A Suite of Models for Dynare Description of Models

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1 A Suie of Models for Dynare Descripion of Models F. Collard, H. Dellas and B. Diba Version. Deparmen of Economics Universiy of Bern

2 A REAL BUSINESS CYCLE MODEL A real Business Cycle Model The problem of he household is max E σ β j ζ c σ +j c,+j j= σ ψ + ν ζ ν h,+j h+ν +j c + i = w h + z k τ k + = i + δ)k The las wo condiions can be combined o give Then he firs order condiions are given by k + = w h + z + δ)k c τ The problem of he firm is max a k α h α which leads o he firs order condiions ζ σ c, c σ = λ ) ψζ ν h, hν = λ w ) λ = βe λ + z + + δ)) 3) w h z k w = α) y h and z = α y k The axes, τ, finance an exogenous sream of governmen expendiures, g, such ha τ = g. All shocks follow AR) processes of he ype loga + ) = a loga ) + ε a,+ logg + ) = g logg ) + g ) logg) + ε g,+ logζ c,+ ) = c logζ c, ) + ε c,+ logζ h,+ ) = h logζ h, ) + ε h,+ The general equilibrium is herefore represened by he following se of equaions and he definiion of he shocks. ζ σ c, c σ = λ ψζ ν h, hν = λ α) y h λ = βe λ + α y ) + + δ) k + k + = i + δ)k y = a k α h α y = c + i + g Deparmen of Economics Universiy of Bern

3 A NOMINAL MODEL WITH PRICE ADJUSTMENT COSTS 3 A Nominal Model wih Price Adjusmen Coss In all wha follows, we will assume zero inflaion in he seady sae. The problem of he household is max E σ β j ζ c σ +j c,+j j= σ ψ + ν ζ ν h,+j h+ν +j B + P c + P i = R B P w h + P z k P τ k + = i + δ)k The las wo condiions can be combined o give B + P c + P k + = R B P w h + P z + δ)k P τ Then he firs order condiions are given by ζ σ c, c σ = Λ P 4) ψζ ν h, hν = Λ P w 5) Λ P = βe Λ + P + z + + δ)) 6) Λ = βr E Λ + 7) The economy is comprised of many secors. The firs secor he final good secor combines inermediae goods o form a final good in quaniy y : y = ) θ y i) θ θ θ di 8) The problem of he final good firm is hen subjec o 8), which rewries max P y {y i);i,)} max P {y i);i,)} which gives rise o he demand funcion P i)y i)di ) θ y i) θ θ θ di P i)y i)di ) P i y i) = y 9) These inermediae goods are produced by inermediaries, each of which has a local monopoly power. Each inermediae firm i, i, ), uses a consan reurns o scale echnology P y i) = a k i) α h i) α ) Deparmen of Economics Universiy of Bern

4 A NOMINAL MODEL WITH PRICE ADJUSTMENT COSTS 4 where k i) and h i) denoe capial and labor. The firm minimizes is real cos subjec o ). Minimized real oal coss are hen given by s x i) where he real marginal cos, s, is given by wih ς = α α α) α. s = zα w α ςa Inermediae goods producers are monopolisically compeiive, and herefore se prices for he good hey produce. However, i incurs a cos whenever i changes is price relaively o he earlier period. This cos is given by j= ϕ p ) P i) P i y The problem of he firm is hen o maximize he profi funcion ) ) E ϕ p P+j i) Φ,+j P +j i)y +j i) P +j s +j y +j i) P +j P +j i y +j ) where Φ,+j is an appropriae discoun facor derived from he household s opimaliy condiions, and proporional o β j Λ+j Λ. The firs order condiion of he problem is given by ) P i θ) y + θ P ) P P i) s P i y P ) P P i) ϕ P i) p P i [ ) ] Λ+ P + P + i) P+ i) + βe Λ P i) ϕ p y + = P i Using Sheppard s lemma we ge he demand for each inpu w = α)s y i) h i) z = αs y i) k i) The axes, τ, finance an exogenous sream of governmen expendiures, g, such ha τ = g In order o close he model, we add a Taylor rule ha deermines he nominal ineres rae logr ) = r logr ) + ) logr) + γ π logπ ) logπ)) + γ y logy ) logy)) ) where π = P /P denoes aggregae inflaion. All shocks follow AR) processes of he ype loga + ) = a loga ) + ε a,+ logg + ) = g logg ) + g ) logg) + ε g,+ logζ c,+ ) = c logζ c, ) + ε c,+ logζ h,+ ) = h logζ h, ) + ε h,+ y Deparmen of Economics Universiy of Bern

5 3 A NOMINAL MODEL WITH STAGGERED PRICE CONTRACTS 5 The symmeric general equilibrium is herefore represened by he following se of equaions ζ σ c, c σ = λ ψζ ν h, hν = λ α)s y h λ = βe λ + αs + y + ) λ+ λ = βr E π + )] + δ k + = θ)y + θs y π ϕ p π ) y + βe [ λ+ λ π + ϕ p π + ) y + k + = i + δ)k y = a k α h α y = c + i + g + ϕ p π ) y logr ) = r logr ) + ) logr) + γ π logπ ) logπ)) + γ y logy ) logy)) ) and he definiion of he shocks. ] 3 A Nominal Model wih Saggered Price Conracs The main difference beween his model and he previous one lies in he specificaion of he nominal rigidiies. Inermediae goods producers are monopolisically compeiive, and herefore se prices for he good hey produce. We follow Calvo [983] in assuming ha firms se heir prices for a sochasic number of periods. In each and every period, a firm eiher ges he chance o adjus is price an even occurring wih probabiliy ξ) or i does no. This is illusraed in he following figure. ξ ξ... P = P ξ P + = P ξ ξ P + = P ξ P + = P + P + = P ime When he firm does no rese is price, i jus applies he price i charged in he las period such ha P i) = P i). When i ges a chance o do i, firm i reses is price, P i), in period in order o maximize is expeced discouned profi flow his new price will generae. In period, he profi is given by ΠP i)). In period +, eiher he firm reses is price, such ha i will ge Deparmen of Economics Universiy of Bern

6 3 A NOMINAL MODEL WITH STAGGERED PRICE CONTRACTS 6 ΠP + i)) wih probabiliy q, or i does no and is + profi will be ΠP i)) wih probabiliy ξ. Likewise in +. The expeced profi flow generaed by seing P i) in period wries max P i) E Φ,+j ξ j ΠP i)) j= subjec o he oal demand i faces: ) P i y i) = y P and where ΠP i)) = P i) P +j s +j ) y +j i). Φ,+j is an appropriae discoun facor relaed o he way he household value fuure as opposed o curren consumpion, such ha j= Φ,+j β j Λ +j Λ This leads o he price seing equaion E βξ) j Λ ) +j P θ) i y +j + θ P ) ) +j P i s +jy +j = Λ P +j P +j i) P +j from which i shall be clear ha all firms ha rese heir price in period se i a he same level P i) = P, for all i, )). This implies ha P = P n P d ) where P n = E βξ) j θ Λ +j θ P +θ j= +j s +jy +j and = E βξ) j Λ +j P+jy θ +j P d j= Forunaely, boh P n and P d admi a recursive represenaion, such ha P n = θ θ Λ P +θ s y + βξe [P+] n 3) P d = Λ P θ y + βξe [P+] d 4) Recall now ha he price index is given by P = ) P i) θ θ di In fac i is composed of surviving conracs and newly se prices. Given ha in each an every period a price conrac has a probabiliy ξ of ending, he probabiliy ha a conrac signed in Deparmen of Economics Universiy of Bern

7 3 A NOMINAL MODEL WITH STAGGERED PRICE CONTRACTS 7 period j survives unil period and ends a he end of period is given by ξ)ξ j. Therefore, he aggregae price level may be expressed as he average of all surviving conracs P = ξ)ξ j P j θ j= θ which can be expressed recursively as P = ξ)p θ + ξp θ ) θ 5) Noe ha since he wage rae is common o all firms, he capial labor raio is he same for any firm: k i) h i) = k j) h j) = k h we herefore have inegraing across firms, we obain ) α k y i) = a h i) h ) α k y i)di = a h i)di h denoing h = h i)di, and making use of he demand for y i), we have Denoe P i) P diy = a k α h α = ) P i di P P = ξ)ξ j j P j= ) P θ = ξ) + P P = ξ) P P = ξ) P P = ξ) P + + ξ + ξ j= ξ)ξ j P j P P ξ)ξ l+ l l= P P P P ) P θ = ξ) + ξπ θ P l= P ξ)ξ l P l P Hence aggregae oupu is given by y = a k α h α Deparmen of Economics Universiy of Bern

8 3 A NOMINAL MODEL WITH STAGGERED PRICE CONTRACTS 8 Hence he se of equaions defining he general equilibrium is given by, where p n = P n /P θ, p d = P d /P θ, λ = Λ P and π = P /P. ζ σ c, c σ = λ ψζ ν h, hν = λ α)s y h λ = βe [ λ + αs + y + ) λ+ λ = βr E π + )] + δ k + p n = θ θ λ s y + βξe [π+p θ n +] p d = λ y + βξe [π+ θ pd +] ) p n θ = ξ) p d + ξπ θ ) p n θ = ξ) p d + ξπ θ k + = i + δ)k y = a k α h α y = c + i + g logr ) = r logr ) + ) logr) + γ π logπ ) logπ)) + γ y logy ) logy)) ) and he definiion of he shocks. Deparmen of Economics Universiy of Bern

9 4 A SMALL OPEN ECONOMY MODEL WITH STAGGERED PRICES 9 4 A Small Open Economy Model wih Saggered Prices The problem of he domesic household is max E β j ζ c σ σ +j c,+j ψ + ν ζ ν σ j= h,+j h+ν +j B h + e B f + P c + P i = R B + e R B f + P χ w h + P z k P τ P e B f ) k + = i ϕ ) ) k i δ + δ)k k Then he firs order condiions are given by ζ σ c, c σ = Λ P 6) ψζ ν h, hν = Λ P w 7) )) i Λ P = Q ϕ k δ 8) k Λ = βr E Λ + 9) Λ + χe B f ) = e + βr E Λ + ) e [ Q = βe Λ + P + z + + Q + δ + ϕ ) ))] k i+ δ ) where Λ and Q denoe respecively he Lagrange muliplier of he firs and second consrain. The reailer firm combines foreign and domesic goods o produce a non radable final good. I deermines is opimal producion plans by maximizing is profi max {x d,xf } P Y P x, x d e P x,x f where P x, and Px denoe he price of he domesic and foreign good respecively, denominaed in erms of he currency of he seller. The final good producion funcion is described by he following CES funcion y = ω x d + ω) x f ) ) where ω, ) and, ). Opimal behavior of he reailer gives rise o he demand for he domesic and foreign goods x d = ) Px, P ωy and x f = e P x, P k + ) ω)y 3) x d and x f are hemselves combinaions of he domesic and foreign inermediae goods according o ) θ x d = x d i) θ θ ) θ θ di and x f = x f θ θ i) θ di 4) where θ, ). Deparmen of Economics Universiy of Bern

10 4 A SMALL OPEN ECONOMY MODEL WITH STAGGERED PRICES Profi maximizaion yields demand funcions of he form: ) x d Px i ) i) = x d, x f P P i) = x i x Px x f A his sage, we need o ake a sand on he behavior of he foreign firms in order o deermine he demand for he domesic good by foreign agens. We assume ha heir behavior is symmerical o he one observed in he domesic economy, such x d i) = Px i) e P x x d Plugging hese demand funcions in profis and using free enry in he final good secor, we ge he following general price indexes P x = P = ωp ) P x i) θ θ di x ) + ω)e Px) 5) 6) 7) The inermediae goods are produced by inermediaries, each of which has a local monopoly power. Each inermediae firm i, i, ), uses a consan reurns o scale echnology x i) = a k i) α h i) α 8) where k i) and h i) denoe capial and labor. The firm minimizes is real cos subjec o 8). Minimized real oal coss are hen given by s x i) where he real marginal cos, s, is given by wih ς = α α α) α. s = zα w α ςa The price seing behavior is essenially he same as in a closed economy. The expeced profi flow generaed by seing P i) in period wries subjec o he oal demand i faces: max P x,i) E Φ,+j ξ j Π P x, i)) P j= ) P i x i) = x wih x = x d + x d and where Π P ) x, i)) = Px, i) P +j s +j x +j i). Φ,+j is an appropriae discoun facor relaed o he way he household value fuure as opposed o curren consumpion, such ha Φ,+j β j Λ +j Λ Deparmen of Economics Universiy of Bern

11 4 A SMALL OPEN ECONOMY MODEL WITH STAGGERED PRICES This leads o he price seing equaion ) θ E βξ) j Λ +j Px, i) P +j Px, i) θ) y +j + θ s +jx +j = Λ P x,+j P x,+j i) P x,+j j= from which i shall be clear ha all firms ha rese heir price in period se i a he same level P i) = P, for all i, )). This implies ha P x, = P n x, P d x, 9) where and Px, n = E βξ) j θ Λ +j θ P +jpx,+js θ +j x +j j= Px, d = E βξ) j Λ +j Px,+jx θ +j Forunaely, boh P n x, and P d x, admi a recursive represenaion, such ha Recall now ha he price index is given by j= Px, n = θ θ Λ P Px,s θ x + βξe [Px,+] n 3) Px, d = Λ Px,x θ + βξe [Px,+] d 3) P x, = ) P x, i) θ θ di In fac i is composed of surviving conracs and newly se prices. Given ha in each an every period a price conrac has a probabiliy ξ of ending, he probabiliy ha a conrac signed in period j survives unil period and ends a he end of period is given by ξ)ξ j. Therefore, he aggregae price level may be expressed as he average of all surviving conracs P x, = ξ)ξ j θ P j= x, j θ which can be expressed recursively as P x, = ) θ ξ) P x, + ξpx, θ θ 3) Noe ha since he wage rae is common o all firms, he capial labor raio is he same for any firm: k i) h i) = k j) h j) = k h we herefore have ) α k x i) = a h i) h Deparmen of Economics Universiy of Bern

12 4 A SMALL OPEN ECONOMY MODEL WITH STAGGERED PRICES inegraing across firms, we obain ) α k x i)di = a h i)di h denoing h = h i)di, and making use of he demand for y i), we have Denoe = = ) Px, i di P x, Px, i) P x, Px, j ξ)ξ j j= = ξ) = ξ) = ξ) = ξ) = ξ) Px, P x, Px, P x, Px, P x, Px, P x, Px, P x, P x, ξ + ξ j= diy = a k α h α ξ)ξ j Px, j P x, l= Px, ξ)ξ l+ Px, l P x, P x, Px, P x, + ξπ θ x, l= ξ)ξ l Px, l P x, Hence aggregae oupu is given by x = a k α h α The behavior of he foreign economy is assumed o be very similar o he one observed domesically. We however assumeha foreign oupu and prices are exogenously given and modelled as AR) processes. We furher assume ha P x, = P. The foreign household s saving behavior is he same as in he domesic economy absen preference shocks), such ha he foreign nomina ineres rae is given by y σ = βr E y + σ P P + We assume ha foreign households do no buy domesic bonds, which implies ha in equilibrium B d =. Deparmen of Economics Universiy of Bern

13 4 A SMALL OPEN ECONOMY MODEL WITH STAGGERED PRICES 3 The general equilibrium is hen given by ζ σ c, c σ = λ ψζ ν p x, x h, hν = λ α)s h )) i λ = q ϕ k δ k y = c + i + g + χ bf 36) x = a k α h α x d = p x d = x, ωy px, rer 33) 34) 35) 37) 38) ) ω)y 39) x f = rer p x,) ω)y 4) x = x d + x d = ωp x, λ = βr E λ + π + 4) + ω)rer p ) 4) λ + χb f ) = e βr + E λ + 44) π [ + p x,+ x + q = βe λ + αs + + q + δ + ϕ ) ))] k i+ δ 45) k + b f = e R π b f + p x,x y 46) k + = i ϕ ) ) k i δ + δ)k 47) k rer = e p π p rer 48) p x, = π x, p x, 49) π logr ) = r logr ) + ) logr) + γ π logπ ) logπ)) + γ y logy ) logy)) ) 5) p n x, = θ θ λ p θ x,s x + βξe [p n x,+π+] θ 5) p d x, = λ p θ x,x + βξe [p d x,+π+ θ ] 5) ) p n θ ) θ θ x, px, p x, = ξ) p d + ξ 53) x, π k + 43) = ξ) p n x, p x, p d x, y σ = βr E y + σ p p + + ξ π θ x, 54) where λ = Λ P, rer = e P /P, p x, = P x, /P, p n = P n /P θ, p d = P d /P θ, π = P /P, 55) Deparmen of Economics Universiy of Bern

14 6 A REAL SMALL OPEN ECONOMY MODEL 4 π x, = P x, /P x,, e = e /e. All shocks follow AR) processes of he ype loga + ) = a loga ) + ε a,+ logg + ) = g logg ) + g ) logg) + ε g,+ logζ c,+ ) = c logζ c, ) + ε c,+ logζ h,+ ) = h logζ h, ) + ε h,+ logy+) = y logy ) + y ) logy) + ε y,+ logp +) = p logp ) + p ) logp) + ε p,+ 5 A Nominal Small Open Economy Model wih Price Adjusmen Coss When price conracs are replaced wih price adjusmen coss, equaions 36) 37) become y = c + i + g + χ bf x = a k α h α + ϕ p π x, ) y and equaions 5) 54) are replaced wih [ ] λ+ θ)p x, x + θs x ϕ p π x, π x, )y + βe π x,+ ϕ p π x,+ )y + λ = 6 A Real Small Open Economy Model All nominal aspecs disappear, such ha he general equilibrium becomes ζ σ c, c σ = λ 56) ψζ ν h, hν = λ α) p x,x λ = q ϕ k i k δ 57) h )) 58) y = c + i + g + χ bf 59) x = a k α h α 6) x d = px, ωy 6) ) x d px, = ω)y rer 6) x f = rer p x,) ω)y 63) x = x d + x d 64) = ωp x, + ω)rer p ) 65) Deparmen of Economics Universiy of Bern

15 6 A REAL SMALL OPEN ECONOMY MODEL 5 λ = βr E λ + 66) λ + χb f ) = βr E λ + 67) [ q = βe λ + α p x,+x + + q + δ + ϕ ) ))] k i+ δ 68) k + ogeher wih he shocks. b f = R b f + p x,x y 69) k + = i ϕ ) ) k i δ + δ)k 7) k k + y σ = βr E y + σ 7) Deparmen of Economics Universiy of Bern

16 7 A NOMINAL COUNTRY MODEL WITH STAGGERED PRICES 6 7 A Nominal Counry Model wih Saggered Prices The problem of he domesic household is max E β j ζ c σ σ +j c,+j ψ + ν ζ ν σ j= h,+j h+ν +j B h + e B f + P c + P i = R B + e R B f + P w h + P z k P τ k + = i ϕ ) ) k i δ + δ)k k Then he firs order condiions are given by ζ σ c, c σ = Λ P ψζ ν h, hν = Λ P w )) i Λ P = Q ϕ k δ k 7) 73) 74) Λ = βr E Λ + 75) Λ = βr e + E Λ + 76) e [ Q = βe Λ + P + z + + Q + δ + ϕ ) ))] k i+ δ 77) where Λ and Q denoe respecively he Lagrange muliplier of he firs and second consrain. The behavior of he foreign household, hereafer denoed by a, is oally symmerical. We furher have he following risk sharing condiion Λ = Λ e The reailer firm combines foreign and domesic goods o produce a non radable final good. I deermines is opimal producion plans by maximizing is profi max {x d,xf } P Y P x, x d e P x,x f where P x, and Px denoe he price of he domesic and foreign good respecively, denominaed in erms of he currency of he seller. The final good producion funcion is described by he following CES funcion y = ω x d + ω) x f ) 78) where ω, ) and, ). Opimal behavior of he reailer gives rise o he demand for he domesic and foreign goods x d = ) Px, Abroad, he behavior is symmerical, such ha P ωy and x f = e P x, P k + ) ω)y 79) y = ω x f + ω) ) x d 8) Deparmen of Economics Universiy of Bern

17 7 A NOMINAL COUNTRY MODEL WITH STAGGERED PRICES 7 and ) x d Px, = ω)y e P and x f = P x, P ) ωy 8) x d and x f are hemselves combinaions of he domesic and foreign inermediae goods according o ) θ x d = x d i) θ θ ) θ θ di and x f = x f θ θ i) θ di 8) where θ, ). Likewise abroad x d = x d i) θ θ di ) θ θ and x f = x f ) θ i) θ θ θ di 83) Profi maximizaion yields demand funcions of he form: similarly abroad ) x d Px i ) i) = x d, x f P P i) = x i x Px x f, x d i) = Px i) P x x d, x f i) = P x i) P x x f Plugging hese demand funcions in profis and using free enry in he final good secor, we ge he following general price indexes P x = P = ωp P = ω ) P x i) θ θ di, P x = x ) Px e + ω)e P x) + ω)p x ) ) ) Pxi) θ θ di 84) 85) 86) The inermediae goods are produced by inermediaries, each of which has a local monopoly power. Each inermediae firm i, i, ), uses a consan reurns o scale echnology x i) = a k i) α h i) α 87) where k i) and h i) denoe capial and labor. The firm minimizes is real cos subjec o 87). Minimized real oal coss are hen given by s x i) where he real marginal cos, s, is given by wih ς = α α α) α. Similarly abroad s = zα w α ςa x i) = a k i) α h i) α 88) Deparmen of Economics Universiy of Bern

18 7 A NOMINAL COUNTRY MODEL WITH STAGGERED PRICES 8 where k i) and h i) denoe capial and labor. The firm minimizes is real cos subjec o 87). Minimized real oal coss are hen given by s x i) where he real marginal cos, s, is given by wih ς = α α α) α. s = z α w α ςa The price seing behavior is essenially he same as in a closed economy. The expeced profi flow generaed by seing P i) in period wries subjec o he oal demand i faces: max P x,i) E Φ,+j ξ j Π P x, i)) P j= ) P i x i) = x wih x = x d + x d and where Π P ) x, i)) = Px, i) P +j s +j x +j i). Φ,+j is an appropriae discoun facor relaed o he way he household value fuure as opposed o curren consumpion, such ha Φ,+j β j Λ +j Λ This leads o he price seing equaion ) θ E βξ) j Λ +j Px, i) P +j Px, i) θ) y +j + θ s +jx +j = Λ P x,+j P x,+j i) P x,+j j= from which i shall be clear ha all firms ha rese heir price in period se i a he same level P i) = P, for all i, )). This implies ha P x, = P n x, P d x, 89) where and Px, n = E βξ) j θ Λ +j θ P +jpx,+js θ +j x +j j= Px, d = E βξ) j Λ +j Px,+jx θ +j Forunaely, boh P n x, and P d x, admi a recursive represenaion, such ha j= Px, n = θ θ Λ P Px,s θ x + βξe [Px,+] n 9) Px, d = Λ Px,x θ + βξe [Px,+] d 9) Deparmen of Economics Universiy of Bern

19 7 A NOMINAL COUNTRY MODEL WITH STAGGERED PRICES 9 Likewise abroad, Recall now ha he price index is given by Px, n θ = θ Λ P Px, θ s x + βξe [Px,+] n 9) Px, d = Λ Px, θ x + βξe [Px,+] d 93) P x, = ) P x, i) θ θ di In fac i is composed of surviving conracs and newly se prices. Given ha in each an every period a price conrac has a probabiliy ξ of ending, he probabiliy ha a conrac signed in period j survives unil period and ends a he end of period is given by ξ)ξ j. Therefore, he aggregae price level may be expressed as he average of all surviving conracs P x, = ξ)ξ j θ P j= x, j θ which can be expressed recursively as P x, = ) θ ξ) P x, + ξpx, θ θ 94) Noe ha since he wage rae is common o all firms, he capial labor raio is he same for any firm: k i) h i) = k j) h j) = k h we herefore have inegraing across firms, we obain ) α k x i) = a h i) h ) α k x i)di = a h i)di h denoing h = h i)di, and making use of he demand for y i), we have Px, i) P x, diy = a k α h α Deparmen of Economics Universiy of Bern

20 7 A NOMINAL COUNTRY MODEL WITH STAGGERED PRICES Denoe herefore = = ) Px, i di P x, Px, j ξ)ξ j j= = ξ) = ξ) = ξ) = ξ) Px, P x, Px, P x, Px, P x, Px, P x, P x, ξ + ξ j= ξ)ξ j Px, j P x, l= Px, ξ)ξ l+ Px, l P x, P x, Px, P x, l= ξ)ξ l Px, l P x, = ξ) Px, P x, + ξπ θ x, Hence aggregae oupu is given by x = a k α h α In a general equilibrium, we will have B d + B d is hen given by = and B f + Bf =. The general equilibrium ζ σ c, c σ = λ 95) ζc, σ c σ = λ 96) ψζ ν h, hν = λ α)p x, s x h 97) x ψζh, ν h ν = λ α)p x,s h 98) )) i λ = q ϕ k δ 99) k )) i λ = q ϕ k k δ ) y = c + i + g ) y = c + i + g ) x = a k α h α 3) x = a k α h α 4) Deparmen of Economics Universiy of Bern

21 7 A NOMINAL COUNTRY MODEL WITH STAGGERED PRICES x d = p x d = x, ωy px, rer 5) ) ω)y 6) x f = rer p x,) ω)y 7) x f = p x, ωy x = x d + x d x = x f + xf = ωp = ωp x, λ = λ rer x, 8) 9) ) + ω)rer p x,) ) + ω) p x, rer ) λ + λ = βr E π [ + p x,+ x + q = βe λ + αs + + q + δ + ϕ ) ))] k i+ δ k + k + q = βe [λ +αs p x,+ x + + k+ + q+ δ + ϕ k i ) ))] + k+ δ k + = i ϕ ) ) k i δ k k + = i ϕ k R = R E e + rer = e π π p x, = π x, p x, π rer i k ) 3) 4) 5) 6) + δ)k 7) ) ) δ + δ)k 8) 9) ) ) p x, = π x, π p x, ) logr ) = r logr ) + ) logr) + γ π logπ ) logπ)) + γ y logy ) logy)) ) 3) logr ) = r logr ) + ) logr) + γ π logπ ) logπ)) + γ y logy ) logy)) ) 4) p n x, = θ θ λ p θ x,s x + βξe [p n x,+π+] θ 5) p n x, = θ θ λ p x, θ s x + βξe [p n x,+π+] θ 6) p d x, = λ p θ x,x + βξe [p d x,+π+ θ 7) p d x, = λ p x, θ x + βξe [p d x,+π θ x,+ ] 8) Deparmen of Economics Universiy of Bern

22 8 A NOMINAL COUNTRY MODEL WITH PRICE ADJUSTMENT COSTS p x, = ) p n θ x, ξ) p d + ξ x, p x, = ξ) ) p n θ x, p d + ξ x, px, π ) θ p ) θ x, π θ θ 9) 3) = ξ) = ξ) p n x, p x, p d x, p n x, p x, pd x, + ξ π θ x, 3) + ξ π x, θ 3) where λ = Λ P, λ = Λ P, rer = e P /P, p x, = P x, /P, p x, = Px,/P, p n = P n /P θ, p d = P d /P θ, p n = P n /P θ, p d = P d /P θ, π = P /P, π = P /P, π x, = P x, /P x,, πx, = Px,/P x,, e = e /e. All shocks follow AR) processes of he ype loga + ) = a loga ) + a loga ) + ε a,+ loga +) = a loga ) + a loga ) + ε a,+ logg + ) = g logg ) + g ) logg) + ε g,+ logg +) = g logg ) + g ) logg) + ε g,+ logζ c,+ ) = c logζ c, ) + ε c,+ logζ c,+) = c logζ c,) + ε c,+ logζ h,+ ) = h logζ h, ) + ε h,+ logζ h,+ ) = h logζ h, ) + ε h,+ 8 A Nominal Counry Model wih Price Adjusmen Coss When price conracs are replaced wih price adjusmen coss, equaions ) 4) become y = c + i + g + ϕ p π x, ) y y = c + i + g + ϕ p π x, ) y x = a k α h α x = a k α h α and equaions 5) 3) are replaced wih [ ] λ+ = θ)p x, x + θs x ϕ p π x, π x, )y + βe π x,+ ϕ p π x,+ )y + λ [ λ = θ)p x,x + θs x ϕ p πx,π x, )y ] + βe + λ πx,+ϕ p πx,+ )y+ Deparmen of Economics Universiy of Bern

23 9 A REAL COUNTRY MODEL 3 9 A Real Counry Model All nominal aspecs disappear, such ha he general equilibrium becomes ζ σ c, c σ ζc, σ c σ = λ = λ 33) 34) ψζ ν x h, hν = λ α)p x, 35) h ψζh, ν h ν = λ α)p x x, h 36) )) i λ = q ϕ k δ 37) k )) i λ = q ϕ k k δ 38) y = c + i + g 39) y = c + i + g 4) x d = px, ωy 4) ) x d px, = ω)y rer 4) x f = rer p x,) ω)y 43) x f = p x, ωy 44) x = x d + x d 45) x = x f + xf 46) x = a k α h α 47) x = a k α h α 48) = ωp x, = ωp x, λ = λ [ q = βe q = βe [ + ω)rer p x,) 49) + ω) p x, rer ) λ + α p x,+x + k + + q + λ +α p x,+ x + k+ i ϕ k k + q + δ + ϕ ) ))] k i+ δ k + δ + ϕ k i ) ))] + δ k + 5) 5) 5) 53) k + = i ϕ ) ) k δ + δ)k 54) k ) ) i k+ = i δ + δ)k 55) Deparmen of Economics Universiy of Bern