Jang-Ting Guo Lecture 1-1. Introduction and Some Basics. The building blocks of modern macroeconomics are

Transcription

1 Jang-Ting Guo Leure - Inroduion and Some Basis The building bloks of modern maroeonomis are () Solow (Neolassial) growh model Opimal (Ramse) growh model Real business le (RBC) model () Overlapping generaions (OLG) model RBC OLG Finie # Infinie # Infiniel-Lived Finie-Lived CE PO CE PO Saddle Deerminae/Unique Sink Indeerminae/Muliple

2 Main Themes () Maroeonomi models are desribed b nonlinear sohasi differene/differenial equaions () These models an be linearized around a sead sae (3) Examine he resriions ha maroeonomi heories plaes on he behavior of hese equaions, and hen onfron wih he daa Common Charaerisis () Dnami (Ineremporal) () Mirofoundaion (3) General Equilibrium (4) Nonlinear Log-linearizaion (5) Sohasi (6) Raional Expeaions (7) Infinie Horizon Disree/Coninuous Time

3 () Dnami U.S. Real GDP Per Person, () Mirofoundaion (A) Sai Opimizaion (a) over wo onsumpion goods Max s.. u( p, p ) Y MU MU MRS, p p (b) over onsumpion and labor (leisure) Max u(, l) s.. p Wn W(T l) p Wl WT MRS MU MU W p l l, w Sine l T n, MU MU l n MRS MU MU n n, w 3

4 Thousands of 996 dollars Time Real GDP per person Trend in real GDP per person Real GDP Per Person in US,

5 (B) Dnami Opimizaion (a) one seor, wo periods and endowmen eonom Max s.. u( p, p ) p p MRS MU r, MU p p (b) one seor, wo periods and produion eonom Max s.. u( ) βu( k k f (k given Nex, form he Lagrangian ( δ)k ) r ), 0 < β < MPK r k [ f (k ) ( δ)k k ] L u( ) βu( ) λ L L k 0 u'( ) λ 0 λ βλ [f '(k ) δ] u'( ) βu'( )[f '(k ) δ] 4

6 (C) Puing hings ogeher Max u(, l ) βu(, l ), 0 < β < s.. n k f (k l ( δ)k,n T ) and r k k w n given (a) Inra-emporal FOC for labor suppl MRS MU n w n, MU MPN (b) Iner-emporal FOC for onsumpion Consumpion-Euler Equaion u '( ) βu'( )[r δ], r MPK (3) General Equilibrium Compeiive Equilibrium Pareo Opimum Firs Welfare Theorem Seond Welfare Theorem 5

7 (4) Nonlinear Log-linearizaion Consider f (x ) A he sead sae f (x) Linearizaion in levels ( ) f '(x)(x x) Linearizaion in perenage deviaions ( ) (x f '(x) x) x x f '(x)x ŷ xˆ ŷ f '(x)x f (x) xˆ Log-Linearizaion Use log( ẑ ) ẑ and ẑ z z z ŷ log( ) log() and xˆ log(x ) log(x) ŷ f '(x)x f (x) xˆ log( ) log() f '(x)x [log(x ) log(x)] f (x) 6

8 (5) Sohasi Impulses: shoks o suppl or demand Propagaion Mehanism: Exogenous or Endogenous (6) Raional Expeaions General Formulaion e (A) f( ), where is an endogenous variable suh as inflaion rae or prie level e (B) f( x ), where and x are boh endogenous variables, e.g., is onsumpion and x is inome Three Was o Model e e (A) Perfe Foresigh: : (B) Adapive Expeaions e e e λ λ( ( λ) e ), e where 0 < λ < e (C) Raional Expeaions: E[ I ] (7) Infinie Horizon Disree/Coninuous Time 7

9 Jang-Ting Guo Leure - Linear Raional Expeaions Models Consider a salar raional expeaions model ) α E[ Ω] x v ) x x, for all 3) E [ v ] 0, for all s> 0 s We will disuss wo ases in whih he soluion remains bounded: < for all Case : he REGULAR model where 0< α < To solve he model, ierae equaion (i) ino he fuure: 4) x v αe{ x v αe x v x α α s 0 s E [ v ] s [ K x 5) v α

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11 Case : he IRREGULAR model where α > ) α E[ Ω] x v I is no longer possible o solve equaion (i) forward. Bu his does no mean ha here is no raional expeaions soluion. In fa here are man! Consider he following sohasi differene equaion: 6) x v ε α α α ε where is a random variable wih a ime ondiional mean of zero, whih an be inerpreed as self-fulfilling beliefs of agens (sunspos, animal spiris) B onsruion, (6) represens a soluion o equaion (). Noie ha sine α>, he soluion remains bounded Chek: ake ondiional expeaions on boh sides of (6), we obain ha E x v, whih is exal α α α he same as equaion () 3

12 Now, onsider a veor raional expeaions model 7) Y AY U Eigenvalues and Eigenveors n x Ax x R Ax λx λ is a salar A λi 0 or x 0 Use n as an example where A a a a a Charaerisi polnomial ( ) P( λ) λ a a λ ( a a a a ) P( λ) λ ( r A) λ de A 0 0 λ, λ are roos or eigenvalues of he marix A λ λλ λ Trae( A) a a De( A) a a a a λ i is sable if λ < (inside he uni irle) i λ i is unsable if λ (ouside he uni irle) i > 4

13 λ One sable and one unsable eigenvalues 5

14 7) Y AY U where Y R. Also U U wih bounded suppor, and a finie unondiional mean Diagonalizaion of A a a a a M M M M λ 0 0 λ Le Q A (, ) (, )Λ, where Λ λ 0 0 λ, ), and assume ha Q is non-singular ( AQ QΛ A Q Q Λ Y AY QΛQ Y Q Y ΛQ Y 6

15 8) z Le Z Q Y z hen Z ΛZ W where z z λ w z λ z w and W w w Q U i where z evolve independenl, i,. Tha is, we ransform he original veor sohasi differene equaion (7) ino a ssem of wo independen (unoupled) sohasi salar differene equaions. Sabili of he sead sae Le Y* be he sead sae of equaion (7) ) If all eigenvalues of A are inside he uni le, Y * is said o be asmpoiall sable and i is a sink ) If a leas one eigenvalues is ouside he uni le, hen Y * is unsable. If his holds for all eigenvalues, Y* is a soure, oherwise a saddle 3) If no eigenvalue of A is ouside he uni le bu a leas one is on he boundar (has modulus ), hen Y * mabe sable, asmpoiall sable, or unsable. In his ase, bifuraion will arise 7

16 An Example 9) ω, 0 [ ] A E Bx given Le A Q Q Λ Le z Q z φ Q Bx φ u Q u ω 0) z λ E z φ u z λ E z φ u q z q z where Q q q q q marix of he eigenveors.

17 For a unique raional expeaions equilibrium # of roos of A ouside he uni irle equals # of predeermined iniial ondiions In his x ase wih one predeermined variable his ondiion an be wrien as λ < < λ a saddle Deermina Solve he sable roo forward o obain he unique REE s ) z E λ ( φ u ) s s s 0 For muliple raional expeaions equilibria # of roos of A ouside he uni irle greaer han # of predeermined iniial ondiions In his x ase wih one predeermined variable, < λ < λ a sink Indeermina A oninuum of saionar sunspo equilibria 9

18 Definiion: Hperboli Equilibrium Le x be a sead sae of he non-linear dnami ssem x f (x ). x is hperboli if none of he eigenvalues for he Jaobian marix of he parial derivaives Df (x) falls on he uni irle. Tha is, no eigenvalue has modulus exal equal o Theorem: Harmen-Grobman Le x be a hperboli sead sae of he non-linear dnami ssem x f (x ). There is a neighborhood U of x in whih he non-linear dnami ssem is opologiall equivalen o he approximaed linear ssem x x Df (x)(x x) Topologiall equivalen ssems exhibi he same qualiaive dnami properies Loal sabili of a hperboli sead sae x an be deermined b he eigenvalues of he Jaobian marix Df (x), as in page 7 0

19 Jang-Ting Guo Leure -3 Solow and Opimal Growh Models Three Slized Fas abou U.S. Eonomi Growh () The wage share of naional inome has remained onsan over long periods of ime Labor's share The share of wages as a fraion of domesi inome Time Under he neolassial assumpion of onsan reurn-osale, his fa implies ha he produion funion mus be Cobb-Douglas wih elasiiies of apial and labor ha are equal o heir respeive faor shares in naional inome () The raios of onsumpion and invesmen o oupu are approximael onsan in he daa

20 Fraion of gdp Consumpion Invesmen Time Sine oupu, onsumpion and invesmen have all been growing over he pas hundred ears, his fa implies ha oupu, onsumpion and invesmen have grown a equal raes growh is balaned (3) Real GDP and real GDP per apia (sandard of living) have boh grown a roughl a onsan rae in he U.S. over he las hundred ears ( Y / N) Y N ( Y/ N) Y N growh rae of growh rae of growh rae GDP per person GDP of populaion.7% 3.%. 4%

21 0,000 5,000 0,000 (lef sale) Real gdp per apia (.7 % growh) 6,000 4,500 3,000 billions of of 987 dollars 987 dollars 5,000 (righ sale) Real gdp (3. % growh), Time This fa implies ha oupu growh has reahed a sead sae wih a onsan growh rae Combining () and (3) implies ha U.S. oupu, onsumpion and invesmen have grown a he same onsan rae in he las enur The Solow growh model explains hese fas b building hem ino he assumpions of he model: () The onsan of labor share in naional inome is buil ino he assumpion ha oupu is produed wih a Cobb-Douglas produion funion wih he value of labor's and apial s share of /3 and /3, respeivel () The idea of balaned growh is buil ino he assumpion of a onsan savings rae so ha 3

22 S Y s I C and s Y Y (3) The growh of real GDP per apia follows from an exogenous ehnial progress, ha is, from he growh in produivi or he Solow residuals Tehnolog in he Solow Growh Model The produion funion of oupu is Y K α L α S Define effiien unis of labor as follows: α E L α S Hene, he produion funion exhibis CRS in K and E Y K α E α The Solow Growh Model wihou Tehnial Progress Assumpions () In a losed eonom, households save a fixed fraion (s) of heir inome I S () There is no populaion growh and he sae of ehnolog (S) is kep unhanged 4

23 ( δ) α sy s x K E α K K, savings savings x oupu Invesmen rae ( δ) α K K sk E nex period's apial lef over a fraion "s" apial afer subraing of oupu devoed depreiaion o new invesmen α K K K This solid urve is he sum of he wo dashed urves K s K α α E (- δ) K K* s K α α E This dashed urve represens saving (- δ) K This dashed urve represens apial lef over from he previous period K* K 5

24 The Solow Growh Model wih Tehnial Progress Denoe he rae of growh of effiien unis of labor as g. Then he eonom an be desribed b α K ( δ) K sk E E ( g) E I follows ha he sead-sae raio of K/E is given b * K E s g δ Sine E is inreasing in ever period and he fa ha K/E onverges o a onsan implies ha K mus be inreasing in ever period b he same proporion as E. Tha is, growh in K and E are balaned I follows ha oupu and onsumpion per person a he sead sae are given b: α α * Y N K E * α α S and * C N * Y ( s) N Wih exogenous ehnial progress, per-apia oupu, apial and onsumpion all grow forever a he same onsan rae as effiien labor (g) 6

25 Opimal Growh Model ) Max E [ β log ], 0 < β, < 0 0 α ) k ( δ) k sk n α 3) n (Inelasi labor suppl) 4) k k, s s Nex, form he Lagrangian 5) [ ] ( ) α k L β {log λ s k ( δ) k } 0 Firs-order ondiions 6) (i) α βe { [ αs k ( δ)]} α (ii) k ( δ) k sk 7

26 Sead Sae Le s E( s ). The sead sae ( k), is he soluion o he following ssem of equaions β α α sk δ α k ( δ) k sk Now, log-linearize 6) (i) and (ii) around he sead sae 6)(i') a ~ E [ a ~ ~ ak as ~ 3 4 ] ~ ~ 6)(ii') bk bk bs ~ b ~ 3 4 Regarding he ehnolog shok s s s ρ ν, 0 ρ <, s s0, 0 ν i. i. d. random var iable wih E[ ν ] ν V [ ν, ν ], 0< ν < ν <,,,... 8

27 Noie ha X X Expeaions Error, and we an pu hese equaions ogeher as follows: 7) a b b b k s a a a b k s a a a w w w k s ρ ν ~ ~ ~ ~ ~ ~ ~ where w E [x ] x, for x,k,s x. Pre-mulipl (7) b he marix, we obain a b b b ρ 8) given. s~, k ~, w w w ~ B s~ k ~ ~ A s~ k ~ ~ 0 0 s k ν 9

28 Following he proedure we have disussed, 9) ~ ~ k s~ A Bu, A QΛQ Q ΛQ Q Bu Transform (9) ino hree independen salar equaions 0) z Q Λz Φ, Φ Q Bu i i i i for row i z λz φ i 3,,. In his 3x3 ase wih predeermined variables, he ondiion for a unique raional expeaions equilibrium is λ < λ < λ < 3 Sine λ is he sable roo, solve i forward: ) z φ λ { φ λ [ φ

29 Taking ondiional expeaions on boh sides of () leads o z 0, for all. I follows ha 3 ) z q ~ ~ q k q ~ s 0 for all where ( q, q, q 3 ) forms he firs row of Q - Equaion () defines a linear resriion on he veor ha relaes he value of he free variable,, o he values of he predeermined variables, k and s. Tha is, () desribes he sable branh of he saddle pah, whih haraerizes he unique raional expeaions equilibrium k 0