Graduate Macro Theory II: A New Keynesian Model with Price Stickiness

Transcription

1 Graduae Macro Theory II: A New Keynesian Model wih Price Sickiness Eric Sims Universiy of Nore Dame Spring Inroducion This se of noes lays and ou and analyzes he canonical New Keynesian NK model. The NK model akes a real business cycle model as is backbone and adds o ha sicky prices, a form of nominal rigidiy ha allows purely nominal shocks o have real effecs, and which alers he response of he economy o real shocks in a way ha gives rise o a non-rivial role for acive sabilizaion policy. To ge price-sickiness in he model, we have o have firms as price-seers, which means we need o move away from he perfecly compeiive benchmark. To do so we inroduce monopolisic compeiion in he way similar o before. We spli producion ino wo secors, where he final goods secor is perfecly compeiive and aggregaes inermediaes ino a final good for consumpion. This generaes a downward-sloping demand for inermediaes. There are a coninuum of inermediae goods producers who can se heir own prices, bu ake all oher prices as given. All he acion in he model is a he level of he inermediae producers. We assume ha hey are no freely able o adjus heir prices each period. In paricular, he Calvo 1983 assumpion posis ha each period firms face a fixed probabiliy of being allowed o change heir price. This seems a lile ridiculous in erms of is realism, bu his assumpion faciliaes aggregaion, and his is why i is so popular. Wih any price rigidiy, any firm s price becomes a sae variable. Wih a coninuum of inermediae goods firms, we d have a coninuum of sae variables. The Calvo 1983 assumpion allows us o aggregae ou his heerogeneiy. Even hough i seems somewha bizarre on is surface, i has some normaive implicaions ha seem prey reasonable in paricular, price sabiliy ends up being an imporan normaive goal. The basic New Keynesian model ha I ll lay ou below and which is laid ou in Woodford 23 and Gali 27 exbook reamens has no invesmen or capial. This simplifies he analysis quie a bi and permis us o ge beer inuiion. I is no a compleely innocuous omission, and we ll laer look a how he inclusion of capial in he model affecs hings. 1

2 2 Households There is a represenaive household ha consumes, supplies labor, accumulaes bonds, holds shares in firms, and accumulaes money. I ges uiliy from holding real balances. Is problem is: max C,N,B +1,M E β C 1 σ 1 σ ψ N 1+η 1 + η + θ ln = M Here I have gone ahead and assumed ha uiliy from real balances is logarihmic. As long as real balances are addiively separable from consumpion and labor, money in he uiliy funcion doesn do much ineresing here. 1 The nominal flow budge consrain is: P P C + B +1 + M M 1 W N + Π P T i 1 B Here money is he numeraire, and P is he price of goods in erms of money. B is he sock of nominal bonds a household eners he period wih, and hey pay ou known as of 1 nominal ineres rae i 1. The household also eners he period wih a sock of money, M 1. Noe ha I m no being super consisen wih iming noaion here: M 1 and B are boh known a 1. The reason I wrie i his way is because he aggregae supply of money in period, M, is no going o be predeermined bu raher se by a cenral bank. W is he nominal wage denominaed in unis of money, no goods. Π denoes nominal profis remied by firms, and T is a lump sum ax paid o a governmen he governmen will have he role of seing he money supply and remiing any seignorage revenue back o he household lump sum. L = E A Lagrangian for he household is: = [ β C 1 σ 1 σ ψ N 1+η 1 + η + θ ln The FOC are: M P ] + λ W N + Π P T i 1B P C B +1 M + M 1 L = C σ = λ P C L = ψn η = λ W N L = λ = βe λ i B +1 L = θ 1 = λ βe λ +1 M M We can eliminae he muliplier and re-wrie hese condiions as: 1 If we were o assume ha cenral bank policy focuses on an ineres rae raher han a moneary aggregae, as we will do below, hen we could ignore money alogeher so long as uiliy from money is separable. This is someimes referred o as a cashless economy. 2

3 ψn η = C σ w 1 C σ θ = βe C+1 σ P P = i C σ 1 + i 3 M P 3 Producion For he producion side of hings we spli ino wo. There is a represenaive compeiive final goods firm which aggregaes inermediae inpus according o a CES echnology. To he exen o which he inermediaes are imperfec subsiues in he CES aggregaor, his generaes a downwardsloping demand for each inermediae variey, which gives hese inermediae producers pricing power. There are a coninuum large number of inermediaes, so hese producers behave as monopolisically compeiive hey rea all prices bu heir own as given. These firms produce oupu using labor and are subjec o an aggregae produciviy shock. They are no freely able o adjus prices each period, in a way ha we will discuss in more deph below. 3.1 Final Good Producer The final oupu good is a CES aggregae of a coninuum of inermediaes: Y = ɛ Y j ɛ 1 ɛ 1 ɛ dj Here ɛ > 1. The profi maximizaion problem of he final good firm is: 4 max Y j P The FOC for a ypical variey of inermediae j is: P This can be wrien: ɛ 1 ɛ 1 ɛ Y j ɛ 1 ɛ 1 1 ɛ dj P jy jdj ɛ Y j ɛ 1 ɛ dj ɛ 1 1 ɛ 1 ɛ Y j ɛ 1 ɛ 1 = P j Or: 1 Y j ɛ 1 ɛ 1 ɛ dj Y j 1 ɛ = P j P ɛ Y j ɛ 1 ɛ 1 P j ɛ dj Y j = P ɛ Making noe of he definiion of he aggregae final good, we have:: 3

4 P j ɛ Y j = Y 5 This says ha he relaive demand for he j h inermediae is a funcion of is relaive price, wih ɛ he price elasiciy of demand, and is proporional o aggregae oupu, Y. To derive a price index, define nominal oupu as he sum of prices imes quaniies: P Y = Plugging in he demand for each variey, we have: P Y = P P jy jdj P j 1 ɛ P ɛ Y dj Pulling ou of he inegral hings which don depend on j: P Y = P ɛ Y P j 1 ɛ dj Simplifying, we ge an expression for he aggregae price level: P = 1 P j 1 ɛ 1 ɛ dj Inermediae Producers A ypical inermediae producer produces oupu according o a consan reurns o scale echnology in labor, wih a common produciviy shock, A : Y j = A N j 7 Inermediae producers face a common wage. They are no freely able o adjus price so as o maximize profi each period, bu will always ac o minimize cos. The cos minimizaion problem is o minimize oal cos subjec o he consrain of producing enough o mee demand: min N j W N j s.. P j ɛ A N j Y The muliplier on he consrain here will have he inerpreaion as marginal cos how much will coss change if you are forced o produce an exra uni of oupu. A Lagrangian is: P 4

5 P j ɛ L = W N j + ϕ j A N j Y P The FOC is: Or: L N j = W = ϕ ja ϕ = W A 8 Here I have dropped he j reference: marginal cos ϕ is equal o he wage divided by produciviy, boh of which are common o all inermediae goods firms. Real flow profi for inermediae producer j is: Π j = P j Y j W N j P P From 8, we know W = ϕ A. Plugging his ino he expression for profis, we ge: Π j = P j Y j mc Y j P Where I have defined mc ϕ P as real marginal cos. Firms are no freely able o adjus price each period. In paricular, each period here is a fixed probabiliy of 1 φ ha a firm can adjus is price. This means ha he probabiliy a firm will be suck wih a price one period is φ, for wo periods is φ 2, and so on. Consider he pricing problem of a firm given he opporuniy o adjus is price in a given period. Since here is a chance ha he firm will ge suck wih is price for muliple periods, he pricing problem becomes dynamic. Firms will discoun profis s periods ino he fuure by M +s φ s, where M +s = β s u C +s u C is he sochasic discoun facor. Noe ha discouning is by boh he usual sochasic discoun facor as well as by he probabiliy ha a price chosen in period will sill be in use in period + s. The dynamic problem of an updaing firm can be wrien: max P j E s= βφ s u C +s u C P j P +s P j ɛ P j ɛ Y +s mc +s Y +s P +s P +s Here I have imposed ha oupu will equal demand. Muliplying ou, we ge: max P j E s= βφ s u C +s u C The firs order condiion can be wrien: P j 1 ɛ P ɛ 1 +s Y +s mc +s P j ɛ P ɛ +sy +s 5

6 1 ɛp j ɛ E βφ s u C +s P+s ɛ 1 Y +s + ɛp j ɛ 1 E βφ s u C +s mc +s P+sY ɛ +s = Simplifying: s= P j = ɛ ɛ 1 s= E βφ s u C +s mc +s P+sY ɛ +s s= E s= βφ s u C +s P ɛ 1 +s Y +s Firs, noe ha since nohing on he righ hand side depends on j, all updaing firms will updae o he same rese price, call i P #. We can wrie he expression more compacly as: Here: X 1, P # = ɛ 9 ɛ 1 X 2, X 1, = u C mc P ɛ Y + φβe X 1,+1 1 X 2, = u C P ɛ 1 Y + φβe X 2,+1 11 If φ =, hen he righ hand side would reduce o mc P = ϕ. In his case, he opimal price ɛ would be a fixed markup, ɛ 1, over nominal marginal cos, ϕ. 4 Equilibrium and Aggregaion To close he model we need o specify an exogenous process for A, some kind of moneary policy rule o deermine M, and a fiscal rule o deermine T. Le he aggregae produciviy erm follow a mean zero AR1 in he log: ln A = ρ a ln A 1 + ε a, 12 Le s suppose ha he money supply follows an AR1 in he growh rae, where ln M = ln M ln M 1 : ln M = 1 ρ m π + ρ m ln M 1 + ε m, 13 I have wrien his process where he mean growh rae of money is equal o π, which will be he seady sae level of inflaion his is because real balances will be saionary, so M and P mus grow a he same rae in he seady sae. ε m, is a moneary shock. 6

7 Since he governmen prins money, i effecively earns some revenue. Righ now we re absracing from he governmen doing any spending, and for simpliciy I m going o assume ha he governmen does no operae in bond markes his is innocuous since i raises revenue only hrough a lump sum ax. The governmen s budge consrain in nominal erms is hen: P T + M M 1 In oher words, he change in he sock of money, M M 1, is nominal revenue for he governmen. Since i does no spending, a equaliy lump sum axes mus saisfy: T = M M 1 P So if money growh is posiive, e.g. M > M 1, hen lump sum axes will be negaive he governmen will be rebaing is seignorage revenue o he household lump sum. In equilibrium, bond-holding is always zero in all periods: B =. Using his, plus he relaionship beween he lump sum ax and money growh derived above, he household budge consrain can be wrien in real erms: C = w N + Π P Real dividends received by he household are jus he sum of real profis from inermediae goods firms he final good firm is compeiive and earns no profi: This can be wrien: Π P = Π 1 = P P j P Y j W N j dj P P j Y j w P N jdj Now, marke-clearing requires ha he sum of labor used by firms equals he oal labor supplied by households, so N jdj = N. Hence: Π 1 = P P j Y jdj w N P Throwing his ino he household budge consrain, he w N erms drop ou, leaving: C = Plug in he demand funcion for Y j: C = P j Y jdj P P j 1 ɛ P ɛ 1 Y dj 7

8 Bring suff ou of he inegral: C = P ɛ 1 Y P j 1 ɛ dj Now, since P 1 ɛ = P j 1 ɛ dj, he erms involving P s drop ou, leaving: C = Y 14 Now, wha is Y? From he demand for inermediae variey j, we have: P j ɛ Y j = Y Using he producion funcion for each inermediae, his is: Inegrae over j: P P j ɛ A N j = Y A N jdj = P P j ɛ Y dj Take suff ou of he inegral, wih he excepion of he price level on he righ hand side: Now define a new variable, v p, as: P P j ɛ A N jdj = Y dj P v p = P j ɛ dj 15 P This is a measure of price dispersion. If here were no pricing fricions, all firms would charge he same price, and v p = 1. If prices are differen, one can show ha his expression is bound from below by uniy. Using he definiion of aggregae labor inpu, we can herefore wrie: Y = A N v p 16 This is he aggregae producion funcion. Since v p 1, price dispersion resuls in an oupu loss you produce less oupu han you would given A and aggregae labor inpu if prices are disperse. This ends up being he gis of why price sabiliy is a good goal. The full se of equilibrium condiions can hen be wrien: C σ = βe C σ i P P ψn η = C σ w 18 8

9 M P = θ 1 + i i C σ 19 mc = w A 2 C = Y 21 Y = A N v p 22 v p = P j ɛp dj 23 P 1 ɛp = P P j 1 ɛp dj 24 X 1, P # = ɛ p 25 ɛ p 1 X 2, X 1, = C σ mc P ɛp Y + φ p βe X 1,+1 26 X 2, = C σ P ɛp 1 Y + φ p βe X 2,+1 27 ln A = ρ a ln A 1 + ε a, 28 ln M = 1 ρ m π + ρ m ln M 1 + ε m, 29 ln M = ln M ln M 1 3 This is 14 equaions in 14 aggregae variables C, i, P, N, w, M, mc, A, Y, v p, P #, X 1,, X 2,, ln M. There are wo sochasic shocks he usual produciviy shock as well as he moneary shock. 4.1 Re-wriing he equilibrium condiions There are a couple of issues wih how I ve wrien hese condiions. Firs, I haven goen rid of he heerogeneiy I sill have j indexes showing up. Second, I have he price level showing up, which, as I menioned above, may no be saionary. Third, I have he nominal money supply showing up, which is no saionary he way I ve wrien he process in erms of money growh. Hence, I wan o re-wrie hese condiions i only in erms of inflaion, eliminaing he price level; and ii geing rid of he heerogeneiy, which he Calvo 1983 assumpion allows me o do; and iii in erms of real money balances, m M P, insead of nominal money balances. Define inflaion as π = P P 1 1. The Euler equaion can be re-wrien: C σ = βe C σ i 1 + π The demand for money equaion is already wrien in erms of real balances: m = θ 1 + i i C σ 32 9

10 Now, we need o ge rid of he heerogeneiy in he expression for he price level and price dispersion. The expression for he price level is: P 1 ɛ = P j 1 ɛ dj Now, a fracion 1 φ of hese firms will updae heir price o he same rese price, P #. The oher fracion φ will charge he price hey charged in he previous period. Since i doesn maer how we order hese firms along he uni inerval, his means we can break up he inegral on he righ hand side as: This can be wrien: P 1 ɛ = φ P #,1 ɛ dj + P 1 j 1 ɛ dj 1 φ P 1 ɛ = 1 φp #,1 ɛ + P 1 j 1 ɛ dj 1 φ Now, here s he beauy of he Calvo assumpion. Because he firms who ge o updae are randomly chosen, and because here are a large number coninuum of firms, he inegral sum of individual prices over some subse of he uni inerval will simply be proporional o he inegral over he enire uni inerval, where he proporion is equal o he subse of he uni inerval over which he inegral is aken. This means: 1 φ P 1 j 1 ɛ dj = φ P 1 j 1 ɛ dj = φp 1 ɛ 1 This means ha he aggregae price level raised o 1 ɛ is a convex combinaion of he rese price and lagged price level raised o he same power. So: P 1 ɛ = 1 φp #,1 ɛ In oher words, we ve goen rid of he heerogeneiy. + φp 1 ɛ 1 The Calvo assumpion allows us o inegrae ou he heerogeneiy and no worry abou keeping rack of wha each firm is doing from he perspecive of looking a he behavior of aggregaes. Now, we sill have he issue here ha we are wrien in erms of he price level, no inflaion. To ge i in erms of inflaion, divide boh sides by P 1 1 ɛ, and define π# = P # P 1 1 as rese price inflaion: 1 + π 1 ɛ = 1 φ1 + π # 1 ɛ + φ 33 We can also use he Calvo assumpion o break up he price dispersion erm, by again noing ha 1 φ of firms will updae o he same price, and φ firms will be suck wih las period s price. Hence: 1

11 v p = φ P # ɛ dj + P 1 φ P 1 j ɛ dj This can be wrien in erms of inflaion by muliplying and dividing by powers of P 1 where necessary: 1 φ v p = P # P 1 ɛ P 1 We can ake suff ou of he inegral: P ɛ dj + 1 φ P P 1 j ɛ ɛ P 1 dj P 1 v p = 1 φ1 + π# ɛ 1 + π ɛ π ɛ 1 φ P P 1 j ɛ dj By he same Calvo logic, he erm inside he inegral is jus going o be proporional o v p 1. This means we can wrie he price dispersion erm as: P 1 v p = 1 φ1 + π# ɛ 1 + π ɛ π ɛ φv p 1 34 In oher words, we jus have o keep rack of v p, no he individual prices. Now, we need o adjus he rese price expression. Firs, define wo new auxiliary variables as follows: x 1, X 1, P ɛ x 2, X 2, P ɛ 1 Dividing boh sides of he rese price expressions by he appropriae power of P, we have: x 1, = C σ X 1,+1 mc Y + φβe P ɛ x 2, = C σ Y + φβe X 2,+1 P ɛ 1 Muliplying and dividing he + 1 erms by he appropriae power of P +1, we have: x 1, = C σ X 1,+1 mc Y + φβe P ɛ +1 ɛ P+1 P Or, in erms of inflaion: x 2, = C σ Y + φβe X 2,+1 P ɛ 1 +1 ɛ 1 P+1 P 11

12 x 1, = C σ mc Y + φβe 1 + π +1 ɛ x 1,+1 35 x 2, = C σ Y + φβe 1 + π +1 ɛ 1 x 2,+1 36 Now, in erms of he rese price expression, since we divided X 1, by P ɛ and divided X 2, by P ɛ 1. This means ha X 1, x X 2, = P 1, x 2,. The rese price expression can now be wrien: define: P # = ɛ ɛ 1 P Now, simply divide boh sides by P 1 o have everyhing in erms of inflaion raes: x 1, x 2, 1 + π # = ɛ ɛ π x 1, x 2, 37 Now, we need o re-wrie he processes involving money in erms of real balances. We can This is of course equal o: ln m ln m ln m 1 ln m = ln m ln m 1 = ln M ln P ln M 1 + ln P 1 = ln M ln M 1 π Hence: ln M = ln m + π This means we can wrie he process for money growh in erms of real balance growh as: ln m = 1 ρ m π π + ρ m ln m 1 + ρ m π 1 + ε m, 38 This means he re-wrien full se of equilibrium condiions is: C σ = βe C σ i 1 + π ψn η = C σ w 4 m = θ 1 + i i C σ 41 mc = w A 42 C = Y 43 Y = A N v p 44 12

13 v p = 1 φ1 + π# ɛ 1 + π ɛ π ɛ φv p π 1 ɛ = 1 φ1 + π # 1 ɛ + φ π # = ɛ ɛ π x 1, x 2, 47 x 1, = C σ mc Y + φβe 1 + π +1 ɛ x 1,+1 48 x 2, = C σ Y + φβe 1 + π +1 ɛ 1 x 2,+1 49 ln A = ρ a ln A 1 + ε a, 5 ln m = 1 ρ m π π + ρ m ln m 1 + ρ m π 1 + ε m, 51 ln m = ln m ln m 1 52 This is he same se of equaions as above, bu I have replaced P wih π, M wih m, P # wih π #, and X 1, and X 2, wih x 1, and x 2,. 5 The Seady Sae Le s solve for he non-sochasic seady sae of he model. I m going o use variables wihou a subscrip o denoe non-sochasic seady sae values. Seady sae A = 1. Since oupu and consumpion are always equal, i mus also be ha Y = C. Seady sae inflaion is equal o he exogenous arge, π. From he re-wrien AR1 in growh raes for real balances, in seady sae we have: ln m = 1 ρ m π 1 ρ m π + ρ m ln m 1 ρ m ln m = ln m = This means ha real money balances are saionary in he seady sae. From he Euler equaion, we have: 1 + i = π β In approximae erms, his would say i ρ + π, where β = 1 1+ρ, so ρ has he inerpreaion as he discoun rae whereas β is a discoun facor. From he price evoluion equaion, we can derive he seady sae expression for rese price inflaion: 1 + π 1 + π # 1 ɛ φ = 1 φ 1 1 ɛ If π =, hen π # = π because he righ hand side is jus 1. If π >, hen π # > π, and if 13

14 π <, hen π # < π. Given his, we can solve for seady sae price dispersion: 1 + π ɛ π ɛ φ v p = 1 φ 1 + π # If π =, hen we have v p = 1. If π, hen v p > 1. In he figure below, I plo in he lef panel seady sae rese price price inflaion, π #, as a funcion of seady sae inflaion, π. For comparison I plo seady sae inflaion agains iself as well dashed line. We can see ha seady sae rese price inflaion is less han seady sae inflaion for negaive seady sae inflaion and greaer han seady sae inflaion for posiive seady sae inflaion. In he righ panel I plo seady sae price dispersion as a funcion of rend inflaion. This booms ou a one, bu is greaer han one for seady sae inflaion no equal o zero hough price dispersion increases faser as seady sae inflaion moves away from zero on he posiive end han on he negaive end. I compued his graph using ɛ = 1 and φ = Rese Price Inflaion 1.12 Price Dispersion.1.5 π # π * π * π * Given all his, we can solve for he seady sae raio of x 1 /x 2 as: Now, we also know ha: x 1 = 1 + π# ɛ 1 x π ɛ x 1 1 φβ1 + πɛ 1 = mc x 2 1 φβ1 + π ɛ This means we can solve for seady sae marginal cos as: mc = 1 φβ1 + πɛ 1 + π # ɛ 1 1 φβ1 + π ɛ π ɛ Real marginal cos is equal o he inverse price markup. If π =, his is jus equal o ɛ 1 ɛ. In oher words, if seady sae inflaion is zero, hen he seady sae markup will be wha i would be if prices were flexible. If π, hen mc < ɛ 1 ɛ, which means ha he seady sae markup will be higher han i would if inflaion were zero. The figure below plos he seady sae real marginal cos and seady sae price markup as a funcion of seady sae inflaion. Here I used a value of 53 14

15 β =.99 along wih he values of ɛ and φ used above..92 SS MC 1.22 SS Markup π * π * Once we know seady sae marginal cos, hen we know he seady sae real wage: w = mc. The lower is marginal cos, he bigger is he wedge beween he wage and he marginal produc of labor i.e. he more disored he economy is. Take his o he saic labor supply condiion, imposing he equaliy beween Y = C: ψn η = Y σ mc Here I have imposed ha A = 1. Now, from he producion funcion, we know ha Y = N/v p. Plugging his in and simplifying, we can solve for N: ψn η = N σ v p σ mc 1 1 N = ψ vp σ η+σ mc Given his, we now have Y we can solve for seady sae m: m = θ 1 + i Y σ i 6 The Flexible Price Equilibrium A useful concep ha will come in handy, paricularly when hinking abou welfare, is a hypoheical equilibrium allocaion in which prices are flexible, which corresponds o he case when φ =. Because here is no endogenous sae variable in his model when prices are flexible, we can acually solve for he flexible price equilibrium by hand. I use superscrip f o denoe he hypoheical flexible price allocaion. When φ =, we have π # = π regardless of wha π is. Then going o he price dispersion expression, when φ = we have: v f,p = 1 + π # ɛ = π 15

16 In oher words, if prices are flexible, all firms charge he same prices, and price dispersion is a is lower bound of 1. By combining he rese price inflaion erm wih he auxiliary variables x 1, and x 2,, we ge ha mc f = ɛ 1 ɛ, and is herefore consan. Since marginal cos is he inverse price markup, his jus says ha if prices are flexible, firms will se price equal o a fixed markup over marginal cos which we ve already seen before in a flexible price case wih monopolisic compeiion. This means ha w f = ɛ 1 ɛ A. Plugging his ino he saic labor FOC along wih he marke-clearing condiion ha Y = C, we see: Using he fac ha Y f η ψ N f = = A N f, we have: Y f σ ɛ 1 A ɛ η ψ N f = A σ N f N f = This means ha flexible price oupu is: Y f 1 = ψ 1 ψ σ ɛ 1 ɛ ɛ 1 A 1 σ ɛ A 1 σ+η 1 ɛ 1 σ+η 1+η A σ+η 54 ɛ There is somehing ineresing here which is worh menioning. If σ = 1, hen N f is a consan and no a funcion of A. In oher words, if prices were flexible and σ = 1 log uiliy over consumpion, labor hours would no reac o flucuaions in A. Wha is driving his is ha, if σ = 1, hen preferences are consisen wih King, Plosser, Rebelo 1988 preferences, in which he income and subsiuion effecs of changes in A exacly offse. When here is capial in he model, his offse only occurs in he long run, so ha labor hours are consan in he long run, bu no in he shor run as capial adjuss o seady sae. Wihou capial, he cancellaion of income and subsiuion effecs holds a all imes. Noe also ha flexible price oupu does no depend on anyhing nominal. This is because, wih flexible prices, nominal shocks have no real effecs. 7 Quaniaive Analysis I solve he model quaniaively in Dynare using a firs order approximaion abou he seady sae. I use he following parameer values more on his laer: φ =.75, σ = 1, η = 1, ψ = 1, ɛ = 1, θ = 1, ρ a =.95, ρ m =., and π =. I assume ha he sandard deviaion of boh shocks are.1. Impulse responses o he produciviy shock are shown below. 16

17 .15 A 7 x 1 3 Y 2 x 1 3 π i 2 x 1 3 r 5 x 1 3 mc x 1 3 N 7 x 1 3 m 3 x 1 3 P There are a couple of ineresing hings o poin ou here. Oupu responds very lile on impac, and significanly less han he increase in A. Indeed, we acually see a fairly large decline in N when A goes up. Inflaion falls. The response of he price level which I compue by cumulaing he response of inflaion is roughly he mirror image of he oupu response. The nominal ineres rae does no move a all ay any horizon, hough he real ineres rae increases. Real marginal cos falls, which suggess ha he real wage rises by less han A effecively, firms charge bigger markups. 1 x 1 3 Y f x 1 3 X Above I plo he impulse response of he flexible price level of oupu and a new variable I 17

18 call he oupu gap, defined as ln X = ln Y ln Y f. Because oupu responds significanly less han he flexible price level of oupu o he produciviy shock, we see a large negaive oupu gap opening up following he posiive produciviy shock. Wha s going on here? If φ =, we see ha oupu would respond significanly more o he produciviy shock han in he baseline case I used where φ =.75. Wha is going on here? When prices are sicky, oupu becomes parially demand-deermined, and wih exogenous money supply he way I have i here, price rigidiy prevens demand from rising sufficienly when supply increases, so oupu rises by oo lile relaive o wha would happen wih flexible prices. An easy way o see his is o look a he money demand relaionship. In logs, we have: ln m = ln θ + ln1 + i ln i + σ ln Y To he exen o which he nominal ineres rae doesn move which i in fac doesn here 2, he movemen in oupu mus be proporional o he movemen in real balances. Since I ve assumed ha M is se exogenously, he only way m can move is hrough changes in P. Hence, as we can see in he IRFs, he oupu movemen ends up jus being he mirror image of he movemen in P. And since prices are sicky, P can move enough relaive o wha i would do under price flexibiliy. Hence, m fails o increase sufficienly, and Y can rise as much as i would if prices were flexible. There is anoher way o see how price rigidiy effecively limis he demand increase, resuling in a response of oupu ha is oo small relaive o wha would happen in he absence of price rigidiy. If prices were flexible, in he period of he shock, P would immediaely fall so m could rise, bu would hen sar o rise. This means ha expeced inflaion would acually rise. Given a fixed nominal ineres rae via he logic above, his means ha he real ineres rae would fall if prices were flexible. 3 Wih price sickiness, in conras, inflaion falls, and says persisenly low basically, waves of firms come each period and cu heir prices, so inflaion says low for a while. This means ha expeced inflaion falls, no rises as i would if prices were flexible. This means ha he real ineres rae rises when A increases, which works o choke off demand. Nex, consider a shock o he money supply. Since I have assumed ha ρ m =, nominal money follows a random walk, so he shock resuls in a one ime permanen level shif in M. Here, we 2 To see why his happens, go back o he FOC from he household s problem, you can wrie: λ = θ 1 M +βe λ +1. θ Solving his forward, you d ge: λ = E j= β M +j. If M doesn respond o a shock, hen λ can eiher. Bu from he firs order condiion for bonds, λ = βe λ i. If λ and λ +1 don reac o he produciviy shock which hey won if M is fixed, hen i canno reac o he shock. Noe ha his resul would no hold generally for specificaions of uiliy from real balances which are no logarihmic or which are non-separable from he oher argumens of uiliy. To see his, suppose ha uiliy over real balances were sill separable bu insead showed up M as θ + βe T λ +1. If ζ = 1 log uiliy, hen he P erm drops ou, and you ge he resul ha λ can only move if M moves or is expeced o move, which means ha i canno change if you ge a produciviy shock and he money supply is held consan. Bu if ζ 1, hen P shows up; since P change, hen λ and E λ +1 will boh change, which will mean ha i will poenially change. 3 In a log-linear version, i is sraighforward o show ha he hypoheical real ineres rae if prices were flexible someimes called he naural rae of ineres would jus be proporional o expeced produciviy growh, which is negaive afer a produciviy shock given ha I have assumed he process for produciviy is a saionary AR1. P 1 ζ 1 1 ζ. Then he FOC for he choice of M would be: λ = θm ζ P ζ 1 18

19 observe ha Y, N, and π all rise. There is a emporary rise in m. mc rises, which means ha w rises since A is fixed: his is necessary o ge workers o work more. The real ineres rae falls, hough again he nominal ineres rae doesn move. 4 Evidenly, having sicky prices allows he nominal moneary shock o have real effecs..5 M.1 Y 4 x 1 3 π i 1 2 x 1 3 r mc N m P Wha is going on here? There are again a couple of ways o see his. Focusing on he money demand relaionship, we again have he resul ha, for a fixed nominal ineres rae, real balances and real GDP move ogeher. When M increases, if prices were flexible P would increase by he same amoun, so real balances wouldn change, and hence Y wouldn change. Bu wih sicky prices, P can increase sufficienly, so m rises, and herefore so oo does oupu. Anoher way o see wha is going on is by focusing on he real ineres rae. If prices were flexible, he one ime increase in M would be me by a one ime permanen increase in P, so E P +1 = P, and herefore expeced inflaion would no reac. Wih expeced inflaion fixed, and he nominal rae fixed, here would be no effec on he real ineres rae. Bu wih price sickiness, because no all firms can immediaely adjus heir prices, he aggregae price level adjuss slowly, and in paricular E P +1 > P, so expeced inflaion rises. Higher expeced inflaion wih a fixed nominal rae means 4 We can see why he nominal ineres rae doesn move again via he firs order condiions. We again mus have: λ = E j= β θ M +j. Since M follows a random walk, λ and λ +1 will boh fall when M goes up, bu by he same amoun. Since λ = βe λ i, his again implies ha i will no reac. If ρ m >, λ and λ +1 would reac differenly because E M +j M, and he nominal ineres rae would move. The same cavea would apply as in he above foonoe if uiliy over real balances were no logarihmic. 19

20 a lower real ineres rae, which simulaes expendiure and resuls in he oupu increase. Below I show he impulse response of he flexible price level of oupu and he oupu gap o he moneary policy shock. Since he flexible price level of oupu does no reac, he response of he gap is idenical o he response of oupu. 1 Y f 8 x 1 3 X Log-Linearizaion I is very common o see he basic New Keynesian model presened in log-linear form. The equaions urn ou o be prey inuiive. I ends up being a decen amoun of work, bu here are some imporan payoffs o going hrough he hard work of linearizing he equaions by hand. I urns ou ha life is much easier if we do he linearizaion abou a seady sae wih π = i.e. a zero inflaion seady sae. Sar wih he Euler equaion, going ahead and imposing he accouning ideniy ha C = Y. We have: σ ln Y = ln β σe ln Y +1 + i E π +1 σỹ = σe Ỹ +1 + ĩ E π +1 Where Ỹ = Y Y Y, ĩ = i i, and π = π π. In oher words, he variables already in rae form ineres rae and inflaion are expressed as absolue deviaions, and variables no already in rae form as percen log deviaions. We can re-wrie his as: Ỹ = E Ỹ +1 1 σ ĩ E π +1 This is someimes called he New Keynesian IS Curve. This is a bi of a misnomer: in old Keynesian models, he IS curve sands for Invesmen = Saving, and here is no invesmen in his model. Neverheless, he idea is o show ha here exiss an inverse relaionship beween demand for curren spending and he real ineres rae. 55 This expression is New in he sense 2

21 ha i is forward-looking: curren demand depends no jus on he real ineres rae bu also on expeced fuure income. Nex, log-linearize he saic labor demand specificaion. This expression is already log-linear, and works ou o be: From he marginal cos relaionship, we can eliminae he wage: ηñ = σỹ + w 56 Plugging his in: w = mc + Ã 57 Log-linearize he producion funcion: ηñ = σỹ + mc + Ã Ỹ = Ã + Ñ ṽ p Now, wha is ṽ p? Le s ake logs and go from here: ln v p = ln 1 φ1 + π # ɛ 1 + π ɛ π ɛ φv p 1 Now, from our discussion above, we know ha v p = 1 when π =. Toally differeniaing: ṽ p = 1 1 ɛ1 φ1 + π # ɛ π ɛ π # π # + ɛ1 φ1 + π # ɛ 1 + π ɛ 1 π π ɛ1 + π ɛ 1 φv p π π π ɛ φv p 1 vp Using now known facs abou he seady sae, his reduces o: ṽ p = ɛ1 φ π# + ɛ1 φ π + ɛφ π + φṽ p 1 This can be wrien: ṽ p = ɛ1 φ π# + ɛ π + φṽ p 1 Now, log-linearize he equaion for he evoluion of inflaion: 1 ɛπ = ln 1 φ1 + π # 1 ɛ + φ 1 ɛ π π = 1 + π ɛ 1 1 ɛ1 φ1 + π # ɛ π # π # 21

22 In he las line above, he 1 + π ɛ 1 shows up because he erm inside parenheses is equal o 1 + π 1 ɛ evaluaed in he seady sae, and when aking he derivaive of he log his erm ges invered evaluaed a ha poin. Using facs abou he zero inflaion seady sae, we have: Or: 1 ɛ π = 1 ɛ1 φ π # π = 1 φ π # 58 In oher words, acual inflaion is jus proporional o rese price inflaion, where he consan is equal o he fracion of firms ha are updaing heir prices. This is prey inuiive. Now, use his in he expression for price dispersion: ṽ p = ɛ π 1 φ π # + φṽ p 1 Bu from above, he firs erm drops ou, so we are lef wih: ṽ p = φṽp 1 59 If we are approximaing abou he zero inflaion seady sae in which v p = 1, hen we re saring from a posiion in which ṽ p 1 =, so his means ha ṽp = a all imes. In oher words, abou a zero inflaion seady sae, price dispersion is a second order phenomenon, and we can jus ignore i in a firs order approximaion abou a zero inflaion seady sae. Given his, he log-linearized producion funcion is jus: Ỹ = Ã + Ñ 6 Now, plug his in o eliminae Ñ from he log-linearized saic labor supply condiion from above: η Ỹ Ã = σỹ + mc + Ã Simplifying a lile bi, we ge: σ + ηỹ 1 + ηã = mc From above, we had solved for an expression for he flexible price level of oupu as: Y f 1 = ψ This is already log-linear, so we have: 1 ɛ 1 ɛ σ+η A 1+η σ+η 22

23 This means we can wrie: Ỹ f = 1 + η σ + η Ã 61 Plugging his in above, we ge: Ã = σ + η 1 + η Ỹ f mc = σ + η Ỹ Ỹ f 62 In oher words, real marginal cos is proporional o he oupu gap, X = Ỹ Ỹ f. Recall ha real marginal cos is he inverse price markup. So if he gap is zero oupu is equal o wha i would be wih flexible prices, hen markups are equal o he desired fixed seady sae markup of ɛ ɛ 1. If he oupu gap is posiive, hen real marginal cos is above is seady sae, so markups are lower han desired equivalenly, he economy is less disored. The converse is rue when he gap is negaive. Now, le s log-linearize he rese price expression. This is muliplicaive, and so is already in log-linear form. We have: have: π # = π + x 1, x 2, 63 Now we need o log-linearize he auxiliary variables. Imposing he ideniy ha Y = C, we Toally differeniaing: ln x 1, = ln Y 1 σ mc + φβe 1 + π +1 ɛ x 1,+1 x 1, x 1 x 1 = 1 x 1 1 σy σ mcy Y + Y 1 σ mc mc + ɛφβ1 + π ɛ 1 x 1 π +1 π + φβ1 + π ɛ x 1,+1 x 1 Disribuing he 1 x 1 and muliplying, dividing where necessary o ge in o percen deviaion erms, and making use of he coninued assumpion of linearizaion abou a zero inflaion seady sae, we have: x 1, = 1 σy 1 σ mc Ỹ + Y 1 σ mc mc + ɛφβe π +1 + φβe x 1,+1 x 1 x 1 Now, wih zero seady sae inflaion, we know ha x 1 = Y 1 σ mc 1 φβ. This simplifies he firs wo erms: 23

24 Now, log-linearize x 2, : x 1, = 1 σ1 φβỹ + 1 φβ mc + ɛφβe π +1 + φβe x 1,+1 64 Toally differeniaing: ln x 2, = ln Y 1 σ + φβe 1 + π +1 ɛ 1 x 2,+1 x 2, x 2 x 2 = 1 x 2 1 σy σ Y Y + ɛ 1φβ1 + π ɛ 2 x 2 π +1 π + φβ1 + π ɛ 1 x 2,+1 x 2 Disribuing he x 2, muliplying and dividing by appropriae erms, and making use of he fac ha π =, we have: 1 σy 1 σ x 2, = Ỹ + ɛ 1φβE π +1 + φβe x 2,+1 x 2 Since x 2 = Y 1 σ 1 φβ, his can be wrien: x 2, = 1 σ1 φβỹ + ɛ 1φβE π +1 + φβe x 2,+1 65 Now, subracing x 2, from x 1,, we have: x 1, x 2, = 1 φβ mc + φβe π +1 + φβe x 1,+1 x 2,+1 From above, we also know ha: x 1, x 2, = π # π Bu π # = 1 1 φ π, so we mus also have: Make his subsiuion above: Muliplying hrough: x 1, x 2, = φ 1 φ π φ φ 1 φ π = 1 φβ mc + φβe π +1 + φβe 1 φ E π +1 Or: π = 1 φ1 φβ φ mc + 1 φβe π +1 + φβe π +1 24

25 1 φ1 φβ π = mc + βe π φ This expression is someimes called he New Keynesian Phillips Curve. I is new in he sense of being forward-looking. I is a Phillips Curve in he sense of showing a relaionship beween some real measure, mc, and inflaion, π. I is also common o see he Phillips Curve expressed in erms of he oupu gap, using he relaionship beween real marginal cos and he gap ha we derived above: π = 1 φ1 φβ σ + η Ỹ φ Ỹ f + βe π Using he erminal condiion ha inflaion will reurn o seady sae evenually, we can solve he NKPC forward o ge: π = 1 φ1 φβ φ β j mc +j 68 j= In oher words, curren inflaion is proporional o he presen discouned value of expeced real marginal cos. This expression is acually prey inuiive. Real marginal cos is he inverse price markup. In he model wihou price rigidiy, firms desire consan markups. If expeced fuure marginal cos is high, hen firms will have low markups. Firms given he opion of updaing prices oday will ry o increase price oday since hey may be suck wih ha price in he fuure o hi heir desired price markup and vice-versa, puing upward pressure on curren inflaion and vice versa. The slope of he Phillips Curve is decreasing in φ: when φ is large, he coefficien on marginal cos or he gap is small, suggesing ha real movemens pu lile upward pressure on inflaion. When φ is small, he Phillips Curve is seep. In he limiing case, as prices become perfecly flexible φ, he Phillips Curve becomes verical, which means mc = and Ỹ = Ỹ f e.g. we would be a he flexible price allocaion. The expressions for A and money growh are already log-linear, so we have: à = ρ a à 1 + ε a, 69 m = π + ρ m π 1 + ρ m m 1 + ε m, 7 m = m m 1 71 Lasly, we need o log-linearize he money demand expression: ln m = ln θ + i ln i + σ ln Y Toally differeniaing: 25

26 m = ĩ 1 i ĩ + σỹ Recall ha ĩ = i i, where i = 1 β his as: 1 since seady sae inflaion is zero. Hence we can wrie Y. Or: m = ĩ m = β 1 β ĩ + σỹ 1 β ĩ + σỹ 72 1 β This is inuiive. Demand for real balances is decreasing in he real ineres rae and increasing This means we can re-wrie he complee hough reduced, because I ve eliminaed a lo of he exraneous variables log-linearzied sysem of equaions as: π = Ỹ = E Ỹ +1 1 ĩ E π +1 σ 73 1 φ1 φβ σ + η Ỹ φ + βe π Ỹ f = 1 + η σ + η à 75 à = ρ a à 1 + ε a, 76 m = π + ρ m π 1 + ρ m m 1 + ε m, 77 m = m m 1 78 m = 1 β ĩ + σỹ 79 1 β This is seven equaions in seven variables Ỹ, ĩ, π, Ỹ f, Ã, m, m. We have an aggregae demand expression given by he linearized Euler equaion, an aggregae supply relaionship given by he Phillips Curve, a produciviy shock, a money supply relaionship, a money demand relaionship, and wo auxiliary expressions defining growh in real balances and he flexible price level of oupu. Noe ha you won ge exacly he same oupu as Dynare will give you from his linearizaion. I will be very close. The reason is ha I approximaed ln1 + i = i, whereas when Dynare does he approximaion i won use ha exra erm in doing he log-linearizaion. 9 A Taylor Rule Formulaion In he model as laid ou so far, wheher in log-linearized form or no, I have characerized moneary policy wih an exogenous rule for money growh. This doesn seem o square paricularly well wih 26

27 acual cenral bank pracice, where cenral bankers end o hink of policy in erms of ineres raes, no moneary aggregaes per se. For reasons ha will soon become clearer, an exogenous ineres rae rule will lead o an indeerminacy in he model. An ineres rae specificaion of policy needs o feaure nominal ineres raes reacing, and reacing sufficienly, o endogenous variables like inflaion and oupu. The mos popular ineres rae rule is somewha generically called a Taylor rule, afer John Taylor. I akes a form similar o he following: i = 1 ρ i i + ρ i i ρ i φ π π π + φ x ln X ln X + ε i, 8 i is he seady sae ineres, π is an exogenous seady sae inflaion arge, ln X is he oupu gap, ln X is he seady sae oupu gap, and ε i, is a moneary policy shock, analogous o he ε m, in he money growh specificaion. ρ i is a smoohing parameer, and φ π and φ x are non-negaive coefficiens. Assume ha φ π > 1 we ll alk abou why laer. The policy rule is one of parial adjusmen i says ha he curren nominal rae is a convex combinaion of he lagged nominal rae and he curren arge rae, where he curren arge rae is a linear funcion of he deviaions of inflaion and he oupu gap from arge where I have implicily assumed ha he arges are he long run seady sae levels. Noe ha here is no menion of money in his policy rule specificaion. I can effecively replace he money growh process above wih his rule. Given he chosen nominal ineres rae, he cenral bank will implicily prin he requisie amoun of money o mee money demand a ha ineres rae. Given he specificaion for money we have used where money eners he uiliy funcion in an addiively separable fashion we could acually no alk abou money a all, and consider he economy o be cashless. The full se of equilibrium condiions can be wrien: C σ = βe C σ i 1 + π ψn η = C σ w 82 m = θ 1 + i i C σ 83 mc = w A 84 C = Y 85 Y = A N v p 86 v p = 1 φ1 + π# ɛ 1 + π ɛ π ɛ φv p π 1 ɛ = 1 φ1 + π # 1 ɛ + φ π # = ɛ ɛ π x 1, x 2, 89 27

28 x 1, = C σ mc Y + φβe 1 + π +1 ɛ x 1,+1 9 x 2, = C σ Y + φβe 1 + π +1 ɛ 1 x 2,+1 91 ln A = ρ a ln A 1 + ε a, 92 i = 1 ρ i i + ρ i i ρ i φ π π π + φ x ln X ln X + ε i, 93 Here I have goen rid of he money growh specificaion, and since I ve done ha I no longer need o keep ln m as a variable, so his is acually one fewer equaion and one fewer unknown han I previously had. As I said above, I really don even need o keep rack of m anymore eiher, bu I ll keep i in because ha urns ou o be insrucive. One poin ha I should menion up fron. One migh be emped o hink ha he seady sae log oupu gap is zero, meaning ha Y = Y f. This will only be he case if π =, oherwise Y < Y f. From above, we know ha: Y f = For he sicky price economy, we have: 1 ψ 1 ɛ 1 σ+η ɛ 94 We know ha: N = 1 1 ψ vp σ η+σ mc 95 Seady sae oupu is hen: Y = N v p We also know: Y = 1 1 σ+η v p ψ η η+σ mc 1 η+σ 96 mc = 1 φβ1 + πɛ 1 + π # ɛ 1 1 φβ1 + π ɛ π ɛ Now, if π =, hen mc = ɛ 1 ɛ and v p = 1, so his reduces o he same expression as Y f, so we ll have Y = Y f. Bu if π >, you can show ha mc < ɛ 1 ɛ and we know ha v p > 1. Since he exponen on mc is posiive, and he exponen on v p negaive, his means ha π > will mean Y < Y f, which means he seady sae oupu gap will be negaive, ln X = ln Y ln Y f <. I solve he model in Dynare using he coefficiens ρ i =.8, φ π = 1.5, and φ x =. Below are he impulse responses o a echnology shock under he Taylor rule formulaion: 28

29 .15 A.1 Y x 1 3 π x 1 4 i 1 x 1 4 r x 1 3 mc x 1 3 N.6 m P If you compare hese o wha we had earlier, you ll noice ha hey are subsanively differen. In paricular, he impac increase in oupu under he Taylor rule is much larger han under he exogenous money growh rule; hence we also see a smaller drop in hours on impac, a smaller increase in he real ineres rae, and a smaller drop in inflaion. We also see he nominal ineres rae moving. Also, he response of he price level here seems o be more or less permanen, whereas in he money growh rule case i seemed o be mean-revering. To ge a beer sense of hese differences, below I plo he impulse response of he nominal supply o he echnology shock. Noe ha he nominal money supply didn respond in he previous case..55 M

30 Here we see ha he nominal money supply rises raher significanly. In oher words, under he Taylor rule he money supply is effecively endogenous, and he cenral bank reacs o he increased produciviy by accommodaing i and increasing he money supply. This increase in he money supply helps real balances increases now we don have o simply rely on prices falling o ge real balances o go up, so oupu can expand by more han i would if he money supply were fixed. This endogenous response of he money supply is wha allows oupu o rise by more han i did under he exogenous money rule process earlier. Accordingly, we see a smaller drop in he oupu gap in response o he echnology shock. 1 x 1 3 Y f x 1 3 X Nex, consider a posiive shock o he Taylor rule, which raises he nominal ineres rae. This coincides wih a decline in he money supply, an increase in he real ineres rae, and a decline in economic aciviy. The channels a play for why his nominal shock has real effecs are he same as above when we hough abou he nominal shock in erms of he money supply. There are wo ways o hink abou. Firs, he decrease in he money supply is mached by a less han proporional decrease in he price level because of price sickiness; his means ha real balances decline, which via he basic logic above necessiaes a decline in oupu. I also has effec of raising he real ineres rae. The nominal rae rises, and because of price sickiness expeced inflaion does no rise enough, so he real rae rises, which leads o a reducion in demand. 3

31 M 5 x 1 3 Y x 1 3 π x 1 3 i 4 x 1 3 r mc N.2 m 2 x 1 3 P Below are he impulse responses of he flexible price level of oupu as well as he oupu gap Y f 2 x X In log-linear erms, he Taylor rule is jus: We can he wrie he linearized model as: ĩ = ρ i ĩ ρ i φ π π + φ x X + ε i, 97 Ỹ = E Ỹ +1 1 σ ĩ E π

32 π = 1 φ1 φβ σ + η Ỹ φ Ỹ f + βe π Ỹ f = 1 + η σ + η à 1 à = ρ a à 1 + ε a, 11 ĩ = ρ i ĩ ρ i φ π π + φ x X + ε i, 12 Someimes you will see differen ways of wriing he model ou, and i is useful o review hem here. Firs, noe ha we can eliminae à and jus wrie he model in erms of Ỹ f 1+η. Le ω = σ+η : Ỹ f = ωã = ω ρ a à 1 + ε a, 1 = ω ρ a ω Ỹ f 1 + ε a, Ỹ f Ỹ f Ỹ f = ρ a Ỹ f 1 + ωε a, 13 In oher words, we can hink abou he flexible price level of oupu is effecively being exogenous, obeying he same AR1 process as Ã, bu wih he shock scaled by he facor ω. Now, le s wrie he Euler/IS equaion in erms of he oupu gap, X, insead of oupu. We can do his by subracing Ỹ f and E Ỹ f +1 from boh sides: Ỹ Ỹ f E Ỹ f +1 = Ỹ f + E Ỹ +1 E Ỹ f +1 1 ĩ E π +1 σ X = E X+1 + E Ỹ f +1 Ỹ f 1 ĩ E π +1 σ Now, from he Fisher relaionship, we know ha r = ĩ E π +1. Now, le s consider a hypoheical allocaion in which prices are flexible e.g. φ =. Then we know by consrucion ha X =. This means we can solve for a hypoheical flexible price real ineres rae someimes called he Wicksellian naural rae of ineres as: = E Ỹ f +1 Ỹ f 1 σ rf r f = σ E Ỹ f +1 Ỹ f 14 In words, he naural rae of ineres is proporional o he expeced growh rae of he flexible price level of oupu. We can use his o wrie he Euler equaion as: 32

33 X = E X+1 1 σ ĩ E π +1 r f 15 In oher words, he curren oupu gap equals he expeced fuure oupu gap minus 1 σ imes he real ineres rae gap he gap beween he acual real ineres rae and he flexible price real ineres rae. Holding E X+1 fixed, if he real ineres rae gap is posiive acual real rae oo high hen he gap will be negaive, and vice-versa. Since E Ỹ f = ρ a Ỹ f, his furher reduces o: r f = σρ a 1Ỹ f 16 Now, plug in he AR1 process for Ỹ f ha we jus derived: r f = σρ a 1 ρ a Ỹ f 1 + ωε a, r f = σρ 1 a 1 ρ a σρ a 1 rf 1 + ωε a, Wih his, we can reduce he enire log-linearized sysem o: r f = ρ a r f 1 + σρ a 1ωε a, 17 X = E X+1 1 σ ĩ E π +1 r f 18 1 φ1 φβ π = σ + η φ X + βe π ĩ = ρ i ĩ ρ i φ π π + φ x X + ε i, 11 r f = ρ a r f 1 + σρ a 1ωε a, 111 This is someimes called he hree equaion New Keynesian model. This may look odd, since here are acually four equaions, bu only hree of hese equaions describe endogenous variables: he firs equaion is he Euler/IS/demand relaionship, he second is he Phillips Curve, and he hird is a policy rule. These hree equaions are wha make up he hree equaion model. The fourh equaion is an exogenous process for r f produciviy. which, again, we derived from a process for 1 Sligh Deour: The Mehod of Undeermined Coefficiens Consider he small scale model above. I feaures wo forward-looking jump variables π and X and wo sae variables one endogenous, he ineres rae, ĩ, and one exogenous, he naural rae of ineres, r f. We could solve for he policy funcions mapping he saes ino he jump variables 33

34 using he mehodology we laid ou in class. Dynare will do his for us. Anoher soluion mehodology, which is more inuiive, is o use he mehod of undeermined coefficiens. This involves posulaing really, guessing ha he policy funcions are linear, imposing ha, and hen solving a sysem of equaions for he policy rule coefficiens. In a small scale model wihou many sae variables, his is ofen prey sraighforward and will allow us o ge analyical policy funcions, which is nice. Le s ry ha ou here. Consider he hree equaion model wih he exogenous process for r f. To make life easy, le s assume ha ρ i =. This means ha ĩ is no longer a sae; indeed, i becomes redundan and can be subsiued ou enirely. Doing so, he remaining sysem can be wrien as follows: X = E X+1 1 σ φ π π + φ x X E π +1 r f 112 π = κ X + βe π r f = ρ a r f 1 + σρ a 1ωε a, 114 To keep noaion igh, I have defined κ = 1 φ1 φβ φ σ + η as he slope on he Phillips Curve. The only sae variable is r f. Le s guess ha he policy funcions are linear: X = λ 1 r f π = λ 2 r f Plug hese in o he Euler/IS and Phillips Curves: λ 1 r f = λ 1ρ a r f 1 φ π λ 2 r f σ + φ xλ 1 r f λ 2ρ a r f rf σλ 1 r f σλ 1ρ a r f + φ πλ 2 r f + φ xλ 1 r f λ 2ρ a r f rf = σλ 1 σλ 1 ρ a + φ π λ 2 + φ x λ 1 λ 2 ρ a 1 r f = σλ 1 σλ 1 ρ a + φ π λ 2 + φ x λ 1 λ 2 ρ a 1 = λ 2 r f = κλ 1 r f + βλ 2ρ a r f λ 2 r f κλ 1 r f βλ 2ρ a r f = λ 2 κλ 1 βλ 2 ρ a r f = λ 2 κλ 1 βλ 2 ρ a = Here I have made use of he fac ha E r f +1 = ρ a r f. The above amouns o wo equaions in wo unknowns λ 1 and λ 2. We can solve for hese coefficiens. From he second expression, we have: 34

35 Simplify he firs expression somewha: λ 1 = 1 βρ a λ 2 κ σ1 ρ a + φ x λ 1 + φ π ρ a λ 2 1 = Plug in for λ 1 : σ1 ρ a + φ x 1 βρ a 1κ + φ π ρ a λ 2 = 1 So: This can be re-wrien: λ 2 = σ1 ρ a + φ x 1 βρ a 1κ 1 + φ π ρ a Then we have: λ 2 = κ σ1 ρ a + φ x 1 βρ a + κφ π ρ a 115 λ 1 = So he policy funcions are: 1 βρ a σ1 ρ a + φ x 1 βρ a + κφ π ρ a 116 X = π = 1 βρ a σ1 ρ a + φ x 1 βρ a + κφ π ρ a rf 117 κ σ1 ρ a + φ x 1 βρ a + κφ π ρ a rf 118 You can verify ha you ge he same hing by sicking he linearized equaions ino Dynare and leing i do he work for you. 11 Anoher Deour: Calibraing he Calvo Parameer φ How does one come up wih a reasonable value for φ? This is a really imporan parameer in he model he bigger i is he sickier are prices, he bigger will be he effecs of nominal shocks and he more disored will be he response of variables o real shocks. I urns ou ha here exiss a close mapping beween φ and he expeced duraion of a price change. Consider a firm ha ges o updae is price in a period. In expecaion, how long will i be suck wih ha price? The probabiliy of geing o adjus is price one period from now is 1 φ. The probabiliy of adjusing in wo periods is φ1 φ: φ is he probabiliy i doesn adjus afer one period, and 1 φ is he probabiliy i can adjus in wo periods. The probabiliy of adjusing in hree periods is φ 2 1 φ: φ 2 is he probabiliy i ges o he hird period wih is 35