The International Diversification Puzzle when Goods Prices are Sticky: It s Really about Exchange-Rate Hedging, not Equity Portfolios
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1 The Inenaional Diveificaion Puzzle when Good Pice ae Sicky: I eally abou Exchange-ae edging, no Equiy Pofolio by CALES ENGEL AND AKITO MATSUMOTO Appendix A. Soluion of he Dynamic Model An equilibium aifie he fi ode condiion, budge conain and make cleaing condiion. Fi we define an equilibium fomally. Then we will li he lineaized fi ode condiion and edefine equilibium in lineaized fom. Definiion A An equilibium i a e of equence { C, L,, δ, γ, Ch., Cf., Ch. ( i, Cf. ( i, Pflex, h,, P, P, P, P, P, P, Q, V,,,, Π, γ, γ } flex, f, pee, h, pee, h, h, f, h, f, = and hei foeign counepa and { S, F}, which olve he yem of 5 equaion coniing of (8, (, (, (43, (44, (47-(6, and hei foeign counepa plu 3 ae make cleaing condiion, 3 given ochaic equence and iniial condiion = A, M γ =. { A, A, M, M} A = M, γ =, and A. Appoximaed Syem In hi ecion, we deive a log-linea veion of he model, unde he aumpion ha he ochaic diving vaiable (poduciviy and money ae lognomally diibued. Many of he equaion of he model ae linea in log (wihou any appoximaion. Bu ome of he equaion in he model (he budge conain fo houehold, he definiion of pofi fo he fim, and he make cleaing condiion ae log-lineaized aound uncondiional mean. I i immediaely appaen ha Engel: Depamen of Economic, Univeiy of iconin, 8 Obevaoy Dive. Madion, I 5376 ( cengel@c.wic.edu; Maumoo: eeach Depamen, Inenaional Moneay Fund, 7 9h See, N.. ahingon, DC 43 ( amaumoo@imf.og. e ae vey gaeful o Mak Gele, Olivie Jeanne, Kaen Lewi, Mauy Obfeld, Paolo Peeni, Cedic Tille, Eic van incoop, Fank anock, wo anonymou efeee, he edio, Olivie Blanchad, and emina paicipan a eveal iniuion fo helpful commen. Engel acknowledge he uppo of he Naional Science Foundaion hough a gan o he Univeiy of iconin. The view expeed in hi pape ae hoe of he auho and do no neceaily epeen hoe of he Fedeal eeve Syem, he IMF o IMF policy. Ealie daf of hi pape wee ciculaed unde he ile ome Bia in Equiie unde New Open Economy Macoeconomic, Pofolio Choice and ome Bia in Equiie in a Moneay Open-Economy DSGE Model., and Pofolio Choice and Inenaional ik Shaing in a Moneay Open-Economy DSGE Model. Thee ae 4 vaiable. The numbe of equaion hould be 5, bu one i edundan by ala Law. 3 γ γ =, γ γ =, and Fδ = δ. h, h, f, f,
2 ou aumpion of aionay poduciviy pocee and uni-oo moneay pocee imply ha nominal vaiable have uni oo and eal vaiable ae aionay. So we log-lineaize aound he uncondiional mean of he log of eal vaiable. 4 In ome of he log-lineaized equaion below, he algeba i implified conideably if we ue he eul ha ph p =. (In ou noaion, x epeen he uncondiional mean of x. hile we could poceed wih he deivaion wihou uing hi eul, and hen veify in he oluion ha hi eul i ue, i i eaie o demonae hi fi and ue i in ome of he log-lineaizaion. Fi, in he definiion of pofi fo he home fim, divide boh ide of equaion (6 by, hen evaluae he equaion a he poin of expanion fo he log-lineaziaion: ( h ( h exp( π p = exp( c exp ( ω( p p exp ( ω( p p exp( w p l. P ee we have ued ymmey o give u c = c and p p =. Divide he budge conain (4 by P, hen evaluae he equaion a he poin of expanion fo he log-lineaizaion: exp( π p = exp( c exp( w pl. In deiving hi expeion, we have ued ymmey o give u q q =, π π =, and f =. e have alo ued γ, γ, = and M = M T. f h Now compaing he wo equaion we have deived, we mu have ( ω ph p ( ω ph p exp ( ( exp ( ( =. Thi can be wien a ( ω ph p ( ω ph p exp ( ( exp ( ( =, whee we have ued ymmey o give u ha p p = p p, and lineaized (8 o ge ph p=( pf p. I hen follow ha ph p =, which i he eul we will ue below o implify ome of he log-lineaizaion. h A few moe noaional convenion: e denoe x a he deviaion fom he condiional mean ha i, x x E x and Ex ln ln = E x E x vaiable a x w x x and he elaive vaiable a x x x. f. e will alo denoe he wold 4 e could eaily accommodae uni-oo pocee in poduciviy. Then eal vaiable expeed in efficiency uni would be aionay. oweve, hee i no eal gain fom hi genealizaion, o we mainain aionay poduciviy hock o implify he algeba.
3 A.. The fi ode condiion fo houehold Suppeing conan em and aking log, he fi ode condiion fo conumpion (54 can be wien a c = ( m p. (A. ρ Uing equaion (A., equaion (43 can be expeed a l = m w. (A. Some of he equaion of he model ae log-linea (uch a (A. and (A., and heefoe, in he peence of lognomal diibuion, offe exac oluion. Bu ohe (uch a he budge conain, he make cleaing condiion, and he expeion fo a fim pofi equie appoximaion. Becaue all hock ae lognomal, he oluion of he appoximaed model will ake on a lognomal diibuion. e can ue equaion (54 o expe (44 a E ( va ( cov ( m, = f, (A.3 E ( ( m m cov ( m, va ( va ( m = (A.4 E ( ( m m va ( va ( m va ( cov ( m, cov (, cov ( m, = (A.5 A.. The budge conain e log-lineaize he budge conain (53 o ge p c ( ζ v ζh = ( ζ v γ( ζ( h δ( f (A.6 exp( w p l δf ee, ζ, and δ exp( m pc. In deiving hi expeion, we exp( c M have ued he fac ha by ymmey, v p= q p, and hen ue equaion (47 o deive exp( q p = exp( π p. Similaly, fom equaion (49, we ge exp( h p = exp( w p l. Then, evaluaing he budge conain a he poin of expanion, we have exp( c = exp( w p l exp( π. 3
4 A..3 The fi ode condiion fo fim Fim e hei pice opimally. The fi ode condiion can be wien a p = w a (A.7 flex, h,, p = w a, (A.8 flex, f, ( ( λ ω ω ppee, h, = E ( w a va ( w a cov w a, d ( ph p c (A.9 ppee, h, = E ( w a va ( w a va ( ( w a d λ ω p ωp c cov, ( h (A. Noe ha he condiional econd momen in (A.9 and (A. ae all conan ove ime, and will be eaed a conan em in ubequen lineaizaion. Thu, he pice of each caegoy of good (59 and 6 can be expeed a following: ph, = τ ppee, h, ( τ pflex, h,, (A. pf, = τ ppee, f, ( τ pflex, f,. (A. Combining hee wo and uppeing he conan, we ge he expeion fo pice index: =. (A.3 p ph, p f, A..4 Good make cleaing The good make cleaing condiion, equaion (6 can be lineaized a l = ω( ph, p c ω( ph, p c a. (A.4 A..5 Ohe definiion u In ewiing he budge conain (53, we inoduced human capial. Lineaizing (49 give h = E w. (A.5 ( = l wie Uing he definiion of in equaion (5, and he oluion fo ( = ( = = ( E π ( E π = ( ( E = Q in equaion (47, we can π. (A.6 The log of home fim pofi come fom lineaizing (6: 4
5 π = c p ( ω( ph, p ( ω( ph, p ( w l ζ ζ. Similaly, = = ( E ( w l. (A.7 A. Definiion of Appoximaed Equilibium Definiion B An appoximaed equilibium i a e of equence { c, l, w,,, δ, γ, p, v, h} and hei foeign counepa, and {, f } ha olve he yem of equaion (A.-(A.6, (A.4-(A.7, and hei foeign counepa, given equence {, m m, a, } and iniial condiion a =, m =, and γ = γ = a. An appoximaed equilibium i a educed fom of Definiion A. Mo omied pa can be eaily veified and hould no be confuing. e peen he oluion fo x and x in he fom of oluion fo x and x o faciliae he demonaion ha hee aify he equilibium condiion. A.3 Equilibium Allocaion e conjecue ha he following allocaion i an equilibium. l ω( τ ωτ = a E ω( τ ω( τ ω ω( τ ωτ = a ϑa ω( τ ω( τ ω ρ ( l = ( τ ρ a τm τe a ρ ( τ ρ ρ ( = ( τ ρ a τm τ ϑ a m ρ ( τ ρ a m (A.8 (A.9 w ( τ( ω τ ωτ = a ϑ a ω( τ ω( τ ω m (A. ρ( ρ w = ( τ ρ a τ ϑa m ρ ( τ ρ m (A. ρ ( τ 5
6 p = τm ( τ m (A. p ρτ ρ ( ρ ρ ( = ϑ a m ( τ a m ρ ( τ ρ ρ ( τ ρ (A.3 ( c = τ m m (A.4 ρ ( c τ ρ ϑa ( m m ( τ = ρ ( τ ρ a (A.5 ρ ( τ = m. (A.6 f = (A.7 m. ( τ( ω τ τ ω ϑ = ( ( a m ω( τ ζ ω( τ ω ϑ (A.8 ( τ( ρ ζ τρ ρ ϑ = ( ( ρ ( τ ζ ρ ( τ ρ ϑ ζ ( ( τ ( ( ρ τ m ζ ρ ( τ ζ ρ ( τ a (A.9 ( τ( ω τ ω ϑ = ( ( a m (A.3 ω( τ ω ϑ τ ρ ρ ϑ ( ( τ = ( ( a m ρ ( τ ρ ϑ ρ ( τ (A.3 δ δ = ( τ (A.3 ρ τ ϑ ( ω ω ( τ ω ϑ γ γ = γ = τζ τ ϑ ( ζ( ω ω( τ ω( τ ω ϑ (A.33 h h ω ϑ = ( ω ϑ a m (A.34 ρ ϑw = ( a m (A.35 ρ ϑw v ζ ω ϑ = ( ζ ω ϑ a m (A.36 6
7 aded: v ρ ϑ = ( a m ρ ϑ (A.37 Noice ha hi allocaion eplicae he allocaion when a full e of ae-coningen bond i ρ( c c = p p. (A.38 A.4 Poof e will how hi allocaion aifie he equilibium condiion. A.4. Fundamenal Vaiable e now pove ha he fi ode condiion fo fundamenal vaiable and labo make cleaing condiion ae in fac aified. I i immediae o confim ha equaion (A.8 (A. aify equaion (A.. Likewie i i aighfowad o check ha (A. (A.5 aify (A.. e can alo veify ha (A.8, (A. and (A.6 aify he elaive veion of he labo make cleaing condiion (A.4: l ( ( ( = τωw a τωe w a a. (A.39 I i ediou bu aighfowad o veify ha (A.9 and (A. aify he wold veion of labo make cleaing condiion (A.4: l = c a. (A.4 Uing equaion (A. and (A.3, and uing (A. and (A.6, we can how p = E w a w a (A.4 ( ( ( τ τ p = τ E ( τ (A.4 ae aified. Noe ha he vaiance and covaiance em in (A.9 and (A. ae conan, fom he oluion above. Subiuing equaion (A.7 (A. ino (A.3, and uppeing conan em, we ee ha (A.37 and (A.38 ae he oluion o he wold and elaive veion of (A.3. So fa, we have poved equaion (A., (A., (A.3, and (A.4 ae aified. A.4. eun on ae In ode o how ha hi allocaion in fac aifie he fi ode condiion fo ae holding, we wan o calculae he ae of eun on ae human capial and equiie. Since w l = ( ( l l m m, he eun on he human capial i 7
8 E l l m m ( = ( ( = = ( ( ( τ ρ a τm ρ ( τ ρ ϑ ( τ( ω τ ω ϑ ( ( a a ρ ϑ ω( τ ω ϑ (A.43 ( m m. Subacing he foeign counepa, we ge equaion (A.3. Adding he foeign counepa give u he oluion o. Following imila ep a in he eun on human capial, we ge he eun on equiy: ( τ( ρ τρζ ρ ϑ = ( ( a ρ ( τ ζ ρ ( τ ρ ϑ ( τ( ω τ τ ω ϑ a ω( τ ζ ω( τ ω ϑ ζ ( ( τ ( ( ρ τ m ζ ρ ( τ ζ ρ ( τ m (A.44 Subacing he foeign counepa, we ge (A.8, and adding he foeign counepa give u ha (A.9 i he oluion fo. So, we have confimed (A.6 and (A.7. A.4.3 Ae Allocaion Since we eplicae complee make, hee allocaion hould aify he fi ode condiion fo he ae allocaion a expeed in equaion (44 and (55. e will pove ha lineaized veion of hem (A.3 (A.4 ae aified. Fom (A.6 and (A.7, we ee f = E. So, fo equaion (A.3 o be aified, we need which follow ince = m. cov ( m, va (, = (A.45 Since fom (A.8 and (A.9, i i.i.d., we have E ( ( m m i conan. Likewie, uing (A.6, E m m ( ( i conan. e can olve diecly fo hee expecaion fom equaion (A.4 and (A.5, uing he covaiance and vaiance implied by ou oluion in (A.8 (A.33. Bu he following eicion link (A.4 and (A.5: 8
9 cov ( m, va ( = cov ( m, va (. (A.46 e veify hi by uing =, and ewie (A.46 a cov ( m m, m va ( m va ( =. (A.47 e uilize ohogonaliy beween wold hock and elaive hock o implify he fi em: cov ( m m, m = cov ( m, m. (A.48 The econd and hid em can be expeed a va ( m va ( = va ( m cov ( m, (A.49 cov (, = m m e confim ha hi allocaion in fac aifie he fi ode condiion fo ae allocaion. So (A.3 (A.5 ae aified. A4.4 uman ealh To veify ha (A.34 and (A.35 povide he oluion fo human wealh (A.5, we ue (A.8 (A. o wie h E ( w l = = = ( l l m m = (A.5 ρ ω = ( a a m m ϑ ϑ = ρ ω ρ ϑ ω ϑ = ( a a m m ρ ϑ ω ϑ Then ubacing he foeign counepa of (A.5, we ge (A.34, and adding he foeign counepa give u (A.35. A.4.5 Budge Conain Fi, wold budge conain expeed in home cuency i he following: p c v h v h {( ζ ζ } = ( ζ( ζ( (A.5 9
10 whee we have ued γ = γ. e have alo ued δ ( f δ ( f =, which equie δ = δ. Thi equie ome explanaion. The home cuency eaning, expeed in home cuency, fom he fowad make ae δ ( S F. Tha mean ha he foeign cuency eaning fo he foeign couny ae F δ S, which can be wien a δf. So, he foeign budge S F conain, ymmeically o he home budge conain, will conain he em δ, whee S F = F. Uing hi elaionhip, we can eablih δ δ δ δ F m p c m p c δ = e = e δ FM M =, (A.5 whee we have ued (A.7, and m p = m p and c = c. The wold budge conain hold wih any ealizaion of and ince equaion (A.5 imply indicae ha oal wold wealh caied ove ino he nex peiod i equal o he value of peviou wealh, plu eun, le wold conumpion. Moe explicily, becaue a m v h E w l E p c = ( π = ( = =, (A.53 boh ide of he equaion ae he um of fuue conumpion. Finally, we examine elaive budge conain: p c ( ζ v ζh = ( ζ v ( γ γ ( ζ( h δ down ino ep. (A.54 Diec ubiuion fom he oluion veifie hi equaion, bu i i helpful o beak hi Uing γ = γ = γ, and he oluion fo, ( τm ( ζ v ζh m ρ c p, and, we can wie = ( ζ ( ] ( m γ m ζ m m δ (A.55 ( ζ v ζh m Uing elaive eun (A.8 (A.3, we ge
11 ( τ δ m ( ζ v ζh m ρ ( τ( ω τ τ ω ϑ = ( γ ( ( ζ a ω( τ ζ ω( τ ω ϑ ( τ( ω τ ω ϑ ζ ( a ω( τ ω ϑ [( ζ v ζh m ] (A.56 By ubiuing expeion fo δ and γ fom (A.3 and (A.33. ino (A.56, we ge. (A.57 ( ζ v ζh m = ( ζ v ζh m Bu (A.34 and (A.36 give u o (A.57 hold. ( ζ v ζh m =, (A.58 e have veified ha equaion (A.-(A.6 and (A.4-(A.7 ae aified.
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