Mark-up Fluctuations and Fiscal Policy Stabilization in a Monetary Union: Technical appendices not for publication

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1 Mark-up lucuaions and iscal Policy Sabilizaion in a Moneary Union: Technical appendices no for publicaion M. W. J. B Universiy of Amserdam and CEP J Universiy of Copenhagen, CEP and EPU July, 003 Mailing address: Deparmen of Economics, Universiy of Amserdam, oeerssraa 11, 1018 WB Amserdam, The Neherlands. Phone: ; fax: ; .m.w.j.beesma@uva.nl. Mailing address: Insiue of Economics, Universiy of Copenhagen, Sudiesraede 6, DK-1455 Copenhagen K, Denmark. Phone: ; fax: ; enrik.jensen@econ.ku.dk. Web:

2 Appendix A. The underlying model in deail A.1. Uiliies and privae consumpion There are wo counries labeled (ome) and (oreign). These counries form a moneary union. The populaion of he union is a coninuum of agens on he inerval [0, 1]. The populaion on he segmen [0,n) belongs o counry, while he populaion on [n, 1] belongs o counry. In period, he uiliy of he represenaive household j living in counry i is given by U j = E s= β s U C j s + V G j s v y j s ; ξ i s, 0 < β < 1, (A.1) where Cs j is consumpion, Gj s is per-capia public spending, and yj s is he amoun of goods ha household j produces. The funcions U and V are sricly increasing and sricly concave, while v is increasing and sricly convex in ys. j urher, ξ i s is a disurbance affecing he disuiliy of work, which will hroughou be inerpreed as a produciviy shock. 1 The consumpion index C j is defined as C j C j n C j 1 n n n 1 n, (A.) (1 n) where C j and Cj are he Dixi and Sigliz (1977) indices of he ses of imperfecly subsiuable goods produced in counries and, respecively: C j 1 n 1/ n 0 c j 1 (h) 1 dh, 1/ 1 C j 1 c j 1 (f) 1 df, 1 n n (A.3) where c j (h) and c j (f) are j s consumpion of ome- and oreign-produced goods h and f, respecively, and > 1 and > 1 are sochasic elasiciies of subsiuion across goods produced wihin a counry, boh wih mean. The ime variaion in he elasiciy of subsiuion will ranslae ino ime variaion in he mark up, as we will see below. 1 In he log-linearized model below, we consider only bounded flucuaions of a leas order O (ξ), where ξ is he vecor of all disurbances in he economies. 1

3 The price index of counry i is given by P i =(P i )n (P i )1 n where P i = 1 n p i (h) 1 n 0 1 dh 1, P i = 1 1 p i (f) 1 1 n n 1 1 df, and where p i (h) and p i (f) are he prices in counry i of he individual goods h and f produced in ome and oreign, respecively. Because purchasing power pariy holds, in he sequel, we will herefore drop he counry superscrip for prices. The erms of rade, T,isdefined as he raio of he price of a bundle of goods produced in counry and a bundle of goods produced in counry. Thais,T P /P. The allocaion of resources over he various consumpion goods akes place in hree seps. The ineremporal rade-off, analyzed below, deermines C j.givenc j, he household selecs C j and Cj so as o minimize oal expendiure PCj under resricion (A.). This yields: 1 C j = n P C j = nt 1 n C j, P 1 C j P = (1 n) C j =(1 n) T n C j. P (A.4) (A.5) Then, given C j and Cj, he household opimally allocaes spending over he individual goods by minimizing P C j and P C j under resricion (A.3). The implied demands for individual good h, produced in counry, and individual good f, produced in counry, are, respecively, p (h) p (f) c j (h) = T 1 n C j, c j (f) = T n C j. (A.6) P We assume ha public spending is financed eiher by deb issuance or lump-sum axaion. Per capia public spending in counries and is given by he following indices, respecively: P G = 1 n 1 g (h) 1 dh, n 0 G = 1 1 n 1 n 1 g (f) 1 df, (A.7) where g (h) and g (f) are public spending on individual goods h and f produced in ome and oreign, respecively. Minimizaion of P G and P G under resricion (A.7) yields he governmens de-

4 mands for he individual goods h and f: p (h) g (h) = P, G, g (f) = p (f) P, G. (A.8) ence, combining (A.6) and (A.8), he oal demands for he goods h and f are p (h) y (h) = P, T 1 n C W + G p (f), y (f) = P, T n C W + G, (A.9) where C W 1 0 Cj dj, is aggregae consumpion in he union. ollowing Benigno and Benigno (001), we assume ha financial markes are complee boh a he domesic and a he inernaional level. urhermore, each individual s iniial holding of any ype of asse is zero. These assumpions imply perfec consumpion risksharing wihin each counry and equalizaion of he marginal uiliies of consumpion beween counries: U C C = UC C. (A.10) ence, in he absence of exogenous disurbances o he marginal uiliy of consumpion, C = C = C W. Therefore, from now on, we ignore superscrips and denoe consumpion by C. urher, he Euler equaion is U C (C )=(1+ ) βe [U C (C +1 )(P /P +1 )], (A.11) where is he nominal ineres rae on an inernaionally-raded nominal bond. The nominal ineres rae is aken o be he union cenral bank s policy insrumen. inally, using he appropriae aggregaors, aggregae demand in boh counries is found as = T 1 n C + G, = T n C + G. (A.1) A.. irms Individual j is he monopolis provider of good j. The srucure of price seing is assumed o be of he Calvo (1983) form. In each period, here is a fixed probabiliy (1 α i ) ha producer j who resides in i can adjus his prices. This producer akes accoun of he fac ha a change in he price of his produc affecs he demand for i. owever, because he is infiniesimally small, he neglecs any effecs of his acions on aggregae variables. ence, if individual j has he chance o rese his price in period, he chooses his price, denoed 3

5 ˇp (j), o maximize E k=0 α i β k λ i +k ˇp (j) y,+k (j) v y,+k (j);ξ i +k, where y,+k (j) is given by (A.9), assuming ha ˇp (j) sill applies a + k, andλ i +k U C C i +k /P+k is he marginal uiliy of nominal income. or a producer in counry his resuls in he following opimaliy condiion: E k=0 α β k +k v y y,+k (h);ξ +k y,+k (h) ˇp (h) = E k=0 (α β). (A.13) k +k 1 λ +ky,+k (h) ealizing ha, in equilibrium, each producer in a given counry and a given period will se he same price when offered he chance o rese is price, one can show ha P, = P, = 1 α ˇp (h) 1 1 α ˇp (f) α P 1 1, 1, (A.14) + α P 1, (A.15) B. Inefficien seady-sae equilibrium We now derive he seady sae, which is aken o be he equilibrium ha is aained when prices are flexible and shocks are a he mean values, and when here are average monopolisic disorions. Under flexible prices, (A.13) is replaced by ˇp (j) = i v y y, (j);ξ i. (B.1) i 1 λ i Becauseeachageninagivencounrychooseshesameprice,wehavehaˇp (j) =P, for all j living in ome, so ha U C (C )= 1 T 1 n v y T 1 n C + G ; ξ, (B.) and ha p (j) =P, for all j living in oreign, so ha U C (C )= 1 T n v y T n C + G ; ξ. (B.3) ence, he seady-sae values for consumpion and he erms of rade, condiional on 4

6 ome and oreign public spending, follow upon seing he shocks and o heir (common) mean and he oher shocks o zero in (B.) and (B.3). Before we coninue, we inroduce some noaion. ollowing Benigno (003), we denoe wih a superscrip W a world aggregae and wih a superscrip arelaive variable. ence, for a generic variable X, define X W nx +(1 n) X and X X X. urher, we inroduce he following addiional definiions, using an upperbar on a variable o denoe is seady sae in he presence of monopolisic disorions: ρ U CC C C/UC C > 0; ρg V GG G G/VG G > 0; η vyy ;0 /vy ;0 > 0 (because of symmery, = ); and S i (i =, )isdefined such ha v yξ ;0 ξ i = v yy ;0 S i. ence, S i is proporional o he produciviy shock. inally,wedenoeby0 < C/Ȳ <1 he seady-sae consumpion share of oupu. We consider a seady sae in which he fiscal auhoriies coordinae heir policies. ence, he seady-sae values for public spending follow upon maximizing over G s and G s s= : s= E s= β s n U (C s )+V G s v s ; ξ s +(1 n) U (C s )+V G s v s ; ξ s, (B.4) wih ξ s and ξ s se o zero for all s and aking ino accoun he privae secor firs-order condiions in a seady sae: (1 φ) U C (C )=T 1 n v y T 1 n C + G, 0, (B.5) (1 φ) U C (C )=T n v y T n C + G, 0. (B.6) where φ 1/. Seing he derivaive of equaion (B.4) wih respec o G s= = 0. β s nu C (C s ) C s β s nv y s= s= s G β s (1 n) v y +(1 n) U C (C s ) C s G ;0 (1 n) Ts n T s C s G s ;0 ( n) Ts (n+1) o zero, we have: + nv G G + T 1 n s C s T s G C s G + T n s nv y ;0 C s G 5

7 This expression can be rewrien as: = nu C (C ) C G nv y (1 n) v y s=+1 + s=+1 s=+1 +(1 n) U C (C ) C G ;0 (1 n) T n T C G + nv G G C + T 1 n ;0 ( n) T (n+1) T C G β s n (1 n) v y s β s nu C (C s ) C s G G + T n ;0 Ts 1 v y nv y C G s ;0 Ts n +(1 n) U C (C s ) C s G C s T s G β s nv y s ;0 Ts 1 n +(1 n) v y s ;0 Ts n C s. G ;0 Using ha in seady sae, T s =1,C s = C, s = C C nu C +(1 n) U G C G nv y (1 n) v y nc T s=+1 G = s =,ec., s, wehave: + nv G nv y (1 n) C T + C G β s v y,0 UC C C s G, G + C G which, by (B.5) applied o he inefficien seady sae, can be simplified furher o: U C C G v y C G + nv G nv y = φ s=+1 β s U C C C s G. (B.7) rom he firs-order condiion wih respec o G,wederiveasimilarcondiion: U C C G v y C G +(1 n) V G (1 n) v y = φ s=+1 β s U C C C s G. (B.8) We now differeniae (B.5) wih respec o G and G, respecively, and (B.6) wih respec o G and G, respecively. This yields he following four condiions (where we 6

8 already use he fac ha we are evaluaing a he inefficien seady sae): C (1 φ) U CC G C (1 φ) U CC G C (1 φ) U CC G C (1 φ) U CC G = (1 n) v y T G = (1 n) v y T G = nv y T G = nv y T G + v yy (1 n) C T G + v yy (1 n) C T G + v yy nc T G + v yy nc T G Now, add n imes (B.9) and (1 n) imes (B.11) o give: (1 φ) U CC C G = v yy C G + C G + C G Similarly, add n imes (B.10) and (1 n) imes (B.1) o give: (1 φ) U CC C G = v yy C G We rewrie (B.13) and (B.14), o give, respecively: + C G +1, (B.9) + C, G (B.10), (B.11) +1. (B.1) + n. (B.13) +(1 n). (B.14) C G C G = = nv yy, (1 φ) U CC v yy (B.15) (1 n) v yy. (1 φ) U CC v yy (B.16) We now also differeniae (B.5) in period s>wih respec o G and G, respecively, and (B.6) in period s>wih respec o G and G, respecively. This yields he following four condiions (where we already use he fac ha we are evaluaing a he inefficien seady sae): C s (1 φ) U CC G C s (1 φ) U CC G C s (1 φ) U CC G C s (1 φ) U CC G = (1 n) v y T s G = (1 n) v y T s G = nv y T s G = nv y T s G + v yy (1 n) C T s G + v yy (1 n) C T s G + v yy nc T s G + v yy nc T s G + C s G + C s G + C s, (B.17) G + C s, (B.18) G, (B.19). (B.0) 7

9 Now, add n imes (B.17) and (1 n) imes (B.19) o give: (1 φ) U CC C s G = v yy C s G. (B.1) We noe ha by (B.13) (1 φ) U CC = v yy. ence, C s / G (B.18) and (1 n) imes (B.0) o give: =0. Similarly, add n imes (1 φ) U CC C s G = v yy C s G. (B.) ence, C s / G =0.ence, (B.7) and (B.8) become, respecively: U C C G U C C G v y C G v y C G + nv G nv y = 0, (B.3) +(1 n) V G (1 n) v y = 0. (B.4) Using (B.5) in inefficien seady sae, we can wrie (B.3) as: U C C G C (1 φ) U C G C φu C G + nv G n (1 φ) U C = 0, + nv G n (1 φ) U C = 0, C nv G = n (1 φ) U C φu C, G nv G = U C n (1 φ) φ C, G φv yy V G = U C (1 φ) (1 φ) U CC v yy where we have subsiued from (B.15). urher rewriing, we ge (1 φ) U CC V G = U C 1 φ, (1 φ) U CC v yy (1 φ) ρu C /C V G = U C 1 φ (1 φ) ρu C /C + ηv y / ρv y /C V G = U C 1 φ, ρv y /C + ηv y / ρ V G = U C 1 φ ρ + η,,. (B.5) 8

10 The roue via (B.16) yields he same oucome. C. Efficien seady sae and flex-price equilibrium ere, we derive an approximaion o he efficien flexible price equilibrium. The efficien equilibrium obains when here are no monopolisic disorions, which in his model framework can be represened by he case where producers have no marke power, i.e., i corresponds o leing in (B.) and in (B.3). We log linearize he efficien equilibrium around he associaed efficien seady sae. urher, we will refer o he oucomes of he variables in he efficien flex-price equilibrium as he (sochasic) efficien raes. C.1. Efficien seady sae We denoe he equilibrium values of variables in his seady sae by a sar superscrip. ence, he efficien seady-sae values C and T, condiional on G and G, are implicily defined by: U C (C )=(T ) 1 n v y (T ) 1 n C + G ;0, and U C (C )=(T ) n v y (T ) n C + G ;0, from which i follows ha in a symmeric equilibrium T =1. rom (B.5) wih φ =0we obain he seady-sae values G and G for public spending as: V G G = v y ;0, V G G = v y ;0. (C.1) Because =,wehavehag = G G. inally, we obain he efficien seady-sae nominal (=real) ineres rae from (A.11) as 1+ =1/β. C.. Derivaion of relaionships beween efficien and inefficien seady saes We derive c ln C/C and g ln G/G,whichweshalluseinhesequel.To his end, recall he seady-sae relaions derived earlier: (1 φ) U C C = vy C + G;0, (C.) ρ V G G = UC C 1 φ, (C.3) ρ + η 9

11 remembering ha T =1and G = G = G W G. Takeafirs-order Taylor approximaion o (C.) evaluaed around he efficien seady sae: φ + ρc = η [ c +(1 ) g ] η (1 ) g +(ρ + η ) c = φ, (C.4) wherewehaveusedhac / = + O (ξ), because φ is O (ξ). Also, ake a firs-order Taylor approximaion of (C.3) around he efficien seady sae: Subsiue his ino (C.4), o give: ρ g g ρ = φ + ρc ρ + η g = ρ φ + c, (C.5) ρ g ρ + η η (1 ) ρ φ + c +(ρ + η ) c ρ g ρ + η = φ η (1 ) ρφ + ηρ(1 )(ρ + η ) c + ρ g (ρ + η ) c = φρ g (ρ + η ) (ρ + η ) ηρ(1 )+ρ g (ρ + η ) c = φ ρ η + ρ g + ηc ρg ρ (ρ + η ) ρ η + ρ g + ηc ρg ρ c = φ ρ η + ρ g + ηc ρg ρ c = φ ρ + η φ = (ρ + η ) c. ence, we have c = φ/ (ρ + η ), (C.6) g = 0. (C.7) C.3. Efficien flexible price equilibrium Log-linearizing (B.5), wih / 1 =1imposed around he efficien seady sae, we have: ρc =(1 n) T + η (1 n) T + C +(1 ) G ηs, (C.8) 10

12 and an analogous equaion for he oreign counry: ρc = nt + η n T + C +(1 ) G ηs. (C.9) Taking a weighed average (wih weighs n and 1 n) of hese equaions, we obain ρc = η C +(1 ) G W ηs W. ence, C = η S W ρ + η (1 ) G W. (C.10) Subracing (C.9) from (C.8) we obain 0= T + η T (1 ) G + ηs and hus T = η (1 ) G S 1+η. (C.11) urher, because = (1 n)(t ) 1 n C T +(T ) 1 n C C + G G /,wecan also wrie (C.8) as ρ C =(1 n) T +η ηs and (C.9) as ρ C = n T +η ηs. Takingaweighedaverage(wihweighsn and 1 n) of hese wo equaions, we hen obain ρc = η W ηs W. (C.1) Combining his wih (C.10), we find ha: W = η S W + ρ (1 ) G W. (C.13) ρ + η ρ + η We solve now for G and G, hereby compleing he soluion of he efficien flex-price equilibrium. Above we found he seady sae values for public spending as he soluions o (C.1). urher, we have ha V G G = vy ; ξ. (C.14) To show his, se he derivaive of (B.4) wih respec o G E E E s= β s nu C (C s ) C s G β s s= s= nv y ; ξ s β s (1 n) v y s ; ξ s s +(1 n) U C (C s ) C s G (1 n) Ts n T s C s G o zero: + nv G G + T 1 n s nts (n+1) T s C s G C s G + T n s nv y ; ξ C s =0. (C.15) G 11

13 Using (B.) wih,(b.3)wih and (A.10), we have ha T s v y s ; ξ s = v y s ; ξ s, for all s. ence, (C.15) becomes: E E s= β s nu C (C s ) C s s= G β s nv y s ; ξ s +nv G G nvy ; ξ = E s= β s nu C (C s ) C s +nv G G nvy which is equal o which is equal o E β s s= +(1 n) U C (C s ) C s G (1 n) Ts n (1 n) v y s, ξ s G ; ξ, +(1 n) U C (C s ) C s nu C (C s ) C s v y s ; ξ s +nv G G nvy ; ξ = E β s s= +E s= G T 1 n s G n U C (C s ) T 1 n v y ; ξ s β s (1 n) U C (C s ) T 1 n s +nv G G nvy E s= ; ξ s, β s n U C (C s ) T 1 n v y +E β s (1 n) U C (C s ) T n s= +nv G G nvy ; ξ = nv G G nvy ; ξ, s T C s s G + T 1 n s ( n) Ts n T C s s G C s G v y s, ξ s T 1 n s +(1 n) U C (C s ) C s G n C s +(1 n) C s G G s s s C s G v y ; ξ s ; ξ s s C s G C s G v y s ; ξ C s s G + T 1 n s C s G C s G where in he final sep we have used again (B.) wih and (B.3) wih. We log linearize (C.14) and find ρ g G = η (1 n) T + C +(1 ) G 1

14 ηs,fromwhichweobain G = or oreign spending we similarly find G = η S (1 n) T ρ g + η (1 ) + C. (C.16) η S nt ρ g + η (1 ) + C. (C.17) Togeher wih (C.10) and (C.11), we hen have four equaions in four unknowns: G, G, T and C. These equaions are solved o yield C = G W = G = ηρ g ρ ρ g + η (1 ) + η ρ g S W, (C.18) ηρ ρ ρ g + η (1 ) + η ρ g S W, (C.19) η ρ g (1 + η )+η (1 ) S, (C.0) ηρ g T = ρ g (1 + η )+η(1 ) S. (C.1) The above expressions have been derived as follows. Using (C.16) and (C.17), we ge G η = ρ g +η(1 S + c ) T. By subsiuing his ino (C.11), one hen recovers (C.1). Nex, combining (C.16) and (C.17) wih weighs n and (1 n), respecively, yields G W η = ρ g +η(1 S W c ) C. Combining his wih (C.10), and solving, give (C.19). Subsiuing (C.19) back ino (C.10) and working ou yield (C.18). inally, assuming ha he inflaion rae in he flex-price equilibrium is zero, we derive he efficien nominal rae of ineres from (A.11) as = ρe C+1 C. (C.) D. The model under sicky prices Log linearizing (A.11) and using (C.), i is sraighforward o derive (D.3) below, where for a generic variable X, X =ln X/X. Log linearizing (A.1), we derive (D.4) and (D.5), below, and by log linearizing he definiion of he erms of rade, T P /P,weobain (D.8). Mos compuaionally inensive is he derivaion of he Phillips curves, (D.6) and (D.7) below, which we provide now. 13

15 We can rewrie (A.13), for i = and j = h, as 0=E k=0 α β k λ +kp (h)+v y y,+k (h);ξ +k y,+k (h). Afer subsiuing for λ +k we obain E k=0 α β k +k 1 U C (C +k ) p (h) T n 1 +k P,+k +k v y y,+k (h), ξ+k y,+k (h) =0. (D.1) To log linearize his condiion around he seady sae, we firs need o log linearize y,+k (h): aking logarihmic changes on boh sides of (A.9) and evaluaing around he seady sae, we obain: p (h) +k d ln y,+k (h) = d ln + d ln T 1 n +k P C +k + G +k,+k P dp (h) P dp,+k ŷ,+k (h) = P + C (1 n) dt +k + dc +k + dg +k ŷ (h) = ˆp (h) ˆP,+k + (1 n) ˆT +k + Ĉ+k +(1 ) Ĝ +k ŷ (h) = ˆp,+k (h)+ (1 n) ˆT +k + Ĉ+k +(1 ) Ĝ +k. Using his expression, he log-linearized version of (D.1) around he seady sae is: 0=E α β k p,+k (1 n) T +k ρ C +k k η p,+k + (1 n) T +k + C +k +(1 ) G +k S +k k=0 where p,+k ln (p (h) /P,+k ). We rewrie his expression, using ha p,+k = p, k s=1 π +s, as: p, 1 α β = E α β k k= E α β k k k=0 s=1 1+η (1 n) T 1+η +k + ρ+η C 1+η +k 1 1+η +k + η (1 c 1+η ) G +k S +k. π +s, Log-linearizing (A.14), we obain p, = α π 1 α, which we use o simplify he previous 14

16 expression: π α 1 α β 1 α = E α β k k=0 +E α β k k= π +k 1 α β. 1+η (1 n) T 1+η +k + ρ+η C 1+η +k 1 1+η +k + η (1 c 1+η ) G +k S +k inally, we hen obain π = (1 α β)(1 α ) α 1+ηc 1+η (1 n) T + ρ+η C 1+η + η(1 ) 1+η η η 1+η S G + βe π +1. (D.) Combine (C.1) and (C.13) o find ha C = η ρ+η S W η(1 ) ρ+η G W. Using his expression and (C.11), i is sraighforward o show ha (1 + η )(1 n) T (ρ + η ) C η (1 ) G = ηs. ence, (D.) can be rewrien as (D.6) below. In an analogous fashionwederive(d.7)below. Summarizing, he log-linearized sysem under sicky prices is given by E C+1 C +1 = C C + ρ 1 E π W +1, (D.3) = (1 n) T + C +(1 ) G, (D.4) = n T + C +(1 ) G, (D.5) π = βe π +1 + κ T (1 n) T T + κ C C C +κ G G G + u, π = βe π +1 κ T n T T + κ C C C +κ G G G + u, (D.6) (D.7) T T = T 1 T 1 + π T T 1, (D.8) 15

17 which is he sysem (1)-(6) in he paper, where κ T κ (1 + η ), κ T κ (1 + η ), κ C κ (ρ + η ), κ C κ (ρ + η ), κ G κ η (1 ), κ G κ η (1 ), wih and where κ 1 α β 1 α 1 α β 1 α, κ, α (1 + η) α (1 + η) u κ 1 1, u κ 1 1, refer o inflaion variaions caused by flucuaions in producers marke power. or any of he cases wih equal rigidiies considered in he sequel, we define: α α = α, κ κ = κ, κ T κ T = κ T, κ C κ C = κ C, κ G κ G = κ G. E. Derivaion of he micro-founded loss funcion ere, we derive he uiliy-based loss funcion. The per-period average uiliy flows of he households belonging o counries and, respecively, are: w = U (C )+V G 1 n v y (h);ξ n 0 dh, (E.1) w = U (C )+V G 1 1 v y (f);ξ df. (E.) 1 n n The welfare crierion of he auhoriies (he common cenral bank and he coordinaing fiscal auhoriies) is a populaion weighed average of households uiliies: W C = E 0 j=0 β j nw +j +(1 n) w +j. (E.3) We sar by making compuaions for ome. The compuaions for oreign are analogous and, herefore, no shown explicily. Afer his, we combine he expressions for ome and oreign o obain W C. 16

18 E.1. The erm U C Take a second-order expansion of U (C ) around he seady-sae value C: U (C )=U C + U C C C + 1 U CC C C + O ξ 3, (E.4) where O ξ 3 sands for erms of hird or higher order (remember ha all variables are, in equilibrium, funcions of he shock vecor, which exhibis bounded flucuaions of order ξ). Noe ha a second-order log-expansion of C around C yields: C = C 1+ C + 1 C + O ξ 3. (E.5) Subsiue (E.5) ino (E.4) o give: and hus U (C )=U C + U C C C + 1 C + 1U CCC C + O ξ 3, U (C ) = U C C C + 1 C + 1U CCC C +.i.p. + O ξ 3 U (C ) = U C C C + 1 C + 1 U CC C C +.i.p. + O ξ 3 U C U (C ) = U C C C + 1 (1 ρ) C +.i.p. + O ξ 3, (E.6) where.i.p. sands for erms independen of policy. E.. The erm V G We approximae in an analogous way V G.Thisyields: V G = VG G G ρg G +.i.p. + O ξ 3. (E.7) Using (B.5) and assuming ha φ is of a leas order O (ξ), we can wrie(e.7) as: V G ρ = UC G 1 φ G ρg G +.i.p. + O ξ 3, (E.8) ρ + η having used ha φ G is of order a leas O ξ 3. 17

19 E.3. The erm v y (h);ξ Similarly, we ake a second-order Taylor expansion of v y (h);ξ around a seady sae where y (h) = for each h andaeachdae, andwhereξ =0a each dae. We obain: v y (h);ξ = v ;0 + vy y (h) + v ξ ξ + 1v yy y (h) +v yξ ξ y (h) + 1 ξ vξξ ξ + O ξ 3. Then noe ha a second-order logarihmic expansion of y (h) gives: y (h) = 1+y (h)+ 1 y (h) + O ξ 3. Using his expression, we simplify v y (h);ξ = vy y (h)+ 1 v yy y (h) + vyξ ξ y (h)+.i.p. + O ξ 3 o v y (h);ξ = vy y (h)+ 1 y (h) + 1 v yy y (h) + v yξ ξ y (h) +.i.p. + O ξ 3, v y v y or v y (h);ξ = vy y (h)+ 1+η y (h) + v yξ ξ y (h) +.i.p. + O ξ 3. v y Then, recalling he definiion v yξ ξ = v yy S for S v y (h);ξ = vy y (h)+ 1+η y (h) ηs,wefinally arrive a y (h) +.i.p. + O ξ 3. (E.9) ecall ha (1 φ) U C C = vy ;0. ence, using his, we can wrie (E.9) as: v y (h);ξ = UC (1 φ) y (h)+ 1+η y (h) ηs y (h) +.i.p. + O ξ 3, wherewehaveusedhaφy (h) and φs y (h) are of order a leas O ξ 3. 18

20 This las expression should be inegraed over he ome populaion, o find 1 n v y (h);ξ dh n 0 = U C (1 φ) E h y (h)+ 1+η Varh y (h)+[e h y (h)] ηs E hy (h) +.i.p. + O ξ 3. We hen ake a second-order log-expansion of he aggregaor o obain: whereweobserveha and ln 1 1 = 1 = E h y (h)+ 1 1+ln 1 Var h y (h)+o ξ 3 = E h y (h)+ 1 1 Var hy (h)+o ξ 3, 1 / 1 + ln 1 / 1 is of order a leas O (ξ), soha 1 Var h y (h) = 1Var hy (h)+o ξ 3. Inser he implied value for E h y (h) ino he previous expression: 1 n n 0 = U C v y (h);ξ dh (1 φ) 1 1 Var hy (h)+ 1+η +.i.p. + O ξ 3 = U C = U C (1 φ) + 1+η (1 φ) + 1+η Var h y (h)+ / 1 + O ξ 3, ηs η Var h y (h) ηs η Var h y (h) ηs +.i.p. + O ξ 3 +.i.p. + O ξ 3. 19

21 E.4. Combining he erms Combining (E.6), (E.8) and he previous expression, he relevan ome welfare crierion is w = U C C C + 1 (1 ρ) C ρ +U C G 1 φ U C ρ + η (1 φ) + 1+η +.i.p. + O ξ 3, G ρg G η Var h y (h) ηs or w = U C C C + 1 (1 ρ) C 1 φ (1 φ) + 1+η +.i.p. + O ξ 3. ρ ρ+η G ρg G η Var h y (h) ηs or oreign, we similarly find: w = U C C C + 1 (1 ρ) C + 1 c ρ 1 φ 1 ρ + ηc (1 φ) + 1+η +.i.p. + O ξ 3. G ρg G η Var f y (f) ηs 0

22 Now, ake a weighed average of w and w wih weighs n and 1 n, respecively: w = U C C C + 1 (1 ρ) C + 1 c ρ 1 φ G W ρg n G +(1 n) G ρ + η 1 (1 φ) W + 1+η n +(1 n) + 1 [ 1 + η][nvar h y (h)+(1 n) Var f y (f)] ηns η (1 n) S +.i.p. + O ξ 3. (E.10) E.4.1. Expansion of Before coninuing, we expand approximae his as: = W + W = 0+ = T 1 n C + G 1 = T 1 n C T 1 n C.Define he funcion W + 1 W + O ξ 3 T 1 n C + G + G G + O ξ 3 =ln /.We 1 T 1 n C + G T 1 n C + G + O ξ 3 1 T n 1 C T 1 n C + G G + O ξ 3. (E.11) Now define Z (T,C ) T 1 n C. Taking a second-order Taylor expansion of Z (T,C ) around he poin T,C gives: Z (T,C ) = Z T,C + Z T T T + 1Z TT T T + ZC C C + 1Z CC C C + ZTC T T C C + O ξ 3 = T 1 n C +(1 n) T n C T T 1 (1 n) nt (n+1) C T T +T 1 n C C +(1 n) T n T T C C + O ξ 3 = T 1 n C +(1 n) T 1 n T T C 1 T (1 n) nt 1 n T T C T +T 1 n C C C +(1 n) T 1 n T T C C C + O ξ 3. C T C 1

23 ence, T 1 n C T 1 n C Ȳ T T = (1 n) T + C C C Ino his expression, subsiue: 1 (1 n) nc T T T C C +(1 n) T T T C = C 1+ C + 1 C + O ξ 3, T = T 1+ T + 1 T + O ξ 3, C + O ξ 3. so ha he righ-hand side becomes: (1 n) c T + 1 T + C + 1 C 1 (1 n) nc T + 1 +(1 n) T + 1 T T C + 1 C + O ξ 3 = (1 n) T + C + 1 (1 n) c T 1 (1 n) nc T + 1 c C +(1 n) T C + O ξ 3 = (1 n) T + C + 1 (1 n) T + 1c C +(1 n) c T C + O ξ 3. Using ha G = G 1+ G + 1 G + O ξ 3, we can wrie: = (1 n) T + C + 1 (1 n) T + 1 C +(1 n) T C +(1 ) G + 1 G + O ξ 3.

24 ence: = c = c (1 n) T + C + 1 (1 n) T + 1 C +(1 n) T C + (1 ) +(1 ) (1 n) T + C + 1 (1 n) T + 1 C +(1 n) T C G + 1 G + 1 G + O ξ 3 (1 n) T + C +(1 n) T C +(1 ) G + (1 )(1 n) T G + (1 ) C G + O ξ 3. G ence, 1 = (1 n) T + C +(1 ) G + 1 (1 n) T + 1 C +(1 n) T C + 1 (1 c ) G 1 c (1 n) T 1 c C (1 n) c T C 1 (1 c ) G (1 )(1 n) T G (1 ) C G + O ξ 3. or = = 1 (1 n) T + C +(1 ) G + O ξ (1 n) (1 ) T + 1 (1 ) C + 1 (1 ) +(1 n) (1 ) T C (1 )(1 n) T G (1 ) C G + O ξ 3. G 3

25 In a similar way we derive he corresponding expression for he foreign counry: = = 1 + O ξ 3 n T + C +(1 ) G + 1 n (1 ) T + 1 (1 ) C + 1 (1 ) G n (1 ) T C + (1 ) n T G (1 ) C G + O ξ 3. E.4.. Coninuaion of approximaion To furher work ou he approximaion of he welfare loss funcion, i is useful o compue some numbers, before acually making he subsiuions. We have: W = n +(1 n) = C +(1 ) G W + 1 c (1 ) n C G + 1 n (1 n) (1 ) T +(1 n) C G + (1 ) n (1 n) T G + O ξ 3. ence, (1 φ) W = (1 φ) + 1 (1 ) C +(1 ) G W n C G + (1 ) n (1 n) T G + O ξ 3, + 1n (1 n) c (1 ) T +(1 n) C G again using ha φ is of order a leas O (ξ). urher, = = (1 n) T + C +(1 ) G + O ξ 3, n T + C +(1 ) G + O ξ 3. 4

26 ence, n +(1 n) = nc (1 n) T + C +(1 n) c nt + C + n (1 ) (1 n) T + C G +(1 n) (1 ) +n (1 ) G +(1 n)(1 c ) G + O ξ 3 = nc (1 n) T +(1 n) T C + C +(1 n) c + (1 ) C G W n (1 n) T G n T + C G n T nt C + C +n (1 ) G +(1 n)(1 c ) G + O ξ 3. ence, n +(1 n) = n (1 n) c T + c C + (1 ) C G W +(1 ) n G +(1 n) G + O ξ 3 n (1 n) T G = n (1 n) c T (1 ) n (1 n) T G + n c C + (1 ) C G +(1 ) G (1 n) c C + (1 ) C G +(1 ) G + O ξ 3. urher, S S = S = S (1 n) T + C +(1 ) G + O ξ 3, n T + C +(1 ) G + O ξ 3. ence, η ns +(1 n) S = ηns c (1 n) T + C +(1 ) G + η (1 n) S c n T + C +(1 ) G + O ξ 3 = η (1 n) nt S + η C S W +η (1 ) ns G +(1 n) S G + O ξ 3. 5

27 We can now sar o make subsiuions ino (E.10). irs, subsiue he expression for W where and observe ha he linear erms cancel. Thus, we have: w U C C = ρ φ G W ρ + ηc 1 ρg n 1 A 1 1+η + 1 (1 ρ) C G +(1 n) G [nvar h y (h)+(1 n) Var f y (f)] +.i.p. + O ξ 3,(E.1) A φ C +(1 ) G W +n (1 n) (1 ) T +n (1 n) (1 ) T G + (1 ) n C G +(1 n) C G +(1+η) n (1 n) c T (1+η) (1 ) n (1 n) T G +(1+η) n c C + (1 ) C G +(1 ) G +(1+η)(1 n) c C + (1 ) C G +(1 ) G η ns +(1 n) S. Subsiue his back ino (E.1), o give: w U C C = φ C + φ η (1 ) G W ρ + η + 1 (1 ρ) C + n 1 +(1 n) 1 n 1+η (1 n) 1+η 1 ρg G n 1 (1 n) 1 C G C G 1 ρg G c C +(1 ) G +c (1 ) C G 1 n (1 n)(1 ) T c C + (1 ) C G n (1 n)(1 ) T G +(1 ) G 1 (1 + η) n (1 n) c T +(1+η)(1 ) n (1 n) T G + η ns +(1 n) S 1 1+η [nvar h y (h)+(1 n) Var f y (f)] +.i.p. + O ξ 3. 6

28 ence, w U C C = φ C + φ η (1 ) G W ρ + ηc ence, + 1 [1 ρ (1 + η)] C + n 1 +(1 n) 1 1 ρg (1 )(1+η) G (1 + η)(1 ) C G W n 1 1 n (1 n)(1+ηc ) T + η ns 1 +(1 n) S w U C C = φ C + φ η (1 ) G W ρ + η ence, 1 ρg (1 )(1+η) G C G (1 n) 1 + ηn (1 n)(1 ) T G 1+η [nvar h y (h)+(1 n) Var f y (f)] +.i.p. + O ξ [1 ρ (1 + η)] C + n 1 +(1 n) 1 (1 + η)(1 ) C G W 1n (1 n)(1+ηc ) T +ηs W C η (1 n) nt S 1 1 ρg (1 )(1+η) G n 1 1 ρg (1 )(1+η) G C G (1 n) 1 + ηn (1 n)(1 ) T G + η (1 ) ns G +(1 n) S 1+η [nvar h y (h)+(1 n) Var f y (f)] +.i.p. + O ξ 3. w U C C = φ C + φ η (1 ) G W ρ + η 1 (ρ + η) C + 1 (1 c )(1+η) C ρg + η G n 1 + n 1 (1 )(1+η) (1 n) 1 ρg + η G +(1 n) 1 (1 + η)(1 ) C G W n 1 1 n (1 n)(1+ηc ) T +ηs W 1 C η (1 n) n T S C G G (1 c )(1+η) (1 n) 1 + ηn (1 n)(1 ) T G ns G + η (1 ) C G C G G G +(1 n) S 1+η [nvar h y (h)+(1 n) Var f y (f)] +.i.p. + O ξ 3. C G G 7

29 ence, ence, w U C C = φ C + φ η (1 ) G W ρ + ηc 1 (ρ + η) C + n 1 (1 c )(1+η) +(1 n) 1 (1 c )(1+η) C G n 1 ρg + η G (1 n) 1 C G n 1 (1 n) 1 ρg + η G C G C G 1 n (1 n)(1+ηc ) T + ηn (1 n)(1 ) T G +ηs W C η (1 n) nt S + η (1 ) ns G +(1 n) S 1 1+η [nvar h y (h)+(1 n) Var f y (f)] +.i.p. + O ξ 3. w U C C = φ C + φ η (1 ) G W ρ + η 1 (ρ + η) C (1 ) η n ρg + η n C G 1 n (1 n)(1+ηc ) T +ηs W C η (1 n) nt S G +(1 n) G +(1 n) C G + ηn (1 n)(1 ) T G + η (1 ) ns G +(1 n) S 1 1+η [nvar h y (h)+(1 n) Var f y (f)] +.i.p. + O ξ 3. G G 8

30 Now, express everyhing in erms of gaps: w U C C = φ C + φ η (1 ) G W ρ + ηc 1 (ρ + η) C C 1 ρg + η n G G +(1 n) G G + 1 (1 c ) η n C C G G +(1 n) C C G G 1n (1 n)(1+ηc ) T T + ηn (1 n)(1 c ) T T G G (ρ + η) C C n 1 ρg + η G G (1 n) 1 ρg + η G G +n (1 ) η C C n (1 ) η C G n (1 ) η C G + n (1 ) η G G +(1 n)(1 ) η C C (1 n)(1 ) η C G (1 n)(1 ) η C G +(1 n)(1 ) η G G n (1 n)(1+η ) T T + ηn (1 n)(1 ) T G + ηn (1 n)(1 ) T G +ηs W C η (1 n) nt S + η (1 ) ns G +(1 n) S G 1 1+η [nvar h y (h)+(1 n) Var f y (f)] +.i.p. + O ξ 3, 9

31 where producs of he exogenous naural levels of variables have been pu ino he.i.p.. Simplify he previous expression: w U C C = φc + φ η (1 ) ρ + η G W 1 (ρ + ηc ) C C n 1 ρg + η (1 ) G G (1 n) 1 ρg + η (1 ) G G n (1 c ) η C C G G (1 n)(1 ) η C C G G 1n (1 n)(1+ηc ) T T + ηn (1 n)(1 c ) T T G G n 1 (ρ + η ) C C (1 ) η C G W ρg + η (1 ) G G (1 ) η C G W (1 n) 1 ρg + η (1 ) G G n (1 n)(1+η ) T T + ηn (1 n)(1 ) T G + ηn (1 n)(1 ) T G +ηs W C η (1 n) n T S + η (1 ) ns G +(1 n) S 1 1+η [nvar h y (h)+(1 n) Var f y (f)] +.i.p. + O ξ 3, G We can rewrie he laes expression furher as: w U C C η(1 c ) ρ+η G W + = φ C + φ 1 (ρ + ηc ) C C n 1 ρg + η (1 ) G G (1 n) 1 ρg + η (1 ) G G n (1 c ) η C C G G (1 n)(1 ) η C C G G 1 n (1 n)(1+ηc ) T T + ηn (1 n)(1 c ) T T G G +A CW, C + A G, G + A G, G + A T, T 1 1+η [nvar h y (h)+(1 n) Var f y (f)] +.i.p. + O ξ 3, 30

32 where A CW, = (ρ + η ) C (1 ) ηg W + ηs W = ηs W (ρ + η) C +(1 ) η C G W, A G, = n (1 ) η C n 1 ρg + η (1 ) G ηn (1 n)(1 ) T + η (1 ) ns, A G, = (1 n)(1 ) ηc (1 n) 1 ρg + η (1 ) G +ηn (1 n)(1 ) T + η (1 )(1 n) S, A T, = n (1 n)(1+η ) T + ηn (1 n)(1 ) G η (1 n) ns. We shall now evaluae ou hese coefficiens A j,. owever, before doing so, we make use (C.18) and (C.19), so ha C G W = = ηρ g ρ ρ g + η (1 ) S W + η ρ g η ρ g ρ ρ ρ g + η (1 ) S W, + η ρ g ηρ ρ ρ g + η (1 ) + η ρ g S W and 1 ρg + η G W = 1 ρg + η η C G W + η S W ηρ ρ ρ g + η (1 ) S W + η ρ g η ρ g ρ η ρ ρ g + η (1 ) S W + η ρ g 1 ηρ ρg + η = ρ ρ g + η (1 ) + + η ρ g = 1 η ρ ρ g + η (1 ) + η ρ g + η S W η ρ g ρ ρ ρ g + η (1 ) + η ρ g η ρ ρ g + η + η ρ g ρ ρ ρ g + η (1 ) η ρ g S W = 1 η ρ ρ g + η (1 ) ρ ρg + η ηρ ρ ρ + η ρ g + η (1 ) S W g = 1 η ρ ρ g + η (1 ) ρ ρg + η ρ ρ + η ρ g + η S W g = 0. S W 31

33 ence, A CW, = ηs W (ρ + η) C +(1 ) η C G W ηρ = ηs W g (ρ + η) ρ ρ g + η (1 ) η ρ S W g ρ +(1 ) η + η ρ g ρ ρ g + η (1 ) S W + η ρ g (ρ + η) ρ g = η 1 ρ ρ g + η (1 ) (1 ) η ρ g ρ + + η ρ g ρ ρ g + η (1 ) S W + η ρ g η = ρ ρ g + η (1 ) + η ρ g ρ ρ g + η (1 ) + η ρ g (ρ + η) ρ g +(1 ) η ρ g ρ S W η = ρ ρ g + η (1 ) ρρg + η ρ + η ρ g (ρ + η) ρ g +(1 ) ηρ g S W g = 0. and, noing ha G = G W (1 n) G, S = S W (1 n) S and T = ρ g G : 1 A n G, = (1 ) η C 1 ρg + η (1 ) G W (1 n) G +ηρ g (1 n)(1 ) G + η = 1 ρg + η G W (1 ) η (1 ) S W (1 n) S C G W + η (1 ) S W + 1 ρg + η (1 ) (1 n) G + ηρ g (1 n)(1 ) G (1 )(1 n) S = 0+ (1 n)(1 ) ρg + η (1 ) + ηρ c g G ηs = 0+0 = 0, η and, noing ha G = G W + n G, S = S W + ns and T = ρ g G : 1 A 1 n G, = (1 ) ηc 1 ρg + η (1 ) G W + ng ηρ g n (1 ) G + η (1 ) S W + ns ρg = 0 n(1 ) + η (1 ) G + ηρ g G ηs = 0+0 = 0. 3

34 and, noing ha T = ρ g G : A T, = n (1 n)(1+η ) T + ηn (1 n)(1 ) G η (1 n) ns = n (1 n) (1 + η ) T η (1 ) G + ηs = n (1 n) (1 + η ) ρ gg + η (1 ) G ηs = 0+0. ence, all he A j, erms are zero, and we have in conclusion: w U C C = φ C + φ η (1 ) G W ρ + ηc 1 (ρ + ηc ) C C n 1 ρg + η (1 ) G G (1 n) 1 ρg + η (1 ) G G n (1 ) η C C G G (1 n)(1 ) η 1 n (1 n)(1+η ) T T + ηn (1 n)(1 c ) C C G T T G 1 1+η [nvar h y (h)+(1 n) Var f y (f)] +.i.p. + O ξ 3. G G Using (C.6) and (C.7), we can wrie: w U C C =(ρ + η ) c C + η (1 ) c G W ρg + η (1 ) G 1 (ρ + ηc ) C C n 1 G (1 n) 1 ρg + η (1 ) G G n (1 c ) η C C G G (1 n)(1 ) η C C G G 1n (1 n)(1+ηc ) T T + ηn (1 n)(1 c ) T T G G 1 1+η [nvar h y (h)+(1 n) Var f y (f)] +.i.p. + O ξ 3, 33

35 and hus w U C C = 1 (ρ + ηc ) C C c n 1 ρg + η (1 ) G G n (1 c ) η (1 n) 1 ρg + η (1 ) G G (1 n)(1 ) η C C c G G 1 n (1 n)(1+η ) T T + ηn (1 n)(1 c ) C C c G T T G 1 1+η [nvar h y (h)+(1 n) Var f y (f)] +.i.p. + O ξ 3, G G wherewehavepuermsinvolvingc C, c G and c G ino.i.p.. urhermore, we have wrien ou η (1 ) c G W = η (1 ) c n G +(1 n) G. The final sep is o derive Var h y (h) and Var f y (f). Wehaveha var h [log y (h)] = varh [log p (h)] = var h [log p (h)] + O ξ 3. We have var h [log p (h)] = var h [log p (h) p 1 ]=E h [log p (h) p 1 ] ( p ) = α E h [log p 1 (h) p 1 ] + 1 α [log p (h) p 1 ] ( p ) = α var h [log p 1 (h)] + 1 α [log p (h) p 1 ] ( p ), where urher, ence, Using p E h [log p (h)]. p p 1 = 1 α [log p (h) p 1 ]. var h [log p (h)] = α var h [log p 1 (h)] + p =logp, + O ξ, α ( p 1 α ). we have: var h [log p (h)] = α var h [log p 1 (h)] + α 1 α π + O ξ 3. 34

36 ence, var h [log p (h)] = α +1 varh [log p 1 (h)] + and hus where = s=0 α s α 1 α π s + O ξ 3 s=0 α s α 1 α π s +.i.p. + O ξ 3, β var h [log p (h)] = d =0 Similarly, we derive for oreign: d =0 β π α (1 α β)(1 α ). +.i.p. + O ξ 3, β var f [log p (f)] = d =0 =0 β π +.i.p. + O ξ 3, where d α (1 α β)(1 α ). ence, ignoring erms independen of policy as well as erms of order O ξ 3 or higher, he second-order welfare approximaion is given by: β E 0 w C, where =0 w C U C C = 1 (ρ + ηc ) C C c n 1 (1 n) 1 ρg + η (1 ) G G ρg + η (1 ) G (1 c ) η 1n (1 n)(1+ηc ) T T + ηn (1 n)(1 c ) n d π +(1 n) d π. 1 1+η G C C c G W T T G G G W 35

37 ence, w C = 1 U CC (1 + η) / (ρ+η ) C C (1+η) c n(1 )[ρ g +η(1 )] G (1+η) G (1 n)(1 )[ρ g +η(1 )] G (1+η) G (1 )η C C (1+η) c G W G W n(1 n) (1+η ) T T (1+η) + ηn(1 n) (1 ) T T (1+η) G G nd π +(1 n) d π Observing ha nd +(1 n) d = n/κ +(1 n) /κ / (1 + η), we can wrie w C = 1U CC n/κ +(1 n) /κ / (ρ+η ) C C [n/κ +(1 n)/κ ] c n(1 )[ρ g +η(1 )] G [n/κ +(1 n)/κ ] G (1 n)(1 )[ρ g +η(1 )] G [n/κ +(1 n)/κ ] G (1 )η C C [n/κ +(1 n)/κ ] c G W G W n(1 n) (1+η ) T T [n/κ +(1 n)/κ ] + ηn(1 n) (1 ) T T [n/κ +(1 n)/κ ] G G λ π π + λ π π, where nd λ π nd +(1 n) d = n/κ n/κ +(1 n) /κ, λ π =1 λ π. Ignoring an irrelevan proporionaliy facor, he associaed loss funcion is given by L = β E 0 [L ], =0 (E.13) where L = λ C C C c + nλg G G +(1 n) λ G G G +λcg C C c G W G W, (E.14) +λ T T T λtg T T G G +λ π π + λ π π 36

38 where λ C λ CG (ρ + η ) / n/κ +(1 n) /κ, λ G (1 c ) ρg + η (1 ) /, n/κ +(1 n) /κ (1 ) η/ n/κ +(1 n) /κ, λ T n (1 n) (1 + η ) / n/κ +(1 n) /κ, λ TG n (1 n) η (1 ) / n/κ +(1 n) /κ. An alernaive represenaion, which expresses he loss funcion exclusively in erms of underlying parameers, follows by muliplying he above weighs by n/κ +(1 n) /κ : L /Λ = (ρ + η ) C C c +n (1 ) ρ g + η (1 ) G G +(1 n)(1 ) ρ g + η (1 ) G G +n (1 n) (1 + η ) T T + (1 ) η C C c G W G W n (1 n) (1 ) η T T G G + n π (1 n) κ + π κ, which is expression (8) in he paper, where Λ n/κ +(1 n) /κ.. Derivaion of (9)-(1) Take a weighed average wih weighs n and 1 n of (D.6) and (D.7): π W = βe π W +1 +(1+η ) n (1 n) κ κ T T +(ρ + η ) nκ +(1 n) κ C C + nκ η (1 ) G +(1 n) η (1 ) G G + u W. G Wih equal rigidiies we can wrie: π W = βe π W +1 + κ (ρ + η ) C C + η (1 ) G W G W + u W. urher, wriing ou L in (E.14) yields: L = λ C C C + λ CG G W G W c +.i.p. + L S, 37

39 where.i.p. is a erm independen of policy, namely λ C (c ) and where L S = λ C C C + λt T T + nλg G G +(1 n) λ G G G + λπ π + λπ π +λ C CG C G W G W λ T TG T G G. (.1) / Define Ω.Weobservehaλ n/κ +(1 n)/κ C =(ρ + η ) Ω, λ CG = η (1 ) Ω and λ G = 1 ρg + η (1 ) Ω,sohawewrie: L = (Ω/κ) π W βe π W +1 u W c +.i.p. + L S. ence, β E 0 [L ]= (Ω/κ) c =0 =0 β E 0 π W βe π W +1 uw +.i.p. + β E 0 L S. =0 Using ha =0 β E 0 π W βπ W +1 uw = π W 0 βe 0 π W 1 uw 0... = π W 0 =0 β E 0 u W,wecanwriehislasexpressionas: β E 0 [L ]= (Ω/κ) c π W 0 =0 β E 0 u W =0 +.i.p. + + β E0 π W 1 βe 0π W =0 β E 0 L S, + which is equaion (9) in he paper. We noe ha for any generic variable X, he following holds: n X +(1 n) X = X W + n (1 n) X. (.) Using his, we can rewrie L S as: L S = L W + n (1 n) L, (.3) which is equaion (10) in he paper, where L W = λ W C C C +λ W G G W G W + π W +λ W CG C C G W G W, (.4) which is equaion (11) in he paper, and L = λ T T T + π + λ G G G λ TG T T G G, (.5) 38

40 which is equaion (1) in he paper, where λ W C κ C, λw G κ G ρg + η (1 ), λ W CG η κ G, ρg + η (1 ) λ T κ T, λ G κ G G. Opimal commimen policies wih equal rigidiies η, λ TG κ G. Wih equal rigidiies, (D.6) and (D.7) become, respecively: π = βe π +1 +(1 n) κ T T T + κ C C C + κ G G G + u,(g.1) π = βe π +1 nκ T T T + κ C C C + κ G G G + u. (G.) To solve for he opimal policies under commimen we se up he relevan Lagrangian (see, e.g., Woodford, 1999): L = E 0 β L S =0 +φ 1, π βπ +1 κ T (1 n) T T κ C C C κ G G G u +φ, π βπ +1 + κ T n T T κ C C C κ G G G u +φ 3, T T T 1 T 1 π + π + T T 1, where φ 1,, φ,,andφ 3, are he mulipliers on (G.1), (G.), and (D.8), respecively, and L S is given by (.1). Opimizing over C C, T T, π, π, G G,andG G 39

41 yields he following six necessary firs-order condiions for 1, λ C C C + λ CG G W G W φ 1, κ C φ, κ C = 0, (G.3) λ T T T λ TG G G φ 1, κ T (1 n)+φ, κ T n + φ 3, βφ 3,+1 = 0, (G.4) λ π π + φ 1, φ 1, 1 + φ 3, = 0, (G.5) λ π π + φ, φ, 1 φ 3, = 0, (G.6) nλ G G G + nλ C CG C + λ T TG T φ 1, κ G = 0, (G.7) (1 n) λ G G G +(1 n) λ CG C C λ TG T T φ, κ G = 0. (G.8) Use he values of he loss funcion parameers o ge (κ C /) C C +(κ G /) G W G W φ 1, + φ, κc =0, (κ T n (1 n) /) T T (κ G n (1 n) /) G G φ 1, κ T (1 n)+φ, κ T n + φ 3, βφ 3,+1 =0, nπ + φ 1, φ 1, 1 + φ 3, =0, (1 n) π + φ, φ, 1 φ 3, =0, n κ G ρg /η +(1 ) / G G + n (κ G /) C C +(κ G n (1 n) /) T T φ 1, κ G =0, (1 n) κ G ρg /η +(1 ) / G G (κ G n (1 n) /) T T φ, κ G =0. +(1 n)(κ G /) C C 40

42 ence, ( /) C C κg + / G W G κ W C φ 1, + φ, =0, (G.9) (κ T n (1 n) /) T T (κ G n (1 n) /) G G φ 1, κ T (1 n)+φ, κ T n + φ 3, βφ 3,+1 =0, (G.10) nπ + φ 1, φ 1, 1 + φ 3, =0, (G.11) (1 n) π + φ, φ, 1 φ 3, =0, (G.1) n ρ g /η +(1 ) / G G + n ( /) C C +(n (1 n) /) T T φ 1, =0, (G.13) (1 n) ρ g /η +(1 ) / G G (n (1 n) /) T T φ, =0. +(1 n)( /) C C (G.14) Adding he las wo condiions gives ρg /η +(1 ) / G W G W +( /) C C = φ 1, + φ,. Combine his wih he firs equaion: ( /) C C κg + / G W κ C G W = φ 1, + φ,, o ge ρg /η +(1 ) / G W = ( /) C C κg + / κ C G W G W +( /) C C G W from which i follows ha ρg /η +(1 ) G W G W = G W G W =0, κg G W G κ W C, 41

43 unless ρg /η +(1 ) = κ G κ C ρ g /η +(1 )= η (1 ) (ρ + η ) ρ g (ρ + η ) /η +(1 )(ρ + η )=η(1 ) ρ g ρ/η + ρ g +(1 )(ρ + η η )=0 ρ g /η + ρ g /ρ +(1 )=0, which is never he case. I.e., world governmen spending gap is closed under he opimal plan. Adding he hird and he fourh equaion yields π W + φ 1, + φ, φ1, 1 + φ, 1 =0, and, herefore, by (G.9) π W = ( /) C C C 1 C 1. We now urn o he characerizaion of relaive variables. The equaions (G.13) and (G.14) can be rearranged o (by muliplying he firs by (1 n) and muliplying he second by n and hen subracing he firs from he second) (1 n) n ρ g /η +(1 ) / G G (n (1 n) /) T T nφ, +(1 n) φ 1, =0. (G.15) rom he inflaion equaions (G.11) and (G.1) we ge (1 n) nπ + n φ, φ, 1 (1 n) φ1, φ 1, 1 φ3, =0 (1 n) nπ + nφ, (1 n) φ 1, nφ, 1 +(1 n) φ 1, 1 φ 3, =0. Therefore, (1 n) nπ +(1 n) n ρ g /η +(1 ) / G G (n (1 n) /) T T (1 n) n ρ g /η +(1 ) / G 1 G 1 +(n (1 n) /) T 1 T 1 = φ 3,, 4

44 (1 n) nπ +(1 n) n ρ g /η +(1 ) / G G (n (1 n) /) T T T 1 T 1 = φ 3,, G 1 G 1 π + ρ g /η +(1 ) / G G G 1 G 1 ( /) T T T 1 T 1 = φ 3,, or, by use of (D.8), π + ρ g /η +(1 ) / G G ( /) π T T 1 = φ 3,, G 1 G 1 which becomes π (1 /)+ ρ g /η +(1 ) / G +( /) T T 1 = φ 3,. G G 1 G 1 Now examine (κ T n (1 n) /) T T (κ G n (1 n) /) G G φ 1, κ T (1 n)+φ, κ T n + φ 3, βφ 3,+1 =0 (n (1 n) /) T κg T n (1 n) / G G κ T φ 1, (1 n)+φ, n + φ 3, βφ 3,+1 κ T =0. We find nφ, (1 n) φ 1, from (G.15) (1 n) n ρ g /η +(1 ) / G G = nφ, (1 n) φ 1,, (n (1 n) /) T T 43

45 o ge: = 0 (n (1 n) /) T T κg n (1 n) / κ T +(1 n) n ρ g /η +(1 ) / G G (n (1 n) /) T T + φ 3, βφ 3,+1 κ T G G ρg (1 n) n /η +(1 ) / κ G ( /) G G + φ 3, βφ 3,+1 =0. κ T κ T Using ha we ge and hen ence, (1 n) n κ G = η (1 ) κ T (1 + η ), ρ g /η +(1 ) η (1 c ) c G G (1 + η ) + φ 3, βφ 3,+1 =0, κ T φ 3, = βφ 3,+1 κ T (1 n) n φ 3, = κ T (1 n) n ρg /η + ρ g +(1 ) (1 + η ) G G, ρg /η + ρ g +(1 ) β i G +i G +i. 1+η i=0 To sum up, we have π W G W G W =0, = ( /) C C C 1 C 1, π (1 /)+ ρ g /η +(1 ) / G +( /) T T 1 = φ 3,, G G 1 G 1 φ 3, = κ T (1 n) n ρg /η + ρ g +(1 ) β i G +i 1+ηc G +i, which is he sysem (16)-(19) in he paper. Togeher wih he Phillips curves he sysem deermines he six endogenous variables C C, G G, G G, π, π 44 i=0

46 and φ 3,.. Opimal policies under discreion and equal rigidiies We observe ha (G.1) and (G.) can resaed in erms of world and relaive variables exclusively: π W = βe π W +1 + κ C C C + κ G G W G W + u W, (.1) π = βe π +1 κ T T T + κ G G G + u. (.) The problem is o minimize he sream of L S as given by (.3), subjec o he consrains (.1), (.) and (D.8). Since he nominal ineres rae can be adjused freely a no loss, we do no rea equaion (D.3) as a consrain, bu assume ha he consumpion gap is reaed as he moneary policy insrumen direcly, which ogeher wih he world governmen spending gap and he relaive governmen spending gap forms he full se of policy insrumens. aving realized his, par of he discreionary opimizaion becomes simple; namely he choice of world consumpion and world governmen spending. Noice ha hese variables do no affec he relaive inflaion rae, and nor do hey affec he erms of rade direcly; cf. (.) and (D.8). Equally imporan, he variables ener he loss funcion addiively separable from he erms of rade and relaive governmen spending. ence, he opimal choice of he consumpion gap and world governmen spending gap can be cas as a problem of minimizing he discouned sum of L S as given by (.3), aking as given he pah of relaive inflaion raes and he erms of rade, subjec o (.1). This can be labelled as he world par of he problem. One can hen independenly of his deermine he opimal relaive spending gap as he one ha minimizes he discouned sum of L S given by (.3), aking as given he pah of world governmen spending, he world inflaion rae and he consumpion gap, and where he minimizaion is subjec o (.) and (D.8). This can be labelled as he relaive par of he problem. We now urn o solving hese wo pars..1. The world par The world par of he problem reduces o a sequence of saic opimizaion problems of he form min ( C C ),( G W G W ) L s.. (.1) (.3) 45

47 aking as given E π W +1, as he period consumpion gap or governmen spending gap have no dynamic implicaions. Subsiue (.1) ino (.3). Then, he necessary and sufficien firs-order condiions o (.3) are: (κ C /) C C + κ C π W +(κ G /) G W G W =0 (.4) κg ρg /η +(1 ) / G W G W + κ G π W +(κ G /) C C =0 (.5) educing hese equaions slighly, reveals he following: ( /) C C + π W κg c + G W G κ C W ρg /η +(1 ) / G W G W + π W +( /) C C = 0 = 0 ence, world governmen spending follows as G W G W =0, which is he equaion preceding equaion (0) in he main ex, and, hence, π W = C C, which is equaion (0) of he main ex... The relaive par The relaive par of he discreionary opimizaion problem involves, as menioned, he choice of relaive governmen spending. or his purpose, i is imporan o acknowledge ha his choice only affec he relaive inflaion rae and he erms of rade. As hese erms ener addiively in (.3) (and relaive governmen spending only eners muliplicaively wih he erms of rade), he problem reduces o one of minimizing he discouned sum The generaliy of his soluion requires ha which is easy o confirm. κ G ξ c κ C = ρ g /η +(1 ξ c ) /, 46

48 of L = (κ T n (1 n) /) T T analogous o +n (1 n) κ G ρg /η +(1 ) / G G (κ G n (1 n) /) T T G G, + n (1 n) π L = (κ T /) T T + κg ρg /η +(1 ) / G G + π (κg /) T T G G subjec o (.) and (D.8). This problem does no correspond o a sequence of one-period problems, as he choice of G G affecs π, and hus T T wih direc loss implicaions hrough he nex period s erms of rade (by he dynamics of (D.8)]. The period problem is herefore solved by dynamic programming wih pas period s erms-of-rade gap as he sae variable. I.e., he problem is characerized by he recursion V T 1 T 1 (κ T /) T T + κg ρg /η +(1 ) / G G = min ( G G ) E 1 + π (κg /) T T G G + βv T T, where V is he value funcion, and where he minimizaion is subjec o (.) and (D.8). Now, combine hese consrains o π = βe π +1 κ T 1 T T 1 + π T T 1 +κ G G G + u. To proceed wih he soluion, assume ha he relevan driving variables of he sysem of relaive variables, T T 1 and u, boh follow A(1) processes. 3 Thereforeweconjecure ha he soluion o he relaive variables will be linear funcions of he sae and driving 3 As T T 1 and u boh are linear funcions of he underlying naional shocks (produciviy and mark-up shocks, respecively), we could also have assumed ha hese shocks followed A(1) (or more general) processes and formulaed he conjecure in erms of he sae and all hese shocks. This, however, would make he exposiion more messy, wihou affecing he characerizaion of opimal relaive spending gaps we presen in he main ex; cf. below. 47

49 variables. I.e., we conjecure ha π = b 1 T 1 T 1 + b T T 1 + b 3 u, (.6) where b 1, b and b 3 are unknown coefficiens o be deermined. By use of (.6) one obains relaive inflaion as π = b 1 β T T + βe b T +1 T κ T T 1 T 1 + π + b 3 u +1 T T 1 + κ G G G + u, π = b 1 β T 1 T 1 + π T T 1 + βe b T+1 T + b 3 u +1 κ T T 1 T 1 + π T T 1 + κ G G G + u, π (1 + b 1β + κ T ) = (b 1 β + κ T ) T 1 T 1 +κ G G G T T 1 + u + βe b T+1 T + b 3 u +1, π = b 1β + κ T T 1 1+b 1 β + κ T 1 T T 1 T κ G + G G 1+b 1 β + κ u + βe b T+1 T + b 3 u +1 +, T 1+b 1 β + κ T (.7) and he erms-of-rade gap as T T = T 1 T 1 b 1β + κ T T 1 1+b 1 β + κ T 1 T T 1 T κ u G + G G + βe b T+1 T + b 3 u b 1 β + κ T 1+b 1 β + κ T T T 1, 48

50 and, hus, T T = 1 T 1 1+b 1 β + κ T 1 T T 1 (.8) T κ G + G 1+b 1 β + κ G u + βe b T+1 T + b 3 u T 1+b 1 β + κ T One can hen inser (.7) and (.8) ino he value funcion and obain an unconsrained minimizaion problem. The firs-order condiion for opimal G G is T T (κ T /) T T G G + κ G ρg /η +(1 ) / G T T + π G π G (κ G /) (κ G /) G G G = 0, G T T G + 1 T T βv T T G G or, = 0. (κ T /) T T + π κ G + κ G ρg /η +(1 ) / G G 1+b 1 β + κ T T T (κ G /) G G κ G 1+b 1 β + κ T κ G 1+b 1 β + κ T (κ G /) + 1 βv T T κ G 1+b 1 β + κ T (.9) 49

51 Differeniaing he value funcion wih respec o T 1 T 1 yields: 1 V T 1 T 1 = (κ T /) T T T T T 1 T 1 + κ G ρg /η +(1 ) / G G G G T 1 T 1 + π π (κ G /) T G G T T 1 T 1 T 1 T 1 T T (κ G /) G G T 1 T T T βv T T. (.10) T 1 T 1 ByheEnvelopeTheorem,weeliminaeallermsinvolving G G / T 1 T 1 [he explici ones and hose implicily appearing in T T / T 1 T 1 and π / T 1 T 1 ]oge: 1 V T 1 T 1 = (κ T /) T T 1 π b 1 β + κ T 1+b 1 β + κ T 1+b 1 β + κ T (κ G /) G G 1 1+b 1 β + κ T + 1 βv T 1 T. (.11) 1+b 1 β + κ T Muliply on boh sides by κ G o ge κ G V T 1 T 1 = (κ T /) T κ G T π 1+b 1 β + κ κ b 1 β + κ T G T 1+b 1 β + κ T (κ G /) G G κ G 1+b 1 β + κ T + 1 βv T κ G T, 1+b 1 β + κ T 50

52 and add his o (.9): (κ T /) T T κ G + κ G ρg /η +(1 ) / G 1+b 1 β + κ G + T π κ G (κ G /) T 1+b 1 β + κ T (κ G /) G G κ G T 1+b 1 β + κ T + 1 βv T T κ G + κ G 1+b 1 β + κ T V T 1 T 1 = (κ T /) T T κ G π b 1 β + κ T κ G 1+b 1 β + κ T 1+b 1 β + κ T (κ G /) G G κ G b 1 β + κ T βv T T κ G, 1+b 1 β + κ T which simplifies o from which one ges κg ρg /η +(1 ) / G G + π κ G 1+b 1 β + κ T (κ G /) T T + κ G V T 1 T 1 = π κ G b 1 β + κ T 1+b 1 β + κ T, 1 V T 1 T 1 =( /) T T ρ g /η +(1 ) / G G π orward his one period, and use i in (.9) o eliminae he derivaive of he value funcion: (κ T /) T T κ G + κ G ρg /η +(1 ) / G 1+b 1 β + κ G T + π κ G (κ G /) T 1+b 1 β + κ T (κ G /) G G κ G T 1+b 1 β + κ T βκ G + ( /) E T +1 T +1 ρ 1+b 1 β + κ g /η +(1 ) / E G +1 G +1 E π +1 T = 0, and hus κ T T 1+b 1 β + κ T + ρ g /η +(1 ) G G T + π c T 1+b 1 β + κ κ G T G T 1+b 1 β + κ G T β + E T+1 1+b 1 β + κ T +1 ρ g /η +(1 ) E G +1 G +1 E π +1 T = 0, 51

53 or, (1 + b 1 β) T 1+b 1 β + κ κ G T + µ G T 1+b 1 β + κ G + π T 1+b 1 β + κ T β + E T+1 1+b 1 β + κ T +1 µe G +1 G +1 E π +1 T = 0, wih µ ρ g /η +(1 ). This is furher reduced o (1 + b 1 β) T T +(µ [1 + b 1 β + κ T ] κ G ) G G +β E T+1 T +1 µe G +1 G +1 E π +1 = 0. + π This equaion is equaion (1) of he main ex, and will ogeher wih (.7) and (.8) provide soluions for he pahs for G G, T T and π as funcions of he sae and T T 1 and u. Given he assumpion abou he sochasic properies of T T 1 and u he soluion can be characerized by he mehod of underermined coefficiens. The coefficiens found in his sep will be funcions of he unknown parameers b 1, b and b 3. These are hen finally idenified by equaing he coefficiens in he soluion for π wih hose in he conjecure. Noe ha indeed only he undeermined coefficien o he sae variable appears in he characerizaion of he soluion of he sysem of relaive variables as given by equaion (1). ence, had we replaced (.6), by a linear conjecure which depended on he sae and he underlying shocks [and assumed ha hese shocks were A(1) or more general processes], we would have arrived a he same characerizaion of opimal relaive spending gaps as equaion (1) of he main ex. The reason is ha he impac of governmen spending changes on he inflaion differenial and he erms-of-rade-gap only depends on he undeermined coefficien on he sae variable. The coefficiens on he shocks do herefore no affec he firs-order condiion or he envelope condiion [see equaions (.9) and (.11)]. We can herefore wihou loss of analyical generaliy arrive a equaion (1) wih our parsimonious conjecure (.6) as claimed in oonoe 3. 5

54 I. Opimal moneary policy wih consrained fiscal policy under equal rigidiies or convenience, we wrie (G.1) and (G.) ou as π = βe π +1 + κ (1 n)(1+η ) T T + κ (ρ + η ) C C + u, (I.1) π = βe π +1 κn (1 + η ) T T + κ (ρ + η ) C C + u. (I.) The loss funcion is given by (.1) wih he coefficiens given by (because rigidiies equal in he wo counries): λ C κ (ρ + η ) /, λ T κn (1 n) (1 + η ) /, λ G κ (1 ) ρ g + η (1 ) /, λ CG = κ (1 ) η/, λ TG κn (1 n) η (1 ) /, λ π n, λ π =1 n. I.1. Characerizaion of opimal policies under precommimen To solve for he opimal policies under commimen we se up he relevan Lagrangian (see, e.g., Woodford, 1999): L = E 0 β L S =0 +φ 1, π βπ +1 κ (1 n)(1+η ) T T κ (ρ + η ) C C u +φ, π βπ +1 + κn (1 + η ) T T κ (ρ + η ) C C u +φ T 3, T T 1 T 1 π + π + T T 1, where φ 1,, φ,,andφ 3, are he mulipliers on (I.1), (I.), and (D.8), respecively. Opimizing over C W C W, T T, π and π, yields he following four necessary firsorder condiions for 1, λ C C C φ 1, κ (ρ + η ) φ, κ (ρ + η ) = 0, (I.3) λ T T T φ 1, κ (1 n)(1+η )+φ, κn (1 + η )+φ 3, βφ 3,+1 = 0, (I.4) nπ + φ 1, φ 1, 1 + φ 3, = 0, (I.5) (1 n) π + φ, φ, 1 φ 3, = 0. (I.6) 53