Lecture 1 The New Keynesian Model of Monetary Policy. Lecturer Campbell Leith University of Glasgow

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1 Lecure The New Keynesan Model of Moneary Polcy Lecurer Campbell Leh Unversy of Glasgow

2 The New Keynesan model of moneary polcy s becomng ncreasngly sandard n he analyss of moneary polcy. Ths parcular reamen follows Carl Walsh (23), Moneary Theory and Polcy, chaper 5. Anoher good reference s by Rchard Clarda, Jord Galí, and Mark Gerler, The Scence of Moneary Polcy: A New Keynesan Perspecve, Journal of Economc Leraure Vol. XXXVII (December 999), pp Key Feaures: Households consume a baske of goods and supply labour o mperfecly compeve frms. Frms only change prces afer a random nerval of me (.e. prces are scky). Snce prces are scky moneary polcy can have real effecs n he shor-run.

3 Problems we need o analyse: Households Problems: ()allocaon of spendng across goods, and (2)allocaon of spendng across me. Frms Prcng/Producon decson.

4 Households: The uly funcon of he represenave household s gven by, b σ + η C γ M N E β + χ = σ b P+ + η () Where C + s a baske of goods, M/P are real money balances and N s labour supply. The consumpon baske s defned n he followng CES form, θ θ θ θ j C = c dj (2) where θ s he elascy of demand for he ndvdual goods and θ >. Problem - The opmal allocaon of a gven consumpon expendure across he ndvdual goods n he consumpon baske.

5 Ths nal problem amouns o mnmzng he cos of buyng C, mn p c dj j j c (3) j subjec o, θ θ θ θ j C = c dj (4) Form he Lagrangan, L p c dj C c dj θ θ θ θ = j j + ψ j (5) The frs order condon wh respec o good j s, θ θ L θ θ = p j ψ c j dj c j = c j (6) Usng he defnon of he consumpon baske, Rearrangng, p j C θ θ ψ cj = (7) c j p j = ψ θ C (8)

6 From he defnon of he compose level of consumpon hs mples, θ θ θ θ θ θ θ p θ j θ = j ψ = ψ (9) C C dj p dj C Solvng for he lagrange mulpler, ψ θ θ = p j dj P () The lagrange mulpler can be consdered o be he prce ndex approprae for he consumpon bundle. Subsung hs back no he frs order condon, (8) yelds, c j p j = P θ C () As θ we move owards perfec compeon and frms enjoy less marke power. Ths equaon s effecvely he demand curve facng he frm j for s produc.

7 Problem 2 - The Household s Ineremporal Problem: Before maxmzng uly we need o consder he households budge consran. Ths s gven, n nomnal erms by, PC + M + B = W N + M + ( + R ) B + (2) Where Π are he profs from he mperfecly compeve frms redsrbued o households. Dvdng by he prce level P, we can rewre hs n real erms as, M B W M B C + + = N + + ( + R ) + (3) P P P P P P Therefore, formng he Lagrangan for he problem, E = b σ + η C γ M N β + χ σ b P+ + η M B W M B + λ C + + N ( + R ) P+ P+ P+ P+ P+ P+ (4)

8 The frs order condons for consumpon are gven by, Money, β σ ( ) λ C = (5) Labour Supply, b M + P+ γβ + λ+ λ+ + = P+ P+ + W N = η + βχ + λ + P+ (6) (7) Bonds, P+ λ+ λ + + ( + R+ ) = P ++ (8) Usng he equaon of moon for he Lagrange mulpler we can oban he Euler equaon for consumpon, Money, P C = β ( + R ) E C σ σ + P + (9) Labour Supply, M γ P χ N b η = C = C σ σ R + R W P (2) (2)

9 Frms: Frms are prof maxmsers, bu hey face hree consrans. Frsly hey mus work wh a gven producon echnology gven by, c = Z N (22) j j whch s lnear n labour npu (here s no capal) and an aggregae producvy dsurbance Z. The expeced value of Z s. Secondly, he face he downward slopng demand curve gven by, c j p C (23) j = P θ Fnally, we are gong o adop nomnal nera n he form of Calvo (983) conracs. In any perod ( ω) of frms are randomly chosen o be able o change her prce. Therefore n seng prces, frms mus ake accoun of fuure economc condons snce he prce hey se oday may sll be n place omorrow.

10 Snce labour s he only npu n he producve process he real margnal cos of producon s gven by, MC W / P = Z (24) The frm s prcng problem hen becomes, θ θ pj pj E ω, + MC+ C+ = P+ P + (25) Ths problem s he same for all frms able o change her prces n perod. The frs order condon for he opmal prce, p* s gven by, θ θ * * p p E ω, + ( θ) θmc+ C * + = = P+ p P + (26)

11 Flexble Prce Equlbrum: I s helpful o examne he equlbrum when prces are flexble. When frms are able o adjus her prces every perod hen hs reduces o, * p θ = MC P θ (27) θ Where θ reflecs he markup of prces over margnal coss due o he fac ha frms enjoy marke power. Snce, when prces are flexble, all frms wll be seng he same prce here are no relave prce dfferences and usng he defnon of margnal cos hs can be wren as, θ W / P = θ Z (28)

12 Usng he labour supply condon on he par of he household we oban, W Z χ N = = P θ /( θ ) C η σ (29) We now need o consder how o log-lnearse. Take naural logarhms of boh sdes of equaon (29) o oban, Z χ N ln = ln θ /( θ ) C η σ (3) Takng he oal dervave and evaluang a he seadysae yelds, dz = ηdn + σdc Z N C (3) Defnng x ˆ = dx x as beng he percenage devaon of varable x from s seady-sae value allows us o wre hs condon as, ˆ f ˆ f ˆ f Z = ηn + σc (32)

13 where he f superscrp denoes he fac ha we are currenly consderng he flex prce equlbrum. Ths s he loglnearsed verson of (29). Dong he same o he producon funcon yelds, yˆ f ˆ f ˆ f = N + Z (33) Whch snce here s no governmen spendng n he curren model (so ha y ˆ o yeld, yˆ f f c ˆ f = ) we can combne hese + η = Zˆ σ + η f (34) Ths descrbes he varaons n oupu ha emerge due o producvy shocks when prces are flexble. Snce hs reflecs opmal prvae secor responses o producvy shocks, here s no need for polcy o aemp o offse hese oupu flucuaons.

14 Now we reurn o consder he scky prce case. The prce ndex evolves accordng o, ( ) θ ( ) * θ θ ω ω P = p + P (35) Inflaon s hen deermned by hs defnon and he expresson for he opmal prce, θ P + E * ω, + MC+ p θ = P = θ P θ P + E ω, + = P Log-lnearsng he frs equaon yelds, (36) * Pˆ ( ) ˆ ˆ = ω p + ωp (37) Log-lnearsng he second gves, ( ˆ ˆ + + ) * pˆ = E ωβ MC + P (38)] = Ths can be quas-dfferenced o yeld a forward-lookng dfference equaon n he opmal rese prce, * * pˆ ˆ ˆ ˆ = ωβ Ep + + MC + P (39)

15 Combnng he wo equaons gves, Pˆ ˆ ω ˆ P + ω P ˆ ˆ ˆ = ωβ E ωβ P + MC + P ω ω ω ω (4) Solvng for nflaon yelds, where κ = π ˆ = βeπ+ + κmc (4) ( ω)( ωβ ) ω Ths s he New Keynesan Phllps curve embedded n a General Equlbrum model.

16 However, s helpful o manpulae hs fuher. Recall ha margnal cos s gven by, MC W / P = Z (42) Log-lnearsng yelds, MC ˆ ˆ ˆ ˆ = W P Z (43) Usng he labour supply condon(3) gves, MC ˆ = Wˆ Pˆ ( yˆ Nˆ ) + η = ( σ + η) yˆ ˆ Z σ + η (44) Bu from he defnon of he flex prce oupu, hs can be furher rewren as, ˆ f MC ( )( ˆ ˆ = σ + η y y ) (45) Therefore, he NKPC can be re-wren as, f π ˆ ˆ = βeπ+ + κ( y y ) (46) ( ω)( ωβ ) where κ = ( σ + η) κ = ( σ + η) ω and he forcng varable s he oupu gap. Ths measures he exen o whch acual oupu s dfferen from he level ha would occur under flexble prces.

17 The General Equlbrum: Oupu s governed by he Euler equaon n consumpon, yˆ ˆ ˆ = Ey+ ( R Eπ + ) σ (47) Whch can also be rewren n erms of he oupu gap, where + x ˆ = Ex+ ( R E+ π + ) + u (48) σ f f u ˆ ˆ = Ey y whch only depends upon exogenous producvy dsurbances, and x ˆ ˆ f = y y.

18 The New Keynesan Model for Moneary Polcy Analyss: Therefore he dynamc model consss of a descrpon of AD, And AS, x ˆ = Ex+ ( R E+ π + ) + u σ (49) π = βeπ+ + κx (5) and s compleed by a descrpon of moneary polcy (whch de scrbes he seng of nomnal neres raes, R eher drecly or ndrecly hrough conrol of moneary aggregaes).

19 Lecure 2 - Polcy Analyss n he New Keynesan Model: The Moneary Polcy Transmsson Mechansm In hs subsecon we smulae our basc New Keynesan model n order o undersand he basc moneary polcy ransmsson mechansm. Wh moneary polcy followng a smple rule, R = δπ + v (5) ˆ and he polcy shock followng an auoregressve process v =.5v + ε We adop he parameers n Walsh Chaper 5. β =.99, σ = η =, δ =.5 and ω =.8.

20 Inf R x Fg Auoregressve Shock Inf R x Fg 2 d Wh no auoregressve aspec o he polcy shock he movemens n varables are nsananeous.

21 Inf R x Wh relavely less nomnal nera, ω =.6. Now oupu response s less, bu nflaon response s greaer.

22 Consder moneary polcy conduced usng moneary aggregaes Mˆ.5 ˆ = M + ε (52) Ineres raes are hen deermned by he loglnearsed money demand equaon, ˆ ˆ bm ( ) ˆ ˆ P = σ y R + R (53) The mpulse responses o a shock o he money sock are gven below (assumng b= from Walsh page 58), Inf R x M

23 Polcy Objecves: Walsh consders he welfare of our represenave household can be wren as, V = U( Y, Z ) v( y ( ), Z ) d (54) where he frs erm represens he nsananeous uly from consumng he consumpon baske (gven he level of he producvy shock) and he second erm capures he dsuly of supplyng he varous goods n he economy (.e. he dsuly of labour effor s proporonal o oupu).

24 Walsh hen follows Woodford (23) o approxmae he represenave household s uly by he followng quadrac loss funcon, where 2 * 2 β + Ω β π+ + λ( + ) = = (55) E V E x x and ω 2 Ω= YU c ( θ ηθ ) 2 + ( ω)( ωβ) ( ω)( ωβ ) σ + η λ = ω ( + ηθ ) θ (56) (57) We wll no formally derve hs (see Woodford(23) or Walsh(23)) bu s useful o oban some nuon for hs specfcaon. Recall ha x s he gap beween oupu and he oupu level ha would emerge under flexble prces, and x* s he gap beween he seady-sae effcen level of oupu and he acual seady-sae level of oupu.

25 Alhough hs looks lke a sandard quadrac loss funcon here are wo crucal dfferences..the oupu gap s measured relave o equlbrum under flexble prces raher han he smple seady-sae oupu/rend oupu level of oupu. In oher words he flex prce equlbrum ncorporaes he opmal consumpon/lesure and labour supply responses o producvy shocks. 2.The reason for ncludng nflaon s now clear. Scky prces lead o a dsperson of prces and herefore oupu across frms. Ths has wo coss for economc agens: ()Because of dmnshng margnal uly prce dsperson has a drec uly cos (he uly ganed from consumng more of he cheaper goods s less han he uly los from consumng less of he expensve goods).

26 (2)Addonally, he cos of producng more of he cheap goods s also ypcally more han he reduced coss of producng less of he expensve goods (due o dmnshng margnal produc n producon, or dmnshng margnal uly of lesure f consumers/workers are aached o specfc frms). Many auhors assume ha x*= and acheve hs by adopng some knd of fscal subsdy o ensure ha seady-sae oupu s a s effcen level and he dsoron due o mperfec compeon has been elmnaed. In hs case he cenral bank s loss funcon becomes, E 2 2 β π+ + λ( x+ ) = (58) In hs case we have no nflaonary bas, bu we do have a sablzaon bas whch we llusrae below.

27 Opmal Polcy Under Commmen Frsly we consder opmal polcy under commmen. The bank has o mnmze hs loss funcon subjec o he srucural model of he economy, Therefore he dynamc model consss of a descrpon of AD, And AS, x ˆ = Ex+ ( R E+ π + ) + u σ (59) π = βeπ+ + κx (6) Form he Lagrangan, 2 2 { ( ) ˆ E β π+ + λ x + + θ+ x+ x+ + + ( R+ π+ + ) u+ = σ + ψ π βπ κ } [ x e ] (6)

28 The frs order condon n respec of he neres rae s gven by Eθ σ In oher words, + = (62) Eθ + = for >=.e. he lagrange mulpler for he Euler equaon s zero snce does no mpose any real consran on moneary polcy. Ths mples ha he polcy could have been se up as f he cenral bank conrolled he oupu gap raher han he neres rae (see for example Clarda e al, 999). Usng hs condon, he remanng frs-order condons are, for nflaon a me, π + ψ = (63) and for he nflaon n subsequen perods, and he oupu gap, E( π + + ψ+ ψ+ ) = (64) E ( λx + κψ + ) = (65)

29 The poenal me-nconssency of polcy s clear, snce n perod he cenral bank would se nflaon equal o π = ψ and promse o se + + π = ( ψ ψ ). However when perod + arrves he bank would wsh o se π + = ψ +.

30 Tmelessly Opmal Polcy Woodford suggess an alernave meless perspecve where he cenral bank mplemens (64) and (65) even n he nal perod- he bank behaves as f he polcy had always been n place. Dong hs, combnng he wo condons yelds, π λ = κ ( x x ) Summng over he nfne horzon yelds, whch mples, λ κ (66) π + = ( x+ x+ ) (67) = = p p λ = x x κ ( ) (68) In oher words, snce he oupu gap mus evenually be elmnaed, x =, and we can assume he nal value of he oupu gap before a shock h was also zero, x =, hen hs means ha commmen polcy wll ensure, p = p (69).e. commmen polcy wll reurn he prce level o s nal value followng nflaonary shocks.

31 Polcy Under Dscreon: When he cenral bank operaes under dscreon akes nflaon expecaons as gven (snce canno nfluence hem as can under commmen). Therefore has an essenally sac problem o mnmze, π λx + (7) 2 2 subjec o, π = βeπ+ + κx + e (7) whch gves, κπ + λx = (72) Noe ha hs condon s he same as he nal condon a me under commmen (e. In he frs perod of commmen he cenral bank canno affec nal expecaons and so mplemens he dscreonary soluon). There s no promse on he par of he moneary auhores o reurn he prce level o s nal value under dscreonary polcy.

32

33 Inflaon response under precommmen less nfc nfd The Prce Level under commmen Pc Pc

34 xc xd Oupu fall less under commmen, bu more susaned. Inflaon pahs wh correlaed shock wh coeffcen of nfc nfd

35 Agan, oupu cos of sablzng economy nally less under commmen, bu more prolonged xc xd There s a clear welfare mprovemen under commmen gven naure of loss funcon ( λ = for smplcy).

36 Prce Level vs Inflaon Targeng We saw ha he precommmen polcy nvolved sablzng he prce level. Vesn shows ha hs soluon can also be acheved by assgnng a prce level argeng objecve o he cenral bank of he form, p + λ x. 2 2 pl However, he benefs of prce level argeng are no robus o allowng for a backward-lookng elemen n he nflaon adjusmen equaon see Walsh chp. Insrumen Rules An alernave approach o specfyng opmal polcy s o drecly specfy a rule for he polcy nsrumen self. The mos famous of hese s due o Taylor (993), T T R = r + π + axx + a π ( π π ) (73) However, nera n polcy nsrumens ges us closer o he commmen soluon.

37 Conclusons The New Keynesan Model gves a mcro-founded genereal equlbrum model wh suffcen nomnal frcons o make moneary polcy neresng. Is mcrofoundaons also allow he consrucon of a welfare funcon for polcy analyss whch s conssen wh maxmsng he uly of he represenave consumer/worker. Commmen polcy ypcally res o make polcy hsory dependen and n hs case adops a polcy of prce level argeng despe he fac ha hs s no an explc objecve. The desre o make polcy hsory dependen may also explan he observed nera n neres raes.