A dynamic AS-AD Model

Transcription

1 A ynamic AS-AD Moel (Lecure Noes, Thomas Seger, Universiy of Leipzig, winer erm 10/11) This file escribes a ynamic AS-AD moel. The moel can be employe o assess he ynamic consequences of macroeconomic shocks (fiscal policy shocks, moneary policy shocks, an supply shocks). This AS-AD moel can be viewe as a simple business cycle moel. I can explain recurren business cycle flucions in response o ongoing shocks. (Exension: reformulae he moel by using a Taylor rule o escribe moneary policy.) Reference: Shone, R., Economic Dynamics, CUP, 2002, Chaper 11. The moel ü Aggregae eman (IS-LM moel) The goos marke (IS-curve) C C 0 b 1 I I 0 h i e C I G real ineres rae (1) (2) (3) Noaion: C: real consumpion, : real income (oupu), I: real invesmen, i: nominal ineres rae, : expece inflaion rae,... The money marke (LM-curve) m k i m s m p m s m M e k e i M s M P (4) (5) (6) Noaion: m : (naural) logarihm of real money eman, m s : logarihm of real money supply, p: logarihm of he price level,... Solving for gives (he soluion for i is suppresse) C 0 I 0 G h m p h e 1 b 1 hk (7) Defining a 0 : C 0I 0 G 1b 1 hk, a 1 := h 1-b1-+ hk, an a 2 : a 0 a 1 m p a 2 e, a 0,a 1,a 2 0 This equaion represens an aggregae eman (AD) curve. h 1b 1 hk his equaion can be wrien as (8)

2 2 Dynamic_AS_AD_Moel.nb ü Aggregae supply Aggregae supply is escribe by n e, 0 (9) Noaion: p: inflaion rae, : expece inflaion rae, : real oupu, n : real poenial oupu. Noice ha his AS-curve resuls from an augmene Phillips curve, p= - au - U n, ogeher wih Okun s law, U - U n =-b - n, where boh a, b > 0. Alernaively, he following version of he Lucas supply curve = n + hp - P e is also compaible wih his AS-curve. Subracing an aing h P -1 on he RHS gives = n +hp - P -1 - P e - P -1. If P enoes he logarihm of he price level, one can wrie = n +hp -. ü Expece inflaion We assume ha expecaions are forme accoring o an aapive expecaions scheme e e, 0 (10) When he acual rae of inflaion excees he expece rae (p- > 0), expecaions are revise upwars (p e > 0), an vice versa. ü Summarizing: he complee moel a 0 a 1 m p a 2 e (11) n e (12) (13) e e Moel analysis ü Preliminaries: eliminaing one equaion We firs ake he ime erivaive of he AD-equaion o yiel a1 m a 2 e (14) Nex rewrie he AS-equaion as p- =a - n an plug he righ-han sie (RHS) ino he expece inflaion scheule p e =bp-, which gives e n (15) Finally, we plug he RHS of he AS-curve, equ. (12), an he RHS of equ. (15) ino equ. (14). a1 m n e a 2 n a1 m a 1 n a 1 e a 2 n a1 m a 1 a 2 n a 1 e e (16) (17) (18) ü The reuce-form moel: wo ifferenial equaions The reuce-form moel is escribe by (plus wo bounary coniions)

3 Dynamic_AS_AD_Moel.nb 3 e n a1 m a 1 a 2 n a 1 e This sysem can be solve for he seay sae ( e, è ), efine by p e = 0 an = 0. The seay sae reas as follows e m n implying : m (19) (20) (21) (22) To illusrae, below we plo he wo curves p e = 0 an = 0 in (,)-plane. Numerical moel evaluaion ü The seay sae In[1]:= In[2]:= In[3]:= In[4]:= paraminiial C0 10, I0 5, G 5, b 0.8, 0.3, h 0.1, k 0.05, 0.05, 0.1, 1, mo 0.01, n 1; paramfinal C0 10, I0 5, G 5, b 0.8, 0.3, h 0.1, k 0.05, 0.05, 0.1, 1, mo 0.01, n 1; a0, a1, a2 C0 I0 G 1 b 1 hk, h 1 b 1 hk, h 1 b 1 hk AS GraphicsLinen. paraminiial, 0, n. paraminiial, 0.15, AxesLabel, e, PloRange 0, 0.06; a1 a2 AD Plomo n. paraminiial,, 0.5, 1.5, a1 AxesLabel, e, PloRange 0, 0.06; p1 ShowAS, AD, Axes True, AspecRaio 2 3; ; The seay sae (long run equilibrium) is characerize by e m an è = n! ü Dynamic responses A firs, we express he ifferenial equaions in Mahemaica noaion In[7]:= ee e' n; e ' a1 mo a1 a2 n a1 e; In[9]:= In[10]:= esol, sol mo, n. paraminiial; ns NDSolveee, e, e0 1.1 esol, sol. paramfinal, e,,, 0, 100; In[11]:= en_ : Evaluaens1, 1, 2 nn_ : Evaluaens1, 2, 2 In[13]:= _ : nn n en. paramfinal his efines he acual inflaion rae In[14]:= Nees"PloLegens`"

4 4 Dynamic_AS_AD_Moel.nb In[22]:= Ploen,, mo. paramfinal,, 0, 30, AxesLabel, " e blue,re", PloRange 0,, AxesOrigin 0, 0 Plonn, 1,1,, 0, 30, AxesLabel,, PloRange 0.93, 1.1, AxesOrigin 0, 0.93 blue,pre Ou[22]= Ou[23]= In[17]:= AS GraphicsLinen. paramfinal, 0, n. paramfinal, 0.15, AxesLabel, e, PloRange 0, 0.03; a1 a2 AD Plomo n. paramfinal,, 0.5, 1.5, a1 AxesLabel, e, PloRange 0, 0.03; p2 ShowAS, AD, Axes True, AspecRaio 2 3;

5 Dynamic_AS_AD_Moel.nb 5 In[20]:= ra ParamericPlonn, en,, 0, 100, AxesLabel, e, PloRange 0.9, 1.13, 0, 0.03, AspecRaio 2 3; Show ra, p1, p Ou[21]= ü How o moel shocks (1) Moneary policy shock: change m (Ø se of parameers) (2) Fiscal policy shock (permanen increase in G): change iniial coniions (-imension: = a 2 + a 1 m - p + a 0 ; - imension: = cons.) (Ø iniial coniions; increase G in he se of parameers) (2) Supply sie shock: change n (Ø se of parameers) Resuls ü Moneary expansion (coe)

6 6 Dynamic_AS_AD_Moel.nb ü Graphical oupu In[257]:= Showra, p1, p2 imepah imepah Ou[257]= blue,pre Ou[258]= Ou[259]=

7 Dynamic_AS_AD_Moel.nb 7 ü Fiscal expansion (coe) ü Graphical oupu In[432]:= Showra, p1, p2 imepah imepah Ou[432]= blue,pre Ou[433]= Ou[434]=

8 8 Dynamic_AS_AD_Moel.nb ü Negaive supply shock (coe) ü Graphical oupu In[409]:= Showra, p1, p2 imepah imepah Ou[409]= blue,pre Ou[410]= Ou[411]=

9 Dynamic_AS_AD_Moel.nb 9 Phase iagram an economic inuiion The mechanics of he moel can be unersoo by iscussing he phase iagramm, which shows he wo curves p e = 0 an = 0 in (,)-plane. In aiion, he arrows inicae he irecion he economy moves o given a posiion in ( e,)-plane p e = II 0.04 I III 0.01 IV = Consier he following -equaion an he p e -equaion a1 m n e change in AD ue o changes in MP e n n a 2 e change in AD ue o changes in real ineres rae (23) (24) Commens 1. Above (o he righ of) he = 0 locus, we have < 0, i.e. a 1 m -p + a 2 p e < 0. To illusrae, sar on he = 0 locus an increase, holing fix. This ecreases he RHS of (23), via p, by more han i increases he RHS of (23), via p e, provie ha a 1 a>baa 2! Consequenly, falls. Alernaively, increasing (saring on he = 0 locus an holing cons.) increases p an hus reuces AD. 2. To he righ of he p e = 0 locus, - n > 0 implying ha p e > 0; recall: p e =ab - n. An alernaive moel srucure