Macroeconomics (Research) Lecture notes (WS 10/11)

Transcription

1 Maoonomi Rah Lu no WS / Pof. D. Ghad Illing, LM niviy of Munih. ai ool: h Poblm of Opimal onol An inuiiv Guid.. Dynami opimiaion poblm wih inualiy onain Di im max β ] V β / dioun fao wih a im pfn. onain: Law of moion f onol vaiabl; a vaiabl apial. ounday onain: Iniial apial o givn. minal onain:. Fuhmo, ;. haai h opimal im pah { *}; { *},..., Nay / uffiin popi of. and f. o guaan xin of opimum? Aum ] and f o b ily onav funion.

2 Fo opimiaion poblm, w u Lagangian mhod. max ] ] L β f Rip: onu h amilonian:,, β ] f ] ], ing, fomula h Lagangian: max,, ] L,, ].. ow o handl inualiy onain? Kuhn u ondiion! Kuhn u: Sai opimiaion wih inualiy onain. Inodu Lagang mulipli wih omplmnay lan max,..., n].. g j,..., n a j j,..., m Lagangian:,..., n ] m a j g j,..., n j ν j wih opimaliy ondiion: g j m j ν j i i ν j a j g j ; j,..., m omplmnay lan: a j g j,..., n ; ν j ; j,..., m hi way, Lagang mulipli a non-ngaiv! L ] ]

3 3 Exampl: haaiz opimal onumpion bundl wih inualiy onain: Eonomi inuiion fo hadow pi: ], max.. ; ] ] : L ν ν FO: / ; / ν ν ; ; i i i ν ;, ] i i i i ν / / / / d d ν ν Rlaiv hadow pi fo ou onain Eonomi inuiion fo omplmnay lan ], max ß.. ; p ] ] : p L ν β ν: Shadow valu fo h onain ; p β ν ; ; ; ] p p ; ; ; ] ν ν / / / p p p d d ν ν If h hadow valu ν i poiiv, h onain i binding. If h onain i no binding, ν

4 ..3 h Maximum Pinipl: amilonian ondiion Dynami onol poblm W onna on h minal valu onain And igno h oh inualiy onain ; Maximi h Lagangian: max,, ] L ν hi giv h FO amilonian ondiion Maximum Pinipl: A,,,,,, D anvaliy ondiion no: ν wih omplmnay lan: If > ; if > Inpaion: a hadow pi in pn valu m L *. L *. of h opimid pah: ; Lagangian appoah giv u a addiional infomaion h hadow pi of h a vaiabl a ah im, on i dmind. L *. L *.. Fo nonivial poblm, > 4

5 Eonomi inuiion bhind amilonian:,, ] β ] f ] i h um of h di impa of h a vaiabl on h payoff funion a plu h hang in h a vaiabl valuad a pn valu aoding o h law of moion: y o maximi hi um, bu a alo ino aoun ha valu of a vaiabl may hang: L ] : a alid un; ] : Indi impa via apial gain of a vaiabl Maximi, a ah, h um of un and apial gain abiag uaion: KE onomi inigh: : haai h law of moion fo h a vaiabl nay ondiion, i i pa of h bai onain Abiag ondiion: haaiz h pah law of moion fo hadow valu D anvaliy ondiion: Eih h valu of h a vaiabl i poiiv, hn h ndpoin onain i binding fo h a vaiabl; if ndpoin onain no binding, hadow valu mu b zo. 5

6 6..4 No: amilonian in un valu v. pn valu Shadow pi in m of pn valu valuad a iniial ag: ] ] f β Alnaiv: amilonian in un valu m: ] ] ~ f μ Shadow pi μ in m of un valu pi of apial in m of un uiliy! ~ β ] ] f μ β β wih μ β FO: A* ~ in β ~ * ~ μ μ μ ~ β μ β μ β μ β Ining in giv ~ μ β μ * ~ μ D* anvaliy ondiion β μ Saighfowad inpaion of * a abiag ondiion: μ μ μ Δ / ~ *: Valu of apial : i i i i μ β β μ ~ in pn valu m: i i i μ Valu of a vaiabl valuad a ; valu of a vaiabl valuad a iniial ag

7 ..5 oninuou im analyi Apply h am mhod o oninuou im poblm max V u, ] d.. onain: a Law of moion: aniion uaion fo a vaiabl: g,, Inmpoal ou onain b > iniial valu of a vaiabl givn ound: ;. Fixd ndpoin givn. W igno bound oh han bound a ndpoin: Somim impo in ma onomi No Ponzi gam ondiion: Opimal oluion: oninuou pah fo onol * and a vaiabl *. Nay/uffiin ondiion fo xin of opimum: Si onaviy of u. and g.. hniu: u gnalizd Lagangian: L { u, ] g,, ]} d Paial ingaion by pa b d F b d G a G d F b G b F a G a a F d d d L { u, ] g,, } d giv: 7

8 onu amilonian: u, ] g,, Maximum Pinipl: Fi od ondiion fo h amilonian Nay ondiion: A Abiag uaion: Law of moion D ; anvaliy ondiion o lim omplimnay lan: If > ; if > Suffiin ondiion fo Maximum: onaviy ondiion: < ; < Nay ondiion a uffiin uniu global maximum if funion f. and g. a ily onav and ~~ amilonian i ily onav 8

9 9 Again: un valu v. pn valu amilonian p o now: Shadow pi dfind in m of pn valu valuad a iniial ag.. g u Alnaivly: onu amilonian in un valu m: Shadow pi μ pi of apial in m of un uiliy:.. ~ g u μ ~.. g u μ μ FO of h opimal onol poblm: A* ~ in ~ * μ μ ~ au: ~ ; μ μ μ * μ D* anvaliy ondiion: lim μ Abiag ondiion: μ μ μ / ~ Maginal onibuion of apial o uiliy apial gain maginal un on un onumpion

10 ..6 Poof of Opimaliy: ao/sala-i-main o hiang Dynami poblm wih ndpoin onain L { * },{ * } b h opimal pah fo a and onol vaiabl. hn, any mall pubaion of h opimal onol pah { * ε p } wih h oponding pubaion of h pah fo h a vaiabl { * ε } and * ε d hould lav h maximum valu L L unhangd: givn h ndpoin onain ε L., ε {., ε,., ε.., ε } d], ε ν, ε a divaiv w ε : L d ν ε ε ε ε Sin p ; d, w g: ε ε L p d ν ] d ε Mu hold fo abiay mall pubaion pah ; ; o fo ah omponn, h amilon ondiion hav o hold: p d, ν p. S: o lim

11 ..7 Exampl: Opimal xaion pah fo xhauibl ou V u ] d.. onain: a Inmpoal ou onain K b K > iniial apial o; K So: d K onu amilonian: u ] Fi od ondiion: A K K u' K D KK ; anvaliy ondiion: lim K : Abiag uaion: pn valu onan olling ul: un valu mu ina a dioun a: μ μ μ ; μ Rfomula ondiion uing h un valu amilonian: ~ u ] μ!

12 Exampl: alula opimal pah fo ES payoff funion: ; / Exampl : u ] ln ] ; d d K uing h ou onain hu K and K K K ; o ha d K ] anvaliy ondiion hold: > lim K lim lim K d K ] olling ul: un valu hadow pi: μ μ μ ;

13 . h ai Famwo Modn Mao i abou dynami ohai gnal uilibium in a up wih pnaiv, opimiing agn in an onomy wih impf ompiion and poibly inompl a ma., in od o gain h bai inuiion fo inmpoal opimiaion, w a by duing h poblm o h impl a: W onid fi a pu ndowmn onomy.. wih on good onumpion and no unainy. W allow fo inmpoal ad on pf apial ma via al bond wih al in a. In hi up, w analy inniv fo onumpion moohing. La, w xnd hi up by inoduing apial aumulaion, unainy, govnmn aiviy and finally mony ion.3. In h nx p.., w loo a h Ramy gowh modl. I haai h opimal inmpoal pah fo onumpion and apial fomaion { *} { *},,, fo a pnaiv agn, aing wih an iniial apial o. W olv i a opimal onol poblm. Wih a ompl of ma, a dnalid ma onomy an implmn h ffiin alloaion. Sion..3 loo a om appliaion, uh a inoduing govnmn pnding and ax and innaional ad of mall ouni. In boh a, w onna on walh onain and y o figu ou h impliaion fo uainabiliy of publi o innaional db. Sion..4 hn loo a om bai of dynami ohai opimiaion. A pomiing ou fo modn mao i o inga mao and finan. Sohai opimiaion i h aing poin fo hi ah aa. 3

14 .. onumpion Smoohing: h Eul Euaion h pnaiv agn maximiz β.. p piod budg onain: wih and - givn and h nd poin onain No Ponzi Gam onain. - : al un of a invd duing h la piod -. Pu ndowmn onomy: Pah { } i givn. Dfin g a h gowh a of ndowmn Sa wih a wo piod modl g. ß. udg onain p piod:, wo Piod: Saing poin i ; nd poin i! hoo { ; } and { ; 3 } opimally! onain: givn; minal onain 3. A aighfowad way o olv i o mg boh p piod budg onain ino a lif im onain liminaing : 3, and a alo h minal onain 3 ino aoun. 4

15 5 u l u inad u h opimal onol appoah. Fi, dfin h amilonian in pn valu: β Ky FO fo inmpoal opimaliy: A* β * * D* anvaliy ondiion: 3 y ombining A* and *, w div h Eul uaion: E* Dfin h MRS a innal dioun fao and / a ma dioun fao,. hn, w an wi E*, ing, D* giv 3 3,. au > hi implifi o: 3 3,

16 An uivaln fomulaion of h budg onain i in m of walh: If I plan o hav walh W a h bginning of nx piod, I mu pnd oday, W hin of buying zo bond a h pi,. No: W -. W hi giv: W wih h nd poin onain W W; W3. h walh onain p piod an b fomulad a: W W / / W,,, W W. in h amilonian giv u a FO: ~ ~ ~ ~ /,, ~ W ~ No:, anvaliy ondiion: W o W,,3 3 h Eul uaion E* haai inmpoal dmand. Aggga Euilibium: Fo h whol onomy, dmand mu b ual o ndowmn. Euilibium pah dpnd on: a Pfn dioun a, inmpoal laiiy of ubiuion b hnology ndowmn; poduiviy gowh al inmpoal xhang a L u onna on Eul uaion Maginal uiliy ou of onumpion, adjud fo h lvan a of un, ha o b ual ao im Du o onaviy u <, agn pf onumpion moohing. h mo onav h payoff funion, h ong h inniv! W g onumpion moohing p. onumpion iling! 6

17 Elaiiy of maginal uiliy fl h onaviy of h uiliy funion: ow do h lop of h indiffn uv hang wih a hang in h onumpion aio /? d{ u' / u' } / ε d { / } u' / u' Inv o h inmpoal laiiy of ubiuion u" ε fo oninuou a u' Exampl: ES payoff funion: ; > Fo : ln / Eul uaion: o i inaing daing] fo > < ]. h high, h p h pah. Log-linaizaion giv: E 7 ; Δ Δ Dfin Δ, o wih logaihmi appoximaion: Δ / ] hi i h bai laion o povid a miofoundaion fo inmpoal aggga dmand! La, w will u i a h inmpoal IS uv: y E y No : Fo h ES funion, i h inv of h offiin of laiv i avion ; No : Div h ondiion fo oninuou im analyi Eul uaion i: d u' / d u" ; u' u' O, bau u', u"

18 Euilibium: g ; n g g h naual a of in: Euilibium al in a i haaid by uilibium ondiion aggga onumpion ndowmn. ing h gowh a: g, w an wi E* a: n g n n Log-lina appoximaion: g o g. n h naual a of in i high a h high : impain ui ong inniv o av! b h high h gowh a g bau of inniv fo onumpion moohing, high gowh ha o b baland by ong inniv o av! h low h laiiy of ubiuion h mo onav h uiliy, h ong inniv o mooh onumpion fo a givn gowh a! 8

19 .. h Ramy Modl: Opimal Gowh oninuou im Vion Nw wi: Inlud poduion and apial fomaion! Nolaial lina homognou poduion funion FK,N RS N: Labou upply a ; K: apial o a Law of moion: K F K, N δ K K f F K, N F, N N y f ; f ' > ; f " < ; F K, N F N K, N N f f N F K f K, N N N f f K f ' f ' ; Inada ondiion: N n lim f ' ; lim f ' 9 N ; δ a of dpiaion Rpnaiv agn maximi p apia onumpion: V u d wih / N Sily onav payoff funion: u ' > ; u " < ; dioun fao.. onain: a Inmpoal ou onain: apial o p had volv aoding o: f δ n, bau: / K / K N / N o K / K n K / N n ; K / N f δ b Iniial apial o givn; Endpoin onain lim

20 ... amilonian: : u ] f δ n ] FO: A u' f ' δ n] f δ n D lim lim u' FO giv diffnial uaion haaiing h voluion of h a vaiabl / and of h hadow pi / ; f ' δ n] f δ n ing A, an b anfomd ino a diffnial uaion fo h voluion of onol vaiabl / : u" u" u' u u ' ' u' au, w g: u" o A f ' δ n] f δ n bounday ondiion giv a dmind oluion fo h diffnial uaion: and h anvaliy ondiion: lim u'

21 ... Sady Sa and aniion Dynami Digion: Sabiliy analyi ow an w olv a diffnial uaion y f y? I h onvgn o a ady a? Inuiion by gaphial analyi: wo ady a. On abl, on unabl Explii analyial oluion fo lina diffnial uaion: y a y b Sady a y : y b / a

22 Soluion podu: a Solv fo h homognou diffnial uaion y a y y wih a abl ym: a<; unabl ym: a> y a a Gnal oluion: y A wih y a A a y a Paiula oluion: y A b a om paiula oluion fo h non-homognou diffnial uaion: Fo inan, a ady a y wih y b / a b Gnal oluion fo h ompl ym: y A a a h dfini oluion i dmind by a pifi bounday b ondiion: y A y a b b Dmin h uniu oluion: y y ] a a a

23 Sady Sa in h Ramy Modl: h uilibium pah i haaid by h 3 ondiion A, and D. In a Sady Sa: f ' δ n] Modifid goldn ul: f ' * δ n f δ n o f * δ n * * Fo h ady a {*; *}, h anvaliy ondiion i fulfilld in * and u' u' *. I h ady a abl? Gaphial analyi: 3

24 Saddl poin abiliy: f ' δ n * < < > f δ n < > Fo any iniial, xaly on uniu pah of onvgn o h ady a {*; *} aing wih iniial onumpion ˆ - all oh pah a divging. Fo > ˆ : Saving oo low vnually o a om fini : jump in onumpion o If < ˆ : Exiv Saving Ovaumulaion in fini im: violaion of anvaliy ondiion. ompaaiv ai: Wha happn, if om xognou paam hang? Wha i h impa of uh a ho? f ' * * Invmn dud; onumpion go up onumpion pah ild mo owad pn onumpion. 4

25 No fuh adjumn in ady a: * f * δ n * g Ina in al a of in: onumpion ild mo owad fuu onumpion 5

26 ...3 h Dnalid Ma Eonomy h ompiiv uilibium i ffiin in an onomy wih infinily livd agn. fo ading h nx pag, y fi o how hi youlf. Aum ha om fim own h apial o, finand by iuing bond o houhold. ond b pay a un a. h iniial walh of houhold i h valu of h fim b. Labou i upplid inlaially, houhold walh p apia b volv aoding o: b w b n b Fomula h fim and houhold opimiaion poblm. hn, haaiz h inmpoal pi pah. No ha in a on good onomy, only h inmpoal pi h a of un on a, h in a ha o b dmind. h amilonian appoah o haai h FO Finally, how ha h FO fo h opimum a uivaln o h ondiion of h Ramy poblm. in: Dfin ν ] dν} a avag in a up o giving h pn valu fao: Impo h No Ponzi-Gam-onain: n] lim b anvaliy ondiion: hi inualiy mu b binding fo h opimal oluion 3 h fim poblm i no ally dynami. Valu of h fim: b! 6

27 Fim In h abn of adjumn o, fim fa no ally a dynami poblm. Fim n labou and apial a ah poin in im. Aum om fim own h apial o, finand by iuing bond. Aum RS onan un o al. Max N f δ K K w N N, K FO: f ' δ o f ' δ ; f f ' w Zo Pofi: Labou inom: w N ; apial inom p wo:. Wih RS, h valu of h fim i b! ouhold: h pnaiv houhold a wih iniial finanial walh b and an inom fom in on bond and labou inom w: human walh. A p apia fo h houhold volv aoding o: b w b n b w n] b L u follow h andad amilonian appoah: Max V u d u d N.. h law of moion b w b n b w n] b h No Ponzi-Gam-onain lim { b xp ν n] dν} lim { b n] } b Iniial finanial walh: b b 7

28 8 amilonian ] ] ] : b n w u FO: ma u ' m b ] n m b ] b n w b md lim b ' lim b u ompa o h planning oluion: h planning poblm wa: Max d u V.. a Law of moion: n f δ b Endpoin onain } { lim Iniial apial o: wih amilonian: ] ] : n f u δ FO: A u ' δ ] ' n f n f δ D lim ' lim u So n f δ '

29 Euivaln: ma i h am a in h planning oluion. m i uivaln o bau in a ma onomy, f ' δ m i uivaln o f δ n bau f ' δ and b and und RS f w f '. u mo gnally, vn in h abn of RS, h valu of h fim inlud all pofi, o w g h analogou ul alo in ha a. md anvaliy ondiion lim i uivaln o h No Ponzi gam onain bing binding: n] lim { b }, bau b and f ' δ, o fom : n] o n] xp ν n] dν} 9

30 No Ponzi Gam onain and anvaliy ondiion Fom h flow budg onain, ingaion giv voluion of walh ao im: b b xp ν n] dν ] w ] xp ν n] d ν Noaion: xp ν dν wih ν dν n] n] So b b w ] d h inmpoal walh onain i: n] n] n d b b w o fo n] n] d lim b b w No Ponzi-Gam-onain: Fini im: Impo ondiion b fo pnaiv agn: Inuiion: Ohwi ha i if fo om j b j <, om oh agn mu hold poiiv a b i > a h nd of h wold. No Ponzi-Gam-onain wih infini hoizon : n] lim { b xp ν n] dν} lim { b } Pn valu of a mu b aympoially non-ngaiv! Moivaion: la of ounpay - no on i willing o hold a wih poiiv pn valu fov. Ohwi if onain w no impod: i would b opimal o go ino infini db a h aing piod anno b an uilibium. u fom h anvaliy ondiion a pa of h opimaliy ondiion: w now ha hi inualiy mu b binding fo h opimal pah. ] d n] ] d d 3

31 ...4 OLG Modl: Dynami Inffiiny Simpl a: Agn liv wo piod; inom only in fi piod ß.. walh onain: / w FO : o onid : w w w ; Populaion Gowh: Evoluion of apial o: N n N K K F K N w N K N N, ] 3 w w N K N ; onumpion of old: N K K K N w w w N n n N w N w o in p apia m: Fo f ' f ' f f ' ] n α α ; fo f : w α o α α n * α n α α f ' * α f * n α ompa wih goldn ul pah: f ' g n! α Ovaumulaion if α i mall: f ' * < n Empiially lvan? No: Ovaumulaion if al in a i oo low: <ny. Wha a h lvan long m a? Abl/Maniw/Summ/Zhau: Do n apial inom xd n invmn gowh a of h onomy? K > ny K? Ex invmn onibu o onumpion

32 Dynami Inffiiny ~ Raon? Ma failu: miing ma Fuu gnaion do no paiipa in aiv ma oday Inompl ma Mahmaial jo: ol infiniy: on miing bd no poblm, ju a all o mov on bd ahad Paul Samulon: OLG modl a bai fo: a oial uiy ym Ingnaional anf a f lunh b Inodu Mony Sagn/ Walla: OLG modl a miofoundaion fo monay hoy fad no oom fo mony a oon a h i oag; dominad a! ou nd anaion ol fo mony 3

33 ...5 Finanial Walh/ A Piing: Valu of fim ha: ai of a piing Obfld/Rogoff ff; Appndix, h valu of a fim { V } dpnd on h dividnd am { d } and h minal valu. Eah piod, h houhold did abou h numb of ha hld unil nx piod. Sh puha ha of h fim a h valu V -. A im, h ha yild a dividnd d and af dividnd paymn, bfo onumpion hav h valu V. A, addiional ha an b old o bough a h un valu V. Givn h dividnd am, how an w alula h valu of h fim? Inuiion: Inmpoal voluion of h fim valu hould follow h inmpoal ma pi. u Dfin β, a h ma u dioun fao. hn, h a valuaion uaion hould b: V V d, i d i, i V, L u div hi valu funion xpliily. Wih oh inom and pnding w N, availabl onumpion in ah piod i: d V V w N So β u{ d V w N} Fo hooing h numb of ha hld opimally, w mu hav a FO: β u' d V ] β u' V β u' V d V ] d V ] u' udg onain: V d V w N. h andad Lagangian: d V V w N] d V V w N ]. Alnaivly, y amilonian appoah by fomulaing: 33

34 y iad ubiuion, h valu of h fim i h pn valu of h dividnd am diound by h maginal a of ubiuion bwn and oday a ah a ]: V d V d V,, β u' β u' d V u' u' u' anvaliy ondiion: lim V β, o u' V d d, Exnion: Inodu an addiional innaional a wih onan in a : Wih a and a onan a of in, al walh i: W V V V d V V w N β u{ d V w N} FO: V u' V d β u' and u' β u' hu: V d V o V d V d V ing h anvaliy ondiion: V d 34

35 Rma: nd RS: Valu of apial o ~ Valu of Fim: K V d F K, N w N K K, o fim maximi a : d V { F K, N w N K K } wih F w ; F. N nd onan un o al K d K F. F N F K, o F. w N So h a zo pofi and V K K K No ubbl ondiion: In uilibium, i mu b u ha lim V β u'. wih onan, uivaln o lim V bau u' β and u ' > u' Poof: Suppo lim V β u' >. hn, agn would b b off ina pn valu of onumpion by ho-lling ha mainain fov a onan ngaiv valu of h a. u a long a lim V β u' >, no on would b willing o hold h poiiv ounpa all agn would li o do h am o ha anno b an uilibium. Nobody wan o b h viim of a Ponzi gam!! So w impo h no bubbl ondiion: lim V β u'. Opimaliy wih f dipoal impli ha inualiy i binding: anvaliy ondiion: lim V β u' N K 35

36 36 Valu of apial in h a of adjumn o: obin S: ao/sala-i-main, ion 3.6 and Rom, ion 8. In h pn of adjumn o, h valu of apial may dvia fom plamn o. Solv: Max d K I I N w N K F V ], Φ.. K I K δ ; " ; ' ; Φ > Φ Φ Shadow pi: V/K i h pi of h apial o a piod in un valu Ma valu of a fim/ apial o. h wih pn valu amilonian:! un valu amilonian: ] ], ~ K I K I I N w N K F δ Φ FO: ~ N o ] w N F ~ I o ' K I K I K I Φ Φ ~ K o ' K I K I K F Φ δ o: Φ ' K I K I K F δ

37 37 Φ ' K I K I K F δ Run on xiing apial, dflad by o of apial : F maginal duion in adjumn o wih iing K Wihou adjumn o: δ F ' ; ; anvaliy onain moivad by h no bubbl onain: lim K Exampl: Lina adjumn o K I b K I Φ FO: K I b b K I K b K δ b K F ] ' δ onvgn of hadow pi of apial o *δ b? Saddl pah abiliy:

38 ..3 Appliaion:..3. Pmann inom hypohi Milon Fidman agud ha a pnaiv agn onum a onan popoion of hi pmann inom. nd wha ondiion will hi b opimal? ing h anvaliy ondiion, h inmpoal walh onain fo a pnaiv agn i: n] n] d b w d ing h opimaliy ondiion divd in.., w an haai h opimal onumpion pah in a aighfowad way: au of n], w g via ingaion: n] Givn hi pah, i dmind by h inmpoal walh onain. Fo onan w an olv xpliily fo h onumpion pah: n n] { n] ] } d d b W A Spial a : b W ] If h al a of in i onan, hi giv a opimal iniial onumpion lvl: n n ] b W ] Obviouly, only if n : n] n] d b W n o n b W ] n b wp No: n! ady a analyi Modn Vion: onumpion gowh follow andom wal 38

39 ..3. Govnmn Spnding and Riadian uivaln..3.. ai Ida: wo Piod Famwo Wha hang if w inodu govnmn aivii? a h pah of govnmn pnding and ax G ; G; ; a givn. h govnmn an iu bond o finan pnding via db. h govnmn budg onain p piod i: G ; G 3 onid a pu ndowmn onomy. h iniial govnmn db ha o b ual o h iniial walh of piva agn who iniially bough bond. Sin fo piva agn minal ondiion 3 i binding, w hav o impo h No Ponzi gam onain fo h govnmn: 3 uling in h anvaliy ondiion 3 ]. G 3 G h budg onain fo piva agn i now: 3 Riadian uivaln: Govnmn db i no n walh fo piva agn! Fom h govnmn budg onain, w an ubiu fo ax in h piva agn onain: G G So h modifid uilibium ondiion a now: G; G, and h naual al in a i now dmind by: u G, u G 39

40 ..3.. Infini oizon Gowh Modl Aum ha h pnaiv houhold own h apial o and alo hold govnmn bond b boh dfind a p apia]. hi dfin walh. Sh ha o pay ax and may appia g. h gowh a of labou i n. h houhold poblm wih h apial o nd ou o h fim i: Max V u, g d.. law of moion a oal walh Ωb volv aoding o: Ω b wih Ω f n] b n τ n] b No Ponzi-Gam-onain lim { Ω } Iniial walh ou of govnmn bond: b d Iniial apial o: Aum: u, g i addiiv paabl amilonian: : u ] Ω Fom Lagangian: L : u ] Ω - Ω ] So: : u ] f n] b n τ ] FO: A u' Ω b n] and f ' n] 4

41 4 b Ω ω ] n b n f b τ Ω D lim b ' lim b u D lim ' lim u So n n f ' Eual un on apial and govnmn bond: ' f Ra of un on onumpion: Wih fixd labou upply and uiliy addiiv paabl in g g : D i uivaln o: } lim { ] n b D i uivaln o: } lim { ] n ; ] xp ] > ν ν n d n h pah of pnding and axaion anno b abiay: h govnmn ha o a ino aoun i budg onain. I flow onain i: ] b n g b τ Saing wih om iniial lvl of govnmn db b, db volv aoding o: d g b b n n ] ] ] τ o: n n b d g b ] ] ] τ

42 Whn h anvaliy ondiion fo houhold hold, h govnmn i ubj o h No Ponzi gam ondiion: n] lim { b } h anvaliy ondiion giv a inmpoal walh onain fo h govnmn: n] GW b τ g ] d Indpndn of h iming of ax and govnmn pnding, h pn valu of n aning diound ax aning minu govnmn pnding mu ual pn ouanding db. Riadian uivaln: ow do govnmn db poliy aff piva walh? ing h anvaliy ondiion, h houhold walh onain i: n] n] d b f τ ] d Govnmn bond a no n walh: ing GW, w an anfom h houhold walh onain ino: n] n] d f g ] d So h houhold walh onain i indpndn of h ax and db pah. Inuiion: igh govnmn db ul in an uivaln ina in piva aving in od o off fuu ax obligaion. hi ul dpnd on ong aumpion: No impfion of apial ma, piva ual o oial dioun a! di onaind agn: liuidiy onain OLG uu: Dioionay ax: Inniv fo ax moohing Non-opimiing agn Kynian yp agn; ul of humb ~ Alof: inluion of ho nom AER pidnial add 7. 4

43 Suainabiliy of Govnmn Db In h la ion, w haaid h govnmn walh onain. Will a govnmn wih a onan dfii un ino olvny poblm? Do i ma n o impo om upp limi o h dfii aio uh a in h Maaih iia? W hav o diinguih bwn pimay dfii and oal dfii. onid a impl a an onomy wih onan gowh a y and onan al a of in. W aum h in a xd h gowh a: >y only hi aumpion i nial! Whn will h govnmn ay olvn, aing wih om iniial db ouanding bond b >? No: h b/; iuing bond b> i uivaln o db! Wih d P a pimay dfii, h db aio volv aoding o: b g τ y b d P y b giving h diffnial uaion: y y b b g τ ] d o y y b τ g ] d b Impoing h y anvaliy ondiion lim b giv GW y y τ g b d P d τ g ] d y h pn valu diound by h ffiv a -y of aio of pmann fuu pimay uplu o GDP ha o b ual o h un lvl of db /GDP. onid a pah wih a onan pimay dfii a: pd g τ uh ha: b pd y b. Soluion o hi lina diffnial uaion: b a b a omognou pa: b A ; b givn Paiula oluion: b p a 43

44 a Dfini oluion: b b / a] / a Sin a--y< and pd>, w g y b b pd / y] pd / y Obviouly, h govnmn db aio i unabl unl b pd / y u: mpoay dviaion - b pd pd] - do no ham if off la by addiional uplu:3 pd y b pd pd] d y h govnmn ay olvn if pd p max τ max gmin wih τ max < a maximum pmann uainabl ax a and g min > a minimum uainabl pnding a. Sin h a a omwha abiay, a pagmai appoah, alula a uainabl ax a h avag ax a whih would b ndd o finan annuiy valu of fuu xpd pnding and anf, plu h al in a budn on un db: y τ y g d b ] If τ < τ, ax hav o b aid o pnding ha o b u whn db hould b uainabl. h magniud τ τ giv a mau of h adjumn ndd! ow do dlay of adjumn unil im aff h iz of ndd poliy aion? h iming of adjumn i givn by d τ y diffniaing h uaion abov: τ τ d Ky ingdin: h al in a xd gowh a: >y y anvaliy ondiion hold: lim b 3 anvaliy ondiion lim y b impli: y d p d p ] d 44

45 Maaih ay: onan oal budg dfii a h Maaih iia impo a limi fo h nominal db aio b,6 and fo h oal nominal dfii aio d,3. So now w hav o analy an onomy wih nominal db and inflaion. p i nominal GDP. W aum a onan a of inflaion π and h Fih uaion i π. W alo inlud ignioag μ in h govnmn walh onain ion.3 fo dail. ow will h db aio volv whn h oal budg dfii aio inluding db vi ay onan? Will h db aio onvg o om ady a? L g τ μ i b d onan p hn b d y π b. No: d i oal dfii inluding in paymn! Soluion of h Diffnial uaion iniial lvl b : π y b b d] So w hav d b* π y d lim b π π y b d π y b fo any d b d π y π y, onvging o h ady a: y Maaih iia: Fo d,3; y,3 and π. : b onvg o b,6 h low y and π, h high h ady a aio b. Impli: high ax budn on onomy dpnding on al in a -y. h db limi i faibl a long a τ g b < b max min τ max < ; g min > y d π y 45

46 ..3.3 un Aoun Dfii: Opimiing ouni? Funly, onomi analy dynami wih onan in a o gain a b inuiion abou y iu, uh a olvny onain and uainabiliy of db. In hi ion, w onid impl xampl of a pu ndowmn onomy wih infini hoizon o g a fling fo olvny onain and fo h impliaion of anvaliy ondiion. W po h following uion: an w hav an onomy wih opimiing hognou agn, bu a onan al a of in wihou violaion of h anvaliy ondiion? Exampl: ouni wih diffn im pfn in opn onomi? Suainabiliy of un aoun dfii: nd wha ondiion i h db dynami of a mall ouny uainabl? Aum h ouny ha an xognou onan gowh a y wih: / y. Igno apial fomaion and n. Pfn a haaid by a onan laiiy of ubiuion. L b< b h iniial db of h ouny. Will h ouny onvg o a ady a if i an boow and lnd on a pf innaional apial ma a a onan in a? Will h foign db o GDP aio bom unuainabl? Wha happn o h a of onumpion o GDP? Whn i h a i of inolvny? ompa Obfld/ Rogoff, Appndix A. 3 Suainabiliy of un aoun dfii wih nominal a: If w apply uivaln ondiion a in h a of govnmn db, whn do w g onvgn o a ady a? 46

47 W analy h db dynami in a mall ouny wih pfn V u d wih Iniial walh: b / a haai h law of moion fo walh/gdp: Show ha b / volv aoding o: b ]/ y b y b b haai FO uing h amilonian! Show ha onumpion pah fo onan i haaid by: ] nd wha ondiion i h poblm wll-dfind? haaiz h auahy oluion. d Aum h ouny ha iniial db b / < haai ondiion fo h db/gdp aio onvging owad lim and lim y an w hav an onomy wih hognou agn, bu a onan al a of in? h anvaliy onain i aifid fo >n! Fo n<<n: onumpion pofil i daing and boundd Fo n >, h onumpion pofil i inaing. owv, lifim onumpion ha o b boundd by h walh onain n n] d b W h onumpion am onvg only if n n n > o < n S alo nx ion on uainabiliy analyi! / 47

48 Suainabiliy analyi of un aoun dfii. ing h anvaliy ondiion } { lim b, h inmpoal walh onain i: d d y d y h onumpion pah fo onan i haaid by: ] ; ] d h poblm i wll-dfind fo < > ; Ohwi h would b no onvgn of onumpion. So w g: ] ] y ] y y y y } ] { } ] { > < y fo on ]

49 49 A In h a of auahy, w mu hav: y ; ; y y Fo y : on -yb ; on ; lim b If y >, h onumpion pah i ild owad h fuu. h i no onvgn owad a ady a, bu h valu of onumpion will b boundd fo </- If i oo high, inniv o popon onumpion indfinily. Fo y > : ;

50 In ha a, onumpion gow fa han inom, bu iniially h ouny build up a. Raio of n foign b a o GDP hoo off owad infiniy: lim. y No: no abl ady a, bu h anvaliy ondiion hold vn in ha a a long a > ompa Obfld/Rogoff, xi.7! onid now h a ha h wold in a i l han h auahy a < y. ha i h ining a fo olvny iu, in in ha a, h onumpion pah of a mall opn onomy i ild owad h pn laiv o h auahy a > a. So, iniially, h i a un aoun dfii A - ; - A. Wha will happn o a ouny boowing a h wold a ao im? Wh do h db/gdp aio onvg o? 5

51 5 Sady a: g A lim ; y - lim In h long un: y lim ; lim ; lim ; Limi of db/ GDP aio onvg o h ni pn valu of un and fuu oupu db ha o b paid ou of fuu poduion. Poof: ; ; b b b b ; hu: y y b. In ady a: b o lim y ; Fo y < : lim ; y lim uial iu: I h limi lim uainabl?

52 No: a onumpion do no nd o onvg o in gow a a y. u lim fo gowh a of onumpion: < y ady a oluion b Sin boh < y and >y mu hold ohwi infini onumpion would b faibl, w mu hav: <y/y- h gowh a of h ouny wih iniial db ha o xd h avag wold gowh a y*: Inuiion: uilibium wold in a dmind by h ondiion ha fo h wold a a lod onomy n foign a hav o b, o w mu hav: y *. Sin y > y * fo a abl oluion, h ouny gowh a ha o xd wold a! d nalii fau of hi hough xpimn: Aumpion of a onan in and gowh a dpi high gowh onomy. Mo alii: aum ha gowh a onvg o wold gowh a! Ohwi mall ouny aumpion will b violad a la in h long un! 5

53 Mhani of Aouning: Suainabiliy Analyi of un Aoun Dfii - h a of nominal a and inflaion un aoun: uivaln o an ina in foign a A wih A A i y ; π gowh a of al oupu and inflaion Gowh of nominal IP: y π A b b y π b Suainabiliy Simulaion: wha would happn if A/ i uaind fo onan y ; π? Will b aio of n foign a o GDP onvg o a ady a aio b? A / y π A / Diffnial uaion: b b y π y π A b y π b i onan if y π Exampl: y π,5; A /, 5 b* - % Wih onan b, al n xnal a o volv aoding o: d / P π y d P P P Sady a ha of abopion A IG o aa/: A A i d / P A! y ; a y b d P P P P 53

54 a If h i a iial low bounda > a, b > b pn y valu of x poduion ov minimum abopion lvl Obfld/Rogoff: Ppiv on OED Eonomi Ingaion: Impliaion fo.s. un Aoun Adjumn Wha i h long un uilibium? A? aland un aoun? No: Nominally baland un aoun impli whn inflaion i poiiv and wih n foign db an inflaion adjud un aoun uplu! Ra of hang of h onomy al n foign a poiion wih A i: d / P π > fo < inflaion adjud uplu, in d P foign db hin in al m ondiion fo onan al n foign laim: d / P A π π iff A π < d P P P P 54

55 Aum ha abopion ad balan i uh ha i pay a mall faion of h in budn. h will b olld ov ino h nx piod. A ξ, o A A ξ hu, ξ anvaliy ondiion hold: ξ d ξ d ξ ξ u noi ha db and hu in budn gow a a ξ In od fo paymn no o xd oal poduion a om ag A ξ, mu gow a a a abov y y > ξ. wih y<. So ξ > 55

56 ..4 Sohai Opimiaion..4. ond and ma dioun a ow o xnd ma dioun fao o ohai poblm? Walh onain p piod: If I plan o hav walh W a W. h bginning of nx piod, I mu pnd oday,, i h ma dioun fao bwn and. Sin, W W, h walh onain p piod i: W W W /, /, Fom h amilonian, w g h FO: ~ ~ ~ ~ /,, ~ W and h anvaliy ondiion W 56, wo piod: Maximi u β ubj o h p piod budg onain W W3 W ; W ; wih nd poin onain W W; W3. FO: a u, u b,3 W3 anvaliy ondiion Euilibium ondiion:, Impoing uilibium ondiion, i dmind by: u, u Wha i h impa of unainy on onumpion/ uilibium?

57 Sohai Dioun Ra Simpl xampl: Piod; unainy abou h ouom of h ond piod wih poibl a: high o low inom. Aow Dbu onomy: Aum a ompl of i ma h i a pi dfind fo ah a of h wold. h p pobabiliy of high ndowmn; ohwi l p - h p Rpnaiv agn maximi xpd uiliy: ] l l h h p p u β Inmpoal bug onain wih Aow Dbu uii: A pn valu pi, buy a in piod giving a al laim of uni in a nx piod: Wih D laim, walh in a i D W. So D o D FO: ' ' ' ' / / p p p / / β Dfin p Q a ohai dioun fao, ;,,, z z z z pob z z Q Q p / / β Wih ompl ma h ohai dioun fao Q i uniu.

58 58 ow do unainy aff h pi of a i f a h un valu pi fo on uni of onumpion in all a? h ohai dioun fao fo ah a i haaizd by: ; ;,, u u z z Q ξ ξ β h i f a bwn om a and i:,, Q p z z Q E ; ; u u E ξ ξ β Explii analyial oluion an b obaind only fo om xampl wih pial pfn, ombind wih pifi diibuion.

59 Exampl Aum RRA and onumpion gowh i log nomally diibud. In ha a, w had wihou i: o log-linaizd: g Whn fuu onumpion i iy, h Eul uaion in xpd m i: E In h pn of onumpion i, a af a povid inuan fo iy fuu onumpion: i av agn wan o do pauionay aving. Fom Jnn inualiy, w gu ha h af un will b low han a iy onumpion pofil wih am xpd valu inniv o av fo pauion div down h u a of in. Dfin h FO ondiion: ln E Δ ln. If x ~ N E x, x, hn Exp x xp E x x. Auming ha onumpion gowh i log nomally diibud, h w hav: x Δ ln, o w g: Δ xp E ln O, log linaiing, w g: E Δ ln E g Δ ln ln ln. aing log, w an wi 59 Pauionay aving Wih iy onumpion popl av mo in od o inu again i of bing in a a wih low onumpion, diving down h i-f in a. h mo i av popl a h low, h ong hi ff!

60 Exampl: Quadai uiliy funion ainy uivaln: giv: E ] E ; fo : E o ε Wih E ε - onumpion follow a andom wal Exampl 3 Exponnial uiliy funion: xp α α hi giv h Eul uaion: xp α E xp α x x Exp x xp E x No: Fo ~ N E x,. So w g: α xp α xp α E Va α o in log: α αe Va, o: α α E ] Va alula uilibium fo h ohai ndowmn po: ε pin ff of ho; ε ~ N, x 6

61 ..4.3 Sohai Run L u now xnd h analyi o holding abiay a wih andom un ~ i,. Again, aum ompl oningn pi. ~ ing h ohai dioun fao h MRS bwn Q onumpion in a and i haaid by: / Q β. So h Eul uaion giv: / A E ~ / ~ β i, E ~ i, Q / Rwi ondiion A a h podu of xpd vaiabl plu h ovaian ~ ~ ~ E Q ov ~ ; Q. A E i, i, Fo a i-l a wih af un, h ovaian wih h ohai dioun fao and any oh andom vaiabl i zo. hu: ~ E Q, z, z ing in A, w hav fo abiay iy a: ~ ~ ov ~ ; Q E i, i, Euaion i h y laion fo h onumpion APM Modl: h xpd un pmium of an a laiv o i-f a i ngaivly poiivly] popoional o i ovaian wih MRS onumpion]. olaion ovaian bwn h a payoff and onumpion a diiv iion fo holding h a: An a wih high un i aaiv whn onumpion i low Q i high. Dmand fo uh an a will b high; i un will b low han a i f a. 6

62 Appliaion: Evaluaion of iy ha dividnd am Wihou i, h valu of h fim i V d u' h dividnd am i valuad wih, β. o u' Wih, i... i, i,,, ow o valua a iy dividnd am? In a of i nualiy, uiliy i lina in onumpion, o u ' u'. hu, E β o V E d β E d Evaluaion uaion:, Wih i avion, xpd ma dioun fao dpnd on h ovaian bwn dividnd and h dioun fao. o hi wi h Eul uaion a: V u' E β d u' V ] u' u' E β E d V ] ov β d V ] u' u' If dividnd a poiivly olad wih h dioun fao, h pi of h laim, V, i high a poiiv olaion wih h dioun fao man a ngaiv olaion wih onumpion gowh. n, if h dividnd a ngaivly olad wih h yl, V, h pi of h ha, i high, in h laim on h fim povid a good hdg again h vaiabiliy of onumpion. 6

63 Lognomal diibuion and homodaiiy haai h xa oluion fo lognomal diibuion of h andom vaiabl If ~ N E x, Exp x xp E x o log x x, hn x Exp x E x x So if i log nomally diibud, ha i, X log i log E Elog log If i homodai, hn Va log Elog Elog ] Va log Elog ] nomally diibud, Logaihmi Vion: aing h log of A E ~ i, Q, w g: E ~ i, Elog Q Log A Va ~ log ~ i, Va Q ov i,;log Q Fo h il a wih Va and ov ;log Q, h log-vion of i: Log Elog Q Va log Q ] Fo iy a, h log-vion of i: Log E ~ i, Va ~ i, ov ~ i,; log Q ] 63

64 Explii oluion fo RS pfn Wih RS pfn, ohai dioun fao i: Q / / β, h. h inv of i h offiin of laiv i avion. h low, h high h i avion. Dfining g a h go gowh a of onumpion, w g in A: g E ~ β E g~ ov ~, ; g~ i i Log giv fo h i-f a: E g Va g, ] iy i f iy Log giv E ] ov, g Fo h pfn, h Eul-uaion giv a lina laion bwn xpd gowh of onumpion and h xpd un of any a: E g Va g ] o E Δ μ E W, Wih μ dmind by vaian and ovaian bwn onumpion and h a un. If boh E W and μ a onan ao im, gowh of onumpion follow a andom wal. 64