Suggested Solutions to Assignment 1 (REQUIRED)

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1 EC 45 dvaced Macroecoomic Irucor: Sharif F ha Deparme of Ecoomic Wilfrid Laurier Uiveri Wier 28 Suggeed Soluio o igme (REQUIRED Toal Mar: 5 Par True/ Fale/ Ucerai Queio [2 mar] Explai wh he followig aeme i True, Fale, or Ucerai accordig o ecoomic priciple Ue diagram ad / or umerical example where appropriae Uuppored awer will receive o mar I i he explaaio ha i impora Each queio i worh mar Durig he log period of relaivel coa growh i GDP per worer i he pical Weer ecoom, he capial-labor raio ha aed relaivel coa Fale Durig he log period of relaivel coa growh i GDP per worer i he pical Weer ecoom, he capial-labor raio ha grow b approximael he ame rae a GDP per worer I i a lized fac ha durig he log period of relaivel coa growh i GDP per worer i he pical Weer ecoom, labor hare of GDP ha aed relaivel coa Thi fac implie ha he hare of all oher producio facor mu alo ha aed relaivel coa (ice hi laer hare i oe miu labor hare Le u call he oher facor capial Le he oal capial ipu i ear be deoed b If we deoe he rae of reur o r r capial b r, he capial hare i Hece, coac of capial hare implie ha he real rae of reur o capial, r, ad he oupu-capial raio, have bee chagig b he ame rae, mu Page of 6 Page

2 The log-ru daa i he pical Weer ecoom uppor he claim ha he real rae of r reur o capial ha bee relaivel coa If capial hare,, ad he rae of reur o capial, r, have boh bee relaivel coa, he he oupu-capial raio,, ad he capial-oupu raio,, mu alo bee coa We ca rewrie he ( L capial-oupu raio a, where we have deoed he capial-labour L raio, or capial iei, L, b ad deoed GDP per worer, L, b Coac of implie ha he capial-labor raio mu have bee growig a he ame rae a GDP per worer 2 I he baic Solow model, a icreae i he level of oal facor producivi raie he log-ru growh rae of aggregae oupu per worer [Diagram required] Fale I he baic Solow model wih o echological progre, a icreae i he level of oal facor producivi will have o effec o he log-ru growh rae of aggregae oupu per worer Figure 2 how he effec of a icreae i he level of oal facor producivi o he log-ru growh rae of aggregae oupu per worer ume ha he ecoom wa iiiall a a ead-ae equilibrium a poi E wih he level of capial per worer (he capial-labor raio or he capial iei a ad he level of oupu per worer a α hi log-ru equilibrium acual avig, B, equal he brea-eve iveme, (, which i required o eep he level of capial per worer uchaged a The iiial ead-ae capial-labor raio i coa a B α α ad he α α iiial ead-ae oupu per worer i coa a B Tha i, iiiall he log-ru growh rae i capial per worer ad oupu per worer were boh zero α Page 2 of 6 Page

3 ume ha i ear zero he level of oal facor producivi icreae from B o B ad a a i ew higher level hereafer, while o oher parameer of he model chage I ear zero he iiial capial iei will remai uchaged a, becaue i i pre-deermied ad give b he capial accumulaio ad populaio damic i he pa, where he level of oal facor producivi wa B Bu i ear zero oupu per worer will icreae from B ( α o ( α B becaue of he higher level of oal facor producivi The icreaed level of oupu per worer will icreae he level of acual avig from B ( α o B ( α Thee wo effec o oupu per worer ad acual avig per worer are how b upward hif i he oupu per worer curve ad acual avig per worer curve, repecivel i Figure 2 I ear zero, he acual avig will herefore be higher ha he level of brea-eve which i required o eep he level of capial per worer uchaged a I α Figure 2, B ( > (, o capial per worer will icreae from ear zero o ear oe Wih >, he oupu per worer will icreae eve furher i ear oe, B ( ( α > B avig per worer, ( ( α I Figure 2, B ( α >,, ad ou of he icreaed icome per worer will come eve more B > B ( o capial per worer will icreae agai from ear oe o ear wo, 2 >, ad o o I he log ru ucceive icreae i he capial iei ad i oupu per worer will mae F, which i agai a coa value ad will coverge o he ew ad high ead ae value coverge o he ew ad high ead ae value ( B α α α F ( B, which i alo a coa value The poi F i Figure 2 how hi ew ead ae equilibrium Thu, durig he raiio period from poi E o F, here will be a poiive growh rae i he capial iei ad i he oupu per worer Bu oce he ecoom reache he ew log-ru equilibrium a F, here will be o furher poiive growh i he capial iei ad i he oupu per worer I ear zero afer he icreae i he level of oal facor producivi, here will be a jump i he growh rae of oupu per worer from zero o a poiive value Durig he raiio period, he growh rae of oupu per worer will graduall decreae ad will become agai zero oce he ecoom reache he ew ead ae equilibrium a F Therefore, a icreae i he level of oal facor producivi ha o ifluece o he log-ru growh rae of aggregae oupu per worer I ol icreae he growh rae of aggregae oupu per worer for he raiio period I, however, icreae he log-ru level of aggregae oupu per worer α α Page 3 of 6 Page

4 Par B Problem Solvig Queio [3 mar] Read each par of he queio ver carefull Show all he ep of our calculaio o ge full mar B [3 Mar] Coider a Solow ecoom wih o echological progre ume ha he producio fucio i Cobb-Dougla:, < <, where i aggregae oupu, i he oc of aggregae capial, i oal labor ad i he oal facor producivi ume ha grow exogeoul a a coa rae Capial depreciae a a coa rae There i o echological progre The evoluio of aggregae capial i he ecoom i give b where i a coa ad exogeou avig rae (a Derive he baic law of moio, or he raiio equaio, for he aggregae capial per ui of labor Plo he raiio equaio i a diagram Explai how he level of aggregae capial per ui of labor will coverge o he ead ae value from a give poiive iiial value [8 mar] Defie ad deoe capial per worer Thu, deoe deoe oupu per worer ad Producio fucio: ( exogeou growh i labor: ( (2 The evoluio equaio of aggregae capial: (3 Page 4 of 6 Page

5 Dividig boh ide of he producio fucio, equaio (, b we ge: (4 [Uig he defiiio of ad ] Rewrie equaio (3 a, ( (3 Dividig boh ide of equaio (3 b we ge: ( ( ( [Uig equaio (2] ( ( ( [Uig he defiiio of ad ] ( ( [Subiuig uig (4] Therefore, he raiio equaio for he capial per ui of labor i: ( ( (5 Page 5 of 6 Page

6 Figure B(a how he raiio equaio a give b (5 Thi curve ar a (, ad i everwhere icreaig The 45 lie,, ha alo bee draw Differeiaig (5 give: d ( d Thi how ha he lope of he raiio curve decreae moooicall from ifii o ( (, a icreae from zero o ifii The laer lope i poiive ad le ha oe if >, or >, which i plauible empiricall Hece he raiio curve mu have a uique ierecio wih he 45 lie (which ha lope oe, o he righ of ume a iiial level of capial per worer >, a draw i Figure B(a The will be he verical diace from he horizoal axi up o he raiio curve, ad b goig from he aociaed poi o he raiio curve horizoall o he 45 lie, ad he vericall dow, will be ae o he horizoal axi a how ow, 2 will be he verical diace up o he raiio curve The damic evoluio of he capial iei (capial per worer i give b he aircae broe lie I follow ha over ime, will coverge o he pecific value give b he ierecio bewee he raiio curve ad he 45 lie, ad i will do o moooicall, geig cloer ad cloer all he ime ad ever racedig o he oher ide of he ierecio poi Furhermore, he covergece i global i he ee ha i hold for a ricl poiive iiial The ierecio bewee he raiio curve ad he 45 lie i he uique poiive oluio,, which i obaied b eig i (5 ad olvig for Thi i called he ead ae value of aggregae capial per worer Page 6 of 6 Page

7 Page 7 of 6 Page (b Solve for he ead ae equilibrium value of aggregae capial per worer, aggregae oupu per worer, coumpio per worer, he real real rae for capial ad he real wage rae for labor Illurae he eadae equilibrium value of aggregae capial per worer, aggregae oupu per worer ad coumpio per worer i a diagram [8 mar] To olve for he ead ae equilibrium value of aggregae capial per worer we e i (5, ( ( ( ( ( ( ( ( (6 Therefore, he ead ae equilibrium value of aggregae capial per worer i

8 Page 8 of 6 Page The ead ae equilibrium value of aggregae oupu per worer,, i: (7 Coumpio per worer i ( c i a period So, he ead ae equilibrium value of coumpio per worer i: ( ( c (8 The real real rae for capial,, r ad he real wage rae for labor,, w are give b he followig wo profi maximizig codiio of he compeiive firm i hi model: r Margial produc of capial d d ad w Margial produc of labor d d ( ( (

9 Page 9 of 6 Page So, he ead ae equilibrium value of r ad w are: r ( ad w ( ( ( ( ( Figure B(b illurae he ead ae equilibrium level of capial per worer, oupu per worer ad coumpio per worer (c Fid he growh rae of aggregae oupu per labor, aggregae capial per labor, aggregae oupu ad aggregae capial o he balaced growh pah [2 mar] O he balaced growh pah (a he ead ae equilibrium he level of aggregae capial per worer i:, which i a coa So, he growh rae of i zero Similarl, o he balaced growh pah (a he ead ae equilibrium he level of aggregae oupu per worer i:, which i a coa So, he growh rae of i zero Le he approximae growh rae of,,,, ad from period o be deoed b g, g, g, g, ad g Uig he defiiio of we ca expre he aggregae oupu a, (9 Taig log o boh ide of (9, l l l (

10 So, we ca alo wrie, l ( l l Subracig ( from ( we ge, l l l l l l g g g g g g i zero, he growh rae of aggregae oupu, Sice o he balaced growh pah o he balaced growh pah hould be equal o he growh rae of labor, Uig he defiiio of we ca expre he aggregae oupu a, g (2 Taig log o boh ide of (2, l l l (2 So, we ca alo wrie, l (3 l l Subracig (3 from (2 we ge, l l l l l l g g g g g g i zero, he growh rae of aggregae capial, Sice o he balaced growh pah o he balaced growh pah hould be equal o he growh rae of labor, g Page of 6 Page

11 (d Fid he elaici of he ead ae equilibrium value of aggregae oupu per worer wih repec o he avig rae Wha i he ecoomic ierpreaio of hi elaici if he value of [4 mar] 4 The ead ae equilibrium value of aggregae oupu per worer,, i: (7 Taig log o boh ide of (7 give: l l [ l l( ] (4 The elaici of d l 4 d l 3 4 wih repec o he avig rae, Thi mea ha accordig o hi Solow Model, a perce icreae i he avig rae (iveme rae will impl a icreae i people average icome of aroud 333% i he log ru (e Fid he golde rule avig rae ad he golde rule level of aggregae capial per worer Solve for he ead ae equilibrium value of aggregae oupu per worer ad coumpio per worer aumig ha he exogeou avig rae i equal o he golde rule avig rae Illurae he golde rule level of aggregae capial per worer, aggregae oupu per worer ad coumpio per worer i a diagram [8 mar] The ead ae equilibrium value of coumpio per worer i: c (8 ( The golde rule avig rae,, i he ead-ae coumpio maximizig avig rae So, o fid we have o maximize c wih repec o uig (8 We are allowed o ae log before maximizig Taig log o boh ide of (8 give: l c l( l l l( Page of 6 Page

12 Page 2 of 6 Page So, he fir-order codiio of hi maximizaio problem: c ( ( l So, he golde rule avig rae i To fid he golde rule level of capial per worer,, we have o ubiue wih i (6: To fid he golde rule level of oupu per worer,, we have o ubiue wih i (7: To fid he golde rule level of coumpio per worer, c, we have o ubiue wih i (8: ( c Figure B(e illurae he golde rule level of capial per worer, oupu per worer ad coumpio per worer

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