Fiscal and Monetary Policy in a New Keynesian Model with Tobin s Q Investment Theory Features

Transcription

1 MPRA Munich Personal RePEc Archive Fiscal and Moneary Policy in a New Keynesian Model wih Tobin s Q Invesmen Theory Feaures Sylianos Giannoulakis Ahens Universiy of Economics and Business 4 May 2017 Online a hps://mpra.ub.uni-muenchen.de/80892/ MPRA Paper No , posed 21 Augus :11 UTC

2 Fiscal and Moneary Policy in a New Keynesian Model wih Tobin s Q Invesmen Theory Feaures Sylianos Giannoulakis * Absrac The purpose of his aricle is o carefully lay ou he inernal moneary and fiscal ransmission mechanisms in he conex of a New Keynesian model, wih a paricular focus on he role of capial - he mos vial ingredien in he ransiion from he basic framework o he medium - scale DSGE models. The key concep of his paper is he form of he moneary policy: we assume a wo-channel moneary policy, i.e. i is conduced hrough a rule for money supply and a Taylor-ype rule for ineres raes, in order o keep up wih he ECB and Fed s policies. We also adop a simple fiscal policy rule for public consumpion o examine he ineracions beween fiscal and moneary policy. Finally, in order o capure he crisis effecs we inroduce exogenous shocks o boh moneary and fiscal policy rules. JEL Classificaion Code: E37; E52; E62; E63 Keywords: Transmission Mechanisms; New Keynesian Model; Tobin s Q; Two-Channel Moneary Policy *Corresponding auhor: Sylianos Giannoulakis, Deparmen of Economics, Ahens Universiy of Economics and Business, 76 Paission Sree, Ahens 10434, Greece. sgiannoulak@aueb.gr [1]

3 I. Inroducion Dynamic Sochasic General Equilibrium (DSGE) models wih nominal rigidiies ( New Keynesian models) became very popular in he las decades. In his paper we presen a medium-size New Keynesian Dynamic Sochasic General Equilibrium Model (DSGE) wih boh fiscal and moneary policies analysis. The analysis is disinguished from he convenional New Keynesian sudies in hree ways: Firs, we focus on he role of capial - he key ingredien in he ransiion from he basic framework o he medium - scale DSGE models. More specifically, we analyze he accumulaion of capial process in a New Keynesian conex under indeerminacy. Second, we assume a wo-channel moneary policy, which is conduced hrough a rule for money supply and a Taylor-ype rule for ineres raes, in order o keep up wih he ECB and Fed s policies. Boh cenral banks, in order o deal wih he negaive consequences of he 2008 crisis, iniially proceeded o lower ineres raes and hen o an increase of he money supply (wih he form of he Quaniaive Easing -QE). Also, in order o capure he crisis effecs we inroduce exogenous shocks o boh rules. Las bu no leas, in order o examine he efficiency of he fiscal policy and is ineracions wih he moneary one, we adop a simple rule for public consumpion impored in he lieraure by Heer and Maußner (2014). Again, in order o capure he crisis effecs we inroduce an exogenous spending shock. The paper proceeds as follows. In he nex secion a New Keynesian DSGE model wih capial accumulaion is derived. In secion III, we presen he analyic soluion of he model. In Secion IV, he model is calibraed. In secions V, he model is simulaed and is dynamic properies are analyzed. Secion VI concludes. [2]

4 II. The Model We analyze he effecs of capial accumulaion in he conex of a commonly used general equilibrium model wih Calvo-ype price sickiness. More specifically we consider a canonical se-up model in which labor markes are compeiive and he goods markes are monopolisically compeiive. The key concep of our analysis is ha we discern wo kinds of firms: capial producing and final good firms. Capial firms conver consumpion goods ino capial hrough invesmen, and ren his capial o goods producing firms for a renal rae. Final Good firms uses his capial parallel wih labor for producion. Bu le us have a non-formal overview of he model before we lay ou he paricular assumpions explicily: The model economy feaures hree secors, a consumpion secor, a producive secor, and he governmen. Noe ha ime is discree and he planning horizon is infinie and ha he number of he households is equal o he number of he firms. Consumpion secor Households purchase consumpion goods, save via bonds and capial, and supply labor services and capial o he producive secor. They derive uiliy from labor, money and consumpion and are assumed o be represenable by one sand-in agen who maximizes his recursive lifeime uiliy. Producive secor The economy s oupu is produced hrough labor and capial inpus. We discern wo kinds of firms: capial producing and final good firms. Capial firms conver consumpion goods ino capial hrough invesmen, and ren his capial o goods producing firms for a renal rae. Moreover, capial accumulaion is subjec o real adjusmen coss which generae a ime varying real price of capial, Tobin s q. Final Good firms uses his capial parallel wih labor for producion. Monopolisic compeiion in he good s marke gives rise o price seing power which is again consrain by Calvo-ype sickiness. [3]

5 Public secor The governmen conducs he fiscal policy and an independen moneary auhoriy, he Cenral Bank, conducs he moneary one. To be more precise, he governmen exogenously purchases public consumpion financed hrough axes and governmen bonds. The governmen spending follows a very simple auoregressive process. The moneary policy has wo pars: i) he deerminaion of he money supply hrough a simple exogenous money creaion process, and ii) he deerminaion of he bonds nominal ineres rae hrough a Taylor-ype feedback rule. Via he household s Euler equaion for hese bond s real ineres, his rule impacs he real economy due o he presence of he above oulined disorions. 1. Households We suppose ex ane symmery, so we will analyze he behavior of he represenaive household. Is uiliy funcion is given by: E ( 1 =ο 1+ρ ) [ C 1 σ + β 1 σ 1 b (M ) 1 b L 1+λ ] (1) P 1+λ where C is he consumpion, (M/P) is he real money balances and L is he labor supply. The consumpion is consised by many goods, indexed by j, jε[0,1]. The aggregae consumpion across he individual goods is defined in he following CES form, C = [ ε 1 ε c j=0 j dj] ε ε 1 (2) where ε is he demand elasiciy of subsiuion for he individual goods and ε>1. The represenaive household has o deal wih wo problems: 1.1) Allocaion of spending across goods The household in order o deermine his opimal allocaion has o minimize he cos of buying C, 1 min cj p 0 j c j dj (3) [4]

6 s. C = [ ε 1 ε c j=0 j dj] ε ε 1 From he Lagrangian, 1 L = p 0 j c j dj + ψ {C [ ε 1 ε c j=0 j dj] ε 1 ε } (4) F.O.C. : L 1 ε 1 = p C j ψ ([ c j ε dj] j 0 1 ε 1 c j 1 ε ) = 0 => => (2) p j ψ C ε 1 c ε j = 0 => => c j = ( p j ψ ) ε C (5) Subsiuing his o he definiion of aggregae consumpion across he individual goods (2), i follows: C = [ j=0 [( p j ) ε C ψ ] ε 1 ε dj] ε ε 1 = ( 1 ψ ) ε [ p j 1 1 ε dj] ε ε 1 0 C => 1 0 => ψ = [ p j 1 1 ε dj] 1 ε = P (6) The Lagrange muliplier can be considered o be he price index appropriae for he consumpion bundle. And by subsiuing equaion (6) back o he firs order condiion (5), yields, c j = ( p j P ) ε C (7) As ε we move owards perfec compeiion and firms enjoy less marke power. This equaion is effecively he demand curve facing he firm j for his produc. Addiionally, when he household follows his opimal policy i holds ha: 1 p j j=0 c j dj = P C (8) [5]

7 1.2) Allocaion of spending across ime The Budge Consrain of he represenaive household is defined as, 1 p j=0 j c j dj + q I + M i B R K H 1 + B 1 + M 1 + W L H T + Π (9) where p j is he price of good j, W is he nominal wage, i is he nominal ineres rae (which also is he nominal gross bond reurn), B is an one-period bond,t an exernal ransfer from he governmen o he household ( e.g. axes), R is he nominal renal rae on capial, K is he household s capial savings (i.e. he par of he capial sock ha is owned by he household), I he invesmens, q is he price of capial and finally Π denoes profis received from firms owned by households. We assume he following law of moion for he capial sock, I = K (1 δ)k 1 (10) According he relaions (9) and (10) he Budge consrain becomes, P C +q [K H (1 δ)k H 1 ] + M i B R K H 1 + B 1 + M 1 + W L H T + Π (11) Therefore, forming he Lagrangian from he problem, H P +i L = E {( 1 i=0 1+ρ )i [ C 1 σ + β 1 σ 1 b (M ) 1 b L 1+λ ] μ P 1+λ +i(c +i + q +i [ K +i (1 δ) K H +i 1] + M +i + P +i P +i 1 B +i W +i L H 1+i P +i P +i R +i H K +i P +i 1 M +i 1 (1 + i +i P +i ) B +i 1 + +i P +i T +i P +i Π +i P +i )} (12) From he firs order condiions we ge: C σ = 1 Ε 1+ρ {C σ +1 [ (1 δ)q +1+R +1 consumpion q ] P P +1 } (13) -> Euler equaion for [6]

8 L λ = W P C σ (14) Euler for Labor β( M ) b C σ P P ρ C +1 σ P +1 = 0 (15) Euler equaion for money Ε {( C +1 σ P +1 ) } = 1 C P 1 + ρ (1 + i )(16) Euler for bonds Combining equaions (15) and (16) we ge he money demand funcion: i 1+i C σ = β ( M P ) b (17) Money Demand 2. Governmen 2.1) Fiscal Policy In some period,, he governmen collecs real axes T, consumes a quaniy G = gyy which is a percenage of he oal income of he economy, and issues bonds of nominal volume B +1 which pay he predeermined nominal ineres i. I hereby has o resric is aciviy o policies ha saisfy is budge consrain, condiional on no defauling Governmen s BC: Revenues = Expenses B B 1 + T = G + i B 1 + M M 1 (18) As in Heer and Maußner (2014), governmen spending is exogenous. In paricular, we assume he following auoregressive process of firs order where: - ρ G [0,1] - G ss = seady sae level - ε G ~iidn(0,1) G G = (1 ρ G )G ss + ρ G G 1 + ε +1 [7]

9 Public consumpion and axes deermines he sances of he governmen s fiscal policy. To be more precise, he hree main sances of fiscal policy are: - Neural fiscal policy is usually underaken when an economy is in equilibrium. Governmen spending is fully funded by ax revenue and overall he budge oucome has a neural effec on he level of economic aciviy. - Expansionary fiscal policy involves governmen spending exceeding ax revenue, and is usually underaken during recessions. I is also known as reflaionary fiscal policy. - Conracionary fiscal policy occurs when governmen spending is lower han ax revenue, and is usually underaken o pay down governmen deb. Furhermore, he governmen is no allowed o apply a Ponzi scheme o ineremporally finance is expendiures such ha he deb growh rae B+1/B is capped. 2.2) Moneary Policy Moneary policy is conduced by an independen Cenral Bank and i has wo pillars: he money supply which is deermined hrough an exogenous process and he ineres rae which is deermined hrough a Taylor-ype rule ) Money Supply We assume ha he moneary auhoriies conrol money supply hrough he following simple money creaion process: where: M M 1 = 1 + μ (19) μ = ρ m μ 1 + ε m, ε m ~iidn(0,1) The sign of he parameer μ specifically, parly implies he sances of moneary policy. More [8]

10 - If μ < 0, i.e. he governmen reduces he size of he money supply, he moneary policy is conracionary. - If μ > 0, i.e. he governmen increases he size of he money supply, he moneary policy is expansionary. - If μ = 0, i.e. he governmen keeps he size of he money supply consan, he moneary policy is neural ) An Ineres Rae Rule We assume ha he cenral bank follows a Taylor (1993) rule of he form, i = ρ i + φ y y + φ π π + ν (20) where φ y and φ π are posiive coefficiens, and ν is an exogenous sochasic disurbance in he nominal ineres rae. I is worh noing ha because he consan in his rule is equal o ρ, his rule is consisen wih zero seady sae inflaion 1. Τhis rule implies a counercyclical moneary policy. When inflaion is posiive, he cenral bank increases nominal ineres raes in order o reduce i. When employmen is low, i.e. when oupu is lower han is naural level, he cenral bank reduces nominal ineres raes in order o increase employmen and nudge oupu owards is naural level. In addiion, his feedback ineres rae rule does no resul in inflaion and price level indeerminacy if he Taylor principle is saisfied, i.e. if he reacion of nominal ineres raes o inflaion is sufficienly srong. 3. Firms We separae firms ino goods producing and capial producing firms in order o simplify he derivaion of he price seing equaion on he one hand, and he invesmen/q equaion on he oher hand. More specifically, Capial firms conver consumpion goods ino capial 1 See Woodford (2003), for a more exensive and complee analysis. [9]

11 hrough invesmen, and ren his capial o goods producing firms for a renal rae. Final Good firms uses his capial parallel wih labor for producion. In our model, we are going o assume Calvo ype price fixiies. Hence, before we coninue wih he analysis of firms' profi maximizaion problem, i would be helpful o wrie a few words abou Calvo conracs. 3.1) Calvo ype price fixiies In he Calvo saggered conracs model (1983), here is a consan probabiliy 1-γ ha he firm can se a new price. Thus, a proporion 1- γ of firms can rese heir prices in any period, whils he remaining proporion γ keep heir prices consan. This approach has very significan consequences for he moneary policy and business cycles in he basic "new Keynesian" model we are analyzing. According o he above analysis he expeced pricing duraion will be equal o (1 γ) sγ s γ s=0 = 1 γ. Since all firms se he same prices in period, i follows ha, P = (γ(p 1 ) 1 ε + (1 γ)(p ) 1 ε )) 1 1 ε (21) so he dynamic adjusmen of he price level is given by, ) 1 ε = γ + (1 γ) ( P ) 1 ε P 1 P (22) 1 ( P In he seady sae wih zero inflaion i holds ha, P = P 1 = P = P. We can wrie equaion (22) as, f(p, P 1, P ) = ( P ) 1 ε γ (1 γ) ( P ) 1 ε P 1 P = 0 => 1 f(p, P 1, P ) = e (1 ε)[lnp lnp 1 ] γ (1 γ)e (1 ε)[lnp lnp 1 ] = 0 By firs order Taylor log-linear approximaion around he long run equilibrium wih zero inflaion we ge, [10]

12 f(p, P 1, P ) f(p, P 1, P ) LR + f lnp LR (lnp lnp LR ) + f LR (lnp lnp 1 lnp LR 1 ) + f LR (lnp 1 lnp lnp LR ) = + [ e (1 ε)lnp 1 ε P γ)e (1 ε)lnp 1 ε P e (1 ε)lnp 1 ε P (lnp lnp) P 1 ε + (1 γ)e (1 ε)lnp P ] (lnp 1 lnp) + (1 P 1 (lnp lnp) P = P P 1 (1 γ)(p P 1 ) = 0 => P P 1 = (1 γ)(p P 1 ) (17) or P = 1 1 γ P γ 1 γ P 1 (23) The fac ha firms se higher prices han he previous period prices causes he inflaion. 3.2) Capial producing firms Capial firms conver consumpion goods ino capial hrough invesmen, and ren his capial o goods producing firms for a renal rae R. The capial sock evolves according o I = K (1 δ)k 1 (9) and period profis for hese firms are given by D = R K 1 I φ k 2 K 1( I K 1 δ) 2 (24) where he las erm capures convex adjusmen coss o physical capial. [11]

13 The firm wans o maximize is real value, i.e. max V 1 {I,K } = ( s= 1 + ρ )s s= {R s K s 1 I s φ k 2 K s 1( I s K s 1 δ) 2 } s.. I s = K s (1 δ)k s 1 Forming he Lagrangian equaion, we ge: 1 L = ( 1 + ρ )s s= {R s K s 1 I s φ k 2 K s 1 ( I 2 s δ) q K s (K s (1 δ)k s 1 I s )} s 1 FOCs: (I ): q = 1 + φ k ( I K 1 δ) (25) (26) (K ): q = 1 {R 1+ρ +1 φ k 2 (I +1 δ) 2 + φ K k (( I +1 ) 2 δ I +1 ) + (1 δ)q K K +1 } The Lagrange muliplier q plays a cenral role (his is Tobin s q). As any oher Lagrange muliplier, i is a shadow price. In his case, q is he shadow price of capial in place a he end of period. Under he opimal plan, he firm invess such ha he marginal cos of an addiional uni of capial (which equals 1 plus he adjusmen cos) mus equal he shadow price of capial. We can also wrie his as he invesmen equaion ha Tobin (1969) posied: I = ( q 1 φ k + δ)k 1 (27) [12]

14 So invesmen is only posiive when q > 1, i.e. when he shadow price of capial exceeds he price of new capial (before adjusmens coss). Equaion (26) plays he role of an invesmen Euler condiion. The shadow price of capial oday mus equal he discouned value of: - he reurn of capial nex period, - wha you save in adjusmen coss nex period, - he fuure shadow price (since capial can be sold nex period). By using he no-bubble condiion lim q +T T (1+r) we can rewrie equaion (21) as follows, T = 0 (27) and using ieraive subsiuion q = ( 1 s=+1 1+ρ )s {R s+1 + φ k 2 [(I s+1 ) 2 δ 2 ]} (28) K s so q reflecs he NPV of all fuure marginal reurn and reduced adjusmen cos ha you ge from purchasing one uni of capial. 3.3) Final Good firms We assume ha all firms have access o he same echnology. Firms face hree consrains in order o maximize heir profis. Firsly, hey have o work wih a given producion echnology given by, Y = A K α L 1 α (29) which is a Cobb-Douglas producion funcion wih wo inpus, labor and capial, and an aggregae disurbance A (In fac, i is he Toal Facor Produciviy - TFP). The Final Good firms ren he capial sock from he Capial Producing firms. Secondly, firms face he downward sloping demand curve given by, c j = ( p j P ) ε C => y j = ( p j P ) ε Y (8') [13]

15 where Y denoes aggregae demand. Thirdly, we are going o assume Calvo conracs. According o hem in any period a random proporion (1-γ) of firms is able o change heir price. Thus, in order o se prices oday, firms ough o ake ino consideraion he exising fuure economic condiions. Hence, all firms which are able o change heir prices in period, solve he following profi maximizaion problem, max i=0 γ s i E ( ( 1 s=0 1+i +s ) ( p j Y P +i W +i L +i P +i R +i +i K P +i +i )) (30) s.. Y +i = ( p j ) ε Y P +i (8 ) +i where L +i, K +i and Y +i deermined is price in period. are he labor and oupu level in period +i of he firm which Labor Marke We can deermine he real marginal cos of producion by solving he following cosminimizaion problem for firm j: or by forming he Lagrangian equaion, min TC = W L L j P j R K P j s.. Y j = c j = A K j α L j 1 α = Υ (29) L = W P L j R P K j λ (A K j α L j 1 α Υ ) F.O.C.: L = 0 => W = λ(1 α)a L j P ( K j ) α (31) L j L Κ j = 0 => R P = λαa ( K j L j ) α 1 (32) Dividing he above equaions, we ge, [14]

16 W = 1 α R α (K j L j ) (33) Now, subsiuing (32) o (30) or (31) we ge he Lagrange muliplier: λ = ( R α )α ( W 1 α )1 α Of course, he Lagrange muliplier (i.e. he shadow price which shows he change of oal cos for a marginal increase of producion) is he marginal cos, λ = MC = ( R α )α ( W 1 α )1 α (34) Therefore, he profi-maximizaion problem (30) becomes: max E i=0 γ i Δ i,+1 p j [ ( p j ) 1 ε Y P +i λ(1 α)a ( K +i +i W +i P +i L +i ) α L +i + λαa ( K +i R +i P +i ) α 1 K L +i +i ] => max E i=0 γ i Δ i,+1 [( p j ) 1 ε a K Y p j P +i λ{(1 α)a +i + αa K a +i α 1 +i L +i L α 1}] +i => max E i=0 γ i Δ i,+1 [( p j ) 1 ε Y p j P +i MC A K α 1 α L ] +i Y +i => max p j E γ i i=0 Δ i,+1 [( p 1 ε j ) P +i Y +i MC Y +i ] [15]

17 => max p j E γ i i=0 Δ i,+1 ( p 1 ε j ) Y P +i MC ( p j +i [ ε ) P +i Y +i Y +i ] => max p j E γ i i=0 Δ i,+1 [( p 1 ε j ) MC P ( p ε j ) ] Y +i P +i (30 ) +i For he opimal price p * he firs order condiion is given by, E i=0 γ i Δ i,+1 [(1 ε) ( p j ) ε εmc P +i ] ( 1 +i p ) ( p ) ε Y+i = 0 (35) P +i The above equaion describes he opimal pricing policy of a firm j. 3.4) The New Keynesian Phillips Curve By he firs order condiion for he opimal price p * (35), inflaion can be deermined as, ( p ) = ( ε Ε P ε 1 ) E i=0 γi i=0 Δ i,+1 MC +i ( P ε +i P ) γ i Δ i,+1 ( P +i P ) ε 1 (36) By log-linearizing he above relaion i follows, p = E [ γ i β i (MC +i + P +i ) i=0 (36 ) This can be quasi-differenced o yield a forward-looking difference equaion in he opimal rese price, p = γβe p +1 + MC + P (37) We remind ha he log-linear price evoluion index funcion (20) is, [16]

18 By equaions (23) and (37) i follows, P P 1 = (1 γ)(p P 1 ) (23) P 1 γ γ 1 γ P 1 = γβ ( P +1 1 γ γ ) γβ 1 γ P + MC + P (38) and solving for inflaion yields, π = βe π +1 + κ MC (39) where κ = (1 γ)(1 βγ) γ (40) Equaion (39) is he new Keynesian Phillips curve embedded in a General Equilibrium Model. This equaion is nohing else bu an expecaions augmened Phillips curve, which saes ha inflaion rises when he real marginal coss rise. Also, i is nohing else bu an aggregae supply curve for he whole economy. 4. General Equilibrium In a dynamic general equilibrium, all markes in he economy have o be cleared simulaneously wih all agens acing muually opimal a all ime. Firsly, we assume ex pos symmery: - P j = P, firm j - Y j = Y, firm j - L j = L, firm j - K j = K, firm j. Now, we are able o demonsrae each marke s clearing condiion. [17]

19 i) Labor Marke: L H f = L ii) Capial Marke: K H f = K iii) Bonds Marke: B = 0 iv) Money Marke: M = M = cons, given by he governmen. v) Dividends: Π H = Π f = 0 vi) Goods Marke: Adding Households and governmen s budge consrains and using he clearing condiions of he oher markes we ge he desired clearing condiion which is nohing else bu he resource consrain of he economy. Indeed: C +I + M+ 1 1+i B = R K H 1 + B 1 + M 1 + W L H + Π H T (11) B B 1 + T = G + i B 1 + M M 1 (18) Π H = Π f = Y W L R K We conclude: Y = C +I + G (41) Resource Consrain An implici assumpion of his consrain is ha he elasiciy of subsiuion beween individual consumpion goods, σ, is he same as he elasiciy of subsiuion beween individual invesmen goods. Combining he above condiions, imposing symmery beween firms and households he equilibrium of he economy is described by he following equaions: Y = C +I + G (41) Resource Consrain I = K (1 δ)k 1 (9) Law Maion of Capial [18]

20 A) Demand Side C σ = ρ Ε {C +1 σ Z +1 P P +1 } (13) Euler for consumpion where Z +1 = (1 δ)q +1+R +1 q L λ = W P C σ (14) Euler for Labor i 1+i C σ = β ( M P ) b (17) Money Demand M M 1 = 1 + μ (19) Money Creaion Process i = ρ i + φ y y + φ π π + ν (20) Taylor Rule q = 1 + φ k ( I K 1 δ) = 1 + φ k x (25) Tobin s q where x = K K 1 K 1 is he ne invesmen rae. q = 1 1+r {R +1 + φ k 2 [(I +1 K ) 2 δ 2 ] + (1 δ)q +1 } (26) Invesmen Euler Y = A K α L 1 α (29) Aggregae Producion Funcion W P = MC (1 α)a ( K L ) α (31) Demand of Labor R P = MC αa ( K L ) α 1 (32) Demand of Capial Or combining hem, W = 1 α R α (K L ) (33) MC = ( R α )α ( W 1 α )1 α (34) Marginal Cos [19]

21 (1 + i ) = (1 + R )(1 + π ) (42) Fisher Equaion 2 B) Supply Side π = βe π +1 + κ MC (39) NKPC Again, he NKPC plays he role of he economy s aggregae supply. Moreover, AS and NKPC are he wo sides of he same coin. Seady Sae In Seady Sae (SS) our variables are unchanging in ime. Hence, (40) => Y ss = C ss +I ss + G ss (9) => I ss = δk ss (13) => Z ss = 1 + ρ and R ss = δ + ρ 2 The well-known Fisher equaion provides he link beween nominal and real ineres raes. Here (1 + π) is one plus he inflaion rae, i is he nominal ineres rae and R is he real ineres rae. The inflaion rae π+1 is defined as usual as he percenage change in he price level from period o period + 1. π+1 = (P+1 P)/P. If a period is one year, hen he price level nex year is equal o he price his year muliplied by (1 + π): P+1 = (1 + π) P. The Fisher equaion says ha hese wo conracs should be equivalen: (1 + i) = (1 + R) (1 + π). As an approximaion, his equaion implies: i R + π. [20]

22 (14) => L λ ss = W ss R ss C ss σ (17) => (19) => μ = 0 i ss C σ 1 + i ss = β ( M b ss ) ss P ss (24) => q ss = 1 and x ss = 1 (25) => R ss = δ + ρ (29) => Y ss = A ss K ss α L ss 1 α (31) => W ss P ss = MC ss (1 α)a ss ( K ss L ss ) α (32) => R ss P ss = MC ss αa ss ( K ss L ss ) α 1 (33) => W ss = 1 α R ss α (K ss ) => W L ss = = 1 α ss α (K ss L ss ) (δ + ρ) (34) => MC ss = ( R ss α )α ( W ss 1 α )1 α (41) => i ss = R ss = ρ + δ (39) => MC ss = (1 β)π κ Log-Linearizaion We log-linearize he above equaions around seady sae (all variables are expressed as percenage poin deviaions from seady sae): C ss Y ss ĉ + K ss Y ss k K ss Y ss (1 δ)k 1 + G ss Y ss ĝ = ŷ (40 ) [21]

23 Ι = 1 δ [k (1 δ)k 1 ](9 ) c +1 = c + 1 σ (z +1 + (p +1 p )) (13 ) E π +1 where: ẑ +1 = q + α 1 q +1 + α 2 r +1, α 1 = 1 δ, α 1+ρ 2 = ρ δ 1+ρ ŵ p = σĉ + λl (14 ) m p = 1 b (σc i ) (17 ) m m 1 = g m (19 ) q = φ κ (k k 1 ) (23 ) (1 + ρ)q + (1 δ)q +1 + (ρ + δ)r +1 + δφ κ (k +1 k ) δq +1 = 0 (24 ) y = a + ak + (1 a)l (27 ) where: AR(1): α = να 1 + ε, ε ~WN(0, σ ε 2 ) w p = mc + a + (1 a)(k l ) (30 ) r p = mc + a + a(k l ) (31 ) w r = k l (32 ) mc = ar + (1 a)w (33 ) i = r + π (41 ) π = βe π +1 + κ mc (38) [22]

24 III. Analyical Soluion of he Model By simplifying he condiions of general equilibrium we ge he following 16 x16 sysem of equilibrium firs-order difference equaions, ĉ + k (1 δ)k 1 + ĝ = ŷ (40 ) Recource Consrain Î where: AR(1): ĝ = ρ g ĝ 1 + ε g c +1 = c + 1 σ (z +1 + E π +1 ) (13 ) Euler for consumpion ẑ +1 = q 1 δ 1 + ρ q +1 ρ δ 1 + ρ r +1 ŵ p = σĉ + λl (14 ) Euler for Labor m p = 1 b (σc i ) (17 ) Money Demand m m 1 = μ (19 ) Money creaion process where: AR(1): μ = ρ m μ 1 + ε m (1 + ρ)q + (1 δ)q +1 + (ρ + δ)r +1 δq +1 = 0 y = a + ak + (1 a)l (27 ) Producion Funcion where: AR(1): α = ρ α α 1 + ε α, ε ~WN(0, σ ε 2 ) ŵ r = k l (32 ) mc = αr + (1 α)ŵ (33 ) Marginal Cos i = r + π (41 ) Fisher Equaion π = βe π +1 + κ mc (38) NKPC i = ρ i + φ y y + φ π π + ν (20) Taylor Rule [23]

25 We discree our parameers ino: i) Sae/pre-deermined: {k, i, a, g, μ} ii) Conrol/jump: {y, c, L, Inves., m, r, w, π, mc, z, q} In addiion, we can caegorize he model s parameers as follows: - RBC Parameers: { α, ρ, δ, σ, λ, g y } - New Keynesian Parameers: {β, ε, ρ i, φ y, φ π } - Shock Parameers: {ρ g, ρ m, ρ α } By manipulaing he above funcions, we conclude o he following final 13 x 13 sysem of equilibrium firs-order difference equaions, k +1= ŷ ĉ ĝ + (1 δ)k g ĝ +1 = ρ g ĝ + ε +1 c +1 1 σ z +1 1 σ E π +1 = c z +1 + δ 1 + ρ q +1 2δ 1 + ρ r +1 = 0 m + σ b 1 b c + λl w + 1 b r + 1 b π = 0 m +1 μ +1 = m m μ +1 = ρ m μ + ε +1 q +1 + φ κ k +1 = φ κ k (23 ) y a ak (1 a)l = 0 α α +1 = ρ α α + ε +1 [24]

26 ŵ r k + l = 0 βe π +1 = π κ αk κ (1 a)l r + π ρ i φ y y φ π π ν = 0 Now, we have he following 13 variables, Sae/pre-deermined: s = [k, a, g, μ], Conrol/jump: x = [m, y, c, l, r, z, q, w, π]. Now we can give he sae-space form of our sysem, where: A= A [ s +1 x +1 ] = B [ s x ] + Ce σ σ δ 1 + ρ 1 δ 1 + ρ φ κ β [ ] [25]

27 B= 1 δ ρ g σ(1 b) λ [ b ρ m φ κ a (1 a) ρ a κ α κ (1 α) φ y φ π] b b C= [ ] g ε εm and e = εα [ ρ i + ν ] Since marix A is non-singular (de(a)=0) he Blanchard-Kahn s Mehod canno be applied o solve he model. Hence, we adop he QZ decomposiion form. IV. Calibraion Table 1 conains he calibraed parameers. The choice of parameers is one of he main feaures of he analysis as i mus represen economic feaures and o ensure he sabiliy of he sysem. The parameers are separaed ino RBC and New Keynesian parameers. For he laer, we follow he sandard lieraure. To be more precise, new Keynesian parameers [26]

28 are mosly chosen as in Galí (2008) and he recen work by Pouineau, Sobczak and Vermandel (2015). Regarding he Taylor rule, he moneary auhoriies should respond more han proporionally o inflaion developmens (namely, φ π > 1) according o he Taylor principle 3. In his case a rise in inflaion leads o a more han proporional rise in nominal ineres causing an increase in real ineres raes ha affecs agens economic decisions and hus he real macroeconomic equilibrium of he model. In addiion, he inraemporal elasiciy beween inermediae goods is se a 6 which implies a seady sae mark-up of 20 % in he goods marke corresponding o wha is observed in main developed Parameers Explanaion Value α share of capial in oupu 0.36 ρ discoun facor 0.2 δ depreciaion of capial σ risk aversion for 1 consumpion λ labor disuiliy 1 b risk aversion for cash 1 β NKPC, forward erm 0.75 ε elasiciy/ mark-up on prices 6 γ φ κ φ y φ π ρ i ρ g ρ m ρ a porion of firms ha canno change heir prices in capial adjusmen cos parameer moneary policy GDP growh marke moneary policy inflaion growh arge moneary Policy smoohing parameer governmen sending s shock smoohing parameer money supply s shock smoohing parameer produciviy s shock smoohing parameer Table 1: Calibraed Parameers In paricular, he Taylor rule sipulaes ha for each one-percen increase in inflaion, he cenral bank should raise he nominal ineres rae by more han one percenage poin. This aspec of he rule is ofen called he Taylor principle. [27]

29 economies. For he RBC parameers we also follow he sandard bibliography and more specifically we use he values from Cooley and Presco (1995). Regarding he policy shock s smoohing parameers, for he governmen shock s smoohing parameer we follow he work by Heer and Maußner (2014) and he money supply s smoohing parameer is chosen by Sim s (2015) paper. Finally, in order o examine he effec of he capial o he fiscal and moneary policy we allow he adjusmen cos parameer o ake hree possible values: 0, 1.5, and 3. The case of φ κ = 0 corresponds wih he case of he sandard RBC model wihou adjusmen coss. V. Impulse Responses We nex explore he inernal mechanics of he model by ploing some impulse response funcions. Each impulse response repors he effec of a one sandard deviaion shock on he variables of he model, expressed in percen deviaion from heir seady sae level. Figure 1: Produciviy Shock [28]

30 Le's sar our analysis wih impulse responses o a one percen posiive produciviy shock for oupu, consumpion, invesmen, labor, Tobin s q, real wages, nominal ineres rae, inflaion and echnology, which are ploed in Figure 1. Oupu, consumpion, and invesmen all increase on impac. Hours worked decline. This decline is driven by real fricions (in paricular he invesmen adjusmen cos). The pah of he real wage is similar o oupu. In addiion, he pah of capial s shadow price, i.e. Tobin s q, follows an analogous pah o invesmen. Finally, inflaion falls. Figure 2: Ineres Rae Policy Shock Now, le's focus our analysis on he impulse responses o a one percen ineres rae policy shock for oupu, consumpion, invesmen, labor, Tobin s q, real wages, nominal ineres rae and inflaion. We plo he effecs of such a shock in Figure 2. Since his is a posiive shock o he ineres rae rule, i implies conracionary moneary. Oupu falls on impac and follows a hump-shape before revering back o rend. Consumpion and invesmen [29]

31 boh fall. Hours worked again decline due o he invesmen adjusmen cos. In addiion, he pah of capial s shadow price, i.e. Tobin s q, follows an analogous pah o invesmen. Ineresingly, real wage rises on impac a he beginning bu afer some ime i falls. Inflaion falls unil i reurns o he zero level. Figure 3: Money Supply Shock Now, le's focus our analysis on he impulse responses o a one percen negaive money supply shock (which implies conracionary moneary policy) for oupu, consumpion, invesmen, labor, Tobin s q, real wages, nominal ineres rae and inflaion. We plo he effecs of such a shock in Figure 3. As we can see he impulse responses of a negaive money supply shock are idenical wih hose of a posiive ineres rae policy rule. This implies ha he wo ools of moneary policy (namely, ineres rae and money supply) have very similar impac on he real economy. [30]

32 Boh FED and ECB followed an expansionary moneary policy in order o deal wih he negaive consequences of he 2008 crisis. More specifically, hey iniially proceeded o lower ineres raes and hen o an increase of he money supply (wih he form of he Quaniaive Easing -QE). Of course, he adopion of such a policy re-sparkled a grea deal of conroversy beween Keynesian and neoclassical economiss around he issue of he liquidiy rap. The formers claim ha furher injecions of cash ino he privae banking sysem by a cenral bank will fail o decrease ineres raes and hence make moneary policy ineffecive 4. On he oher hand, he neoclassical economiss assered ha, even in a liquidiy rap, expansive moneary policy could sill simulae he economy via he direc effecs of increased money socks on aggregae demand. This essenially was he hope of he cenral banks of he Unied Saes and Europe in , wih heir foray ino quaniaive easing. These policy iniiaives ried o simulae he economy hrough mehods oher han he reducion of shor-erm ineres raes. Figure 4: Governmen Spending Shock 4 See for example Krugman Paul s aricle "How much of he world is in a liquidiy rap?" in The New York Times (17 March 2010). [31]

33 Finally, we examine he impulse responses o a one percen governmen spending shock for oupu, consumpion, invesmen, labor, Tobin s q, real wages, nominal ineres rae, inflaion and governmenal expendiures in Figure 4. This also raises oupu and inflaion. Labor goes up, while he real wage falls. Consumpion and invesmen boh fall. Again, he pah of capial s shadow price, i.e. Tobin s q, follows an analogous pah o invesmen. VI. Conclusions The purpose of his aricle is o carefully lay ou he inernal moneary and fiscal ransmission mechanisms in he conex of a New Keynesian model. More specifically, his paper presens fiscal and moneary policies analysis in a conex of a medium-size New Keynesian Dynamic Sochasic General Equilibrium Model (DSGE) wih Calvo ype price sickiness and capial accumulaion. The analysis is disinguished from he convenional New Keynesian sudies in hree ways. Firs, we focus on he role of capial - he key ingredien in he ransiion from he basic framework o he medium - scale DSGE models. Second, we assume a wo-channel moneary policy, i.e. i is conduced hrough a rule for money supply and a Taylor-ype rule for ineres raes, in order o keep up wih he ECB and Fed s policies. Boh cenral banks, in order o deal wih he negaive consequences of he 2008 crisis, iniially proceeded o lower ineres raes and hen o an increase of he money supply (wih he form of he Quaniaive Easing -QE). Third, in order o examine he efficiency of he fiscal policy and is ineracions wih he moneary one we adop a simple rule for public consumpion impored in he lieraure by Heer and Maußner (2014). Finally, in order o capure he dynamic crisis effecs we inroduce exogenous shocks o boh moneary and fiscal policy rules. Our paper is a furher sep o he effor for he invigoraion of he link beween economic realiy and heory. More specifically, boh ECB and FED conduced a wo sage expansionary moneary policy afer he burs of he 2008 global financial crisis: hey iniially proceeded o lower ineres raes and hen o an increase of he money supply [32]

34 (hrough he Quaniaive Easing -QE). In his paper we adoped a wo channel moneary policy in order o simulae he moneary policy of he above cenral banks: i is conduced hrough a rule for money supply and a Taylor-ype rule for ineres raes. We found ha he wo ools of moneary policy (ineres rae and money supply) have very similar impac on he economy. More specifically, we found ha a conracionary moneary policy (which can be achieved eiher by a higher ineres rae for governmen bonds or a lower money supply) leads o a significan decrease of oupu, consumpion, employmen and invesmen. I also drives o a emporary deflaion. Of course, hese changes are no persisen over ime. The growh of he economy will reurn o is rend and he inflaion o is zero level. Hence, an expansionary moneary policy is expeced o emporary one up he economy (by increasing he oupu, he invesmens and he consumpion) and miigae he negaive effecs of he crisis wihou creaing a persisen inflaion. Thus, our model is compaible wih he recipe of he expansionary moneary for he ackling of he severe financial crisis which was adoped by he aforemenioned cenral banks. Finally, regarding he governmen s fiscal policy we found ha an expansionary spending policy raises oupu and inflaion. In addiion, labor goes up bu he real wages fall. Also, invesmens are decreased a fac ha implies ha such a policy crowds ou he privae secor s spending. Hence, our model is compaible wih he sandard implicaions of economic heory. [33]

35 References Calvo, G., 1983, Saggered prices in a uiliy-maximizing framework, Journal of Moneary Economics, Vol. 12 (3), pp Chrisiano, L.J., Eichenbaum, M., and Evans, C.L., 2005, Nominal Rigidiies and he Dynamic Effecs of a Shock o Moneary Policy, Journal of Poliical Economy, Vol. 113 (1), pp Cooley, T., and Presco, E., 1995, Froniers of Business Cycle Research, chaper 1, Princeon Universiy Press. Del Negro, M., and Sims, C. A., "When does a cenral bank s balance shee require fiscal suppor?," Journal of Moneary Economics, Elsevier, Vol. 73(C), pp Gali, J., 2008, Moneary Policy, Inflaion, and he Business Cycle: An Inroducion o he New Keynesian Framework and Is Applicaions, Princeon Universiy Press. Gali, J., Smes, F., and Wouers, R., 2011, Unemploymen in an Esimaed New Keynesian Model. NBER Working Paper, no Heer, B., and Maußner, A., 2014, Q-Targeing in New Keynesian Models. Mimeo. Universiy of Augsburg. Lucas, R., 1987, Models of business cycles, New York: Basil Blackwell ediions. Pouineau, J. C., Sobczak, K., and Vermandel, G., 2015, The analyics of he New Keynesian 3-equaion Model, Economics and Business Review, Vol. 1 (15), pp Presco, E.C., 1986, Theory Ahead of Business Cycle Measuremen. Federal Reserve Bank of Minneapolis Quarerly Review, No. 10, pp Smes, F., and Wouers, R., 2003, An Esimaed Dynamic Sochasic General Equilibrium Model of he Euro Area, Journal of he European Economic Associaion, Vol. 1 (5), pp Taylor, J., 1993, Discreion versus Policy Rules in pracice", Carnegie-Rocheser Conference Series on Public Policy, Vol. 39, pp Tobin, J., 1958, Esimaion of Relaionships for Limied Dependen Variables, Economerica, Vol. 26 (1), pp [34]