Chapter 5: Coordination and Externalities in Macroeconomics. () e. is defined as. de eb

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1 Chapr 5: Coordinaion and Exrnaliis in Maroonomis I Moivaional Qusions and Exriss: Exris 5: b (a Driv h lasii ( b( givn on p 73 o h xbook b( (b Driv uaion (56 on p 75 o h xbook ( b ( Dmonsra lim ( r b givn on p 75 o h xbook Δ Δ r b Soluions: Subusion (a: B diniion h lasii o b is dind as ( ( d b (5 ( b ( d b Expanding h dirnial ilds h rsuls obaind in h xbook (5 Subusion (b: ( b( ( ( d d b b ( ( ( b b b ( b( Th ingral o uaion (55 has h ollowing soluion: (53 b r ( r b ( r b b d b d b d b r b b r b d r b r b b ( r b ( r b b ( r b Δ ( ( r b r b sin Thus (55 in h x boms Δ (54 b ( r b Δ E ( E ( U r b ( r b Δ b ( r b Δ E( r b b ( E U r b Δ r b U whih is (56 in h x Subusion (:

2 Hospial s Rul provids a mhod or valuaing h limi λ Aording o Hospial s Rul (55 lim g ( λ ( λ lim ( λ g λ λ λ ( Taking h irs drivaivs o h numraor and h dnominaor o lim Δ ( r b Δ ilds: Δ r b (56 r b Δ (57 g Δ r b Thror w hav (58 lim Δ ( ( r b Δ lim Δ g ( Δ ( Δ r lim Δ lim Δ g ( Δ ( Δ ( Exris 5: Hazard Funion (Foono 39 p 7 b r b Δ ( r b r b (a Exmpli h diniion o h hazard union h( (b Driv h( or h xponnial disribuion and h wibull disribuion Soluions: Subusion (a: Th hazard ra is dind as h probabili pr im uni ha a as whih has survivd unil h bginning o h rspiv inrval will ail wihin ha inrval Spiiall i is ompud as h numbr o ailurs pr im unis in h rspiv inrval dividd b h avrag numbr o surviving ass a h mid-poin o h inrval Th hazard union an b xprssd as h raio o h probabili dnsi union ( o h survival union S( i b (59 h( S ( ( ( u du ( F( whr F( is h umulaiv disribuion union Th rlaionship bwn h survival and hazard union is givn b (5 h( u S du Subusion (b: For h xponnial disribuion w obain (5 S h ( ( ( λ λ λ λ

3 Th imporan aur o no is ha h hazard ra is onsan i i dos no dpnd upon im Th wibull disribuion is a gnralisaion o h xponnial disribuion and has wo paramrs λ and γ λ is usuall rrrd o as h sal paramr whil γ is rrrd o as h shap paramr (5 S h ( λγ ( λ ( λ γ γ γ γ γ λ Exris 53: Driv uaion (533 on p 93 o h xbook Soluion: B moving h rm ( s ( ( J V hav (53 r s ( U ( ( ( o uaion (53 in h xbook o h l-hand sid w ( ( J V ( w J& V& ( J V w hav du (54 ( r s ( ( U ( ( w( d ( as an iv disoun ra a and w a Thror as pr h disussion o h Bllman uaion in Chapr W an inrpr r s ( h ollowing inrmporal valu union wih h ollowing orm: [ ( ] (55 ( r τ U w d h immdia pao U an b hararisd as Alrnaivl w an driv h abov uaion hrough a mor indir rou Mulipling boh sids o [ r s ( ( τ ] (54 b ilds (56 ( r s ( ( U ( w( ( ( τ ] d ( ( τ ] ( ( τ ] d du ( I is as o s ha (57 d U d s ( ( τ ] ( ( [ r s ( ( τ ] U ( ( r s ( ( U s ( ( τ ] ( ( τ ] du s ( ( τ ] ( ( τ ] d du ( ( Thus (56 boms 3

4 (58 d U ( ( ( τ ] [ ( ] ( ( r τ w d Ingraing boh sids o h abov uaion rom o inini givs (59 d U U U ( ( ( s ( ( τ ] d ( w( s ( ( τ ] s ( ( τ ] U ( w( s ( ( τ ] d s ( ( τ ] d s ( ( τ ] ( w( s ( ( τ ] d Wih h ransvrsali ondiions ( τ ] U and ( ( τ ] U U w hav s ( ( τ ] (5 U w d Subsiuing U ( ( ( ( ( J V bak ino h abov uaion givs us (533 o h xbook Exris 54: Driv uaions (555 (557 on p o h xbook Soluion: Subsiuing p ino uaion (548 o h xbook givs (5 J ( p p Dirniaing J( (5 J& ( p ( ( ( wih rsp o givs & ( ( ( p ( ( ( p' ( ( & J& p ( ( Th diniion o h lasii o p ( wih rsp o is dnod b (53 η( p( Subsiuing ino (5 ilds ( ( p( ( p ( p( ( dp d (54 J& ( [ η( ( ]& ( p ( ( p ( p( ( & ( 4

5 whih is uaion (555 o h xbook Subsiuing (549 o h xbook J & ( r s J ( w and ( J ino h abov p( ( uaion ilds uaion (556 o h xbook (55 p ( ( η & r s p( ( ( w( ( Moving h rm p ( ( ( η o h righ-hand sid givs (56 ( ( ( ( ( w( ( η & r s p η B subsiuing (554 o h xbook w z ( z uaion (577 o h xbook: (56 Exris 55: & r s p ( ( ( ( ( ( z ( ( z η ( η & r s p ( ( ( ( ( ( z ( η η [ ] Driv uaions (569 (57 on p 9 o h xbook Soluion: For h maximisaion problm (567 o h xbook subj o (568 o h xbook: max h Hamilonian is dnod b [ F( z ] r ino h abov on an obain d d s s d r (57 H [ F( z ] λ s or H F z s r (58 r whr λ rprsns h shadow pri wih rsp o sa variabl Th irs FOC o uaion (58 wih rsp o is dnod b 5

6 (59 ' H r Th sond FOC o uaion (58 wih rsp o is rprsnd b (53 ' ' r r r r r r r r r r s z F r s z F r s z F r s z F r d d d d H λ & & & & & whih is uaion (57 o h xbook Exris 56: Sragi Foundaions o Coordinaion Gams A Rmindr In h sond gnraion modl o urrn risis wo spulaors ar diding whhr h will aak a urrn or no Th aak is sussul i boh simulanousl did o aak Th wo plars (spulaors ar alld and Boh an aak (srag A or rrain rom doing so (srag B I h plars rrain rom aaking hir pao is I boh plars aak ollivl h boh g h pao p I onl on plar aaks hn h aak ails and ha plar rivs a pao o p- Figur 5: Th Paos o h On-Sho Gam A B A p p p- B p- Plar Plar (a Drmin h dominan sragis or (i p > (ii p < and (iii < p < (b Drmin h ash uilibria or (i p > (ii p < and (iii < p < Soluions: Subusion (a: (i I p > A is h dominan srag or boh spulaors (ii I p < B is h dominan srag or boh spulaors 6

7 (iii I < p < hr ar wo dominan sragis in h gam Subusion (b: (i I p > (AA is h dominan srag or boh spulaors (ii I p < (BB is h dominan srag or boh spulaors (iii I < p < (AA and (BB ar ash uilibria and hus a oordinaion gam xiss in whih h opimal srag dpnds upon xpaions Boh spulaors would lik o oordina o aain h uilibrium (AA and suddn and xognous shis in xpaions ma riggr a risis Exris 57: Expaion Traps in Obsld s ( Sond Gnraion Modl o Currn Crisis Th ingrdins o h sond gnraions modl o urrn risis ar as ollows: Th govrnmn is minimising h loss union (53 ( { ηr} subj o h xpaions-augmnd Phillips urv (53 ( whr is oupu is h oupu arg o h govrnmn is naural oupu is h ra o dvaluaion is h xpd ra o dvaluaion is a random suppl shok and R is an indiaor ha aks h valu o i and i Th paramr η masurs h rpuaion loss o abandoning h ixd xhang ra rgim Th modl assums ha h govrnmn is using h ixd xhang ra rgim as a nominal anhor Th rason is ha in his slisd ixd xhang ra sing h a ha hr is no inlaion abroad mans ha hr is no hom inlaion ihr Viwd rom a dirn prspiv h ramwork implis ha h os o dnding h pg inrass wih xpaions o dvaluaion Th bnis o mainaining h pg ar lowr volaili lowr inlaion and nhand rpuaion On h ohr hand h oss o mainaining h pg ar highr inrs ras and lowr oupu Th iming o h gam is suh ha priva agns mov irs sing wihou knowing Th govrnmn movs las sing ar obsrving and knowing (a Drmin h wlar loss o h govrnmn or kping h pg vrsus abandoning h pg (b Drmin h uilibrium or alrnaiv s and η s and illusra h rsul graphiall Soluion: Subusion (a: W irs driv h bs rspons ( o h govrnmn Wih ( w hav (533 ( ηr ( ( ηr Th FOC wih rsp o is hn dnod b 7

8 (534 d d d d Subsiuing h opimal ino h loss union w hav (535 η η η η η lx I insad h govrnmn mainains h pg and ss hn h losss ar (536 pg Subusion (b: In ordr o drmin h shoks ha riggr mulipl uilibria din ( ( and as h lows and highs soluion o lx pg Whnvr ( h govrnmn inds i opimal o mainain h pg and s Whnvr ( h govrnmn prrs o allow h xhang ra o loa In uilibrium (537 G ( ( prob ( prob I an b shown ha and ovr som rang > G > G > G Th impliaion is ha dpnding upon η h modl ihr has a uniu uilibrium or mulipl ixd poins Th graph blow illusras his 8

9 Figur 5: Mulipl Euilibria in h Obsld (994 Modl EG( inrmdia η low η C B 45 high η A A For ihr vr small or vr larg valus o η h rsuling uilibrium is uniu For inrmdia valus o η howvr mulipl uilibria our whih ar rprsnd b A B and C In h "bad" uilibrium C h pg is alwas abandond On h onrar in h "good" uilibrium A h pg is onl abandond or xrm shoks o h irular i: i vrbod is xping C hn i is opimal o aak On h onrar i vron xps A hn no aaking is h opimal hoi Thus hr ar sl-ulilling riss in h modlling ramwork So-alld sunspos whih ma b ompll unrlad o h onom ma hang xpaions and riggr a urrn risis Addiional Rrn: Obsld M (994 "Th ogi o Currn Crisis" Cahir Eonomius Monéairs Obsld M (996 "Modls o Currn Crisis wih Sl-Fulilling Faurs" Europan Eonomi Rviw Exris 58: Invsmn Complmnariis (pp -6: Considr a modlling s-up whr h pao or agn i is givn b V ( E A( E vi i i i whr i is h or lvl whih an b inrprd as invsmn E is aggrga invsmn A( ar h gross rurns o invsmn ( ar h os o invsmn and is an xognous produivi shok W assum V > A > and > Furhrmor w assum ha an invsmn omplmnari xiss This is uivaln o assum AE > and V E > (a In ordr o kp h modl raabl suppos ha ( i Ddu h bs rspons union and h smmri mark uilibrium and illusra h obaind soluion graphiall Prsn h graph in h ( i E spa and inrodu h onps o wak and srong omplmnari (b Prov ha srong omplmnari dlivrs ampliiaion and o-movmns Soluion: Subusion (a: Th bs rspons union is givn as (538 arg maxv ( E i i V i (539 A( E A( E i i 9

10 Th opimal invsmn o agn i inrass wih h aggrga apial sok and h produivi shok In h smmri uilibrium all agns hoos h sam or (invsmn lvl and hus aggrga invsmn is solvd as E A( E h spnss o h bs rspons union Two possibiliis ar illusrad Figurs 53 and 54 blow Figur 53: Th Smmri Euilibria or Wak and Srong Complmnariis i C 45 Srong Complmnari A Wak Complmnari B Figur 54: Bs Rsponss wih Wak and Srong Complmnariis E i i C 45 A A B E E Th as o wak omplmnari orrsponds o h as whn h bs rspond union inrss onl on wih h 45 dgr lin Poin A hn givs h uniu invsmn uilibrium in h onom Srong omplmnari orrsponds o h siuaion whn h bs rspons union inrss hr ims wih h 45 dgr lin giving ris o on unsabl uilibrium (A and wo sabl uilibria (B and C Th orrsponding bs rsponss ar drawn in Figur 54 as dashd lins Th impliaion is ha h urvaur o h bs rspons union drmins h xisn o mulipl uilibria in h onom Subusion (b: From h implii union horm w hav E E( Dirniaing E ilds (54 de A d A E

11 de Wihou omplmnariis A E and hror A d ow suppos omplmnariis xis In an sabl uilibrium A E < Thror srong omplmnariis ampli shoks and h muliplir is > AE Suppos ha produivi is idiosnrai i i A( E i whr i is disribud wih dnsi ( An uilibrium E E( is now (54 E( A( E( z ( z dz E In h absn o omplmnariis h or/invsmn o i dpnds onl on i In onras in h as o srong omplmnariis h invsmn o i also dpnds upon ohr agns produiviis lading o o-movmns in h onom Exris 58: Sragi Complmnariis bwn Human Capial and R&D In h papr "Th ow-skill ow-quali Trap: Sragi Complmnariis Bwn Human Capial and R&D" (Th Eonomi Journal Sphn Rdding modls h omplmnariis bwn human apial and R&D in an ndognous growh modl Rad h papr arull and ondu h ollowing xriss: (a Driv uaion ( on p 46 (b Driv uaion (5 on p 464 ( Driv uaion (6 on p 464 (d Ddu proposiion 3 on p 467 ( Ddu proposiion 4 on p 467 Soluion: Subusion (a: Th maximizaion o uaion ( o Rdding (996 is rprsnd b ρ (54 max ( v ( λ ( v [ ]( γv A m ( δ H whr v is h onrol variabl or individuals diding in priod h raion o im o spnd on shooling or human apial aumulaion and v ρ is im-prrn disoun ra is h Poisson probabili o innovaion λ > dnos innovaion γ > and < < ar h paramrs saling h produivi o h duaion hno δ is h human apial dpriaion ra rprsns h raion o surplus workrs A m apurs h produivi o h hno mplod in priod wih m numbr o innovaions ha hav ourrd and inall H is h aggrga priod sok o human apial o gnraion - Th irs ordr ondiion o h maximizaion problm prsnd abov is dnod b: ρ (543 [ λ ( ] γv A ( δ H Th rm in brak givs m

12 (544 ρ γ v γ v [ λ ( ] [ λ ( ] ρ [ λ ( ] ρ γv Th variabl v mus onorm o h onsrain o v Thror h irs ordr ondiion boms (545 γ v [ λ ( ] γ[ λ ( ] ρ or or γ ρ [ λ ( ] ρ > whih is uaion ( o Rdding (996 Subusion (b: For h opimal rsarh or in h high growh uilibrium i is ruird ha h ollowing rlaion holds (546 ( R V ( whr V > ρ (547 ( R ( ( ' ( v λ ( and [ ]( v γ A m h V (548 ( ( ( v A h A ( γv h Wih V w hn hav v v m m ρ (549 V ( R > V ( [ ]( γv ( ( ( v λ ( > ( ( v A h A ( γv h m ρ ρ m A h m Rmoving h ommon aor (55 ( ( λ ( Colling rms inall ilds A mh rom h abov rlaionship givs v ρ ρ [ ]( γv > ( v ( γv

13 (55 ( v ( λ ( γv ( λ ' ( v > ρ ρ γv > whih is uaion (5 o Rdding (996 Subusion (: For h low growh uilibrium wih v v and w hav (55 V ( R < V ( [ ]( γv ( ( ( v λ ( > ( ( v A h A ( γv h m ρ ρ m A h m Following h sam produr as in (b w hav (553 ( λ ( v ρ < γv whih is uaion (6 o Rdding (996 Subusion (d: Th xpd ra o growh o inal goods oupu is dnod b [s Rdding (996 p 466] (554 Em iy Y ( E A ( i di A ( i m i ( E H ( H m i ( di [ ] δ m i di E ( i A i di Wih E H ( H E γv( i m i ( i di A A w hav { } m [ λ ( ] and A (555 Em iy Y Em iy Y [ λ ( ] ( A ( A [ ] ( δ E γv( i di ( H ( H m i [ ] ( di [ λ ( ] ( δ γv( i For a high growh uilibrium so ha E Y Y m i (556 λ ( v v w hav ( di [ ] ( δ [ γv ] 3

14 For a low growh uilibrium so ha v v and (557 [ ] di v Y Y E i m γ δ From h wo uaions givn abov proposiion 3 an b ddud Subusion (: Proposiion 4 rlas o h growh ra Thror w nd o solv h ingral I is assumd ha v h opimal raion o im invsd in shooling is onsan ar opimizaion and hus no a union o i Ar subsiuing uaion ( rom p 463 h wo uaions o Proposiion 3 an hus b ransormd as ollows: [ ] di γv For h high growh uilibrium w obain (558 [ ] [ ] [ ] [ ] ρ γ λ γ δ λ γ δ λ v Y Y E m i Th orrsponding growh ra or h low growh uilibrium is (559 ρ γ γ δ m i Y Y E 4