Optimal Monetary Policy with the Cost Channel: Appendix (not for publication)

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1 Opimal Moneary Policy wih he Cos Channel: Appendix (no for publicaion) Federico Ravenna andcarlewalsh Nov 24 Derivaions for secion 2 The flexible-price equilibrium oupu (eq 9) When price are flexible, equaions (4) and (7) of he ex imply ha equilibrium requires χn η µ A ξ C w A, () R ΦR Φ /( ) is he markup From he resource consrain, C γ Y, and from eh producion funcion, Y A N Thus, we can rewrie () as χ(y /A ) η ξ (γ Y ) A ΦR Solving for Y yields he expression for equilibrium oupu under flexible prices: Y f which is equaion (9) of he ex Ã! ξ γ A +η +η χφr f, 2 The social planner s problem under flexible prices (eq 2) We solve a simple one-period version of he social planner s problem o show how he fiscal variable γ affecs he efficien level of oupu Deparmen of Economics, Universiy of California, Sana Cruz, CA 9564, USA Deparmen of Economics, Universiy of California, Sana Cruz, CA 9564, USA and Federal Reserve Bank of San Francisco, Marke S, San Francisco, CA 945, USA

2 The social planner s problem is max U(C, N)+λ [AN C G]+µ [( γ)an G] Firs order condiions are U c λ U N + λa + µ( γ)a λ µ Eliminaing he Lagrangian mulipliers implies U N U c λa + µ( γ)a λ λa λ( γ)a λ γa (2) In eh ex, he uiliy funcion is assumed o ake he form ξ +i C +i χ N +η +i +η Evaluaing (2) yields χn η χ(y/a)η ξc ξγ Y γa Y µ ξγ A +η Y f is he flex-price oupu level when Φ R f χ η+ γ η+ Y f, 3 The real ineres rae in he Euler equaion (eq 4) Using he resource consrain, he Euler condiion (3) can be linearized around he seady sae o yield ˆξ Ĉ E ³ˆξ+ Ĉ+ +ˆr Using he resource conrain (C γ Y ), his can be wrien as µ Ŷ E ³ˆξ+ ˆξ +E (ˆγ + ˆγ )+E Ŷ + ˆr Expressed in erms of he oupu gap, he Euler condiion becomes µ Ŷ Ŷ f E ³ˆξ+ ˆξ +E (ˆγ + ˆγ )+E ³Ŷ+ Ŷ f f + +E ³Ŷ + Ŷ f Finally, defining ˆr f ³ E Ŷ f + Ŷ f ³E ˆξ+ ˆξ + E ˆγ + ˆγ, (3) ˆr 2

3 he Euler condiion can be wrien as Ŷ Ŷ f E ³Ŷ+ Ŷ f + ³ r r f, r ˆR E π + This is equaion (4) of he ex Equaion (3) shows ha he real ineres rae in he flexible-price equilibrium will be affeced by produciviy (via E Ŷ f + Ŷ f ), ase, and fiscal shocks, unless hese shocks follow random walk processes 2 Welfare approximaion 2 The loss funcion In his appendix, he approximaion o he welfare of he represenaive household is derived In doing so, we follow Woodford (999) Our model differs from his in hree ways Firs, we assume he uiliy of he represenaive household depends on consumpion and leisure, while Woodford assumes i depends on consumpion and oupu, he role of oupu is o capure he disuiliy of work This change does no affec he resuls The second, more subsanive change, is ha we allow explicily for sochasic variaion in he share of oupu going o he governmen Finally, he seady-sae values in our model are no independen of moneary policy (as hey are in Woodford) because he level of he nominal ineres rae in he seady-sae affecs equilibrium oupu and employmen We make use of he following noaion: X X X f X ˆX Noaion Seady-sae value Efficien level Flex-price equilibrium level X X log X log X Given his noaion, µ X X +log X + µ 2 X log + X 2 X ˆX + 2 ˆX 2 Because one can always wrie X X X X, i follows ha X X ³ ˆX + ˆX 2 2 Uiliy is assumed o be separable in consumpion and leisure and akes he form " β i ξ+i C+i χ N # +η +i (4) +η X E i 3

4 We begin by approximaing he uiliy of consumpion The second order Taylor expansion for U(C,ξ ) is U(C,ξ ) U( C,)+U c ( C,) C + 2 U cc( C,) C 2 +U ξ ( C,) ξ + 2 U ξ,ξ ξ 2 +U c,ξ ξ C We assume he governmen purchases individual goods in he same proporions as households and ha aggregae governmen purchases are proporional o oupu; G ( γ )Y,γ is sochasic and bounded beween zero and one The aggregae resource consrain hen akes he form Y C + G C +( γ )Y,or C γ Y (5) This implies ha C ³ 2 γȳ ˆγ + Ŷ + 2 ˆγ + Ŷ Given he uiliy funcion specificaion (4), he uiliy of consumpion becomes U(C,ξ ) U( C,) + U c ( C,) γȳ 2 U c( C,) γȳ ˆγ + Ŷ + ³ 2 ˆγ 2 + Ŷ ˆγ + Ŷ + ³ 2 2 ˆγ 2 + Ŷ +U ξ ( C,)ˆξ + 2 U ξ,ξˆξ 2 + U c ( C) γȳ ˆξ ˆγ + Ŷ + ³ 2 ˆγ 2 + Ŷ Ignoring erms of order X i for i>2, U(C,ξ ) U( C,) + U c ( C,) γȳ ( + ˆξ ³ ) ˆγ + Ŷ + ³ 2 2 ( ) ˆγ + Ŷ +U ξ ( C,)ˆξ + 2 U ξ,ξˆξ 2 (6) The nex sep is o obain an approximaion for he disuiliy of work The second order Taylor expansion for V (N ) is aggregae employmen is For employmen a firm i, V (N ) V ( N)+V N ( N)Ñ + 2 V NN( N)Ñ 2 (7) Ñ ñ (i) n ñ (i)di ˆn (i)+ 2 ˆn (i) 2 Each firm has a producion echnology given by y (i) A n (i) Hence, ˆn (i) ŷ (i) Â 4

5 Wecanhenwrie Ñ ñ (i)di n ˆn (i)+ 2 2 ˆn (i) di ȳ ŷ (i)di  + ³ 2 ŷ (i) di 2  Subsiuing his ino (7), and ignoring erms of order X 2 and higher powers, V (N ) V ( N)+V N ( N)ȳ ŷ (i)di  + ³ 2 ŷ (i) di 2  V NN( N)ȳ 2 ŷ (i)di  (8) Given he demand funcions facing each individual firm, he aggregae oupu variable Y is defined as This implies Hence, Noe also ha Therefore, Ŷ Y ŷ (i)di ŷ (i)di Ŷ 2 ŷ (i) 2 di ŷ (i) 2 di Ŷ 2 y (i) di µ µ In addiion,  ŷ (i)di ÂŶ µ 2 Using hese resuls, equaion (8) becomes V (N ) V ( N)+V N ( N)ȳ +V N ( N)ȳ var i ŷ (i) 2 var i ŷ (i) Ŷ 2 2 ŷ (i)di + var i ŷ (i) + var i ŷ (i) Ŷ  2 var i ŷ (i) ÂŶ µ var i ŷ (i) 2 ³Ŷ 2 + var i ŷ (i) ÂŶ + 2Â2 di + 2 V NN( N)ȳ 2 ³ Ŷ Â2 (9) 5

6 Combining erms, and using he uiliy funcion (4), V (N ) V ( N)+V N ( N)ȳ Ŷ Â + 2 Combining equaions (6) and (), var i ŷ (i)+ 2 ( + η) ³Ŷ Â 2 () U(C,ξ ) V (N ) U( C,) V ( N) +U c ( C,) γȳ ( + ˆξ ³ ) ˆγ + Ŷ + ³ 2 2 ( ) ˆγ + Ŷ +U ξ ( C,)ˆξ + 2 U ξ,ξˆξ 2 V N ( N) ȳŷ Â + µ var i ŷ (i) ³Ŷ 2 ( + η) Â () Before simplifying his expression, noe ha he seady-sae labor marke equilibrium condiion becomes V N /Ūc w /Φ R Ifwedefine Θ such ha Θ γφ R, hen V N ( N)ȳ can be wrien as U c ( C) γȳ ( Θ) WewillassumeΘ is small so ha erms such as γφ R Ŷ 2 ( Θ) Ŷ 2 become simply Ŷ 2 In his case, we can now wrie equaion (??) as U(C,ξ ) V (N ) U( C,) V ( N)+U c ( C) γȳ ( + ˆξ ³ ) ˆγ + Ŷ + ³ 2 2 ( ) ˆγ + Ŷ U c ( Ŷ C) γȳ ( Θ) Â + 2 ³Ŷ 2 ( + η) Â µ 2 U c( C) γȳ ( Θ) var i ŷ (i)+u ξ ( C,)ˆξ + 2 U ξ,ξˆξ 2 Collecing erms, U(C,ξ ) V (N ) U( C,) V ( N)+U c ( C,) γȳ hθŷ + ˆξ i Ŷ +U c ( C,) γȳ ³ 2 2 ( ) 2 ˆγ + Ŷ ³Ŷ 2 ( + η) Â µ 2 U c( C,)Ȳ var i ŷ (i) +U c ( C,) γȳ h ( + ˆξ i )ˆγ +( Θ)Â + U ξ ( C,)ˆξ + 2 U ξ,ξˆξ 2 This is sronger han he corresponding assumpion made by Woodford (999) He assumes (Φ) is small The presence of γ and a posiive seady-sae nominal ineres rae increase he average disorions in he economy relaive o he case in which he monopoly disorion is he only source of inefficiency 6

7 Define Ẑ " ( + η)â + ˆξ # +( )ˆγ + η and z Θ + η Then he uiliy approximaion can be wrien as U(γ Y,ξ ) V (N ) U(Ȳ,) V ( N) 2 ( + η) U c(ȳ ³Ŷ )Ȳ Ẑ z 2 µ 2 U c(ȳ )Ȳ var i ŷ (i)+isp (2) isp U c ( C,) γȳ h ( + ˆξ i )ˆγ +( Θ)Â + U ξ ( C,)ˆξ + 2 U ξ,ξˆξ 2 2 U c( C,)ȲcẐ2 are erms independen of sabilizaion policy Recalling ha µ µ Ŷ f +η ³ µ Â ˆRf + η + η ˆξ ˆγ + η, Ẑ can be wrien as µ Ẑ Ŷ f ³ + ˆRf +ˆγ + η Wih he assumed uiliy funcion, log y (i) logy (log p (i) log P ) so var i log y (i) 2 var i log p (i) The price adjusmen mechanism involves a randomly chosen fracion ω of all firms opimally adjusing price each period Define P E log p (i) and var i log p (i) Then Woodford (2) shows ha µ ω ω + π 2, ω ω is he fracion of firms ha rese heir price each period If is he iniial degree of price dispersion, hen X X β ω β π 2 + ip, ( ω)( ωβ) 7

8 ip denoes erms independen of moneary policy Combining his wih (2), he presen discouned value of he uiliy of he represenaive household can be approximaed by X β U Ū Ω X β L and L π 2 + λ ³Ŷ Ẑ z 2, Ω 2 U ω cȳ, ( ω)( ωβ) µ ( ω)( ωβ) + η λ ω κ ( + η) The parameer κ is he coefficien on real marginal coss in he inflaion adjusmen equaion 22 The demand shock in eq 26 In equaion (26) of he ex, a demand shock u appears Using he resuls from secion 3 above, and replacing Y f wih Y, µ Ŷ Ŷ E ³Ŷ+ Ŷ + (r r ), ³ ˆr E Ŷ+ Ŷ ³E ˆξ+ ˆξ + E ˆγ + ˆγ From he definiion of Ŷ given in equaion (24), µ h E Ŷ+ Ŷ ( + η)(e  + + η Â) (E ˆγ + ˆγ )+(E ˆξ+ ˆξ )i u Hence, he Euler condiion can be wrien as µ x E x + ( ˆR E π + )+u (3) µ ³ r + η E Ŷ+ Ŷ ( + η)(e  + Â) which is equaion (29) of he ex µ ³ E ˆξ+ ˆξ + E ˆγ + ˆγ ³ η (E ˆξ+ ˆξ )+ η +η (E ˆγ + ˆγ ), 8

9 3 Opimal moneary policy In order o conras resuls wih and wihou he cos channel, i will be useful o inroduce he index variable δ defined as ½ if here is no cos channel δ if here is a cos channel and o wrie he inflaion adjusmen equaion as π βe π + + κ( + η)x + δκ ˆR, (4) 3 Discreion: derivaion of equaion 3 The problem of he cenral banker is o choose a pah for ˆR, and he implied pahs for x and π, o maximize ( U " 2 E X µ π 2 +i + λ x +i #) 2 + η ˆγ subjec o and i x E x + µ ( ˆR E π + )+u π βe π + + κ( + η)x + δκ ˆR, µ +η ³ η u (E Â + + η Â) (E ˆξ+ ˆξ )+ η +η (E ˆγ + ˆγ ) Le χ and ψ denoe he Lagrangian mulipliers associaed wih each of hese consrains a ime Under opimal discreion, he firs order condiions for he cenral bank s problem are: U λ x U π + ψ π ; µ U ˆR δκψ + µ x + η ˆγ χ κ( + η)ψ + χ ; From he las of hese, χ δκψ so ha χ in he absence of a cos channel (δ ) From he second firs order condiion, ψ π Using hese resuls in he firs of he firs order condiions yields µ π λ κ [( δ)+η] x + η ˆγ, (5) which is equaion (3) of he ex for δ The equilibrium behavior of inflaion is found by subsiuing equaion (5) ino he consrains (3) and (4), and solving he resuling wo-equaion sysem 9

10 32 Commimen: derivaion of firs order condiions under commimen The fully opimal commimen policy involves a choice of curren and fuure values of inflaion, he oupu gap, and he nominal ineres rae o maximize X µ 2 E β (π 2 + λ x ˆγ + η 2 +χ x E x + + ( ˆR E π + ) g +ψ h π βe π + κ( + η)x κδ ˆR io, χ and ψ are Lagrangian mulipliers The basic problem of ime inconsisency is illusraed by conrasing he firs order condiions for ime and for fuure periods + i for i> A ime, he cenral bank ses π, x,and ˆR such ha π + ψ (6) λ x + η ˆγ + χ κ( + η)ψ (7) χ δκψ, (8) while for + i>, π +i + µ ψ +i ψ +i χ+i (9) µ λ x +i ˆγ + η +i + χ +i β χ +i κ( + η)ψ +i (2) µ χ +i δκψ +i (2) If δ (no cos channel), χ +i for all i and hese firs order condiions reduce o he case considered by Woodford (2), Clarida, Galí, and Gerler (999), or McCallum and Nelson (2) When δ, henaureofheime inconsisency inheren in his problem shows up in he comparisons of (6) o (9) and (7) o (2) These firs order condiions for ime can be rewrien as and for + i, i>, as π µ λ κη x + η ˆγ π +i ψ +i +(+κ) ψ +i (22)

11 µ λ x +i ˆγ + η +i β κψ +i + κηψ +i (23) Svensson and Woodford (999) have described a policy ha implemens equaions (22) and (23) for all as he imeless perspecive precommimen policy 33 Simulaions To carry ou he simulaions, we wrie he model in sae space form: ˆγ + ˆγ + u γ+ ˆξ+ ˆξ+ u ξ+ â + â + u a+ Ŷ Ŷ e A Ŷ Ŷ e + R +, x x E x + E x β+κ + β E π + E π + β A ρ γ ρ ξ ρ a +η +η η( ρ γ ) +η +η +η +η +η +η +η +η η( ρ ξ ) (+η)( ρ a ) (+η) +η + κ(+η) β β κ(+η) β β The equilibrium under opimal discreion and commimen are obained using he programs of Paul Söderlind, available a hp://homeiscalinech/paulsoderlind/