Notes on New-Keynesian Models

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1 Noes on New-Keynesian Models Marco Del Negro, Frank Schorfheide, FRBNY DSGE Group Sepember 4, 202 Please do no circulae Smes & Wouers Chrisiano, Eichenbaum, & Evans. Define he problem, FOCs, and equilibrium condiions.. Final goods producers The final good Y is a composie made of a coninuum of goods: [ +λf, +λ Y = Y i f, di.. 0 The final goods producers buy he inermediae goods on he marke, package Y, and resell i o consumers. These firms maximize profis in a perfecly compeiive environmen. Their problem is: max Y,Yi P Y 0 P iy idi [ s.. Y = 0 Y +λf, +λ i f, di µ f,..2 CAVEAT EMPTOR: These are noes on New Keynesian DSGE models wrien for personal use by he auhors; hey have no been revised and hence may well conain errors; hey are no for circulaion. The par of he noes mos relevan for he Handbook of Forecasing Chaper on DSGE Model-Based Forecasing begins on page 45, Secion 8. Correspondence: Marco Del Negro: Research Deparmen, Federal Reserve Bank of New York, 33 Libery Sree, New York NY 0045: marco.delnegro@ny.frb.org. Frank Schorfheide: Deparmen of Economics, 378 Locus Walk, Universiy of Pennsylvania, Philadelphia, PA schorf@ssc.upenn.edu. The views expressed in hese noes do no necessarily reflec hose of he Federal Reserve Bank of New York or he Federal Reserve Sysem.

2 The FOCs are: Y P = µ f,..3 Y i Noe ha [... λ f, = Y λ f, +λ f, P i + µ f, + λ f, [... λ f,y i λ f, +λ f, = From he FOCs one obains: +λ f, P λ i f, Y i = Y..5 P Combining his condiion wih he zero profi condiion hese firms are compeiive one obains an expression for he price of he composie good: [ P = 0 P i λf, λ f, di..6 Noe ha he elasiciy is +λ f, λ f,. λ f, = 0 corresponds o he linear case. λ f, corresponds o he Cobb-Douglas case. We will consrain λ f, 0,. λ f, follows he exogenous process: ln λ f, = ln λ f + ɛ λ,, ɛ λ, Inermediae goods producers Inermediae goods producers i uses he following echnology: Y i = max{z α K i α L i α Z Φ, 0},..8 where Z = Z Υ α α, Υ >...9 Call z = logz /Z. z follows he process: z γ = ρ z z γ + ɛ z,, ɛ z, The firm s profi is given by: P iy i W L i R k K i. 2

3 Cos minimizaion subjec o..8 yields he condiions: L i K i V i αz α K i α L i α = W V iαz α K i α L i α = R k where V i is he Lagrange muliplier associaed wih..8. In urn, hese condiions imply: K i L i = α α Noe ha if we inegrae boh sides of he equaion wr di and define K = K idi and L = L idi we obain a relaionship beween aggregae labor and capial: Toal variable cos is given by K = α α W R k W R k. L... Variable Coss K i = W + R k L L i i = W + R k K i L i ỸiZ α Ki L i α, where Ỹi = Z α K i α L i α is he variable par of oupu. The marginal cos MC is he same for all firms and equal o: MC = W + R k K i L i Z α = α α α α W α α Ki L i R k α Z α...2 Profis can hen be expressed as P i MC Y i MC Z Φ. Noe ha since he las par of his expression does no depend on he firm s decision, i can be safely ignored. Prices are sicky as in Calvo 983. Specifically, each firm can readjus prices wih probabiliy ζ p in each period. We depar rfom Calvo 983 in assuming ha for hose firms ha canno adjus prices, P i will increase a he geomeric weighed average wih weighs ι p and ι p, respecively of he seady sae rae of inflaion π and of las period s inflaion π. For hose firms ha can adjus prices, he problem is o choose a price level P i ha maximizes he expeced presen discouned value of 3

4 profis in all saes of naure where he firm is suck wih ha price in he fuure: max Pi Ξ p P i MC Y i + E s= ζs pβ s Ξ p +s P i P i Π s l= s.. Y +s i = πιp +l π ιp P +s Π s l= πιp +l π ιp +λ f,+s λ f,+s Y +s, MC +s Y +s i..3 where β s Ξ p +s is oday s value of a fuure dollar for he consumers Ξp +s is he Lagrange muliplier associaed wih he consumer s nominal budge consrain - remember here are complee markes so β s Ξ p +s is he same for all consumers. The FOC for he firm is: Ξ p +λ f, Pi λ f, P E s=0 ζs pβ s Ξ p +s P i Π s l= πιp +l π ιp λ f, P P i + λ f, MC Y i+ P i Π s l= πιp +l π ιp P +s +λ f,+s λ f,+s Π s l= πιp +l π ιp λ f,+s P +s + λ f,+s MC +s Y +s i = 0..4 Noe ha all firms readjusing prices face an indenical problem. We will consider only he symmeric equilibrium in which all firms ha can readjus prices will choose he same P i, so we can drop he i index from now on. From..6 i follows ha: P = [ ζ p P λ f + ζ p π ιp π ιp P λ f λ f Households The objecive funcion for household j is given by: [ IE β s b +s logc +s j hc +s j ϕ +s L +s j +ν l + χ +s M+s j νm + ν s=0 l ν m Z+sP +s..6 where C j is consumpion, L j is labor supply oal available hours are normalized o one, and M j are money holdings. Noe ha he household is a habi guy for h > 0. ϕ affecs he marginal uiliy of leisure: i is model as a sochasic preference shifer. Real money balances ener he uiliy funcion deflaed by he sochasic rend growh of he economy, so o make real money demand saionary. χ is anoher sochasic 4

5 preference shifer ha affecs he marginal uiliy from real money balances. b is ye anoher sochasic preference shifer ha scales he overall period uiliy. The preference shifers are exogenous processes common o all households ha evolve as follows: ln ϕ = ρ ϕ ln ϕ + ρ ϕ ln ϕ + ɛ ϕ,, ɛ ϕ, ln χ = ρ χ ln χ + ρ χ ln χ + ɛ χ,, ɛ χ, ln b = ρ b ln b + ɛ b,, ɛ b, The household s budge consrain, wrien in nominal erms, is given by: P +s C +s j + P +s I +s j + B +s j + M +s j R +s B +s j + M +s j + Π +s + W +s jl +s j + R k +su +s j K +s j P +s au +s jυ K+s j,..20 where I j is invesmen, B j is holdings of governmen bonds, R is he gross nominal ineres rae paid on governmen bonds, Π is he per-capia profi he household ges from owning firms assume household pool heir firm shares, so ha hey all receive he same profi W j is he wage earned by household j. The erm wihin parenhesis represens he reurn o owning K j unis of capial. Households choose he uilizaion rae of heir own capial, u j, and end up rening o firms in period an amoun of effecive capial equal o: K j = u j K j,..2 and geing R k u j K j in reurn. They however have o pay a cos of uilizaion in erms of he consumpion good which is equal o au jυ K j. Households accumulae capial according o he equaion: K j = δ K j + Υ µ S I j I j I j,..22 where δ is he rae of depreciaion, and S is he cos of adjusing invesmen, wih S > 0, S > 0. The erm µ is a sochasic disurbance o he price of invesmen relaive o consumpion, which follows he exogenous process: ln µ = ρ µ ln µ + ρ µ ln µ + ɛ µ,, ɛ µ,

6 Call Ξ p j he Lagrange muliplier associaed wih he budge consrain..20 he marginal value of a dollar a ime. We assume here is a complee se of sae coningen securiies in nominal erms, alhough we do no explicily wrie hem in he household s budge consrain. This assumpion implies ha Ξ p j mus be he same for all households in all periods and across all saes of naure: Ξ p j = Ξp for all j and. Alhough we so far kep he j index for all he appropriae variables, we will see ha he assumpion of complee markes implies ha he index will drop ou of mos of hese variables: In equilibrium households will make he same choice of consumpion, money demand, invesmen and capial uilizaion. As we will see, wage rigidiy á la Calvo implies ha leisure and he wage will differ across households. We firs wrie he firs order condiions for consumpion and money demand. The FOCs for consumpion, money holdings, and bonds are: C i P b C j hc j..24 βhie [b + C + j hc j = Ξ p..25 M j νm M i χ b Z P Z = Ξ p P βie [Ξ p B i Ξ p = βr IE [Ξ p The firs FOC equaes he marginal uiliy of consumpion a ime, imes he relaive price of money in erms of he consumpion good, o he marginal uiliy of one dollar a ime. FOCs..25 hrough..27 show ha he quaniy of consumpion, money holdings, and bonds will also be he same apparen across households since he lagrange muliplier is he same, so ha we can drop he j index. Separabiliy in he uiliy funcion is key for his resul: if he marginal uiliy of consumpion depended on leisure, hen equalling he marginal uiliy of consumpion across households would no imply equal consumpion, since leisure differs across j depending on wheher hey can change heir wage or no. Now define Ξ = P Ξ p. The FOCs for consumpion, money and bonds can be rewri- 6

7 en as: Ξ = b C hc βhie [b + C + hc..28 νm M R Z = χ b P R Z..29 Ξ Ξ = βr IE [Ξ + π where inflaion is defined as π = P /P. Le us now address he capial accumulaion/uilizaion problem. Call Ξ k he Lagrange muliplier associaed wih consrain..22. The FOC wih respec o invesmen, capial, and capial uilizaion are: I Ξ k Υ µ S I I S I I I I + βie [Ξ k + Υ+ µ + S I + I I + I 2 = Ξ..3 K Ξ k = βie [Ξ + Rk + P u + au + Υ + + Ξ k + δ u Υ R k P = a u..33 The firs FOC is he law of moion for he shadow value of capial. Noe ha if adjusmen cos were absen, he FOC would simply say ha Ξ k Υ µ is equal o he marginal uiliy of consumpion. In oher words, in absence of adjusmen coss he shadow cos of aking resources away from consumpion equals he shadow benefi absracing from Υ µ of puing hese resources ino invesmen: Tobin s Q is equal o one. The second FOC says ha if I buy a uni of capial oday I have o pay is price in real erms, Ξ k, bu omorrow I will ge he proceeds from rening capial, plus I can sell back he capial ha has no depreciaed. Now o he wage/leisure decision. Before going ino he household s problem, more deails on he labor marke are needed. Labor used by he inermediae goods producers L is a composie: [ +λw, +λ L = L j w, di. 0 There are labor packers who buy he labor from he households, package L, and resell i o he inermediae goods producers. Labor packers maximize profis in a perfecly 7

8 compeiive environmen. From he FOCs of he labor packers one obains: +λ w, W λ j w, L j = L..34 W Combining his condiion wih he zero profi condiion one obains an expression for he wage: [ W = 0 W j λ w, di λw,..35 We will se λ w, = λ w 0,. Given he srucure of he labor marke, he household has marke power: she can choose her wage subjec o..34. subjec o nominal rigidiies á la Calvo. However, she is also Specifically, households can readjus wages wih probabiliy ζ w in each period. For hose ha canno adjus wages, W j will increase a a geomerically weighed average of he seady sae rae increase in wages equal o seady sae inflaion π imes he growh rae of he economy e γ Υ α α and of las period s inflaion imes las period s produciviy π e z. For hose ha can adjus, he problem is o choose a wage W j ha maximizes uiliy in all saes of naure where he household is suck wih ha wage in he fuure: max Wj IE s=0 ζ wβ s b +s [ ϕ +s ν l + L +sj νl s and..34 for s = 0,...,, and W +s j = Π s l= π e γ Υ α α ι w π +l e z +l ι w W j for s =,...,..36 where he... indicae he erms in he uiliy funcion ha are irrelevan for his problem. The FOC for his problem are: W j where Ξ L [ λ ww IE s=0 ζ wβ s Ξ +s Lj +s X,s W j P + + λ w b +sϕ +s L +s j ν l +s Ξ = 0. +s..37 if s = 0 X,s = Π s l= π e γ Υ α α ι w π +l e z +l ι w oherwise. In absence of nominal rigidiies his condiion would amoun o seing he real wage equal o raio of he marginal uiliy of leisure over he marginal uiliy of consumpion 8

9 imes he markup + λ w. All agens readjusing wages face an indenical problem. We will again consider only he symmeric equilibrium in which all agens ha can readjus heir wage will choose he same W j, so we can drop he i index from now on. From..35 i follows ha: W = [ ζ w W λw + ζ w π e γ Υ α α ι w π e z ι w W λw λw Governmen Policies The cenral bank follows a nominal ineres rae rule by adjusing is insrumen in response o deviaions of inflaion and oupu from heir respecive arge levels: R R = R R ρr [ π where R is he seady sae nominal rae and Y π ψ ρr ψ2 Y Y e ɛ R,..39 is nominal oupu. The parameer ρ R deermines he degree of ineres rae smoohing. The moneary policy shock ɛ R, is iid: ɛ R, The cenral bank supplies he money demanded by he household o suppor he desired nominal ineres rae. The governmen budge consrain is of he form P G + R B + M = T + M + B,..4 where T are nominal lump-sum axes or subsidies ha also appear in household s budge consrain. Governmen spending is given by: G = /g Y..42 where g follows he process: ln g = ρ g ln g + ρ g ln g + ɛ g,, ɛ g,

10 ..5 Resource consrains To obain he marke clearing condiion for he final goods marke firs inegrae he HH budge consrain across households, and combine i wih he gvm budge consrain: P C + P I + P G +Π + W jl jdj + R k K jdj P au Υ K jdj. Nex, realize ha Π = Πi di = P i Y i di W L R k K, where L = Li di is oal labor supplied by he labor packers and demanded by he firms, and K = Ki di = K jdj. Now replace he definion of Π ino he HH budge consrain, realize ha by he labor and goods packers zero profi condiion W L = W jl jdj, and P Y = P i Y i di and obain: P C + P I + P G + P au Υ K = P Y, or C + I + au Υ K = Y..44 g where Y is defined by... The relaionship beween oupu and he aggregae inpus, labor anc capial, is: Ẏ = Z α K i α L i α di Z Φ = Z α K/L α Lidi Z Φ = Z α K α L α Z Φ,..45 where I used he fac ha he capial labor raio is consan across firms also, since Ki = K/LLi i mus be he case ha Kidi Lidi = K /L = K/L. The problem wih hese resource consrains is ha wha we observe in he daa is Ẏ = Y idi and L = L jdj, as opposed o Y and L. Bu noe ha from..5: Ẏ = Y P = Y P +λ f, λ f, +λ f, λ f, +λ f, λ P i f, di P +λf, λ f,, 0

11 where P = λ f, P i +λ f, +λ f, λ f, di, and L = L jdj = L W = L W +λ w, λ w, +λ w, λ w, +λ w, λ W j w, di Ẇ +λw, λ w,, where Ẇ = λ +λ w, w, +λ w, λ W j w, dj.

12 ..6 Exogenous Processes The model is supposed o be fied o daa on oupu, consumpion, invesmen, employmen, wages, prices, nominal ineres raes, and money. Technology process: le z = ln Z /Z z γ = ρ z z γ + ɛ z,..46 We will probably resric ρ z o zero. Preference for leisure: ln ϕ = ρ ϕ ln ϕ + ρ ϕ ln ϕ + ɛ ϕ,..47 Money Demand: ln χ = ρ χ ln χ + ρ χ ln χ + ɛ χ,..48 Price Mark-up shock: ln λ f, = ln λ f + ɛ λ,..49 Capial adjusmen cos process: ln µ = ρ µ ln µ + ρ µ ln µ + ɛ µ,..50 Ineremporal preference shifer: ln b = ρ b ln b + ɛ b,..5 Governmen spending process: ln g = ρ g ln g + ρ g ln g + ɛ g,..52 Moneary Policy Shock ɛ R,. Equaion for z = z + α α ln Υ z γ α α ln Υ = ρ zz γ α α ln Υ + ɛ z,..53 We will probably resric ρ z o zero. 2

13 .2 Derending and seady sae We derend he variables as in Alig e al. Lower case variables are all derended variables i.e., saionary suff. Specifically: c = C Z, y = Y Z, i = I Z r k = Υ R k P, w =, k = Υ K Z, k = Υ K Z, W P Z, m = M P Z, p = P P, w = W W, ξ = Ξ Z, ξ k = Ξ k Z Υ, z = logz /Z,.2. Denoe wih he seady sae values of he variables. Realize ha a s.s. z = γ and z = γ + α α log Υ..2. Inermediae goods producers We sar by expressing..2 in erms of derended variables: mc = MC P = α α α α w α r k α..2.2 Hence mc = α α α α w α r k α..2.3 Expression 8.8 becomes: ξ λ f, p +λ f, λ f, p + λ f, mc y i +IE s= ζs pβ s ξ +s Π s l= π +l Π p s l= πιp +l π ιp Π s l= π + λ +l f,+s mc +s y +s i = 0 λ f,+s p +λ f,+s λ f,+s Π s l= πιp +l π ιp +λ f,+s λ f,+s.2.4 his implies ha: p = + λ f α α α α w α r k α.2.5 Expression 8.9 becomes: which means ha: = [ ζ p p λ f, + ζ p π ιp π ιp π λ f, λ f,..2.6 p =

14 Equaion.. becomes: and a s.s.: k = k = Recall ha aggregae profis are equal o: α α w r k L..2.8 α w α r k L..2.9 Π = P Y W L R k K. In erms of derended variables we hen have : Π P Z = y w L r k k = k α L α Φ w L α α w L = k α L α w L Φ = α α α w α r k α α w L Φ A seady sae we can use.2.5 o ge ha s. s. profis are: Π P Z = λ f α w L Φ Households Expression..28,..29, and..30 become: respecively. A seady sae: ξ = b c hc e z βhie [b + c + e z + hc,.2. m νm = χ b R R ξ,.2.2 ξ = βr IE [ξ + e z + π +,.2.3 ξ = c he γ Υ α α βhe γ Υ α α h,.2.4 m νm = χ R R ξ,.2.5 R = β π e γ Υ α α

15 Equaion..2 and..22 become: k = u Υ e z k,.2.7 k = δυ e z k + µ S i e z i..2.8 i which deliver he seady sae relaionships: under he assumpion ha Se γ Υ α α = 0. k = u e γ Υ α k,.2.9 i = µ δe γ Υ α k Equaion..3,..32, and..33 become: ξ k µ Υ S i e z S i e z i e z i i i + βie [e z + ξ k + µ + S i + i e z + + e z + 2 = ξ.2.2 i i ξ k = βie [Υ e z + ξ + r+u k + au + + ξ+ k δ.2.22 r k = a u.2.23 which deliver he seady sae relaionships: ξ k µ Se γ Υ α α S e γ Υ α α e γ Υ α α + βe γ Υ α α ξ k µs e γ Υ α α e γ Υ α ξ k = βe γ Υ α ξ r k u au + ξ k δ α 2 = ξ r k = a u.2.26 Under he assumpions ha S e γ Υ α α = 0, u = and au = 0, he above equaions become: ξ k µ = ξ.2.27 r k = µ β e γ Υ α δ.2.28 r k = a u

16 Expressed in erms of derended variables, equaion..37 becomes: [ IE s=0 ζ wβ s Lj +s ξ +s X,s w w + + λ w b +sϕ +s L +s j ν l = 0,.2.30 ξ +s where and X,s = if s = 0 Π s l= π e γ Υ α α ι w π +l e z +l ι w L +s j = Π s l= π +le z +l w w w +s X,s +λw λw L +s. oherwise Equaion..38 becomes: = [ ζ w w λw which imply a seady sae: + ζ w π e γ Υ α α ι w π e z ι w w w π e z λw λw..2.3 w = + λ w ϕlν l,.2.32 ξ w = Resource consrains The resource consrains become: g c + i + au ē z k = y and ẏ = k α L α Φ becomes where ṗ = P P Y = = [ ζ p P P = [ ζ p p +λf, λ f, P P +λ f, λ f, Ẏ +λ f, y = ṗ λ f, ẏ λ f, λ f, + ζ p π P P +λ f, λ f, λ f, +λ f, + ζ p π ṗ π +λ f, λ f, λ f, 6 +λ f,.2.37

17 While L = Ẇ W +λ w, λ w, L becomes +λ w, λ L = ẇ w, L.2.38 where ẇ = Ẇ W = [ ζ w W W = [ ζ w w +λw, λ w, +λ w, λ w, + ζ w π e γ Υ α α + ζ w π e γ Υ α α π Ẇ W +λ w, λ w, λ w, +λ w, e z w w ẇ +λ w, λ w, λ w, +λ w,.2.39 A seady sae we have: g c + i = y and y = k α L α Φ..2.4 and ẏ = y, L = L..2.4 Governmen Policies The Taylor rule..39 becomes: R R = R R [ ρr ψ ρr ψ2 π y e ɛ R,.2.42 π y 7

18 .3 Seady sae Define φ implicily by defining L noe ha you can only o consider policy changes ha leave L unchanged. From.2.28 if µ = : r k = β e γ Υ α δ From.2.5: From.2.9 From.2.4: w = α α α α r k α α + λ f k =.2.44 α w α r k L y = k α L α Φ From.2.9 and.2.20: k = e γ Υ α k,.2.47 i = δe γ Υ α k Hence i follows ha: δ = e γ Υ α i k From.2.40: c = y g i Given π objecive of cenral bank and r rreal ineres rae we have ha.2.4,.2.6, and.2.5 deliver: ξ = c he γ Υ α α βhe γ Υ α α h, = c e z h e z hβ.2.5 R = r π.2.52 β = r e γ Υ α α m νm = χ R R ξ

19 From.2.27: From.2.32: ϕ = ξ k = ξ w ξ + λ w L ν l The definiion of he labor share LS is LS = WL P Y : LS = W L P Y = w L y.2.57 In absence of fixed coss, i.e. F = 0, a seady sae we have: LS = w L y = w L k α L α = α α α α r k α = α/ + λ f = w L k α w α = αmc.2.58 The following derivaions are also obained for F = 0. We wan o ge he s.s. capial oupu raio in his economy. Divide.2.9 by oupu.2.4, and obain: k y = α α α α w r k = + α r λ k f = + α β λ e γ Υ α δ f.2.59 where we used he s.s. values of w and r k compued above. Hence from he definiion of k : k y = e γ Υ α α β + λ e γ Υ α δ f = α + λ f β e γ Υ α δ

20 .4 Log-linearized model Eq..2.2 becomes: mc = α ŵ + α r k..2.6 Eq..2.6 becomes: p = ζ p ζ p π ι p π Eq..2.4 becomes see appendix: p = ζ p β + λ f mc mc + ζ p βλ f mc λf, ι p ζ p β π + ζ p βie [ π + + ζ p βie [ p Combining.2.62 wih.2.63 we obain: π = ζ [ pβ ζ p mc + ι p βζ + λ f λf, p + λ f + ι p + ι p β π + β + ι p β IE [ π Eq..2.8 becomes: k = ŵ r k + L Eq..2. becomes: e z hβe z h ξ = e z e z h b e 2z + βh 2 ĉ Eq..2.2 becomes: +he z ĉ he z z βhe z hie [ b βhe z IE [ĉ + + βhe z IE [z+. ν m m = χ + b R R ξ Eq..2.3 becomes: ξ = R + IE [ ξ + IE [z + IE [ π Eq..2.7 becomes: k = û z + k

21 Eq..2.8 becomes: k = i k z + i k µ + i k î. + i k k.2.70 Eq..2.2 becomes: ξ S e 2z + S e µ 2z ξ S e 2z = z î + + βî βie[z + βie[î Eq becomes: ξ k = IE [z + + r k r k + δ IE [ξ + + Eq becomes: Eq becomes: r k r k + δ IE [r k + + δ r k + δ IE [ξ k r k r k = a u λ + ν w +λ l λ w w + + ζ w βν w l λ w ŵ = ζ w β b + ϕ + ν l L ξ +λ ζ w β + ν w l λ w IE [ι w π + ι w z π + ẑ+ + ζ +λ wβ + ν w l λ w IE [ w + + ŵ Eq..2.3 becomes: ŵ = ŵ π z + ι w π + ι w z + ζ w ζ w w Subsiuing w from.2.75 ino.2.74 we obain: ŵ ŵ + π + ẑ ι w π ι w z = ζw + βie [ŵ + ŵ + π + + ẑ + ι w π ι w z ζ w ζ wβ +ν +λw l λw b + ϕ + ν l L ξ ŵ where ŵ ŵ + π + ẑ is nominal wage inflaion. Eq becomes:.2.76 ŷ = ĝ + c c + i ĉ + i c + i î + rk k c + i û Eq becomes remember ŷ = ẏ : ŷ = α y + Φ y k + α y + Φ y L

22 Eq becomes: R = ρ R R + ρ R ψ π + ψ 2 ŷ + ɛ R,.2.79 In absence of fixed coss, i.e. F = 0, log-deviaions he labor share equals marginal coss in erms of log deviaions from seady sae: LS = ŵ + L ŷ = ŵ + L α k α L = ŵ α k L = ŵ + α r k, ŵ.2.80 = mc 22

23 .5 Measuremen Oupu growh log differences, quarer-o-quarer, in %: 00 ln Y ln Y = 00 ln y + ln Z ln y ln Z = 00 ŷ + ln y ŷ ln y + z = 00 ŷ ŷ + ẑ + 00γ + 00 α α ln Υ.2.8 where ẑ = z γ α α ln Υ and is modeled in he ransiion equaion. Consumpion growh log differences, quarer-o-quarer, in %: 00 ln C ln C = 00 ĉ ĉ + ẑ + 00γ + 00 α α ln Υ.2.82 Invesmen growh log differences, quarer-o-quarer, in %: 00 ln I ln I = 00 î î + ẑ + 00γ + 00 α α ln Υ.2.83 Hours worked log: ln L = ˆL + ln L + ln L adj.2.84 Nominal wage growh log differences, quarer-o-quarer, in %: 00 ln W ln W = 00 ln w + ln w + z + ln P ln P Inflaion quarer-o-quarer, in %: = 00 ŵ ŵ + ẑ + ˆπ + 00 ln π + 00γ + 00 α α ln Υ ln P ln P = 00 ln π = 00ˆπ + 00 ln π Nominal M2 growh log differences, quarer-o-quarer, in %: 00 ln M ln M = 00 ln m + ln m + z + ln P ln P = 00 ˆm ˆm + ẑ + ˆπ + 00 ln π + 00γ + 00 α α ln Υ

24 Nominal ineres rae annualized, in %: 400 ln R = 4 00 ˆR ln R Coinegraing relaionships. Log consumpion - Log oupu in %: 00 ln C ln Y = 00 ĉ ŷ + 00ln c ln y.2.89 Log invesmen - Log oupu in %: 00 ln I ln Y = 00 î ŷ + 00ln i ln y.2.90 Log nominal wage - Log oupu - Log Price in %: 00 ln W ln Y ln P = 00 ŵ ŷ + 00ln w ln y.2.9 Log M2 - Log oupu - Log Price in %: 00 ln M ln Y ln P = 00 ˆm ŷ + 00ln m ln y.2.92 Noe ha he ransiion equaion has no consan. So we can rescale all heˆvariables by 00, and correspondingly make sure ha he sandard deviaions of he exogenous shocks are measured in %. 24

25 2 Log-linearizaion of Eq..2.4 Log-linearizaion of Eq..2.4, which is reproduced here: ξ p + λ f, mc y i +IE s= ζs pβ s Π ξ +s p s l= πιp +l π ιp Π s l= π +l + λ f,+s mc +s y +s i = 0 2. Noe ha a s.s. he erm wihin... namely p Π s l= π + λ f,+smc +l +s is equal o 0, so we need no boher wih all he erms ouside he parenhesis and we can se hem o heir s.s values. Call d ln x = x. Now noe ha: lhs ln p π s = p + s= ζs pβ s p = ζ p β p lhs ln mc = + λ f mc mc lhs Π j=,2,.. ln mc +j = + λ f mc s= ζs pβ s IE [ mc +s lhs ln λ f, = λ f mc λf, lhs Π j=,2,.. ln λ f,+j = λ f mc s= ζs pβ s IE [ λ f,+s lhs Π j=,2,.. ln π +j Puing all ogeher we ge: = s= ζs pβ s IE [ s l= π +l ι p π +l p = ζ p β + λ f mc mc + ζ p βλ f mc λf, + ζ p β [ s= ζs pβ s + λ f mc IE [ mc +s + λ f mc IE [ λ f,+s + IE [ s l= π +l ι p π +l = ζ p β + λ f mc mc + ζ p βλ f mc λf, + ζ p βie [ π + ι p π + ζ p βie [ ζ p β + λ f mc mc + + ζ p βλ f mc λf,+ + ζ p β s= ζs pβ [ s + λ f βmc IE + [ mc ++s + λ f mc IE + [ λ f,++s + IE + [ s l= π ++l ι p π +l or p = ζ p β + λ f mc mc + ζ p βλ f mc λf, + ζ p βie [ π + + ζ p βie [ p Log-linearizaion of Eq Log-linearizaion of Eq..2.30, which is reproduced here: [ IE s=0 ζ wβ s L +s jξ +s X,s w w + + λ w b +sϕ +s L +s j ν l = 0, 3. ξ +s 25

26 where X,s = if s = 0 Π s l= π e γ Υ α α ι w π +l e z +l ι w Π s l= π +le z +l oherwise. A s.s. he erm wihin [... namely X,s w w + + λ w b +sϕ +s L +s j ν l is equal o 0, so we need no boher wih all he erms ouside he parenhesis and we can se hem o heir s.s values. Loglinearizing: [ IE s=0 ζ wβ s w w w ŵ w s l= ιw π +l + ι w π +l π +l ẑ+l +w b +s + ϕ +s + ν l L+s j ξ +s = 0, hence Realize ha in erms of derended variables: L +s j = w w w+s X +λw λw,s L +s, L +s j = + λ w λ w w + ŵ ŵ +s + Subsiuing in 3.2 we obain: ξ +s 3.2 s ι w π +l + ι w ẑ+l π +l ẑ+l + L +s. l= ζ w β + ν l +λw λ w w + ŵ = b + ϕ + ν l L ξ +λ + ν w l λ w ŵ [ +IE s= ζ wβ s +λ + ν w l λ w s l= ιw π +l + ι w ẑ+l π +l ẑ+l + b +s + ϕ +s + ν l L+s ξ +λ +s + ν w l λ w ŵ +s ζ w β + ν l +λw λ w w + ŵ = b + ϕ + ν l L ξ +λ + ν w l λ w ŵ ζ wβ ζ w β + ν l +λw λ w IE [ι w π + ι w z π + ẑ+ +ζ w βie [ b+ + ϕ + + ν l L+ ξ +λ + + ν w l λ w ŵ + + s= ζ wβ s [ s l= ιw π ++l b ++s +... or +λ + ν w +λ l λ w w + + ζ w βν w l λ w ŵ = ζ w β b + ϕ + ν l L ξ +λ ζ w β + ν w l λ w IE [ι w π + ι w z π + ẑ+ + ζ +λ wβ + ν w l λ w IE [ w + + ŵ This expression can be furher simplified as: w = ζwβ +ν l +λw λw b + ϕ + ν l L ξ ŵ +ζ w βie [ w + + ŵ + ŵ + π + + ẑ + ι w π ι w z

27 4 The flexible price/wage version of he model In he flexible price/wage version of he model ζ p = ζ w = 0. The price-seing problem of he inermediae good producer under flexible prices is: The FOC becomes: P i MC Y i +λ f, P λ i f, s.. Y i = Y, max Pi P 4. Ξ p P i P +λ f, λ f, P i + λ f, MC Y i = λ f, P This affecs he equilibrium condiions as follows. Equaion 8.4 becomes: and expression.2.6 becomes: which implies: p = + λ f, mc, 4.3 = p, 4.4 = + λ f, mc, 4.5 The nominal ineres rae and money need no be inroduced, hence we can skip condiion.2.3.same applies o.2.2. The wage-seing problem of he workers under flexible wages is: max Wj The FOC becomes: [ ϕ ν l + L j νl s and..34 for s = and equaion.2.3 becomes: w w = + λ w b ϕ L ν l, 4.7 ξ = w, 4.8 which ogeher imply: w = + λ w b ϕ L ν l. 4.9 ξ 27

28 The seady sae is unchanged. The log-linearized condiions are modified as follows: Eq drops ou and is replaced by: 0 = + λ f mc + λ f λf, 4.0 Eq..2.67,.2.68, and.2.79 drop ou. Expressions.2.74 and.2.75 boh drop ou and are replaced by: ŵ = b + ϕ + ν l L ξ Normalizaions We redefine he shocks as follows: λ f, = ζ p βζ p λ f λf, + λ f 5. µ = µ + βe 2z S 5.2 b = ez he z e 2z + h 2 β b 5.3 ϕ = ζ w β ϕ 5.4 χ = ν m χ Inroducing Capial Producers decenralizing he invesmen decision In his secion we decenralize he invesmen decision by inroducing capial producers who buy goods, ransform hem ino insalled capial, and sell i back o he households a a price Q k. We will see ha Q k is Tobin s Q ha is, he value of insalled capial in erms of consumpion, which previously was equal o Ξ k /Ξ. The household s problem is he same as before excep ha now hey do no decide abou invesmen, bu only on how much capial o buy from he capial goods producers. 28

29 The household s budge consrain, wrien in nominal erms, is given by: P +s C +s j + B +s j + M +s j R +s B +s j + M +s j + Π +s + W +s jl +s j + R k +su +s j P +s au +s jυ K+s j +P +s Q k +s δ K+s j K +s j, 6. where Q k is he price of capial in erms of consumpion goods. Noe ha households a he beginning of period bu afer he realizaion of he shocks sell undepreciaed capial from he previous period δ K j o capial producers and a he end of he period purchase he new sock of capial K j. Their FOC wr K +s j are: K Ξ Q k = βie [Ξ + Rk + P u + au + Υ + + Ξ + Q k + δ Noe ha his FOC is idenical o..32 if we replace Q k wih Ξ k /Ξ. Capial Producers produce new capial by ransforming general oupu, which hey buy from final goods producers, ino new capial via he echnology: x = x + Υ µ S I I. 6.3 I where x is he iniial capial purchased from households a he beginning of he period, and x is he new sock of capial, which hey sell back o households a he end of he period. Their period profis are herefore given by: llπ k = Q k x Q k x I = Q k Υ µ S I I I I. 6.4 Noe ha hese profis do no depend on he iniial level of capial x purchased, so effecively he only decision variable for capial producers is I. profis using he households discoun rae β Ξ, heir FOC wr I are: I Ξ Q k Υ µ S I I S I I I I Since hey discoun + βie [Ξ + Q k + Υ+ µ + S I + I I + I 2 = Ξ 6.5 Noe ha his FOC is idenical o..3 if we replace Q k wih Ξ k /Ξ. 29

30 7 Adding BGG-ype financial fricions as in Chrisiano, Moo, Rosagno 7. Households The objecive funcion for household j is unchanged expression..6. The household s problem is differen as households no longer hold he capial sock, and make invesmen and capial uilizaion decisions. Raher, hey inves in deposis o he banking secor D in addiion o governmen bonds and money, which pay a gross nominal ineres rae R d. Household j s budge consrain is: P +s C +s j + B +s j + D +s j + M +s j R +s B +s j + R+sD d +s j + M +s j + Π +s + W +s jl +s j + T r +s, 7. where he erm T r +s represens ransfers from he enrepreneurs, which we will discuss laer. Households firs order condiions for consumpion, money holdings, bonds, and wages, are unchanged. [deposis FOC? 7.2 Capial Producers There is a represenaive, compeiive, capial producer who produces new capial by ransforming general oupu which is bough from final goods producers a he nominal price Q k ino new capial via he echnology: x = x + Υ µ S I I. 7.2 I where x is he iniial capial purchased from enrepreneurs in period, and x is he new sock of capial, which hey sell back o enrepreneurs a he end of he same period. Their period profis, expressed in erms of consumpion goods, are herefore given by: Π k = Qk P x Qk P x I = Qk P Υ µ S I I I I

31 Noe ha hese profis do no depend on he iniial level of capial x purchased, so effecively he only decision variable for capial producers is I. profis using he households discoun rae β Ξ, heir FOC wr I are: Q I Ξ k P Υ µ S I I S I I I I Since hey discoun + βie [Ξ + Q k + P + Υ + µ + S I + I I + I 2 = Ξ 7.4 Noe ha his FOC is idenical o..3 if we replace Qk P wih Ξ k /Ξ. 7.3 Enrepreneurs There is a coninuum of enrepreneurs indexed by e. Each enrepreneur buys insalled capial K e from he capial producers a he end of period using her own ne worh N e and a loan B d e from he banking secor: Q k K e = B d e + N e where ne worh is expressed in nominal erms. In he nex period she rens capial ou o firms, earning a renal rae R k per uni of effecive capial. In period she is subjec o an i.i.d. across enrepreneurs and over ime shock ωe ha increases or shrinks her capial, where logωe Nm ω,, σ 2 ω, where m ω, is such ha IEωe =.Denoe by F ω he cumulaive disribuion funcion of ω a ime, where he disribuion needs o be known a ime. In addiion, afer observing he shock she can choose a level of uilizaion ue by paying a cos in erms of general oupu equal o aue Υ per-uni-of-capial. A he end of period he enrepreneurs sells undepreciaed capial o he capial producers. Enrepreneurs revenues in period are herefore: or equivalenly { R k ue + δq k P aue Υ } ωe Ke ωe Rk e Q k Ke where R k e = Rk ue + δq k P aue Υ 3 Q k 7.5

32 is he gross nominal reurn o capial for enrepreneurs. From he profi funcion i is clear ha he choice of he uilizaion rae is independen from he amoun of capial purchased or he ω shock, and is given by he FOC: R k P = a ue Υ, 7.6 which is he same condiion as..33. Consequenly we can drop he index from he reurn R k. The deb conrac underaken by he enrepreneur in period consiss of he riple B d e, R d e, ωe where R d e represens he conracual ineres rae, and ωe he heshold level of ωe below which he enrepreneur canno pay back, which is herefore defined by he equaion: ωe Rk Q k Ke = R d e B d e. 7.7 For ωe < ωe he bank moniors he enrepreneurs and exracs a fracion µ e of is revenues R k Q k Ke, where µ e represens exogenous bankrupcy coss. The bank s zero profi condiion implies ha [sae by sae?: [ F ωe R d e B d e + µ e ωe 0 ωdf ω R k Q k Ke = R B d e where R is he rae paid by he bank o he deposiors. If we define leverage as: use he definiions ϱe Bd e Ne, Γ ω ω [ F ω + G ω G ω ω 0 ωdf ω, as well as he definion of ωe, he zero-profi condiion can be rewrien as: [ Γ ωe µ e G ωe Rk R + ϱe = ϱe. 7.8 Enrepreneurs expeced profis before he realizaion of he shock ω can be wrien as: ωe [ωe Rk e Q k Ke R d e B d e df ωe = [ ωe ωe df ωe ωe [ F ωe Rk e Q k Ke = [ Γ ωe R k R [ + ϱe R Ne 32

33 The conrac ha maximizes expeced ne worh for he enrepreneurs is given by: R max E [ Γ ωe k R [ + ϱe R Ne { [Γ {ϱe, ωe } +η ωe µ e G ωe } Rk R [ + ϱe ϱe so ha he FOCs are: R ϱe : 0 = E [[ Γ ωe k R R Ne { [Γ +E [η ωe µ e G ωe } Rk R Γ ωe : η = ωe Γ ωe µe G ωer Ne Subsiuing he second FOC ino he firs we obain: E [[ Γ ω R k R + Γ ω Γ ω µ e G ω { [Γ ω µ e G ω Rk R 7.9 where we omi he he indicaor e since he condiion implies ha ωe only depends on aggregae variables and is he same across enrepreneurs. From he zero profis condiion 7.8 his implies ha leverage ϱe is also he same, hence we can rewrie 7.8 as a funcion of aggregae variables only: V [ Γ ω µ e G ω Rk R = Qk K N Q k K. 7.0 Aggregae enrepreneurs equiy evolves according o: = ω ω Rk Q k Ke df ω [ F ω R d e B d e [ R + µ e G ω R k = R k Q k K Q k K Q k K N Q k K N. 7. A fracion γ of enrepreneurs exis he economy and fracion γ survives o coninue operaing for anoher period. A fracion Θ of he oal ne worh owned by exiing enrepreneurs is consumed upon exi and he remaining fracion of heir neworh is ransfered as a lump sum o he households. Each period new enrepreneurs ener and receive a ne worh ransfer W e. Because W e is small, his exi and enry process ensures ha enrepreneurs do no accumulae enough ne worh o escape he financial fricions. Aggregae enrepreneurs ne worh evolves accordingly as: } = 0. N = γ V + W e

34 7.4 Derending and seady sae We derend he addiional variables inroduced by his exension as follows: q k = Qk P Υ, n = N P Z, v = V P Z, w e = W e P Z. 7.3 All oher variables are derended as in Expressions 7.4, 7.5, 7.8, 7., and 7.2 become ξ q k µ S i e z S i e z i e z i i i + βie [e z + ξ+ q k +µ + S i + i e z + i + i e z + 2 = ξ 7.4 R k = rk u + δq k au q Υ k π 7.5 ω Rk = R d [ Γ ω µ e G ω Rk [ v e z π = R k q k k R + µ e G ω R k r k = a u 7.6 q k k n q k k 7.7 R = qk k n q k k 7.8 q k k q k q k k n k n 7.9 n = γ v + w e Expression 7.9 is already expressed in erms of derended variables. The seady sae relaionships are: ξ q µ k Se γ Υ α α S e γ Υ α α e γ Υ α α + βe γ Υ α α ξ q k µs e γ Υ α α e γ Υ α α 2 = ξ 7.2 which implies since S. = S. = 0 a seady sae ha q k =. We also parameerize a. so ha u = and au = 0. Wih his informaion, and afer some simplificaion, we can rewrie he remaining seady sae equaions as R k = rk + δ π Υ

35 n R k R = Ψ ω, σ ω, µ e 7.23 = [Γ ω µ k e G ω R k 7.24 R } { γ β n Rk = γ β k [ µ e R G ω + we k 7.25 v = γ n w e wih Ψ ω, σ ω, µ e = Γ ω [ Γ ω [Γ ω µ e G ω + Γ ω [Γ ω µ e G ω µ e G ω Γ ω [ Γ 7.27 ω µ e G ω Our sraegy for compuing he seady sae is going o be he following: find a soluion for he real reurn o capial R k π and use 7.22 o find r k : r k = Υ R k π δ Once we have r k we can proceed exacly as in secion.3 o find he seady sae for he oher variables. Recall ha from he Euler equaion he seady sae real rae is given by: R π = β e z. In absence of financial fricion R π and R k π would be idenical, bu fricions induce a spread beween he wo, which we will compue subsequenly as a funcion of he primiives in he economy σ 2 ω,, µ e, γ, w e. We solve for he seady sae according o he following seps:. Se F ω = F 7.29 and define which we can use o wrie z ω ln ω + 2 σ2 ω σ ω = Φ F ω σ ω = exp { σ ω z ω } 2 σ2 ω

36 2. Given he value for he spread for deb conracs, R d /R, we can use equaion 7.7 o wrie R k R = Rd n q k k 7.32 R ω Noe: his second sep can be skipped if insead we calibrae/esimae R k /R direcly. 3. Given R k /R, we can use 7.23 o wrie Rk R which we can use o se = µ e G ω Γ ω [ Γ ω µ e G ω { } G = µ e ω Γ ω [ Γ ω + G ω µ e σ ω = Rk R G ω Γ ω [ Γ ω + G ω and plugging in he exac expressions we ge µ e σ ω = Rk R φz ω { } σ ω F Φ z ω σ ω ω F + Φ z ω σ ω Given he above and equaion 7.24 we ge n k σ ω = { ω [ F + µ e Φ z ω σ ω } Rk R Given he elasiciy of he spread w.r.. leverage, ζ sp,b, derived below in equaion 7.42, we ge he following expression Φz ω σω F ω [ ω + µ e Φzω σω µe φz ω σ ω F + F µ e ω σ 2 ω φz ω F zω [ µe σω φz ω F 2 φz ω F 2 R k R n k = ζ sp,b n k 7.35 which we can solve for σ ω. Once we find his value we can plug back ino he previous expressions, ha depend on σ ω. 36

37 6. Given γ, and using equaion 7.25 we ge w e k = γ β n k γ β { Rk R [ µ e Φ z ω σ ω } 7.36 and from equaion 7.26 v = γ k n we k k We ge r k using equaion 7.22 o wrie r k = Υ R k π δ Log-linearizaion Log-linearizaion of he FOC w.r.. leverage expression 7.9 yields: 0 = E Rk + ˆR + ζ b, ω E ω + + ζ b,σω ˆσ ω, + ζ b,µ e ˆµ e 7.39 wih ζ b,x [{ x [ Γ ω + { [ Γ ω + } Γ ω Γ ω µ e G ω [Γ ω µe G ω Rk R Γ ω Γ ω µe G ω [Γ ω µ e G ω Γ ω Γ ω µ e G ω } Rk R x defined for x { ω, σ 2 ω, µ e}. Log-linearizaion of he zero profi condiion expression 7.8 yields: wih R k ˆR + ζ z, ω ω + ζ z,σω ˆσ ω, + ζ z,µ e ˆµ e = ϱ ˆn ˆq k k ζ z,x 7.40 x [Γ ω µe G ω Γ ω µ e G ω x 7.4 defined for x { ω, σω, 2 µ e}. We can furher wrie ω = ˆn ˆq k k Rk ζ z, ω ϱ ζ ˆR + ζ z,σω ˆσ ω, + ζ z,µ e ˆµ e z, ω and plug his expression ino 7.39 o obain: 0 = E [ Rk + ˆR + ζ b,σω ˆσ ω, + ζ b,µ e ˆµ e [ ζ b, ω ζ z, ω ϱ ˆn ˆq k k + E [ Rk + ˆR + ζ z,σω ˆσ ω, + ζ z,µ e ˆµ e 37

38 hence E [ Rk + ˆR = ζ sp,b ˆq k + k ˆn + ζ sp,σω ˆσ ω, + ζ sp,µ e ˆµ e 7.42 where ζ sp,b ζ sp,σω ζ sp,µ e ζ b, ω ζz, ω ζ b, ω ϱ ζz, ω ζ b, ω ζz, ω ζz,σω ζ b,σω ζ b, ω ζz, ω ζ b, ω ζz, ω ζ z,µ e ζ b,µ e ζ b, ω ζz, ω Log-linearizaion of he expression 7.20, characerizing ne worh, yields: ˆn = γ v n ˆγ + ˆv + we n ŵ e Log-linearizaion of he expression 7.9, characerizing he evoluion of enrepreneurial equiy, is ˆv = ẑ β k n v ˆR π + R k k π e z v µ e G ω Rk π + β n v ˆn Rk π e z µe G ω β k v ˆq k + k µ e R G ω k k π e z v [ˆµ e + ζ G, ω ω + ζ G,σω ˆσ ω, Plugging in he expression for ω we obain ˆv = ẑ β k n v ˆR π Rk π e z µe G ω β k v [ µ e R G ω k k π e z v ζ G, ω ζ z, ωϱ ζ z, ω Rk ˆR + ζ z,σω ˆσ ω, + ζ z,µ e ˆµ e Collecing erms yields ˆv = ẑ + ζ v, Rk + R k k π e z v µ e G ω Rk π + β n v ˆn R k k π e z v [ˆµ e + ζ G,σω ˆσ ω, ˆq k + k µ e G ω ˆn ˆq k k Rk π ζ v,r ˆR π + ζ v,qk ˆq k + k + ζ v,nˆn ζ v,µ e ˆµ e ζ v,σ ω ˆσ ω,

39 wih ζ v, Rk R k k π e z v [ µ e G ω ζ G, ω ζ z, ω [ ζ v,r β k v n ζ v,qk R k k π e z ζ v,n β n v + R k ζ v,µ e µ e G ω ζ v,σω µ e G ω k + µ e G ω R k v [ µ e G ω k π e z R k π e z R k π e z v µ e G ω ζ G, ω k v ζ G, ω R ζ z, ω ζ G, ω ζ z, ωϱ β k v ζ z, ωϱ ζ z, ω ζ G, ω ζ z,µ e k v ζ G, ω ζz,σω ζ z, ω Finally, subsiuing his expression ino 7.43 we ge: v ˆn = γ n ˆγ + we n ŵ e v γ n ẑ Rk π ζ n,r ˆR π + ζ n,qk ˆq k + k + ζ n,nˆn +ζ n, Rk ζ n,µ e ˆµ e ζ n,σ ω ˆσ ω, 7.46 wih [ Rk ζ n, γ Rk + ϱ π e µ e z G ω ζ G, ω ζ [ z, ω ζ n,r γ β + ϱ n + µ k e G ω R k ζ n,qk γ Rk π e z + ϱ [ µ e G ω ζ G, ω R ζ z, ω ζ n,n γ β Rk + γ + ϱ π e µ e z G ω ζ G, ω ζ z, ωϱ ζ n,µ e γ µ e R G ω k + ϱ ζ π e ζ z,µ e z G, ω ζ z, ω ζ n,σω γ µ e R G ω k + ϱ π e ζ z G, ω ζz,σω ζ z, ω ζ G, ω ζ z, ωϱ γ β + ϱ Now normalize he shocks, σ ω, ζ sp,σω ˆσ ω, 7.47 µ e ζ sp,µ e ˆµ e 7.48 γ γ v n ˆγ 7.49 so ha he relevan log-linear equaions, 7.42 and 7.46, become: E [ Rk + ˆR = ζ sp,b ˆq k + k ˆn + σ ω, + µ e 7.50 and ˆn = ζ n, Rk Rk π ζ n,r ˆR π + ζ n,qk ˆq k + k + ζ n,nˆn γ + we n ŵ e v γ n ẑ ζ n,µ e ζ sp,µ e µe ζn,σω ζ sp,σω σ ω, 39

40 and Log-linearizaion of 7.5 and 7.4 yield: S e 2z R k π = r k δ r k + δ ˆrk + r k + δ ˆqk ˆq, k 7.52 ˆq k + S µ = z e 2z î + + βî βie[z + βie[î Log-linear disribuion Consider which has he properies ln ω N m ω, σω E [ω = e mω+ 2 σ2 ω 7.55 In order o ge E [ω = we need o se m ω = 2 σ2 ω 7.56 This implies ha he pdf is f ω = ωσ ω 2π e 2 ln ω+ 2 σ 2 ω σω The CDF is F ω = Φ ln ω + 2 σ2 ω σ ω 7.58 Furher noice ha φ z 2π e 2 z Φ z z 2π e 2 x2 dx 7.60 for which we can use malab funcions normpdf and normcdf. We also need he following expression z = Φ F for which we can use an inverse cdf funcion also available in malab as norminv

41 The parial expecaion obeys 2 E [ω ω > ω = Φ σ2 ω ln ω ln ω 2 = Φ σ2 ω σ ω σ ω which implies ha G ω ω ωf ω dω = 0 ln ω 2 = Φ σ2 ω σ ω 0 ωf ω dω ω ωf ω dω 7.62 Finally we define Γ ω ω ωf ω dω + ω f ω dω 0 ω [ ln ω + 2 = ω Φ σ2 ω + G ω 7.63 σ ω If we define hen we ge z ω ln ω + 2 σ2 ω σ ω 7.64 G ω = Φ z ω σ ω 7.65 and Γ ω = ω [ Φ z ω + Φ z ω σ ω 7.66 In order o compue he derivaives, firs noice ha we can wrie φ z ω σ ω = ωφ z ω 7.67 and Using his resul we can wrie he derivaives as follows: φ z = zφ z, z 7.68 G ω = σ ω φ z ω 7.69 G ω = zω G ω = zω ωσ ω ωσ ω 2 φ z ω

42 and Γ Γ ω G ω ω = = Φ z ω ω 7.7 Γ ω = ω G ω; σ ω = φ z ω ωσ ω 7.72 z ω z ω = σ ω σ ω 7.73 G σω ω = zω σ ω φ z ω σ ω 7.74 G σ ω ω = φ zω σ 2 ω [ z ω z ω σ ω 7.75 Γ σω ω = φ z ω σ ω 7.76 z Γ ω σ ω ω = φ z ω 7.77 σ ω where we use noaion f ω f ω / ω and f σω ω f ω / σ ω, for f {G, Γ}. 7.7 Elasiciies Firs noice ha we have several elasiciies defined as [{ Γ x Γ ω + ω Γ ζ b,x ω µ e G ω [Γ ω µe G ω { Γ ω + which we can rewrie as Γ ω Γ ω µ e G ω [Γ ω µe G ω } Rk Γ R ω Γ ω µ e G ω } Rk R x ζ b,x { Γ ω + Ψ x x Γ ω Γ ω µ e G ω [Γ ω µe G ω } Rk R wih Ψ { Γ ω + = [ Γ ω R k R + Γ } ω Rk Γ ω µ e G ω [Γ ω µe G ω Γ ω R Γ ω µ e G ω [ Γ ω Γ ω µ e G [Γ ω µ e ω G ω R k R 7.78 Elasiciies w.r.. ω 42

43 Firs wrie Ψ ω = Γ ω R k R [ + and simplify o [Γ ω µ e G ω R k R Γ ω + Γ ω µ e G ω Ψ ω = µe { [ Γ ω µ e G ω Rk R [Γ ω µ e G ω R k R = µ e n Γ ω G ω G ω Γ ω k [Γ ω µ e G ω 2 which we can plug ino he elasiciy o ge Γ ω [Γ ω µ e G ω Γ ω [Γ ω µ e G ω [Γ ω µ e G ω 2 } G ω Γ ω G ω Γ ω [Γ ω µ e G ω 2 ζ b, ω = µ e n k Γ ω G ω G ω Γ ω [Γ ω µe G ω 2 { Γ ω + Γ ω Γ ω µe G ω Γ ω µe G ω } Rk R ω 7.79 We also have ζ z, ω Γ ω µ e G ω Γ ω µ e G ω ω 7.80 Noice ha if we plug everyhing ino which becomes ζ sp,b = ζ sp,b = ζ b, ω ζ z, ω ζ b, ω ζ z, ω n k n k { [ ω [ Φz ω Φz ω σω [ ω [ Φz ω + µ e Φz ω σω Φz ω µe σω φzω µ e φzω φz ω ω σ ω 2 z ω [ Φz ω [ Φz ω µe n 2 k σω φzω n k n k Φz ω } Rk R 7.82 Elasiciy of w.r.. σ ω Firs we compue he derivaive Ψ σ ω = Γ σω ω R k R + Γ ω Γ ω µ e G ω [Γ σ ω ω µ e G σω ω R k R + Γ σ ω ω [Γ ω µ e G ω Γ ω [ Γ σ ω ω µ e G σ ω ω [Γ ω µ e G ω 2 43 [ [Γ ω µ e G ω R k R

44 hence Ψ σ ω = µe µ e G σω ω Γ σω ω G ω Γ ω Γ σω ω R k + µ e R n G ω Γ σ ω ω Γ ω G σ ω ω k [Γ ω µ e G ω 2 so ha ζ b,σω = µ e Gσω ω Γσω ω µ e G ω Γ ω Γ σω ω R k R + µ e n k G ω Γ σω ω Γ ω G σω ω [Γ ω µe G ω 2 [ Γ ω R k R + Γ ω Γ ω µ e G ω n k σ ω 7.83 We also have and finally we can wrie ζ z,σω = Γ σ ω ω µ e G σω ω Γ ω µ e σ ω 7.84 G ω ζ sp,σω = ζ b, ω ζ z, ω ζ z,σω ζ b,σω ζ b, ω ζ z, ω 7.85 so ha Elasiciy of w.r.. µ e Firs solve ζ b,x Ψ µ e = Γ ω G ω n [Γ ω µ e G ω 2 Γ ω G ω k Γ ω µ e G ω Γ ω G ω Γ ω µe G ω n k + Γ ω G ω R k [ Γ ω [Γ ω µ e G ω R k R + Γ ω R R k R n k µ e 7.86 We also have Finally we wrie G ω ζ z,µ e = Γ ω µ e G ω µe 7.87 ζ sp,µ e = ζ b, ω ζ z, ω ζ z,µ e ζ b,µ e ζ b, ω ζ z, ω

45 8 SW original model In his secion we describe in deail he Smes and Wouers 2007 model, henceforh SW, and emphasize he differences wih he model presened in Secion of hese noes. 8. Model 8.. Inermediae firms We follow SW and assume he producion funcion o be: Y i = max{e z K i α L ie γ α Φe γ+ α α log Υ, 0}, 8. where z = ρ z z + σ z ɛ z,, ɛ z, N0, 8.2 Noe ha wha SW call γ in our noaion is e γ, and ha hey assume Υ =. SW assume ha produciviy z is saionary. Define Z as follows: ln Z = α z. 8.3 For ρ z 0, he process ln Z is saionary, as in SW. For ρ z = i follows a random walk. This specificaion accomodaes boh. Noe ha we can rewrie he producion funcion as: Y i = max{k i α L iz α Φe α z Z e γ+ α α log Υ, 0}. 8.4 Cos minimizaion subjec o 8.4 yields he condiions: L i K i V i αz α K i α L i α = W V iαz α K i α L i α = R k where V i is he Lagrange muliplier associaed wih..8. In urn, hese condiions imply: K i L i = α α W R k. 45

46 Noe ha if we inegrae boh sides of he equaion wr di and define K = K idi and L = L idi we obain a relaionship beween aggregae labor and capial: Toal variable cos is given by K = α α W R k L. 8.5 Variable Coss K i = W + R k L L i i = W + R k K i L i ỸiZ α Ki L i α, where Ỹi = Z α K i α L i α is he variable par of oupu. The marginal cos MC is he same for all firms and equal o: MC = W + R k K i L i Z α = α α α α W α α Ki L i R k α Z α. 8.6 [TO DO WITH KIMBALL Prices are sicky as in Calvo 983. Specifically, each firm can readjus prices wih probabiliy ζ p in each period. We depar rfom Calvo 983 in assuming ha for hose firms ha canno adjus prices, P i will increase a he geomeric weighed average wih weighs ι p and ι p, respecively of he seady sae rae of inflaion π and of las period s inflaion π. For hose firms ha can adjus prices, he problem is o choose a price level P i ha maximizes he expeced presen discouned value of profis in all saes of naure where he firm is suck wih ha price in he fuure: max Pi Ξ p P i MC Y i + E s= ζs pβ s Ξ p +s P i P i Π s l= s.. Y +s i = πιp +l π ιp P +s Π s l= πιp +l π ιp +λ f,+s λ f,+s Y +s, MC +s Y +s i 8.7 where β s Ξ p +s is oday s value of a fuure dollar for he consumers Ξp +s is he Lagrange muliplier associaed wih he consumer s nominal budge consrain - remember here 46

47 are complee markes so β s Ξ p +s is he same for all consumers. The FOC for he firm is: Ξ p +λ f, Pi λ f, P E s=0 ζs pβ s Ξ p +s P i Π s l= πιp +l π ιp λ f, P P i + λ f, MC Y i+ P i Π s l= πιp +l π ιp P +s +λ f,+s λ f,+s Π s l= πιp +l π ιp λ f,+s P +s + λ f,+s MC +s Y +s i = Noe ha all firms readjusing prices face an indenical problem. We will consider only he symmeric equilibrium in which all firms ha can readjus prices will choose he same P i, so we can drop he i index from now on. From..6 i follows ha: P = [ ζ p P λ f + ζ p π ιp π ιp P λ f λ f Households Household j s uiliy is as opposed o..6: [ IE β s σc C +s j hc +s σc exp L +s j +ν l σ c + ν l s=0 8.0 where C j is consumpion, L j is labor supply. Three observaions are in order regarding his uiliy funcion. Firs, uiliy is increasing in consumpion and leisure regardless of he value of σ c. Second, here are no discoun rae or leisure shocks in he uiliy funcion. Third, SW have exernal as opposed o inernal habi. The household s budge consrain, wrien in real erms, is given by: C +s j + I +s j + + W h +s P +s L +s j + B +sj b +s R +s P +s B +s j P +s R k +s P +s u +s j K +s j au +s jυ K+s j + Π +s T +s, 8. where I j is invesmen, B j is holdings of governmen bonds, R is he gross nominal ineres rae paid on governmen bonds, Π is he per-capia profi he household ges from owning firms assume household pool heir firm shares, T +s is lump-sum axes, so ha hey all receive he same profi W h j is he wage earned by household j. b is a risk premium shock. The erm wihin parenhesis represens he reurn o owning 47

48 K j unis of capial. Households choose he uilizaion rae of heir own capial, u j, and end up rening o firms in period an amoun of effecive capial equal o: K j = u j K j, 8.2 and geing R k u j K j in reurn. They however have o pay a cos of uilizaion in erms of he consumpion good which is equal o au jυ K j. Households accumulae capial according o he equaion: K j = δ K j + Υ µ S I j I j I j, 8.3 where δ is he rae of depreciaion, and S is he cos of adjusing invesmen, wih S > 0, S > 0. The erm µ is a sochasic disurbance o he price of invesmen relaive o consumpion Households are all idenical, so he j subscrip is prey redundan excep for he fac ha we have exernal habis. We will drop he j subsequenly. The FOCs for consumpion, bonds, and labor are: σc C j C hc σc exp L +ν l = Ξ ν l B j Ξ = βr b IE [ Ξ π + σc L j C hc σc exp L +ν l L ν W l h = Ξ ν l P Noe ha households ake W h as given and maximize wih respec o L. The wage sickiness par will be discussed below. Using 8.4 we can rewrie 8.6 as: C hc L ν l = W h P. 8.7 Le us now address he capial accumulaion/uilizaion problem. Call Ξ k he Lagrange muliplier associaed wih consrain 8.3. The FOC wih respec o invesmen, capial, and capial uilizaion are: I Ξ k Υ µ S I I S I I I I + βie [Ξ k + Υ+ µ + S I + I I + I 2 = Ξ 8.8 K Ξ k = βie [Ξ + Rk + P u + au + Υ + + Ξ k + δ u Υ R k P = a u

49 The firs FOC is he law of moion for he shadow value of capial. Noe ha if adjusmen cos were absen, he FOC would simply say ha Ξ k Υ µ is equal o he marginal uiliy of consumpion. In oher words, in absence of adjusmen coss he shadow cos of aking resources away from consumpion equals he shadow benefi absracing from Υ µ of puing hese resources ino invesmen: Tobin s Q is equal o one. The second FOC says ha if I buy a uni of capial oday I have o pay is price in real erms, Ξ k, bu omorrow I will ge he proceeds from rening capial, plus I can sell back he capial ha has no depreciaed. Define Q k = Ξk Ξ. Q k has he inerpreaion of he value of insalled capial relaive o consumpion goods i.e., Tobin s Q. Then condiion 8.9 can be rewrien as: Q k = βie [ Ξ + Ξ R k + P + u + au + Υ + + Q k + δ Governmen Policies The cenral bank follows a nominal ineres rae rule by adjusing is insrumen in response o deviaions of inflaion and oupu from heir respecive arge levels: R R = R R ρr π π ψ Y Y f ψ2 ρ R Y Y f Y Y f ψ3 e rm 8.22 where he parameer ρ R deermines he degree of ineres rae smoohing, R is he seady sae nominal rae and Y f policy reacs o boh level differences beween Y and Y f is oupu under flexible/prices and wages. Noe ha ψ 2 ρ R coefficien as well as growh differences ψ 2 coefficien. Noe also ha he exogenours par of moneary policy is capured by he process r m, which follows an auoregressive process. The cenral bank supplies he money demanded by he household o suppor he desired nominal ineres rae. The governmen budge consrain is of he form P G + B = P T + B b R,

50 where T are nominal lump-sum axes or subsidies ha also appear in household s budge consrain. SW, who assume echnology is saionary, express governmen spending relaive o he deerminisic rend in oupu: G = g y e z 8.24 where y is he seady sae of derended oupu. below by Z, we need o be careful. Define Since we derend everyhing see G g = y Z = g e α z A seady sae g = g. Noe he difference wih DSSW, where g = y c +i >. In SW g 0,. Y Y G and g = 8..4 Resource consrains To obain he marke clearing condiion for he final goods marke firs inegrae he HH budge consrain across households, and combine i wih he gvm budge consrain: P C + P I + P G +Π + W jl jdj + R k K jdj P au Υ K jdj. Nex, realize ha Π = Πi di = P i Y i di W L R k K, where L = Li di is oal labor supplied by he labor packers and demanded by he firms, and K = Ki di = K jdj. Now replace he definion of Π ino he HH budge consrain, realize ha by he labor and goods packers zero profi condiion W L = W jl jdj, and P Y = P i Y i di and obain: C + I + au Υ K + G = Y 8.26 where Y is defined by... The relaionship beween oupu and he aggregae inpus, labor anc capial, is: Ẏ = Z α K i α L i α di Z Φ = Z α K/L α Lidi Z Φ = Z α K α L α Z Φ,