conse s dscon faco c ay o ay no concde e ae dscon faco and ay also be eqal o. Fo noaonal slcy e ado e noaon c and c. c c e oble.-. s a ando ozon dscee-e socasc conol oble ncean ayoffs see Yeng and Peosyan and 4. No consde e case en e conse as led o eod and s eal s and s efeence s. e conse oble can be folzed as e azaon of e dsconed ayoff: E.4 sbec o e bdge consan caacezed by e eal dynacs.5 fo { }. In a socasc dynac faeo saegy sace sae-deenden oey as o be consdeed. In acla gen a e efeence of e conse s f e ses o eod a e-secfed class of ang : X e oey n n { } { } fo { } s e saegy sace and eac of s eleens s a adssble saegy. e defne e ale fncon V and e se of saeges * * fo { } c odes an oal conson solon as follos:

5 Rando ozon socasc dynac Slsy eqaon 75 V a E E [ ] [ ] fo { }..6 e ale fncon V eflecs e eeced ne-eoal ly n esen ale es a e conse ll oban fo eod o e end of s lfe-san. Follong e analyss of Yeng and Peosyan and 4 one can dee an oal solon o e ando-ozon conse oble.-. as follos: * eoe.. A se of conson saeges { fo { } and { }} odes an oal solon o e ando ozon conse oble.-. f ee es fncons V fo { } and } sc a e follong ecse elaons ae sasfed: V V V and { a V [ ] a E V [ ] fo { } and }..7 {

6 76 Dad. K.Yeng Poof. Follong Bellan s 957 ecnqe of dynac ogang e begn e las sage/eod. By defnon e ly of e conse a eod s eefoe V. e fs consde e case en e conse ses n e eod s efeence s and e sae. e oble en becoes a E V [ ].8 sbec o..9 Snce obably V n.8-.9 can be eessed as and e oble V sbec o a V [ ]. and.. No consde e oble en e conse ses o eod and s efeence s goened by.e oble n eod can be eessed as azng E sbec o. fo { } and.. If e ale fncon V n. ess e oble.-. can be eessed as a sngle sage oble:

7 Rando ozon socasc dynac Slsy eqaon 77 V a E ] [ V..4 No consde e oble a e conse ses n eod } { and s efeence s. Follong e analyss aboe e oble n eod becoes e azaon of e eeced ayoff E E.5 sbec o fo } { and..6 Noe a e e E can be eessed as: E.

8 78 Dad. K.Yeng In.7 e e.7 E ges e eeced ne-eoal ly o be azed n eod f e conse s efeence n eod s. If e ale fncon V ess e ae: V a E..8 Usng.8 e oble.5-.6 can be folaed as a sngle sage oble c azes e eeced ayoff E ] [ V..9 If V ess e ae V a E ] [ V fo } { and } {.. Hence eoe. follos. e socasc oal sae aecoy deed fo eoe. s caacezed by:

9 Rando ozon socasc dynac Slsy eqaon 79 [ ] [ ] f efeence s eod n [ 4 ] 4 f efeence s n eod [ ] f efeence s n eod. [ ] * e se o denoe e se of ossble ales of eal a eod along e oal aecoy geneaed by eoe.. eal-deenden Odnay Deand In s secon e consde e al oble of deng eal-deenden odnay deand fncons n c e conse azes s ne-eoal eeced ly sbec o ncean ne-eoal bdge lfe-san and fe efeences. Follong e analyss n Yeng 4 e fs consde e case en e conse ses n e las eod and s efeence s. Le V denoe e conse s eal n eod. Gen a * and. Hence e conse faces e oble a o eas all e eal n s eod sbec o.. Poble. s a sandad sngle eod ly azaon oble. Seng e coesondng Lagange oble and efong e elean azaon one obans a se of fs ode condons. I s ell-non see Ceng and Yeng 995 a f e se of fs ode condons sasfes e

10 7 Dad. K.Yeng lc fncon eoe one can oban e odnay deand as elc fncons of e aaees and a s: fo n } and }.. { { Noe a coesonds o e oal conson saeges n eoe.. o so s e noe a e oble n eod n eoe 5. s a E V [ ] sbec o. Snce e oble a sbec o a and e oble and e oble a sbec o e sae consan yelds e sae conols eefoe =. Sbsng. no. yelds e ndec ly fncon n eod as [ ]. Inong e defnon n.6 eqals e fncon V n eoe.. No consde e case en e conse ses n eod and s * efeence s. If eal eqals n s eod e oble n concen becoes a E sbec o e ne-eoal bdge Le *.. denoe e eal a eod f { } occs and e conse s efeence s eod. Usng e ndec ly fncon n and. e oble facng e conse n eod can be eessed as a sngle-eod oble:

11 Rando ozon socasc dynac Slsy eqaon 7 a..4 Fs ode condon fo a azng solon yelds fo n }.5 { Agan e lc fncon oldng.5 can be soled o yeld e odnay deands n eod as fo n }. { Noe a coesonds o e oal conson saeges n eoe.. o so s e ansfo.4 no a sla oble by llyng e aand by a Moeoe ecallng a eoe. e can eess.6 as and oban..6 eqals e V n a E V ] [ c s e eod condon o be azed n eoe.. Snce e conols fo oble.4 and e ansfoed oble.6 ae e sae e ae. Sbsng no.4 yelds e ne-eoal

12 7 Dad. K.Yeng ndec ly fncon. Inong.6 and eoe. e e coesonds o V n eoe.. Reeang e analyss fo eods o yelds e conse oble a eod } { efeence as: a.7 ee * and s e so fo fo e eco. Fs ode condon fo a azng solon o e obles n.7 can be obaned as:.8 fo } { n and } {. Noe also e condon a n eod good ll be consed e on ee agnal ly of conson eqals es e eeced agnal ly of eal. In acla e eeced agnal ly of eal aes no consdeaon e ando fe ncoe and efeences ls e obably of e conse sng n eod. Solng.8 yelds e odnay deands n eod as: fo } { n and } {..9

13 Rando ozon socasc dynac Slsy eqaon 7 Afe solng e al conse oble c azes eeced ly sbec o nceanes n fe ncoe fe efeences and lfe-san e oceed o dee e Roy s deny esl n a dynac faeo nceanes n fe lfe-san efeences and ncoes. 4 Rando Hozon Socasc Dynac Roy s Ideny nde Uncean Pefeences In s secon e dee e socasc dynac eson of Roy s deny nceanes n fe lfe-san efeences and ncoes. Inong.7 e oban e deny ] [ ] [. 4. Dffeenang e ne-eoal ndec ly fncon n 4. esec o : ] [ n 4. ee and.

14 74 Dad. K.Yeng Inong e fs ode condons n.8 e e nsde e cly baces anses condon 4. en becoes:. 4. e effec of a cange n nal eal on e azed ly can be obaned by dffeenang n 4. esec o : ] [ n. 4.4 Agan nong e fs ode condons n.8 e e nsde e cly baces anses condon 4.4 en becoes:. 4.5 Ddng e g-and-sde of eqaon 4. by e g-and-sde of eqaon 4.5 and e lef-and-sde of 4. by e lef-and-sde of 4.5 yelds: fo } { n. 4.6

15 Rando ozon socasc dynac Slsy eqaon 75 Condon 4.6 odes a ando ozon socasc dynac eson of Roy s deny nolng a cange n cen ces nde ncean fe efeences. en e consde deng e ando ozon socasc dynac Roy s deny nde ncean fe efeences fo a cange n ces n cen and fe eods. eoe 4.. Rando Hozon Socasc Dynac Roy s Ideny nde Uncean Pefeences fo } { n fo } { } { } { n and } { ee ] [ ] [ f efeence s n eod

16 76 Dad. K.Yeng ] [ f efeence s n eod ] [ f efeence s n eod ] [. 4.9 Poof. See e Aend. eoe 4. ges e ando ozon socasc dynac Roy s deny nde ncean fe efeences. Inong A.9 n e oof of eoe 4. n e Aend an alenae fo of e ando ozon socasc dynac Roy s deny can be eessed as: 4. fo } { } { } { n and } {. 5 Daly and eal Coensaed Deand In s secon e noe e daly ncle n conse eoy o consc eal coensaed deand fncons nde an ncean ne-eoal bdge ncean efeences and a ando lfe-san. o do s e consde e dal

17 Rando ozon socasc dynac Slsy eqaon 77 oble of nzng eende coeed by e cen eal sbec o ananng e leel of ly aceed n e al oble. Follong e analyss n Yeng 4 e fs eane e case en e conse ses n e las eod and s efeence s *. Le denoe e conse s eal n eod. Snce eal eqals ncoe n s eod o dee e coensaed deand e follo e sandad sngle eod conse oble of n sbec o aceng e leel of ly =. 5. Seng e coesondng Lagange fncon and efong e nzaon oeaon yelds a se of fs ode condons. e lc fncon eoe oldng fo e fs ode condons one can oban e eal ncoe coensaed deand fncons as fo n }. 5. { Sbsng 5. no 5. yelds e eal-eende fncon. No e oceed o eod and le eal n s eod be * and efeence be. o oban e eal coensaed deand fncon n eod e consde e oble of nzng eende coeed by cen eal n e eod o bng abo e eeced ne-eoal ly eal fo e al oble. Hoee n eod does no only coe conson eende n e eod b also a of e conson eende n eod. o delneae eendes abed o eal n eod e fs noe e dynacal eqaon. and eess as:. 5. Usng e eal-eende fncon n eod and ang eecaon oe e ando aables and ~ one can oban a ccal deny elang eal o cen and eeced fe eendes abable o eal fo 5. as:

18 78 Dad. K.Yeng ] [. 5.4 Usng 5.4 e conse s dal oble n eod can be folaed as nzng eal eende ] [ 5.5 esec o sbec o e consan ] [. 5.6 Snce s a se of eal coensaed deands a leads o e leel of ly so [ eqals. Inong ] [ e consan 5.6 can be eessed as: ] [. Seng e Lagange fncon and efong e elean ozaon oeaon sla o e analyss n Yeng 4 yelds a se of fs ode condons. e lc fncon eoe oldng e eal coensaed deand fncons can be obaned as: fo } { n. 5.7 Sbsng e eal coensaed deand fncons n 5.7 no 5.5 yelds e eal-eende fncon n eod :

19 Rando ozon socasc dynac Slsy eqaon No e oceed o eod } { and le eal be * and efeence be n e eod. Agan sng. e can eess eal n eod as. Inong e eal-eende fncons n eod and ang eecaons oe e ando aables and ~ one can oban e deny 5.9 ee s e so fo fo. e conse s eal eende nzaon oble can be eessed as: n 5. sbec o ] [ 5. fo } { and *. Seng e Lagange fncon and deng e fs ode condons one can oban e eal coensaed deand fncons e lc fncon

20 7 Dad. K.Yeng eoe oldng as fo { } n } and {. * Slaly e eal-eende fncon can be obaned as:. 5. e eal coensaon deand fncons and eal-eende fncons deed n s secon eesen e dal esls of e al oble n Secon. 6 Rando Hozon Socasc Dynac Slsy Eqaons In s secon e dee e Slsy eqaons nde a dynac faeo nceanes n e conse s fe ncoe fe efeences and lfe-san. Inong e daly esls n Secon and Secon 5 e ae and }. { fo * and and { } n } and { Sbsng deand fncon yelds e deny: by no e eal-deenden odnay [ ] 6. fo { n} and { }. One can dee a eoe concenng e elaonss beeen e ce effec of e deand of a coody and e e sbsaon effec and e eal effec n a ando ozon socasc dynac faeo ncean fe efeences as follos.

21 Rando ozon socasc dynac Slsy eqaon 7 eoe 6.. Rando Hozon Socasc Dynac Slsy Eqaon nde Pefeence Unceany 6. fo } { } { } { n } { n and } {. Poof. Dffeenang e deny 6. esec o yelds: ] [ ] [ 6. fo } { n } { and } {. Inong one can eess 6. as:. 6.4

22 7 Dad. K.Yeng o dee e e fs noe a bog abo by a cange n e oally dffeenae d d o oban: n a oe eadly coable fo e. o dee e effec on n d beng eld consan. 6.5 d and d fo all n } and { } ece d eqaon 6.5 becoes { d d c yelds d d. 6.6 Inong and sng 6.6 one can eadly oban. 6.7 Sbsng 6.7 no 6.4 and nong e ando ozon socasc dynac Roy s deny n eoe 4. one obans 6.. Hence eoe 6. follos. e ando ozon socasc dynac Slsy eqaon nde ncean efeences n 6. genealzes e classc Slsy eqaon o a l-eod faeo nceanes n fe ncoe e conse s lfe-san and efeences. In acla e effec of a ce cange on e deand of a coody can be decoosed no a e sbsaon effec and a eal effec. e lef and sde of eqaon 6. eesens o e deand fo good a eod canges n esonse o a cange n ce and e fs e on e g and sde of e eqaon ges e cange n deand cased by a cange n ce oldng ly fed a. e second e on e g and sde

23 Rando ozon socasc dynac Slsy eqaon 7 of eqaon 6. s e odc of e cange n deand en eal canges and e eqed cange n eal bog abo by a cange n ly e fed a. s e cange n e deand of a good cased by a ce cange can be decoosed no a e sbsaon effec and a eal effec. 7 Concldng Reas s ae eends e conenonal conse analyss o a ando ozon socasc dynac faeo n c ee ae nceanes n e conse s fe ncoes fe efeences and lfe-san. e eenson ncooaes ealsc and essenal caacescs of e conse no conenonal conse eoy. e ae dees o of e os ccal fondaons n conse eoy Roy s deny and Slsy eqaon n a ando ozon socasc dynac faeo ncean fe efeences. e analyss adances e coeconoc sdy on oal conson decson o a ando ozon socasc dynac faeo ncean conse efeences. Fe eseac deeloen and oagaons c eloe fe econoc lcaons of e esls n s ae ae n ode. 8 Aend: Poof of eoe 4.. Inong.7 e oban e deny ] [ ] [ ] [

24 74 Dad. K.Yeng ] [. A. Dffeenang A. esec o yelds: ] [ n n ] [

25 Rando ozon socasc dynac Slsy eqaon 75. A. Usng 4.8 e ae and. A. Sbsng A. no A. yelds ] [ n

26 76 Dad. K.Yeng n ] [. A.4 Inong 4.5 e oban:

27 Rando ozon socasc dynac Slsy eqaon 77. A.5 Usng A.5 e es nsde e sqae baces n A.4 can be en as ] [. A.6 Inong e fs ode condons n.8 e e nsde e sqae baces n A.6 anses and eefoe A.4 becoes:. A.7 Usng A.5 one as

28 78 Dad. K.Yeng. A.8 Sbsng A.8 no A.7 yelds. A.9 Inong A.5 one obans. A. Ddng A.9 by A. yelds anoe fo of e ando ozon Roy s deny as:

29 Rando ozon socasc dynac Slsy eqaon 79 A. fo } { } { } { n and } {. Hence eoe 4. follos. Q.E.D. Acnoledgeens. Fnancal so by HKSYU s gaeflly acnoledged. Refeences [] R.G.D. Allen Pofesso Slsy s eoy of Conses Coce. Ree of Econoc Sdes [] R.G.D. Allen e o of Egen Slsy. Econoeca [] R. Bellan Dynac Pogang. Pnceon Pnceon Unesy Pess 957. [4] M.. Ceng and D..K. Yeng Mcoeconoc Analycs Ne Yo Pence Hall 995. [5] P.C. Dooley Slsy s Eqaon Is Paeo s Solon. Hsoy of Polcal Econoy [6].. Es eal Effecs and Slsy Eqaons fo Asses. Econoeca [7] J.R. Hcs and R.G.D. Allen A Reconsdeaon of e eoy of Vale. Pas -. Econoca Ne Sees & Rened In Hcs

30 74 Dad. K.Yeng [8] R. Roy La Dsbon D Reen Ene Les Des Bens. Econoeca [9] H. Sclz Ineelaons of Deand Pce and Incoe. Joal of Polcal Econoy [] E.E. Slsy Slla eoa Del Blanco Del Consaoe. Gonale Degl Econos anslaed as On e eoy of e Bdeg of e Conse n G. Sgle and K. Boldng eds. Readng n Pce eoy Pblsed fo e Assocaon by Rcad D. In Inc. Hoeood IL 95. [] H. Vaan Mcoeconoc Analyss d ed... Noon Ne Yo 99. [] D..K. Yeng Oal Conson nde an Uncean Ine-eoal Bdge: Socasc Dynac Slsy Eqaons. Vesn S Peesbg Unesy: Maeacs -4. [] D..K. Yeng Oal Conson nde Unceanes: Rando Hozon Socasc Dynac Roy s Ideny and Slsy Eqaon focong n Aled Maeacs 4. [4] D..K. Yeng and L. A. Peosyan Sbgae Conssen Cooeae Solon of Dynac Gaes Rando Hozon. Jonal of Ozaon eoy and Alcaons [5] D..K. Yeng and L. A. Peosyan Sbgae-conssen cooeae solons n andoly fcang socasc dynac gaes. Maeacal and Coe Modellng [6] D..K. Yeng and L. A. Peosyan Sbgae-conssen Cooeae Solons fo Randoly Fcang Socasc Dynac Gaes Uncean Hozon focong n Inenaonal Gae eoy Ree 4. Receed: Ags 4