and substitute in to the 1 st period budget constraint b. Derive the utility maximization condition (10 pts)
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1 Eon 50 Sping 06 Midem Examinaion (00 ps S.L. Paene Ovelapping Geneaions Model (50 ps Conside he following he Ovelapping Geneaions model whee people live wo peiods. Eah geneaion has he same numbe of people. Pefeenes ae ln( ln(. Eah agen is endowed wih unis of ime when oung and.5 unis of ime when. Poduion is done b ompeiive fims aoding o he ehnolog Y =AH, whee H is oal ime inpu.. Suppose hee is a lump-sum ax/ansfes when he agen is oung equal o Tx and a lump-sum ax/ansfe implemened when he agen is equal o Tx. The fis peiod budge onsain is s w h Tx. The seond peiod budge onsain is w h Tx ( s whee s is savings, w is he eal wage, and is he eal inees ae. a. Deive he Ineempoal budge onsain (0 ps Take seond peiod budge onsain and solve fo s. ( w h Tx s and subsiue in o he s peiod budge onsain olg ( w h Tx wh Tx and wih some algeba we aive a he ineempoal budge onsain b. Deive he uili maximizaion ondiion (0 ps U / U / /(. Use (a and (b o deive he savings funion. Veif ha savings will no deease if he eal inees ae ineases. Assume.5A-Tx 0 (0 ps (b implies ha. Now subsiue ino he ineempoal budge onsain o ge w h Tx. ( w h Tx and solve fo w h Tx 3. [ wh Tx ] w h Tx 4. Finall, s wh wh Tx s wh Tx 5. Take he deivaive wih espe i 0 (. Solve ou he equilibium whee hee is no govenmen poli. Veif ha he ompeiive equilibium is No Paeo Opimal. (0 ps When hee is no
2 govenmen, A and. 5A. The inees ae is deemine b he uili maximizing ondiion U / U /.5 /( i i The equilibium is No Paeo Opimal if he inees ae is negaive. When people ae bee endowed when oung, he despeael wan o save. Howeve, hee is no one o lend o. Thus he inees ae in equilibium mus be suffiienl unaaive (i.e. negaive so ha he oung will be happ wih saving zeo in equilibium. 3. Assume he govenmen implemens a pa as ou go pension ssem ha allows eah geneaion o enjo he same level of onsumpion when oung and, i.e., =. Solve ou he equilibium fo his eonom (0 ps If hen =0. The govenmen budge onsain is Tx =-Tx. Goods make leaing ondiion is A. 5A. Sine =, i follows ha.5a /. Now go o he househ budge onsains whee s=0. A Tx and. 5A Tx, Tx. Fom he fis peiod budge onsain and ou soluion fo, we know ha A A. 75A.5A]. You an hek ha fom seond peiod househ onsain ha Tx. 5A Neolassial Gowh Model (50 ps. Deive he elaion beween he eal inees ae and he enal pie of apial using a noabiage agumen unde he following senaios. (Assume apial depeiaes a ae δ a. Househs pa a ax a ae τ k on ne apial inome, i.e. he diffeene beween apial inome and apial depeiaion. (0 poins Answe: Househs pa a ax a ae τ k on ne apial inome, i.e. he diffeene beween apial inome and apial depeiaion. (0 poins Savings uni of oupu +i omoow Buing apial uni of oupu one uni of apial omoow + +(-δ- τ k( + -δ Theefoe, i ( ( k b. Househs pa a ax on inees inome equal o τ i. (0 poins Househs pa a ax a ae τ i on inees inome, Savings uni of oupu +(- τ i i omoow Buing apial uni of oupu one uni of apial omoow + +(-δ- Theefoe, i ( i. Take he Neolassial Gowh Model wihou an axes o populaion gowh given b he 3 equaions below. Desibe how ou would assign paamee values fo β, α, A, δ, θ, and γ A in Sep 4 of he alibaion. Be sue o sae whih equilibium ondiions and
3 obsevaions ou would use o assign he value of eah paamee. (Noe: ou do NOT need o ome up wih he paamee values. (0 Poins [ln 0 ln( h ] Househ Pefeenes x Ak [( A h ] Poduion Funion and Uses of Oupu k ( k x Capial Sok Law of Moion (i A nomailized o. (ii γ A in use bpg ha gows a + γ A wih US obsevaion fo gowh ae fo. (iii k k/ use obsevaion fo apial inome shae and equilibium esul ha θ= k k/ (iv δ use apial sok law of moion equaion wih obsevaion fo K/Y and X/Y (v β use FONC beween + and wih impued i fom + =i + δ and gowh ae of onsumpion (vi α use FONC beween and h wih obsevaion fo h, / and Pofi max ondiions w=(- θ /h 3. In a home poblem se, ou used he alibaed neolassial gowh model o deemine he effe of hanges in he uen US ax ssem on sead sae hous, apial pe peson, oupu pe peson, and onsumpion pe peson (h, k, and. One poli hange ou evaluaed was an eliminaion of he ax on labo inome. This poli hange was assumed o be evenue neual so ha he eliminaion of he ax on labo inome equied an inease in he onsumpion ax ae o keep govenmen expendiues and ansfes unhanged. You found ha h, k, and all ineased. Give some inuiion fo wh hese vaiables ineased. (0 poins Basiall, he ansfe is kep he same. If we jus had a sai model, he budge line fo he leisue onsumpion indiffeene uve would T be l. So if ou ae keeping T fixed and eliminaing w ( h w ( h he axh on labo inome, his is a pue wealh effe. Causing he househ o feel pooe. So he wok moe. FONC pofi is =θ(k/h θ- and no abiage is sill i ( k (, and hene i follows ha does no hange and so k/h does no hange. Sine h ineases, k mus inease, so o mus inease. C mus also inease fo wh would ou wok moe and save moe if ou ae less? BUSINESS CYCLES (00 ps. Business Cle Regulaiies a. Desibe he H-P file being aeful o idenif he inpus, he oupus and he manne b whih he oupus ae deemined in he algoihm. (0 poins Take eal gdp, quael daa and hen logs. Find end whih is soluion o following algoihm T T ( [( ( ] wih λ=600 3
4 b. Given he oupus of he file, explain how o alulae he volaili measue of a maoeonomi vaiable x and is o-movemen wih eal GDP. (0 poins volaili is sandad deviaion of deviaion o end wheeas omovmen is oelaion oeffiien.. Wha ae he ineesing business le fas wih egad o hese volaili measues and he o-movemen measues? (0 poins hous ae abou /3 as volaile as oupu. Invesmen is mos volaile. Consumpion less. Blah blah blah. Real Business Cle Theo. a. Explain in ve geneal ems he alibaion exeise undeaken b Kdland and Peso (98. Wha answe did he aemp o answe and how did he go abou ing o answe i? NOTE: I am no asking ou fo a deailed desipion of he seps, paiulal sep 3 o 4. In ohe wods, I do no wan ou o ell me wha speifi obsevaion and equilibium ondiion is used o deemine he value of eah paamee. (0 ps Kdland and Peso answe he quesion how muh of he business les an poduivi shoks aoun fo? Anohe wa o pose hei quesion, is wha ae he business le impliaions of he (neolassial gowh model? In his efomulaion, i is eas o see ha he ae going o use he neolassial gowh model and add leisue o uili. Defining onsisen measues as well as paameeizaion of he all he pefeene and ehnolog paamees exep fo he TFP paamees is done aoding o he long-un us obsevaions. In ohe wods, all paamees exep he shoks ae esied so ha he model mahes he US long un obsevaions. The poduivi shoks ae esimaed via a Solow gowh aouning exeise o impue he ime seies of US TFP. One he poess fo TFP has ben esimaed, he shoks ae fed ino he model and he equilibium is ompued. This is epeaed a lage numbe of imes so ha hee is a sample of ime seies. Then aveage volaili and oelaion saisis ae ompued fo hese simulaions and hese aveages ae ompaed wih he daa. b. Peso (990 agues ha a model ha fis he daa bee ma be a eason o eje a heo. Explain wha Peso mean b his in he onex of he suesses and failues of he alibaed RBC model. (0 poins Kdland and Peso s model onl aouned fo /3 of he oupu volaili. I inoel pedis a oelaion beween hous and labo poduivi o wages equal o. (In he daa his oelaion is lose o zeo. If K&D had geneaed 00 peen of he volaili, hen hei heo ould be ejeed beause i sill is a odds wih he obsevaion of almos zeo oelaion beween hous and aveage poduivi. B no aouning fo 00% of he volaili, K&P allow fo he possibili ha hee ae ohe shoks ha aoun fo he emaining /3 of he volaili and whih ae ausing a shif in he labo suppl o ounea he shif in he labo demand uve assoiaed wih poduivi shok. Namel, an inease in demand fom he posiive poduivi shok would be assoiaed wih a oesponding inease in he labo suppl due o his unnamed shok. 3. Monea Business Cle Theo a. Eonomiss who wok wihin he New Kenesian paadigm use impulse esponse funions o doumen he business le fas. Explain wha an impulse esponse funion is and how one goes abou finding i? (5 ps 4
5 An impulse esponse aes ou he pah of a vaiable assoiaed wih a paiula size inease in a fundamenal shok (piall one sandad deviaion. To obain he impulse esponse one needs o eplae he foeasing eo in he VAR wih he fundamenal shoks. Wih he oeffiiens of he vaiables esimaed in he VAR, we an hen inodue a supise o one of he fundamenal shoks in he equaion, and en simpl un he ssem fowad assuning no subsequen fundamenal shoks. b. The simples New Kenesian Model is defined b he following hee equaions: (i i (ii [ i ( p p ] ( (iii E ln (a Wha is he name of eah of hese hee equaions? (5 poins; Monea (Talo Rule, IS Cuve, Phillips Cuve. (b Fo eah equaion, povide a vebal geneal desipion of how his equaion is deived o aived a? (0 poins Monea ule is assumpion abou wa Fed has se mone suppl poli. IS- FONC wih espe o + and plus fishe elaion, and goods make leaing. Phillips Cuve- Saggeed pie seing, Monopolis opimal pie seing, HH FONC wih espe o and h and labo make leaing. ( Povide inuiion fo wh a supise in monea poli has a eal effe in his model. (0 poins Fis i ineases demand beause i will lead o a hange in he eal inees ae beause no all fims an adjus hei pies b assumpion. This is eviden fom he IS uve, paiulal he middle em. I will inease he Suppl beause of he pie sikiness. Sine some fims anno ese hei pie, an inease in he mone suppl will ause he esees o inease hei pie and will heefoe inease he pie index/level. Those fims who anno ese have elaivel heape goods. The suppl of hese good will inease. Also based on he deivaion of he Phillips uve, sine i uses he uili maximizing ondiion, he monea shok will have an impa on he numbe of hous supplied in he eonom as he eal wage will inease (pesumabl sine he nominal wage an adjus bu no all pies of he inemediae goods will. This will esul in an inease in suppl. 5
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