1. Consider an economy of identical individuals with preferences given by the utility function
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1 CO 755 Problem Se e Cbrer. Cosder ecoomy o decl dduls wh reereces e by he uly uco U l l Pre- rces o ll hree oods re ormled o oe. Idduls suly ood lbor < d cosume oods d. The oerme c mose d lorem es o oods d res / d / o rse dollrs o mee s reeue reuremes. Ob eresso or he drec uly uco s uco o he res. Sole or he cosumer's demds d wre dow he oerme bude cosr. b Fd he rs order ecessry codos or he oml res whe. Are he res eul? c Suose he cosumer hs l edowme o ood. Do he rele < res deed o he s o or o he s o? Ge y uo or hs resul you mh he. Pre- rces ormled Tes: d / ued Cosumer roblem: m U l l s.. r: l l U urees ; e eror soluo > [] [] [] [] Sub [] us o [] d []; sole or d : Sub hese o []; sole or : < o 6
2 o 6 Sub o e cosumer demds erms o res: Sole or U o e l l l l Goerme Bude Cosr: Plu eulbrum lues or d : b l l m s.. r: l l
3 Te FOC or ; ull ou o brces d e commo deomor Combe erms; e commo deomor or secod d hrd erms Sole or FOC or mrrors hs dero: I we he hese erms mus be eul: Ccel he rom boh erms her h me cy mhemcl rumes I lued hs o cel us d. The rh wh shows he oly wy s or howeer or y > s ossble o he wh. I hese suos he res wll he oose ss. he bude cosr s o bd or > d he res he he sme s he he res re eul. I he rem cse > d he res he dere ss obously he res re dere o 6-5
4 c oo he cosumer's demds rom r < he s mos lely ee does' he o be bu wll be less h >. Ths rees u resources he cosumer's bude cosr whch bsed o he cosumer's obece uco wll be u o beer use o ood ssum >. The o whch MS MS wll be deermed by he derece bewee d. Ths wll deerme he demds whch he wll dre he res order o ssy he oerme's bude cosr. Thereore he rele res deed o he s o.. Cosder Dmod d Mrrlees's A ers o oml commody o. Oml o d roduce ececy re led oeher s rcles o ereo. Suose ule he rdol model h boh oods d re uble--s ror resrco. To ee smle cosder oe-cosumer ecoomy wh oods rom he resrcos. Sr rom her orl roblem euo 5 d wor hrouh o e omo roblem loous o 7. Wh mc does he cosr h he? b Show h roduce ececy s o desrble hs ecoomy. 5 Orl roblem: m s.. y mre cler y r m y s.. y y y re roduco ececy y ublc roduco ececy o o ood Use mre cler o sole or re roduco ues: y Sub d o mre cler cosr or ood : y y y y y y y rom he rs se: Sub he lues or Pre roduco ececy d y e so we he y becomes New roblem: o 6
5 5 o 6 m s.. s.. The resrco h mes here s ddol cosr whch esselly elmes decso rble. b r: [ ] [ ] µ [ ] µ Assum eror soluo we c sole or : µ I we ssume he re roduco uco s wce dereble d oler y so he / ess d s he mrl re o rsormo re d ublc secors re o eul so we do o he ree roduco ececy.. Go bc o Ky's er d dere he oml ormuls by mm ecess burde where he uly leel he eedure uco s drec uly. l why us he comes ro hs roch wll o yeld he sme resuls. cess burde us eule ro: u u Subsue drec uly: Assume re- rces re ed Hmlo old us o Assume oerme bude cosr s euly le Ky dd Problem: m s.. r: Noe: se u wh rher h he sdrd order o e he Dmod d Mrrlees oml commody rule Noe couresy o mcro oes: c 7
6 6 o 6 Sub d hs s he sme s '' o.5 o he Ky rcle: cess burde us comes ro d drec uly: Problem: m s.. r: Ths me c Us comesed ro does o yeld he sme resul becuse he drec uly s elued re- rces d wll o che wh resec o os- rces.e. hey're deede o he. Documeo. Prob. I se u r o my ow d o cormo rom eeryoe h ws correc. Josh showed me beer wy o smly he demds o me r b eser. JC d I red o e soluo us Mhemc bu dd' wor. The rc r b o sol or d cme rom Nc by wy o Chrse d Ke. I led o eeryoe bou r c bu eer relly el comorble wh wh I herd. I red rous umercl soluos cel bu he roblem s ery usble d smll ches rmeers cused hue ches so I ws' ble o e "eel" or wh ws o o. Prob. Pro. Hmlo cormed he bscs o he roblem o rom 5 o 7 ws esselly uched d we us he ew cosr. Chrse oed ou h he ree roduco ececy resul cme rom he dere wr. I checed my resuls JC d Josh. Prob. Chrse d Pro. Hmlo boh oed me o he oml ormuls '' he Ky rcle. Pro. Hmlo sd o use reulr demds or ll he reeue comuos. He rcclly dd he whole roblem or us becuse we bued hm wh so my uesos.
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