Technical Notes for Discussion of Eggertsson, What Fiscal Policy Is Effective at Zero Interest Rates? Lawrence J. Christiano

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1 Tecnical Noes for Discussion of Eggersson Wa Fiscal Policy Is Effecive a Zero Ineres Raes? Larence J Crisiano Te model a as simulaed for e discussion is presened in e firs secion Te simulaion sraegy is described in e nex secion Houseold Te uiliy funcion of e j ouseold is: Tebudgeconsrainis: U (C j )logc A φ j φ P C B R B W j j Π j ere Π j denoes lump sum profis and lump sum axes Ineremporal Condiion Te discoun rae from o as e folloing represenaion: or β r µ βˆβ dr r ˆβ βdr Te firs order condiion associaed i bonds is: β C ( R ) C π Linearizing around a zero inflaion seady sae: Ĉ C βdr \ ( R ) ˆπ or Ĉ C βdr βdr ˆπ Ĉ Ĉ β (R r )ˆπ

2 Houseold Wage/Employmen Decision Te ouseold selecs e age rae o opimize: " # E ( ) i υ i W ji ji A φ ji φ ere υ i denoes e muliplier on e ouseold budge consrain in e Lagrangian represenaion of is problem Te ouseold reas is objec as an exogenous consan Eac ouseold a opimizes is age cooses e same age rae W : " E ( ) i φ # υ i W i A i ( φ) υ i ere i denoes e level of employmen in period i of a ouseold a opimizes is age in period : Ã! λ W λ i H i W i and H i denoes aggregae employmen Also W denoes e aggregae age rae Imposing e requiremen a e ouseold is alays on e labor demand curve implies: Ã! ³ λ λ W λ (φ) E ( ) i υ i W λ W W i H φ i H i A W i ( φ) υ i Differeniae i respec o W : µ E ( ) i υ i λ λ E µ ( ) i υ i λ λ µ W λ λ λ λ Hi λ A W i λ W λ λ λ λ (φ) µ W i W λ λ (φ) λ λ Hi λ A λ ³ ³ λ λ (φ) W i H φ i υ i λ λ (φ) W i H φ i υ i E ( ) i υ i W λ λ φ µ W i λ λ Hi λ A ³ λ W i λ (φ) H φ i υ i

3 E ( ) i υ i W λ λ φ Ã W λ λ W W i! λ λ H i λ A W λ λ (φ) ³ W W i λ υ i λ (φ) H φ i E Ã ( ) i υ i W W W i! ³ λ λ W λ W i H i λ A λ (φ) H φ i υ i or or E E " ( ) i φ # υ i W i λ A i υ i " W ( ) i υ i φ # ip i λ A i P i P i υ i " W E ( ) i φ # υ i ip i λ A i P i P i υ i Given our uiliy funcion e ave υ i U ci P i P i C i Noe a φ MRSi A i AC i φ P i υ i i ere MRSi denoes e marginal rae of subsiuion beeen consumpion and leisure in period i for a person a reopimizes in period and does no reopimize beeen and i Subsiue is ino e firs order condiion: " # E ( ) i i W λ MRSi C i P i No consider e folloing scaling: W W ½ π i π i i W P i Noe a i is definiion i P P i all i 3

4 Ten e firs order condiion reduces o: E ( ) i i i λ MRSi C i Noe ˆ i ˆπ ˆπ i We are ineresed in e case ere β is ime varying Aloug in e end e ime varying β as no impac on e reduced form age equaion i is useful o esablis is Tus β βˆβ r µ dr r β dr ˆβ βdr We rie ou e firs order condiion like is: β λ MRS C β β ξ λ MRS C β β β ξ λ MRS C β β β β 3 ξ λ MRS3 C 3 Log-linearly expanding is expression and aking ino accoun a e objec in square brackes is zero in seady sae: β b ŵ (λ MRS) \MRS i C β ³ ³ ξ b ŵ ˆ λ MRS MRS \ i C ³ ³ b ŵ ˆ MRS \ i β 3 ξ C β 4 ξ 3 C λ MRS ³ b ŵ 3 ˆ ³ λ MRS MRS \ 3 i 4

5 (noeoeimevaryingβ disappeared) noing λ MRS β b C ŵ \MRS i β ξ C b ŵ ˆ \MRS i β 3 ξ C b ŵ ˆ \MRS i β 4 ξ 3 C b ŵ ˆ 3 \MRS i 3 dividing roug by β : C Ten b ŵ \MRS b ŵ ˆ \MRS β ξ b ŵ ˆ \MRS i β 3 ξ 3 b ŵ ˆ 3 \MRS i i 3 b ŵ \MRS b ŵ ˆπ \MRS β ξ b ŵ ˆπ ˆπ \MRS i β 3 ξ 3 b ŵ ˆπ ˆπ ˆπ 3 \MRS i 3 i and b ŵ \MRS ˆπ \MRS i β ξ ˆπ ˆπ \MRS i β 3 ξ 3 ˆπ ˆπ ˆπ 3 \MRS i 3 5

6 b ŵ ˆπ ( ) ˆπ () MRS \ \MRS ( ) \MRS i Noe \MRS i Ĉ i φĥ i Ĉ i φĥ i φ ³ĥ i Ĥ i Recall i à µ µ µ W W i W W i W! λ λ H i λ λ Hi W W Wi W W i π π i λ λ Hi λ λ Hi for i : so a Ten Subsiuing Ã! λ λ W H ( ) λ λ H i H i ½ ĥ i Ĥi W ³ λ λ π π i i> ( ) λ λ i λ λ (ŵ ˆπ ˆπ i ) i> λ \MRS i Ĉi φĥi φ ³ĥ i Ĥi λ ŵ i Ĉi φĥi φ λ λ (ŵ ˆπ ˆπ i ) 6

7 for i> Ten \MRS \MRS ( ) \MRS Ĉ φĥ φ λ ŵ λ Ĉ φĥ φ λ (ŵ ˆπ ) λ ( ) Ĉ φĥ φ λ (ŵ ˆπ ˆπ ) λ ( ) 3 Ĉ 3 φĥ3 φ λ (ŵ ˆπ ˆπ ˆπ 3 ) λ \MRS \MRS ( ) \MRS i ( ) i Ĉ i φĥ i φ λ ŵ φ λ λ λ ( ) i ˆπ i i Subsiuing e previous expression ino () e conclude a e firs order condiion for ages looks as follos: b ŵ ( ) i ³ Ĉ i φĥ i ( ) i ˆπ i () i φ λ λ ŵ φ λ λ We no deduce e implicaions of e aggregae resricions across ages: ³ λ W ( ξ ) W λ ξ (W ) λ Divide by W and use e scaling noaion: µ ( ξ )( ) λ ξ π λ Log-linearize is abou seady sae: ( ξ ) () λ ŵ ξ λ λ (π ) λ π ˆπ aking ino accoun π : ŵ ξ ξ ˆπ 7 ( ) i ˆπ i i

8 Subsiuing is ino (): ξ ξ ˆπ b ³ ( ) i Ĉ i φĥi ( ) i ˆπ i i φ λ λ ξ ξ ˆπ φ λ λ ( ) i ˆπ i i Muliply by κ ( )( ξ ) ξ κ ξ ξ b ˆπ ξ ξ ( ) i ˆπ i i ³ ( ) i Ĉ i φĥi φ λ λ ˆπ ξ ξ φ λ λ ( ) i ˆπ i i Noe S ( ) i ˆπ i (3) i ˆπ ( ) ˆπ ( ) 3 ˆπ 3 ˆπ ˆπ ( ) ˆπ 3 ˆπ S S ( ) i ˆπ i ˆπ S S o i ³ ( ) i Ĉ i φĥi Ĉ φĥ S o So e expression for e age can be rien Lead and muliply by : ξ ξ b ˆπ ξ S ξ κs o φ λ λ ˆπ ξ ξ φ λ λ S ξ ξ b ˆπ ξ ξ S κ S o φ λ λ ˆπ ξ ξ φ λ λ S 8

9 Subrac e second from e firs and make use of (3) Collecing erms: ξ b ξ Noe ξ ξ b ξ ξ b ˆπ ˆπ ξ ξ ˆπ κ ³Ĉ φĥ φ λ ˆπ φ λ λ λ ˆπ ξ ξ φ λ λ ˆπ ξ ξ b ξ ξ b ˆπ ˆπ ξ ξ ˆπ κ ³Ĉ φĥ φ λ ˆπ φ λ λ ξ λ ˆπ κ ³Ĉ φĥ φ λ λ ˆπ ξ ξ ˆπ b b b ˆπ ˆπ b b ˆπ ˆπ λ φ λ ξ ˆπ so a en ξ b ξ κ ³Ĉ φĥ φ λ λ κb φ λ λ κ ³Ĉ φĥ φ λ ˆπ λ ³ κ b Ĉ φĥ ˆπ ξ ξ b ˆπ ˆπ ξ ξ ˆπ λ φ λ ξ ˆπ λ φ λ ξ ˆπ ξ ξ ˆπ φ λ λ ξ ˆπ 9

10 or No divide by e erm on ˆπ : φ λ ˆπ λ ³ µ κ b Ĉ φĥ φ λ ξ λ φ λ ˆπ λ ³ κ b Ĉ φĥ ˆπ κ φ λ λ 3 Goods Producion and Price Seing ξ ˆπ µ φ λ βˆπ λ ³ b Ĉ φĥ βˆπ Suppose a a final good Y is produced using a coninuum of inpus as follos: Z λ Y Y f diλf i λ f < (4) Te good is produced by a compeiive represenaive firm ic akes e price of oupu P and e price of inpus P i as given Te firs order necessary condiion associaed i opimizaion is: µ λ f P λ f Y Y i (5) A useful resul is obained by subsiuing ou for Y i in (4) from (5): P i Z P (λ f ) (P i ) λ f di (6) Eac inermediae good is produced by a monopolis using e folloing producion funcion: Y i A i Te equilibrium condiion associaed i price seing is afer log-linearizing abou seady sae: p ξp ˆπ βˆπ ŝ ξ p Marginal cos in is model is s W P

11 so a ŝ b Te resource consrain is: C p H ere p denoes e Tak Yun disorion ic is uniy o a firs order approximaion 4 Equilibrium Condiions Te sysem as 6 unknons: ˆπ b Ĥ ˆπ R Z and e folloing equaions Te equaions a caracerize e privae economy are: b b ˆπ ˆπ p ξp ˆπ βˆπ b ξ p κ ˆπ b φ λ λ dτ τ Ĥ Ĥ β (dz dr )ˆπ ( φ) Ĥ βˆπ and moneary policy: dz ρ R dr ( ρ R ) i r πˆπ r y Ĥ β ³ dz dz β dr zero bound no binding ³ oerise zero bound binding β Te laer capures e fac a R ic means R R /β We rie is in marix form as follos Supppose e zero bound is no binding so a dr dz (7)

12 Tisgivesussixequaionsinesixunknons: dr β ˆπ ( p)( ξ p ) ξ p β Ĥ b κ(φ) κ φ λ φ λ λ λ ˆπ β β ( ρ dz R ) r β y ( ρ R ) r β π dr ˆπ µ Ĥ b dr κ φ λ τ λ dτ ρ R ˆπ β dz α z α z α z β s β s ere e definiions of e marices are obvious and Le dr ˆπ µ z Ĥ dr b s dτ ˆπ dz ere Te linearized sysem en e zero bound is binding is as follos: d α z α z α z β s β s d β and α is α i is 6 6 elemen replaced by zero Tis sysem is simply e previous one i (7) replaced by: µ dr β

13 Simulaing e Model Te general algorim appears in e ird subsecion belo I capures e feaure of our seing a e equaions a caracerize equilibrium cange during e simulaion Before e discuss e algorim in is full generaliy e provide o examples o illusrae feaures of e algorim no relaed o e equaion sicing Simple Example We use a slig perurbaion on a sandard sooing algorim Te perurbaion is designed so a e algorim is required o i a specific argeaaspecific dae as opposed o e usual sooing in ic a arge is reaced asympoically Because e general algorim involves oer complicaions i is useful o poin ou e perurbaion a e use in isolaion from e oer complicaions Suppose a e sysem obeys e folloing scalar difference equaion: α z α z α z d for T Here z is given For T Wriing e equaions ou explicily: z Az α z α z α z d α z T α z T α z T d T Given α 6 and given an arbirary z R e can use ese equaions o compue z z T Bu i as o be e case a z T Az T So e algorim is o adjus z unil e above equaion is saisfied Algorim Based on QZ Decomposiion Te problem e ave o confron in applying e simple algorim in e previous secion is a α is no inverible One ay o adap e algorim applies e QZ decomposiion Tus le Muliply (??) byq : Qα Z H Q α Z H Z z γ Qα ZZ z Q α ZZ z Qα ZZ z Qd Qβ s Qβ s 3

14 Summarizing for T Wrie H G H H D Q [d α z ] z } { H γ H γ Qα Zγ Q[d β s β s ] H H γ H γ D G H H d γ µ γ γ Z z µ L z L z ere H and H are upper riangular and e diagonal elemens of G are non-zero ile e l diagonal elemens of H are all zero I is necessary o verify numerically a all e elemens of H are zero We assume a e diagonal elemens of H are all non-zero Also d d β s β s Ten e sysem is rien G H γ γ G H γ 3 γ 3 G H γ T γ T G H γ T γ T G H H G H H G H H G H H γ γ γ γ γ T γ T γ T γ T D D D D D T D T D T D T ereeaveimposedh We ave found a numerically is is a propery of our model To simulae is sysem forard noe firs a D is deermined because γ Fix avalueforγ and compue: γ H D For : For T : D Q (d Qα Zγ ) γ H D γ G H γ G γ H γ D D Q (d α Zγ ) γ H D γ G H γ G γ H γ D 4

15 We no ave γ T γ T D T Recall a in period T so a afer muliplying by Z : z T Az T γ T Ãγ T Ã Z AZ µ γ T γ T Ã Ã γ T We mus sill saisfy e T equilibrium condiions: G H γ T G H γ γ T T H γ T D T D T Noe oever a e boom se of equaions are saisfied because of e ay γ T as cosen and because γ T does no ener ese equaions Te firs se of equaions need no be saisfied oever and so e use e requiremen a ese be saisfied o pin don γ In paricular e adjus γ unil e folloing expression is saisfied: G γ T H γ T G γ T H γ T D T Noe a is is a number of equaions equal o e dimension of γ 3 Exending e Algorim We no address e possibiliy a and T Ta is e loer bound sars o bind in some period afer e discoun rae goes negaive and before i urns posiive again Tus e loer bound is no binding for i is binding for and i is no binding for > Because e assume T e can apply a sraigforard adapaion of e algorim in e previous secion Firs e d sequence needs o be adjused so a e consan vec d in (??) is only urned on for Second e require e QZ decomposiion of e sysem bo for e ime en e loer bound is binding and e ime en i is binding: Qα Z H Qα Z H Qα Z H Q α Z H γ Z z γ Z z 3Te Iniial Non-Binding Regime For : α z α z [α z β s β s ] 5

16 To solve ese equaions proceed as before Afer applying e QZ decomposiion: G H γ G H γ γ D H γ D G H γ 3 G H γ γ D 3 H γ D G H γ G H γ γ H D γ D G H γ G H γ γ H D γ D Te basic idea of e simulaion is a e sar a given period i (γ γ ) and use e period equaion o solve forard o obain γ γ Tis ould be compleely sraigforard and sandard ifeleadmarixineqzdecomposiion ofedynamic equaion ere inverible I is no So o do e simulaion e compue γ using e equaion Ten ere is only γ o compue using e period equaion Tis compuaion is possible because e relevan block in e lead marix is inverible To begin e simulaion noe firs a D is deermined because z Fix a value for γ and compue: For : γ H D D Q (d α Zγ ) (consan erm in period equaion) γ H D (solving period equaion for γ ) γ G H γ G γ H γ D (using period equaion o find γ ) Weproceedinisayineacperiod : D Q (d α Zγ ) γ H D γ G H γ G γ H γ D Period requires special reamen because e subsequen period s equaion belongs o a differen regime We firs solve for γ using e equaion Te consan erm in e equaion is: µ µ γ D Q d α Z γ Noe a e Q ere belongs o e binding regime ile Z belongs o e non-binding regime We require e Z from e non-binding regime because a is needed o consruc e consan erm is z andaeacuallyaveinandaispoinisγ Z z Wi D in and e compue γ : ³ γ H D 6

17 We no reurn o e equaion o seek γ Te equaion is: α z α z [α z β s β s ] α Z γ α z [α z β s β s ] No apply e QZ decomposiion relevan for e non-binding regime: Qα ZZ Z γ Qα ZZ z D D [α z β s β s ] Le H Z Z γ H γ D Z Z M M {z} (m l) (m l) M {z} l (m l) M {z} (m l) l M {z} l l ere m denoes e leng of z and m l is e rank of α Ten e previous sysem can be rien: G H {z} M {z} M {z} γ {z} (m l) (m l) (m l) l (m l) (m l) (m l) l {z} (m l) {z} {z} M {z} M {z} γ {z} G H γ H D γ D l (m l) l l l (m l) l l l G M H M G M H M γ γ G H H γ γ D D We are no in a posiion o solve for γ Wriing ou e firs of e above equaions: G M H M γ G M H M γ G γ H γ D so a γ G M H M G M H M γ G γ H γ D 3Te Binding Regime We no urn o e period en e loer bound consrain on e ineres rae is binding Teequaionobesolvedis: α z α z [α z d β s β s ] Muliply by Q H γ H γ D 7

18 G H γ γ G H γ γ G H γ γ G H γ γ G H H G H H G H H G H H γ γ γ γ γ 3 γ 3 γ γ D D D D D 3 D 3 D D We ave γ in and Consider firs Asbeforeemuscompue γ using e period equaion Tus D Q ³d α Z γ γ ³ H D Ten using e period equaion: γ G H γ G γ H γ D i We proceed in is ay in eac period : D Q ³d α Z γ γ γ ³ H G D H γ G γ H γ D i Te equaion requires special adjusmens analogous o e ones used a e end of e non-binding regime because solving e equaion requires orking i e equaion firs Te consan erm in e equaion is: µ µ γ D Q d α Z γ As before Q is par of e QZ decomposiion relevan o e non-binding regime bu Z belongs o e binding regime because e ave γ in and and is mus be convered o z Wi D in and e compue γ as follos: γ H D We no reurn o e equaion: α z α Z γ α Z γ d β s β s i 8

19 afer muliplying by Q : H Z Zγ H γ D ere D is available from e previous compuaions Wriing is ou more carefully G H M M γ G M M γ H γ D H γ D ere M M M Z M Z ( M ) M Wriing ou e firs of e above equaions: G M H M i γ G M H M i γ G γ H γ D γ G M H M i ³ G M H M γ G γ H γ D i 33Te Final Non-Binding Regime Given γ γ e no solve e equaions in e non-binding regime T Consider period firs: α z α z [α z β s β s ] Muliplying by Q and applying e QZ decomposiion: Qα ZZ z Qα ZZ z Q [α z β s β s ] H γ H γ D G H γ γ G H H γ γ D D To solve for γ γ e firs obain γ using e equaion: D Q d α Zγ (consan erm in period equaion) γ H D (solving period equaion for γ ) Ten γ G H γ G γ H γ D For T : D Q (d α Zγ ) γ H D γ G H γ G γ H γ D 9

20 We no ave γ T γ T D T Recall a s for T so a afer muliplying by Z : z T Az T γ T Ãγ T Ã Z AZ µ γ T γ T Ã Ã γ T We mus sill saisfy e T equilibrium condiions: G H γ T G H γ γ T T H γ T D T D T Noe oever a e boom se of equaions are saisfied because of e ay γ T as cosen and because γ T does no ener ese equaions Te firs se of equaions need nobesaisfied oever and so e use e requiremen a ese be saisfied o pin don γ T In paricular e adjus γ unil e folloing expression is saisfied: G γ T H γ T G γ T H γ T D T Noe a is is a number of equaions equal o e dimension of γ