Econ 546 Lecture 4. The Basic New Keynesian Model Michael Devereux January 2011

Transcription

1 Econ 546 Lecure 4 The Basic New Keynesian Model Michael Devereux January 20

2 Road map for his lecure We are evenually going o ge 3 equaions, fully describing he NK model The firs wo are jus he same as before: The Euler equaion for oupu (consumpion) growh as a funcion of he ineres rae The ineres rae rule followed by he moneary auhoriy The hird one describes he pah of inflaion, as a funcion of sae of economy To see, look ahead o 3 Equaion New Keynesian.. slide

3 Road map for his lecure The only essenial difference from Lecure is ha prices are slow o adjus When we ge o he end, make sure ha you can show ha if you look a he limi of his model where prices are fully flexible, you resore he model of Lecure Aside from he deails of his Lecure, he main hing is o ge familiar wih he 3 equaion model

4 Basic NK model Exend he previous model in wo ways so as o make i more realisic and useful for policy evaluaion s : Assume firms canno change heir prices freely Menu coss Informaion coss Cusomer resisance Saggered price seing compounds price sickiness 2 nd : Allow households o consume a variey of differen `brands of goods a) a non rivial heory of price seing b) Saggering implies a disribuion of prices across firms

5 Essenial Elemens of model Again, as before, a large number of idenical, compeiive households Households consume a large number of differen varieies of goods Each variey is supplied by a separae firm, who is a monopolis in ha marke Model of Monopolisic Compeiion Perfec compeiion is no a legiimae working assumpion in an economy wih price sickiness

6 Price seing assumpions Assume ha each firm (producing a given brand) can rese is price wih probabiliy θ in a given period No maer how long ago in he pas ha i fixed is price This is unrealisic, bu i faciliaes a very racable model a he aggregae level, because i allows for aggregaion from he micro o he macro `Calvo model of price seing Noe, his is a `ime dependen pricing assumpion Price seing based on dae only Alernaive is `sae dependen pricing Price seing based on sae of he world (e.g. High demand, cos episodes) such models are harder o analyze From las lecure, we know Calvo is lierally incorrec, bu may be a reasonable approximaion

7 Household uiliy funcion Households maximize expeced uiliy E 0 0 U( C, N ) C and N are consumpion and labour supply Use he same funcional forms as before C N E 0 0

8 Produc Differeniaion There is a coninuum of differeniaed goods (on line 0 ) Elasiciy of subsiuion beween each good is ε C C() i di 0 Each good has price Pi ()

9 New household Budge Consrain Household purchases all goods 0 PiC () () idi QB B WNT Given C, we can compue individual demand for each variey (each firm faces elasiciy ε ) Pi () C() i C P Where P is he aggregae price index P P() i di 0

10 I mus be ha Household purchases all goods 0 PiC () () idi PC Sum of expendiure on varieies adds up o oal consumpion expendiure, so a level of aggregae consumpion we ge exacly previous b.c. PC Q B B W N T

11 Tha means ha household s oher opimaliy condiions are same as before Opimal Labour Supply Opimal savings U U N C W P QU E U C C P P

12 So as before, given he funcional form we arrive a Log linear labour supply w p c h Approximae iner emporal Euler equaion c Ec ( i E )

13 Each firm ihas same producion funcion as before Firm producing variey `i : Y() i AN () i Noe his is consan reurns o labour Somewha unrealisic, bu easier han more general case Easier case han in Gali s ex Calvo pricing srucure firm ican revise is price wih probabiliy θ, in each period, no maer how long ago i ses is price Once i re ses is price, i can rese i again in each period in he fuure, wih probabiliy θ

14 Behaviour of he Aggregae Price Level By he law of large numbers Which implies ha *( ) ( ( ) ) P P P * P ( ) P Aggregae inflaion rae is driven by firms re seing prices (log approximaion is): * ( )( p p )

15 Firm s price seing If he firm had fully flexible prices (Ѳ=0), hen i would se he price as a mark up over marginal cos ˆ * () () MC i, A P i This is because monopolis s price is /( /elasiciy) imes MC, where elasiciy is ε, and MC=W/A Then, in log erms W * ˆ ( ) log( ) p i w a

16 When he firm has o se is price (See Gali for deails) Take firm ha ges o se is price in any period. I follows parial adjusmen rule (log) Price is se as a linear funcion of opimal desired price under fully flexible prices, and nex year s opimal price p () i ( ) pˆ () i E p () i * * * * * p () i pˆ () i Ask yourself: why would i no jus se? Make sure you undersand difference beween he ha variable and he ohers

17 Then, noe following seps:. All firms have same opimal price in flexible price environmen: pˆ () i * * 2. Then, each firm seing is new price in Calvo model chooses he same price as any oher firm p ( )(log( ) w a ) Ep * * pˆ 3. Wih previous condiion, his becomes (will show his), * (log( ) w a p) E, ( )( )

18 More seps:. Bu from labour supply and producion funcion: 2. So we ge, w a p ( ) y ( ) a ( ) a log( ) ( ) y E,

19 Las sep: Noe when prices flexible, so fully flexible 0 price oupu is (check wih Lecure ): yˆ lim ( ) ( ) a log( ) So herefore, from previous page, we have ( y yˆ ) E y E Inflaion is a funcion of he oupu gap and expeced fuure inflaion

20 Derive he DIS equaion Subsiue in for c=y y Ey ( i E ) Add and subrac away ŷ and E ŷ + from each side o ge y E y ( i E r ) ˆ Where r is he naural rae of ineres ( ) rˆ E ( yˆ yˆ ) Ea

21 Big difference from previous classical model Oupu is no longer independen of he pah of inflaion Given he gap beween he acual and naural real ineres rae, he pah of he oupu gap is deermined Given he oupu gap, he pah for inflaion is deermined Need an ineres rae policy o complee he model

22 Assume a simple rule which depends only on inflaion The same feedback rule as before (recall i affeced only inflaion and no oupu or he real ineres rae) i In general, we migh expec his would depend on oupu gap as well Combine his wih he NKPC and he DIS equaion o solve for inflaion and he oupu gap

23 3 Equaion New Keynesian Model DIS equaion Taylor Rule (wih added shock o ineres rae) NKPC equaion y Ey ( i E r ) ˆ i v y E

24 Look a i.i.d. shock o a, v, and σ= From DIS equaion From NKPC equaion y 0 ( v 0 a ) y Soluions for oupu gap and inflaion y ( a v ) ( a v)

25 Conclude Shock o a reduces oupu gap oupu falls below he naural rae Because i is deflaionary Shock o ineres rae reduces oupu gap Because i raises he real ineres rae above he naural real ineres rae Clearly, he moneary policy rule is no longer irrelevan for real oucomes