Understanding the Gains from Wage Flexibility: the Exchange Rate Connection. Online Appendix

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1 Undersanding he Gains from Wage Flexibiliy: he Exchange Rae Connecion. Jordi Galí, CREI, Universia Pompeu Fabra and Barcelona GSE. Tommaso Monacelli, Universià Bocconi and IGIER. Online Appendix July 22, 216 We are hankful o Michele Fornino and Francesco Giovanardi for heir excellen research assisance. 1

2 A Medium Scale DSGE Model of a Small Open Economy We describe a medium-sized DSGE model of a small open economy inhabied by a represenaive household and by a coninuum of firms each producing a differeniaed variey. The household invess in financial asses and accumulaes physical capial. In his version of he model all goods are raded. See Appendix C for an exension o a wo-secor srucure wih boh raded and non-raded goods. A.1 Households The household feaures a coninuum of members, indexed by j [, 1]. Each household member is specialized in a differeniaed labor service, which she supplies in an amoun N j. Household preferences are given by: { } E β U C, {N j}; Z 1 where C j C j hc 1 measures exernal habi-adjused consumpion, h [, 1], and Z is an exogenous preference shifer. We specialize uiliy o ake he following expression: U C, {N j}; Z The consumpion index is defined by C log C ϕ 1 υ 1 η C 1 1 η H, + υ 1 η C 1 1 η F, N j 1+ϕ dj Z, η η 1 where C H, is an index of domesic goods consumpion given by he CES funcion C H, ɛp 1 C H,i ɛp 1 ɛp 1 ɛp di, wih i [, 1] denoing he good variey, and CF, is he quaniy consumed of a composie foreign good. Parameer ɛ p > 1 denoes he elasiciy of subsiuion beween varieies produced domesically. Parameer υ [, 1] can be inerpreed as a measure of openness. The invesmen index can be defined in a compleely analogous way. The opimal allocaion of consumpion beween domesic and impored goods requires: C H, 1 υ /P η C ; C F, υp F, /P η C, 3 where P 1 υp 1 η H, + υp 1 η 1 1 η F, is he consumer price index CPI, for shor. Analogous expressions hold for he invesmen good. 2 2

3 The sequence of budge consrains assumes he following form expressed in unis of he aggregae consumpion baske, and absracing from he specificaion of sae coningen asses: C + B H,/P + I N + R k, K 1 + τ + B H, 1 + Γ i 4 R P P P P where B H, denoe Home holdings of a riskless bond denominaed in domesic currency, R 1 is he price of ha bond, τ are governmen ne ransfers of domesic currency, R k, is he nominal renal rae of capial, and Γ i are he profis of monopolisic firm i, whose shares are owned by he domesic residens. The accumulaion of capial obeys: ] I K +1 1 δk + I [1 Ω 1 5 I 1 where Ω is increasing and convex, and such ha Ω Ω and Ω. Equilibrium condiions Le λ and λ ψ be he Lagrange mulipliers on consrains 4 and 5 respecively. Hence λ denoes he shadow value of one uni or real income. Firs order condiions wih respec o N, C, B H,, I, K read: C hc 1 σ N ϕ Z C hc 1 σ λ λ βr E { P 6 P λ +1 P +1 [ ] { ψ 1 Ω Ω I I λ +1 1 βe ψ I 1 I +1 1 λ where r k,+1 R k,+1 /P +1. } I+1 { } λ+1 [ ] ψ βe rk, δψ λ +1 I 2 } Ω I+1 I The porfolio choice by households in Foreign implies he fol- Foreign Households lowing Euler condiion: U c, βr E {U c,+1 P P+1 } 3

4 A.2 Producion and Price Seing Each monopolisic firm i in Home produces a homogenous good according o he CRS producion funcion: Y i A N i 1 α K α i 7 where A is a labor produciviy shifer common across firms. The cos minimizing choice of labor and capial inpu implies: i MC α i A K i 1 α 8 N i R k, i MC 1 α i A N i α 9 K i where M C denoes he nominal marginal cos. Noice ha he above condiions imply MC W 1 α R α k, 1 α α 1 α 1 α A Hence, and due o he CRS assumpion, he nominal marginal cos is he same across firms. In equilibrium, his implies ha also he capial-labor raio is common across firms. Opimal Pricing Each domesic firm can revise is price a random inervals. Le 1 θ p be he probabiliy ha a firm can reopimize is price a any given ime. The firs order condiion wih respec o for profi maximizaion reads: E θ k p { ν,+k Y +k } } {{ } LHS M p E { } ν,+k Y +k MC +k θ k p } {{ } RHS 11 where from equilibrium ν,+k β k U c,+k P U c, P +k, and MC +k is he nominal marginal cos a + k of a firm ha las rese is price a ime. Noice ha, using 1, i holds MC +k MC +k 4

5 The above equivalence is an implicaion of he assumpion of consan reurn o scale in producion. Dividing hrough by we can wrie he LHS of he above equaion as follows: 1 εp { εp} LHS E θ k P pν,+k Y +k Π H,+s H, Consider nex he RHS of 11: RHS M p 1 E M p 1 E θ k p θ k p { { εp PH, M p E ν,+k ν,+k θ k p PH, +k PH, +k εp Y +kmc +k} εp } Y +kmc +k Π H,+s 1+εp ν,+ky +k mc +k Π H,+s. Equaing LHS and RHS and rearranging we finally obain: M p E E θ k p εp θ k pν,+k Y +k Π H,+s 1+εp ν,+ky +k mc +k Π H,+s Recursive represenaion of he pricing block Define K p, E Z p, E θ k p θ k pν,+k Y +k ν,+ky +k mc +k Π H,+s εp 1+εp Π H,+s Express recursively as: K p, Y + θ p β U c,+1 U c, Π +1 } {{ } ν,+1 Π εp H,+1 K p,+1 5

6 Similarly We also have: Z p, Y mc + θ p β U c,+1 Π 1+εp H,+1 U c, Π Z p, εp P 1 θ p Π H, εp 1 H, + 1 θ p. 12 A.3 Terms of Trade and Exchange Rae Pass-Through The erms of rade is he relaive price of impored goods: Under he assumpion P S P F, 13 PF,, he real exchange rae is defined as Q E P E P where E is he nominal exchange rae he price of uni of foreign currency expressed in unis of domesic currency, and where he second equaliy holds under he assumpion ha he res of he world is an approximaely closed economy. The erms of rade can be relaed o he CPI-PPI raio as follows: wih q s, qs / S >. P [1 υ + υs 1 η ] 1 1 η qs, 14 Law of-one-price gap Nominal sickiness in impor prices and he presence of local disribuion coss modeled below moivae deviaions from he law of one price. Le he law-of-one-price gap be denoed by: Φ F, E P F,. The expression for he real exchange rae becomes: Q E P 15 Φ F, S qs In he case of complee pass-hrough, Φ F, 1 for all. 6

7 A.4 Opimal impor pricing Each variey produced in he res of he world is disribued o he final consumer by a local imporer. Disribuing C F unis of impored variey f o he local consumer requires combining M F, unis of a homogeneous impored inpu wih labor, according o he following consan reurn o scale producion funcion: C F, f N f 1 α F M F, f α F 16 where M F, f and N f denoe he quaniy of impored inpu and of labor respecively employed by he inermediae local imporer f. Le PF, f be he "dock price" of he impored inpu expressed in unis of foreign currency, and le P F, f be he local currency price of he disribued variey. The local currency price of he disribued impored variey, P F, f, can be changed only a random inervals wih probabiliy 1 θ F,p. The cos minimizing choice of impored inpus and labor requires: P F, f MC αf F, P F, f 1 α MF, f F 17 N f Φ F, f MC 1 αf F, P F, f α N f F 18 M F, f where MC F denoes he nominal marginal cos of local imporer f. The above condiions imply: MC F, W 1 α F E PF, αf α α F 1 α F 1 α F F Hence, and due o he CRS assumpion, he nominal marginal cos is he same across local imporers. The inermediae local imporer solves: max E θ k F,pν,+k {[ P F, f MC F,+k ] CF,+k f } s.. 51, and o he opimal demand funcion for variey f : 19 εp P F, f C F,+k f C F,+k. 2 P F,+k 7

8 The firs order condiion wih respec o P F, f reads: E εp E ε p 1 θ k p { ν,+k C F,+k P F, f } 21 θ k p {ν,+k C F,+k MC F,+k } The real marginal cos for he local inermediae imporer reads: mc F, MC F, P F, w qs S 1 αf Φ α F F, where w /P is he real CPI wage. Dividing hrough by P F, in 21, and using 2 he above pricing condiion can be wrien: 1 εp { P F, εp} E θ k F,p ν,+k C F,+k Π F,+s P F, { εp 1 εp PF, E θ k F,p ν,+k C F,+kMC F,+k} ε p 1 P F, P F,+k { εp 1 εp } PF, E θ k F,p ν,+k C F,+kmc F,+kP F, Π F,+s ε p 1 P F, P F, εp εp PF, E ε p 1 θ k F,p P F,+k 1+εp ν,+kc F,+k mc F,+k Π F,+s. Simplifying: { P F, εp} E θ k F,p ν,+k C F,+k Π F,+s P F, εp 1+εp E θ k F,p ε p 1 ν,+kc F,+k mc F,+k Π F,+s. 8

9 Expressed in recursive form he above condiion reads: P F, K F, M p Z F, where Furhermore: P F, K F, C F, + θ F,p β U c,+1 Π εp F,+1 U c, Π K F,+1 +1 Z F, C F, [ w gs S 1 αf Φ α F F, ] βuc,+1 + θ F,p Π 1+εp F,+1 U c, Π Z F,+1 +1 A.5 Expor Demand 1 εp P 1 θ F,p Π F, εp 1 F, + 1 θ F,p. We assume ha aggregae expor demand, X, akes he following form assuming P 1 for all : P F, P F, X υ where he second equaliy has used 15. A.6 Wage Seing PH, E η Y υ S Φ F, η Y Le N i be he labor demand by firm i. Each firm i employs all differeniaed labor ypes. Hence oal labor demand by firm i can be wrien: N i εw 1 εw N i, j εw 1 εw dj where N i, j is demand by firm i of labor ype j. Opimal demand for labor ype j by firm i reads: εw W j N i, j N i 22 9

10 Inegraing across domesic good producing firms, we can derive he equilibrium oal demand for each labor ype j using 22 above: N j }{{} oal demand for labor ype j N i, jdi } {{ } inegraing across firms εw W j N idi εw W j 1 N idi εw W j N The above expression would hold in he absence of labor disribuion coss for local imporers. The opimal demand for labor ype j by he inermediae imporer f reads: 23 εw W j N f, j N f 24 Hence he oal demand for each labor ype j reads: N j N i, jdi + N f, jdf }{{} }{{} inermediae good producers in Home inermediae local imporers εw W j 1 εw W j N idi + N fdf εw [ W j 1 N idi + εw W j N ] N fdf 25 where now N N idi + N fdf. Opimal wage seing problem for household j: Nex, consider he opimal wage seing problem max E βθ w k U C +k j, N +k j 1

11 where N +k j is ime + k labor supply by household ype j who las rese her wage in ime. A he chosen wage j, household ype j is assumed o supply enough labor o saisfy demand. The consrain reads, using 23: N +k j }{{} oal supply of labor ype j N+k j d }{{} oal demand for for labor ype j εw j N +k +k Noice ha N +k bears he index + k and no + k because i corresponds o aggregae or average labor demand. The addiional household s consrain is he budge consrain: P +k C +k j + E { ν+k,+k+1 B +k+1 } B+k + jn +k j T +k where we now make explici ha he households can pool labor income risk hrough sae coningen asses B. Each household j reopimizing he wage a a given ime will choose he same opimal wage. I is herefore convenien o absrac from index j. Household problem problem is The relevan porion of he Lagrangian of he household s L w E βθ w k {U C +k, N +k λ+k [ P+k C +k N +k ] }. 26 where λ +k is he shadow value of one uni of nominal income a + k. The FOC of he problem wih respec o is: Noice: { βθ w k N +k E U N,+k + λ +k } N +k N +k + εw 1 N +k N+k ε w +k +k 1 ε w N +k 11

12 Hence we can wrie: { } βθ w k 1 E U N,+k ε w N +k + λ +k N +k ε w 1 Under complee markes and separable uiliy we have U c,+k C +k, N +k U c,+k C +k. In addiion, equilibrium implies U c,+k λ +k P +k since λ +k is he shadow value of one uni of nominal income a + k. Recall ha under our calibraion U c, Z C σ hc 1 λ λ P. Hence we have: { } βθ w k E U N,+k N +k M w + U c,+k N +k where M w ε w /ε w 1. The above expression can be rewrien: { [ W βθ w k E U c,+k N +k + U N,+k P +k U c,+k M w P +k ]} 27 Recursive represenaion Condiion 27 reads: E βθ w k N +k U c,+k P +k }{{} LHS Using he opimal labor demand condiion E βθ w k N +k M w U N,+k } {{ } RHS we can wrie he LHS as follows: εw N +k N +k, 28 +k LHS εw εw 1 εw W N P U c, + βθ W+1 w P +1 Π ε w 1 +1 N +1 U c,+1 + P + βθ w 2 εw +2 Π+1 P +2 Π +2 εw 1 N +2 U c, εw 1 w 1 εw E βθ w k w εw +k Π +s N +k U c,+k, where w /P. 12

13 Nex consider RHS: This can be wrien εw W RHS P RHS w εw E P εw N M w U N, +βθ w W+1 P +1 εw N+1 Π εw +1M w U N, βθ w k w εw +k εw Π +s N +k M w U N,+k Under he assumpion ha U N is homogenous of degree ϕ in N we have using 28: εwϕ U N,+k U N,+k N +k Subsiuing: RHS w εw1+ϕ E +k Combining LHS and RHS we obain: w 1+εwϕ E εwϕ εwϕ /P Π +s U N,+k N +k +k /P +k βθ w k w εw1+ϕ +k N +k M w βθ w k w εw +k Π +s εw1+ϕ εw 1 Π +s N +k U c,+k } {{ } K w U N,+k N +k εw1+ϕ M w E βθ w k w εw1+ϕ +k N +k Π +s U N,+k N +k }{{} Z w We can rewrie recursively: K w, w εw N U c, + βθ w Π εw 1 +1 K w,+1 Z w, w εw1+ϕ N U N, N + βθ w Π εw1+ϕ +1 Z w,+1 Hence he firs order condiion can be wrien in compac form: w 1+εwϕ K w, M w Z w, 13

14 A.7 Price Dispersion, Wage Dispersion, and Equilibrium Marke clearing for each individual domesic variey implies: A K α in i 1 α }{{} supply of variey i εp PH, i Y P } H, {{} demand of variey i where N i denoes he oal amoun of labor employed by firm i. Rearranging: [ PH, ] εp i Y N i A K /N α where we used he fac ha, in equilibrium, all firms choose he same capial labor raio. Inegraing across all producers: [ 1 PH, ] εp i Y N idi di 3 A K /N α Y 1 εp PH, i Y di A K /N α A K /N p,, 31 α where p, εp 1 PH, i P H, di measures he dispersion of relaive prices across domesic producers. In a more compac form: 29 where N N idi. Hence we can finally wrie: N Y A K /N α p,, Expressing p, in recursive form: p, 1 θ p εp PH, i di εp di + A K α N 1 α Y p, 32 PH, 1 εp εp 1 θ p + θ p Π εp H, p, 1 θ p εp PH, 1 i di 1 14

15 Marke clearing Toal demand for domesically produced goods reads: Hence condiion 6 becomes: A K α N 1 α Y C H, + I H, + X 33 1 υqs η C + I + υ S Φ F, η Y [1 υqs η C + I + υ S Φ F, η Y ] p, 34 Le N j denoe labor supply by each differeniaed household. Since each household is assumed o saisfy labor demand a he given posed wage, equilibrium in he labor marke requires: N j N j Aggregaing across each household j one obains, using 23: N N jdj N j εw W j dj N where N is an index of aggregae labor supply. By defining w, εw 1 Wj as an index of wage dispersion, he above equaion becomes. Noice ha by subsiuing 62 ino 6 one obains: N w, N Y 1 α p, N w, A K α which shows ha he relaionship beween aggregae employmen N and aggregae oupu Y depends on boh price and wage dispersion. Evoluion of LOOP gap and erms of rade Φ F, Φ F, 1 E /E 1 Π F, Π F, S S 1 E /E 1 Φ F, /Φ F, 1 Π H, 15 36

16 A.8 Wage dispersion and welfare Each household is a monopolisic supplier of a specialized labor ype. While households can pool consumpion uncerainy so ha he marginal uiliy of nominal income is he same across households, hey canno pool employmen uncerainy. Hence labor supply is heerogenous in equilibrium. Under he assumpion of separable preferences, and in paricular of isoelasic disuiliy from labor, he ineremporal uiliy for household j [, 1] reads: V j E β {U k C+k j N } +kj 1+ϕ 1 + ϕ where, under consumpion pooling, U C j U C for all j 37 We wish o evaluae an aggregae measure of household s welfare: } V V jdj E β {U k N +k j 1+ϕ C+k 1 + ϕ dj, where we have made use of 37. In equilibrium, using 23: Thus we can wrie: V E N j N j εw W j N for all β k {U C+k N 1+ϕ +k 1 + ϕ W+k j +k Le he welfare relevan measure of wage dispersion be: } εw1+ϕ dj, Thus we can wrie: where 1+ϕ w, V E β k {U W j εw1+ϕ dj C+k Ñ } 1+ϕ +k, 1 + ϕ 16

17 Ñ N 1+ϕ w, for all is he aggregae employmen index ha is relevan for aggregae average welfare. Noice ha in he case ϕ, i.e., of linear disuiliy of labor, w, w,. A recursive expression for reads: 1+ϕ w, 1 θ w W j 1 θ w 1+ϕ w, εw1+ϕ dj εw1+ϕ dj + εw1+ϕ w w W 1 + θ w w Π w 1 εw1+ϕ θ w εw1+ϕ 1+ϕ w, 1 W 1 j 1 εw1+ϕ dj where w /P. Noice ha, for he measure 1 θ w of reopimizing households, j. A.9 Price seing problem wih indexaion In his secion we lay ou he problem of domesic good producers in he case of parial indexaion. A similar srucure applies o he price seing problem of domesic imporers in he case of disribuion coss and incomplee exchange rae pass-hrough. We assume ha hose domesic producers which canno reopimize heir price choose o raise prices by a fracion χ p [, 1] of he domesic price level. Formally: +1 j Π χ p H, j If he firm is unable o reopimize for k periods, his will become: +k j j We use he convenion of seing: The problem of he reopimizing firm becomes: [ max E θ k p {ν,+k Π χ p H,+s 1 1 Π χ p H,+s 1 17 Π χ p H,+s 1 Y +k MC +k Y +k ]},

18 subjec o he demand consrains: Y +k k Πχ p H,+s 1 +k The associaed firs order condiion is given by: { } E θ k p 1 E ν,+k Y +k θ k p Π χ p H,+s 1 M p E εp Y +k. θ k p { ν,+k Y +k MC +k } Dividing boh sides by and subsiuing ou he demand for firm i condiional on no readjusing he price for k periods yields: 1 M p E ν,+k θ k p Noice ha we can wrie: k Πχ p H,+s 1 +k ν,+k k Πχ p H,+s 1 +k εp Y +k Π χ p H,+s 1 εp Y +k MC +k +k +k PH,+1 +k 1 +s +s 1 Π H,+s We follow he convenion of seing: We are lef wih: 1 εp M p E θ k p E ν,+k θ k p Π H,+s 1 ν H,,+k +k εp Y +k +k εp Y +kmc +k Π χ p H,+s 1 Π χ p H,+s 1 1 εp εp Π H,+s 18

19 Nex, muliply he LHS by 1 P εp εp H, so ha we can wrie: 1 εp εp E θ k p ν PH,+k,+k Y +k M p E θ k p ν,+k +k εp Y +kmc +k Following similar seps for he RHS we obain: E θ k p ν,+k Π H,+sεp Now le: M p E F p, E K p, E θ k p ν,+ky +k mc +k θ k p θ k p ν,+k ν,+ky +k mc +k Π χ p H,+s 1 Π χ p H,+s 1 Π χ p H,+s 1 1 εp Π χ p H,+s 1 Y +k εp 1 εp εp Π H,+s Π H,+sεp εp Π χ p H,+s 1 This implies ha we can rewrie he firs order condiion as: F p, M p K p, Consider F p, and he following facs: ν, β U c, U c, P P 1 1+εp Π H,+s 1 εp Π χ p H,+s 1 Y +k 1+εp Π H,+s Moreover, θ p 1 Π H,+s 1 U c,+k U c, U c,+s U c,+s 1 19

20 ν,+k β k U c,+s U c,+s 1 1 k Π +s The above resuls ogeher imply: F p, Y + E θ k p ν,+k Π H,+sεp k1 Y + E ν,+1 θ p Π εp H,+1 θ k 1 p k1 U c,+1 1 Y + E βθ p Π εp U c Π +1 θ k p H,+1 ν +1,+k Π χ p H, 1 εp k+1 εp ν +1,+1+k Π H,+s s2 }{{}}{{} k Π H,+1+s εp s2 Π χ p H,+s ν,+1 ν +k 1,+k 1 εp Π χ p H,+s 1 Y +k s2 Π H,+sεp s2 1 εp k Πχ p H,+s 1 εp Finally, we can use he law of ieraed expecaions and wrie: A similar argumen shows: Y +1+k U c,+1 1 F p, Y + βθ p Π εp χ H,+1 Π p 1 εp U c Π H, Fp,+1 +1 U c,+1 1 K p, Y mc + βθ p Π 1+εp χ H,+1 Π p εp U c Π H, Kp, εp Π χ p H,+s Y +k Finally, we have assumed ha hose firms ha are no allowed o reopimize will se heir prices according o j Π χ p H, 1 1 j Hence: P 1 εp H, θ p Π χ p 1 εp H, 1 P 1 εp H, θ p P 1 εp H, This can be rewrien as: 1 θ p Π χ p 1 εp H, 1 Π εp 1 H, + 1 θ p 1 εp 2

21 p, The index of price dispersion of domesic good prices follows: ε PH, i di 1 θ p 1 θ p Therefore we have: p, 1 θ p θ k p θ k p εp + θ p 1 θ p 1 Π χ p H, 1 k k k Πχ p H, k+s 1 k Πχ p H, k+s 1 εp 1 θ p + 1 θ p k1 εp 1 θ p + 1 θ p θ k p θ k+1 p k ε p ε p εp + θ 2 p 1 θ p k Πχ p H, k+s k 1 εp 1 θ p + θ p Π εp χ H, Π p εp H, 1 1 θp θ k p s2 k Π χ εp p H, 1 Πχ p H, 2 + ε p Π χ p H, k+s εp Π χ p H, k+s 1 εp } {{ } p, 1 Hence: εp p, 1 θ p + θ p Π εp χ H, Π p εp H, 1 p, 1 A.1 Impor pricing problem wih indexaion We ha he domesic imporers who canno reopimize raise heir price by a fracion χ pf [, 1] of he PPI of he imporer. Formally, he price of imporer j a +1, if no reopimized, will be: P F,+1 j Π χ pf F, P F, j In general: P F,+k j P F, j Π χ pf F,+s 1. 21

22 We use he convenion of seing: Π χ pf F,+s 1 1 The problem of he reopimizing imporer becomes: [ ]} max E θ k F,p {ν,+k P F, Π χ pf F,+s 1 C F,+k MC F,+k C F,+k P F, subjec o: and he demand consrains: C F, f N f 1 α F M F, f α F C F,+k f P F, k Πχ pf F,+s 1 P F,+k The associaed firs order condiion is given by: { } E θ k F,p 1 P F, E ν,+k C F,+k P F, θ k F,p Π χ pf F,+s 1 εp C F,+k M p E θ k F,p {ν,+k C F,+k MC +k } Dividing boh sides by P F, and subsiuing ou he demand for firm i condiional on no readjusing he price for k periods, we obain. We are lef wih: 1 P F, M p E ν P F,,+k θ k F,p k Πχ pf F,+s 1 P F,+k ν P F,,+k The real marginal cos for he imporer reads: Also, we can wrie: k Πχ pf H,+s 1 P F,+k mc F, MC F, P F, εp C F,+k P F, Π χ pf F,+s 1 εp C F,+k MC F,+k P F,+k P F,+k PF,+1 P F, P F, P F,+k 1 P F, P F,+s P F,+s 1 P F, Π F,+s 22

23 We follow he convenion of seing: Then, we are lef wih: M p E P F, 1 εp P F, θ k F,p E ν,+k θ k F,p P F, P F,+k Π F,+s 1 ν,+k P F,+k εp C F,+k εp C F,+kmc F,+k Π χ pf F,+s 1 Π χ pf F,+s 1 Nex, muliply he LHS by 1 P εp εp F, so ha we can wrie: 1 εp P F, εp E θ k F,p P F, ν,+k Π F,+s C F,+k εp P F, M p E P F, Hence, simplifying: Now le: P F, P F, E M p E K F, E Z F, E θ k F,p θ k F,p θ k F,p ν,+kc F,+k mc F,+k Π χ pf F,+s 1 εp ν,+k Π F,+s C F,+k ν,+kc F,+k mc F,+k θ k F,p θ k F,p Π χ pf F,+s 1 εp ν,+k Π F,+s C F,+k ν,+kc F,+k mc F,+k Π χ pf F,+s 1 1 εp εp Π F,+s Π χ pf F,+s 1 εp Π χ pf F,+s 1 εp Π χ pf F,+s 1 εp 1 εp 1 εp 1+εp Π F,+s 1+εp Π F,+s 1 εp 1+εp Π F,+s 23

24 This implies ha we can rewrie he firs order condiion as: K F, M p Z F, We can wrie K F, and Z F, recursively as: K F, C F, + θ F,p β U c,+1 U c, Π +1 Π εp [ ] 1 ξ g S Z F, C F, w Φ ξ F, S F,+1 Πχ pf 1 εp F, K F,+1 + θ F,p β U c,+1 U c, Π +1 Π 1+εp F,+1 Π χ pf εp F, Z F,+1 We have assumed ha hose firms ha are no allowed o reopimize will se heir prices according o P F, j Π χ p F, 1 P F, 1 j Hence: P 1 εp F, θ F,p Π χ p 1 εp F, 1 P 1 εp F, θ F,p P 1 εp F, This can be rewrien as: 1 θ F,p Π χ p 1 εp F, 1 Π εp 1 P F, F, + 1 θ F,p P F, A.11 Wage seing problem wih indexaion 1 εp Households who do no reopimize are no allowed o change heir wage and updae i according o he following rule: +1 j Π χ w j Noice ha Π is CPI inflaion. Mor generally, his reads: +k j j Π χ w +s 1 We use he convenion of seing: Π χ p +s 1 1 The problem reads absracing from consumpion habis: { max E βθ p k U [ ]} C +k, N +k λ+k P +k C +k Π χ w +s 1 N +k 24

25 The demand for labor of ype j condiioning on he household no reopimizing for k periods reads: N +k k Πχ w +k +s 1 εw N +k Hence he firs order condiion reads: { } βθ w k N +k E U n,+k + λ +k N +k + Π χ N+k w +s 1 W Noice ha: N +k ε w βθ w k E k Πχ w +k +s 1 Using he las expression and rearranging: ε w 1 U n,+k ε w N +k 1 k Πχ w N +k 1 ε w N +k W k +k Πχ w +s 1 + λ +k N +k ε w 1 +s 1 Dividing hrough by ε w 1, muliplying by, and using U c,+k λ +k P +k, i yields: βθ w k E U 1 W n,+k N +k M w k + U c,+k N +k Πχ w P +k +s 1 Finally, rearranging: βθ w k E U c,+kn +k U n,+k 1 M w U k + W c,+k Πχ w P +k +s 1 Rewrie he above condiion as: E βθ w k U c,+k N +k P +k E Recall he opimal labor demand condiion N +k βθ w k N +k M w Un,+k 1 k Πχ w +s 1 k Πχ w +k +s 1 εw N +k 25

26 So, he LHS can be wrien as LHS E βθ w k U c,+k k Πχ w +k +s 1 εw N +k P +k 1 εw εw εw E βθ w k W+k U c,+k N +k Π +sεw 1 Π χ w P P +s 1 +k Nex, he RHS can be expressed as: εw εw εw RHS E βθ w k W+k Π +sεw Π χ w P P +s 1 N +k M w Un,+k +k Under he assumpion ha U n is homogeneous of degree ϕ in N we have ha: U n,+k Subsiuing, we obain: P εw1+ϕ RHS M w E P k Πχ w +s 1 +k k Πχ w +s 1 +k P +k Hence he whole opimaliy condiion reads: ε wϕ U n,+k N +k ε wϕ εwϕ Π +s U n,+k N +k 1+ϕεw εw1+ϕ βθ w k W+k N +k Π +sεw1+ϕ Π χ w P +s 1 +k 1+ϕεw F w, M w K w,, P where F w, E εw εw βθ w k W+k U c,+k N +k Π +sεw 1 Π χ w P +s 1 +k K w, E 1+ϕεw εw1+ϕ εw1+ϕ βθ w k W+k N +k Π +s U n,+k N +k Π χ w P +s 1 +k 26

27 The above condiions can be wrien recursively as: εw W F w, N U c, + βθ w Π εw 1 +1 Π χ w εw F w,+1 K w, W P P εw1+ϕ N U n, N + βθ w Π 1+ϕεw +1 Π χ w εw1+ϕ K w,+1 A household who does no reopimize will se he real wage equal o: w +1 j +1 j P +1 Πχw P +1 j P j P P Π χw Π w j Πχw Π The average real wage in any given period depends on he previous period average real wage and on he reopimized one: W P 1 εw W 1 Π χw 1 εw θ w + 1 θ w P 1 Π P 1 εw The dispersion in wages effecively drives a ime-varying wedge beween he supply and demand of labor: 1 εw N i N d W j 1 εw i, j dj N d i dj N d W j i dj W }{{ } w, Clearly, we can inegrae across all firms, o obain: N N i di w, N d i di w, N d The wage dispersion index w, can hen be wrien: w, εw W j dj 1 θ w 1 θ w θ k w εw + θ w 1 θ w 1 Π χ w 1 k k Πχ w ε w +s 1 εw + θ 2 w 1 θ w 2 Π χ w 1 Π χ εw w This is rue for all labor ypes ha are hired by each firm i. Indeed, all firms hire all ypes of labor. 27

28 B A small open economy wih raded and non-raded goods In his secion we exend our baseline DSGE model o allow for raded and non-raded goods. Le consumpion demand be given by: C 1 γ 1 ξ C 1 1 ξ N, + γ 1 ξ C 1 1 ξ T, ξ ξ 1 ξ > 38 where C N, is a composie index of consumpion of non-raded goods, and C T, is a composie index of raded goods, in urn given by: C T, 1 υ 1 η C 1 1 η H, + υ 1 η C 1 1 η F, η η 1 η >, 39 ɛp 1 where C ι, C ι,i ɛp 1 ɛp 1 ɛp di is an index of domesic goods consumpion in he domesic secor ι H, N,wih i [, 1] denoing he good variey. C F, C F,f ɛp 1 ɛp 1 ɛp 1 ɛp df is he quaniy consumed of a composie foreign good. Parameer ɛ p > 1 denoes he elasiciy of subsiuion beween varieies produced domesically wheher raded or non-raded. Opimal consumpion demand for variey i in secor ι H, N requires: Opimal demand for impored variey f reads: Also: C ι, i P ι, i/p ι, εp C ι, ι H, N 4 C F, f P F, f/p F, εp C F, 41 C H, 1 υ /P T, η C T, ; C F, υp F, /P T, η C T, 42 where C N, 1 γp N, /P ξ C ; C T, γp T, /P ξ C 43 P ι, is he uiliy based price index in secor ι H, N, P T, 1 υp 1 η H, 1 P 1 εp 1 εp ι, i di + υp 1 η 1 1 η F, 28

29 is he uiliy-based price index of raded goods and P 1 γp 1 ξ N, is he uiliy-based aggregae CPI index. Using he above equilibrium condiions i holds: 1 + γp 1 ξ 1 ξ T, P N, C N, + P T, C T, P C C H, + P F, C F, P T, C T, P N, ic N, idi P N, C N, ic H, i + P F, ic F, idi P T, C T, Terms of rade, relaive price of radables and real exchange rae he erms of rade as he relaive price of he impored good We define S P F,. We also define: wih q S >. The relaive price of radables is : We also define P T, [1 υ + υs 1 η ] 1 1 η qs 44 T P T, P N, P P N, [1 γ + γt 1 ξ ] 1 1 ξ ht 45 wih h T >. Since P PF,, and under he assumpion P F, 1, he consumpion real exchange rae reads: Q E. P 29

30 B.1 Opimal pricing Each monopolisic firm i in secor ι i.e., domesic radable and nonradable produces a differeniaed good according o he CRS producion funcion: The cos minimizing choice of labor implies: Y ι, i N ι, i, ι N, H 46 P ι, i MC ι, P ι, i where MC ι denoes he nominal marginal cos in secor ι. Opimal pricing in secor j implies: p ι, M p E { k θk p,ιν,+k Y ι,+k mc ι,+k Π ι,+s { k εp } E θk p,ιν,+k Y ι,+k Π ι,+s 1+εp } 47 where p ι, P ι, /P ι, is he opimally chosen individual price in secor j expressed as a raio o he average price in secor ι, and mc ι, is he real marginal cos expressed in unis of goods produced in secor j. Define: K pι, E Z pι, E θ k p,ιν,+k Y ι,+k θ k p,ι ν,+ky ι,+k mc ι,+k Therefore he opimal pricing condiion reads: Π ι,+s εp 1+εp Π ι,+s p ι, M p Z pι, K pι, Expressing recursively as: { K pι, Y ι, + θ p,ι E β U } c,+1 Π εp U c, Π ι,+1k pι,+1 +1 Similarly { Z pι, Y ι, mc ι, + θ p,ι E β U } c,+1 Π 1+εp U c, Π ι,+1z pι,

31 We also have: 1 θ p,ι Π ι, εp θ p,ι p 1 εp ι,. 48 Finally, and leing w /P denoe he CPI real wage, noice ha: { w ht if ι N mc ι, w ht qs /T if ι H B.2 Expor Demand We assume ha oal expor demand for domesically produced raded good i, X, akes he following form recalling P PF, 1 for all : X υ PH, E υ S η Y η Y B.3 Equilibrium wih radables and nonradables Equilibrium in he marke for each differeniaed variey i in secor ι requires: εp Pι, i N ι, i Yι, d where Y d ι, is oal demand in secor ι H, N. Inegraing i yields: where pι, Pι,i P ι, Y d ι, B.4 Equilibrium condiions P ι, N ι, idi N ι, pι,y d ι, εp di denoes price dispersion in he secor ι, and where { 1 γht ξ C if ι N ξ 1 υqs η γ ht C + υs η Y T if ι H Le he pricing block in secor ι H, N be defined by he se of processes: P ι, { } p ι,, K pι,, Z pι,, pι, 31

32 An equilibrium in he sicky-price model wih radables and nonradables is a se of processes {P ι,, mc ι,, Π ι, N ι,, N, w, S, T, Π,Π T,, C, R } which solve: Pricing block in secor ι N, H p ι, K pι, M p Z pι, { } K pι, N ι, + θ p,ι E β Cσ +1 Π εp C σ Π ι,+1k pι,+1 +1 Z pι, N ι, mc ι, + θ p,ι E { β Cσ +1 C σ Π +1 1 θ p,ι Π ι, εp θ p,ι p 1 εp ι,. } Π 1+εp ι,+1z pι,+1 pι, 1 θ p,ι p ι, εp + θp,ι Π εp ι, pι, 1 { w ht if ι N mc ι, w ht qs /T if ι H Goods marke equilibrium: nonradables Goods marke equilibrium: radables N N, pn,1 γhs ξ C [ ] ξ N H, ph, 1 υqs η ht γ C T + υs η Y Labor marke equilibrium Risk-sharing N }{{} N ι, labor supply ιn,h C σ C S T qs ht 32

33 Consumpion Euler Consumpion-leisure C σ +1R Π +1 E { C σ } C σ N ϕ w CPI inflaion Traded good inflaion Relaive price of radables Moneary policy rule Π ht ht 1 Π N, Π T, qs qs 1 Π H, T T 1 Π T, Π N, R fπ B.5 Deviaions from he law of one price In he presence of nominal sickiness in impor prices and/or local disribuion coss he law of one price in radables ceases o hold. Le he law-of-one-price gap be denoed by: Φ F, E P F, Nex we show how o relae he consumpion real exchange rae o he hree key relaive prices of our seup: he erms of rade S, he relaive price of radables S T, and he law of one price gap Φ F,. From he definiion of consumpion real exchange rae: Noice ha can wrie: S Q Φ F, 49 P / P P P N, P N, P T, P T, ht T qt 33

34 Combining we can finally wrie: S Q qs }{{} erms of rade Φ }{{} F, lop gap T ht }{{} relaive price of radables QS, Φ F,, T 5 The above expression shows how movemens in he consumpion real exchange rae can be decomposed, respecively, ino movemens of he erms of rade capured by he erm S /qs, deviaions from he law of one price in radables capured by he erm Φ F,, and movemens in he relaive price of radables capured by he erm T /ht. Noice ha in he case of law of one price holding a all imes, Φ F, 1 for all. Also, in he case in which all goods are radable, T ht 1. B.6 Disribuion coss in radables Each variey produced in he res of he world is disribued o he final consumer by a differeniaed local imporer. Disribuing C F unis of impored good o he local consumer requires combining M F, unis of a homogeneous impored inpu wih labor, according o he following consan reurn o scale producion funcion: C F, f N F, f 1 α F M α F F, f 51 where N F, f and M F, f denoe he quaniy of impored inpu and of labor respecively employed by he local imporer f. Le PF, 1 be he "dock price" of he impored inpu expressed in unis of foreign currency, and le P F, f be he local currency price of he disribued impored variey. We assume ha he impor prices are flexible also a he consumer level. Condiional on 51, he cos minimizing choice of impored inpus and labor requires: P F, f MC αf F,f MF, f 1 α F 52 P F, f N F, f Φ F, MC 1 αf F,f P F, f α NF, f F, 53 M F, f where MC F denoes he nominal marginal cos of he local imporer. The above condiions imply: MC F, f W 1 α F α E F α α F F 1 α F 1 α ιf MC F, 54 Hence he nominal marginal cos is common across local imporers. Condiional on he opimaliy condiion 54, he local imporer solves: 34

35 max E ν,+k {[P F,+k f MC F,+k ] C F,+k f} subjec o he opimal demand funcion for he impored good: εp PF,+k f C F,+k f C F,+k 55 P F,+k The firs order condiion of his problem requires: Expressing in real erms: P F, f P F, ε p ε p 1 MC F, mc F, MC F, P F, w qs ht S T }{{ } 1 if all goods radables Hence, besides he real consumpion wage, movemens in boh he erms of rade and he relaive price of radables affec he imporer s marginal cos. The presence of disribuion coss implies ha aggregae expor demand should be wrien: 1 α F Φ α F F, Marke clearing for variey i: X υ PH, E υ S η Φ η F, Y η Y Inegraing εp Pι, i N ι, i Yι, d P ι, N ι, N ι, idi pι,y d ι, 35

36 where { 1 γht ξ C if ι N Yι, d ξ 1 υqs η γ ht C + S η Φ η F, Y T if ι H B.7 Full model wih capial and nonradables We assume ha physical capial is employed in he non-raded secor only. The producion funcion in secor ι reads: Y ι, i A ι, K αι ι, in ι, i 1 αι 56 where α ι ζα ι, and wih ζ 1 if ι N, and ζ if ι H. Marke clearing for each individual domesic variey i in secor ι H, N implies: Rearranging: N ι, i Y ι, i }{{} supply of variey i in secor ι [ Pι, i P ι, Pι, i P ι, εp Y d ι, } {{ } demand of variey i in secor ι ] εp Yι, d A K ι, /N ι, αι where we used he fac ha, in equilibrium, all firms in secor ι choose he same capial labor raio. Inegraing across all producers in secor ι: N ι, idi [ Pι,, i Y ι, P ι,, A ι, K ι, /N ι, αι ] εp Yι, d A ι, K ι, /N ι, αι εp Pι,, i di P ι,, 57 di 58 Y A ι, K ι, /N ι, αι p ι,,, 59 where pι, εp 1 Pι,,i P ι, di measures he dispersion of relaive prices across domesic producers. In a more compac form: N ι, Y d ι, A ι, K ι, /N ι, αι p ι,,, where N ι, N ι,idi. Hence we can finally wrie, for each secor ι: 36

37 Expressing p, in recursive form: A ι, Kι, αι Nι, 1 αι Yι, d pι, 6 εp P ι, pι, 1 θ p,ι + θ p,ιπ εp ι,, pι,, 1 P ι, Marke clearing Marke clearing in secor ι H, N where A ι, Kι, αι Nι, 1 αι Yι, d pι, 61 { 1 γht ξ C + I if ι N Yι, d ξ 1 υqs η γ ht C + I + S η Φ η F, Y T if ι H Labor marke equilibrium Toal demand for each labor ype j reads: N j ι H,N N ι, i, jdi [ εw W j εw W j N. + N f, jdf N ι, idi + ι H,N N fdf ] where N ι H,N N ι, + N F,, and N ι, N ι,idi, N F, N fdf. Le N j denoe labor supply by each differeniaed household. Since each household is assumed o saisfy labor demand a he given posed wage, equilibrium in he labor marke requires: N j N j Aggregaing across each household j one obains, using 23: N N jdj N j 37 εw W j dj N

38 where N is an index of aggregae labor supply. By defining w, εw 1 Wj as an index of wage dispersion, he above equaion becomes. B.8 Full se of equilibrium condiions Pricing block in secor ι N, H N w, N. 62 p ι, K pι, M p Z pι, { } K pι, Y ι, + θ p,ι E β Cσ +1 Π εp C σ Π ι,+1k pι,+1 +1 Z pι, Y ι, mc ι, + θ p,ι E { β Cσ +1 C σ Π +1 1 θ p,ι Π ι, εp θ p,ι p 1 εp ι,. } Π 1+εp ι,+1z pι,+1 pι, 1 θ p,ι p ι, εp + θp,ι Π εp ι, pι, 1 Firms effi ciency condiions in each secor w ht qs T mc H, A H, 1 α ι N H, αι 1 63 w ht mc N, A N, 1 α ι αι KN, 64 N N, r k, ht qs T mc H, A H, α ι 1 αι NN, 65 K N, i Producion funcion Y ι, A ι, Kι, αι Nι, 1 αι ι H, N Invesmen effi ciency condiions 38

39 [ ] { ψ 1 Ω Ω I I λ βe ψ I 1 I +1 1 λ I+1 { } λ+1 [ ] ψ βe rk, δψ λ +1 Goods marke equilibrium: nonradables Goods marke equilibrium: radables Y N, pn,1 γht ξ C + I [ ] ξ Y H, ph, 1 υqs η ht γ C + S η Φ η F, Y T I 2 } Ω I+1 1 I Labor marke equilibrium N }{{} N ι, + N F, labor ιn,h supply Risk-sharing C σ C S qs Φ F, T ht Consumpion Euler Consumpion-leisure C σ +1R Π +1 E { C σ } C σ N ϕ w CPI inflaion Traded good inflaion Π ht ht 1 Π N, Π T, qs qs 1 Π H, 39

40 Relaive price of radables Moneary policy rule T T 1 Π T, Π N, R fπ, E Deviaions from law of one price Φ F, Φ F, 1 E Π F, Pricing in he impor secor flex prices 1 αf w qs ht Φ α F F, S T ε p 1 ε p Evoluion of he erms of rade Choice of labor in impor secor Choice of impored inpu S S 1 Π F, Π H, w ht qt T S mc F, 1 α F αf MF, N F, Marginal cos in he impor secor 1 αf NF, Φ F, mc F, α F, M F, mc F, 1 αf w qs ht Φ α F F, S T 4

41 B.9 Calibraion The baseline calibraion is idenical o he one employed in he one-secor version of he model see he main ex and Appendix B. Wha remains o be specified is he value of a few addiional parameers: he preference share of raded goods in he consumpion aggregaor γ, he elasiciy of subsiuion beween raded and non-raded goods ξ, and he secoral degree of price sickiness θ p,ι, ι H, N. We se γ.6, which corresponds o he GIPS average share of goods as opposed o services in he HICP price index in 29 source Eurosa. Following Mendoza 1991 and Corsei e al. 28, we se he elasiciy of subsiuion beween raded and non-raded goods ξ.74. We se he degree of price rigidiy in he domesic raded secor equal o θ p,h.8 consisen wih our baseline calibraion, which srikes a balance beween micro and macro-based empirical sudies. Concerning he non-raded secor, he micro-based evidence of Alvarez e al. 26 suggess ha, on average in he Euro area, he frequency of price changes in he service secor is almos half he one in he indusrial goods secor our proxies for non-raded and raded goods respecively. For he GIPS, his difference is however less sark, implying ha if, in he baseline seing, prices are on average sicky for five quarers, hey should remain sicky in he non-raded secor for 5.75 quarers, which requires o se θ p,n.83. Our resuls are, however, largely insensiive o he choice of he secoral relaive degree of price sickiness. We calibrae he shock process for secoral produciviy as follows. We compue, for each secor ι H, N and each counry i GIP S, he produciviy measure a i ι, yι, i.75n i ι,, where yι, i is he log of hp-filered real gross value added afer deflaing by he naional GDP deflaor, and n i ι, is he log of hp-filered employmen housands of hours worked. 2 This measure assumes ha he labor share is equal in boh secors, which is consisen wih he evidence in he GIPS from OECD daa. We hen esimae he AR1 process for each counry i and secor ι: a i ι, ρ i a,ι a i ι, 1 + ε i a,ι,, and se ρ a,ι and σ a,ι he sandard deviaion of he innovaion in he model equal o he esimaed average value across i GIP S wihin each secor. Those values are repored in Table A1. Our seings are summarized in Table C1 below. 2 All daa are from Eurosa Quarerly Naional Accouns, basic breakdown of main GDP aggregaes and employmen by indusry. The secoral breakdown is based on he classificaion of economic aciviies NACE Rev.2. In our measure, he radable secor is manufacuring C. The non-radable secor is an aggregae of: consrucion F + wholesale and reail rade+ ranspor, accomodaion and food service aciviies G-I + informaion and communicaion J + financial and insurance aciviies K + real esae aciviies L + professional, scienific and echnical aciviies + adminisraive and suppor service aciviies M-N + public adminisraion, defence, educaion, human healh and social work aciviies O-Q+ ars, enerainmen and recreaion + aciviies of household and exra-erriorial organizaions and bodies R-U. 41

42 Table C1. Calibraion Parameer Descripion Value DSGE Model wih Traded and Non-Traded Goods γ Share of raded goods in consumpion baske.6 ξ Elasiciy of subsiuion beween raded and non raded goods.74 θ p,h Calvo index of price rigidiies: raded secor.8 θ p,n Calvo index of price rigidiies: non-raded secor.83 ρ H,a Persisence of echnology process: raded secor.58 ρ N,a Persisence of echnology process: non-raded secor.82 σ H,a S.dev. of innovaion of echnology process: raded secor.19 σ N,a S.dev. of innovaion of echnology process: non-raded secor.9 B.1 Resuls Noe: all remaining parameer values as in he baseline calibraion. See main ex. Figures 1 and 3 describe, for he case of a currency union φ e 1 and in he exended DSGE model wih raded and non-raded goods, he effec on welfare losses of variaions in he degree of wage rigidiy, θ w, condiional on all shocks and on each individual shock respecively. In all cases, welfare losses are expressed as a raio o is value under he baseline wage rigidiy θ w.8. The price rigidiy parameer is kep unchanged a is baseline seing of θ p.8. As i is clear, he main message of our previous analysis is largely confirmed. Figure 3 displays he effecs on welfare of varying, simulaneously, boh he wage rigidiy and he price rigidiy parameer. Also in his case, he resuls of our baseline DSGE model are largely confirmed. C Welfare and Wage Flexibiliy in Large Recessions In his secion we are ineresed in assessing he effec on welfare of varying he degree of wage rigidiy condiional on he economy being subjec o a negaive shock of paricular magniude. This exercise speaks o he following quesion: how advanageous is i, for a small open economy belonging o a currency area facing a large adverse shock, o enjoy a higher degree of wage flexibiliy? To address his poin, we compue he consumpion compensaing variaion ha, condiional on he same negaive realizaion of he exogenous sae variables, would make he domesic household indifferen in he following wo economies: one wih a given degree of wage rigidiy θ w, and one where θ w is se o is baseline value θ w.8. Throughou his exercise we employ our baseline DSGE model presened in he main ex. Formally, le S denoe he sae vecor condiional on a wo-sandard deviaion 42

43 Figure 1: Wage Rigidiies and Welfare in a Currency Union: DSGE Model wih Traded and Non-Traded Goods all shocks 43

44 Figure 2: Wage Rigidiies and Welfare in a Currency Union: DSGE Model wih Traded and Non-Traded Goods conidional on shock a he ime. 44

45 Figure 3: Nominal Rigidiies and Welfare in a Currency Union: DSGE Model wih Traded and Non-Traded Goods all shocks. 45

46 negaive realizaion of he shocks, and le: { V θw S, λ θw E U Cθ w 1 + λ θw, {N θw } j} S denoe he condiional expeced presen discouned value of uiliy our measure of welfare associaed o a given degree of wage rigidiy θ w. We compue he value of λ θw ha solves: V θw S, λ θw V θw S, In pracice, we need o compue, under alernaive values of θ w, he impulse response of he variable V condiional on a wo-sandard deviaion negaive realizaion of a generic componen k of he exogenous sae vecor. 3 Since, in a second-order approximaion of he model, he impulse response of V is sae dependen, we numerically proceed as follows. We le he sysem begin in he deerminisic seady sae, and draw a series of random shocks e for T periods. Based on a given draw, we derive wo simulaions for he vecor Y of he endogenous variables. The firs is a baseline simulaion called Y 1, T e ; he second, Y 2, T e + ε k, is a simulaion obained by adding a deerminisic negaive impulse ε k of size wo sandard deviaions o he componen k of vecor e in period T p + 1. The impulse response o shock k is compued as Y 2, Y 1,. We repea his exercise for Z imes, compue he average, and drop he firs T p observaions. De faco, his procedure amouns o compuing an average i.e., generalized impulse response o a wo sandard deviaion innovaion when he sae vecor is iniialized a is ergodic mean via a suiable choice of T. Figure 4, 5 and 6 display he value of λ θw expressed in percenage for alernaive values of θ w in he case of a domesic demand shock, expor shock and world ineres rae shock respecively. By consrucion his measure is equal o zero in he baseline case of θ w.8. The figure repors he compued λ θw condiional on a wo sandard deviaion negaive realizaion of each ype of innovaion. A posiive value on he verical axis herefore corresponds o he household s welfare loss expressed in unis of consumpion compensaing variaions of having a degree of wage rigidiy θ w relaive o he baseline θ w when he economy is hi by a negaive shock of paricular magniude. We clearly see ha, saring from θ w.8, reducing he degree of wage rigidiy generaes a welfare loss. This is in line wih our cenral resuls previously obained for uncondiional welfare measures. Absolue welfare losses from higher wage flexibiliy can be paricularly large in he case of demand shocks - as high as 1.5 percen of consumpion relaive o he allocaion under he baseline degree of wage rigidiy, alhough hey remain relaively small in he case of expor and world ineres rae shocks respecively up o.2 and.1 percen of consumpion relaive o he baseline allocaion. 3 We choose o repor he impac response, which means ha welfare V is evaluaed condiional on he value of he endogenous sae vecor being a is ergodic mean. 46

47 Figure 4: Wage rigidiy and welfare loss condiional on a wo-sandard deviaion negaive realizaion of a domesic demand shock. The welfare loss is measured by he consumpion compensaing variaion λ θw in % unis necessary o make he household as well off under a generic θ w as under he baseline θ w.8. 47

48 Figure 5: Wage rigidiy and welfare loss condiional on a wo-sandard deviaion negaive realizaion of an expor demand shock. The welfare loss is measured by he consumpion compensaing variaion λ θw in % unis necessary o make he household as well off under a generic θ w as under he baseline θ w.8. 48

49 Figure 6: Wage rigidiy and welfare loss condiional on a wo-sandard deviaion negaive realizaion of a world ineres rae shock. The welfare loss is measured by he consumpion compensaing variaion λ θw in % unis necessary o make he household as well off under a generic θ w as under he baseline θ w.8. 49

50 References Álvarez Luis J, Emmanuel Dhyne, Marco M. Hoeberichs, Claudia Kwapil, Hervé Le Bihan, Parick Lünnemann, Fernando Marins, Robero Sabbaini, Harald Sahl, Philip Vermeulen and Jouko Vilmunen 25: "Sicky Prices in he Euro Area: A Summary of New Micro Evidence", ECB Working Paper n. 563 December Corsei, Giancarlo, Luca Dedola and Sylvain Leduc 28. "Inernaional Risk Sharing and he Transmission of Produciviy Shocks". Review of Economic Sudies, 752, pp Mendoza, Enrique G "Real Business Cycles in a Small Open Economy". American Economic Review, 814, pp