Appendix: No for Publicaion Currency Misalignmens and Opimal Moneary Policy: A eexaminaion Charles Engel Universiy of isconsin July 8,
Appendix A Model Equaions Aa Households The represenaive household in he home counry maximizes σ + φ (A U( h =Ε β C+ ( h N+ ( h = σ + φ, σ >, φ C ( h is he consumpion aggregae e assume Cobb-ouglas preferences: ν ν ( ( ( H F (A ( ( C h = C h C h, ν In urn, CH ( h and CF ( h are CES aggregaes over a coninuum of goods produced in each counry: ξ ξ ξ ξ = H ξ ξ ξ ξ (A CH ( h C ( h, f df and C ( (, F h = CF h f df N ( h is an aggregae of he labor services ha he household sells o each of a coninuum of firms locaed in he home counry: (A4 N( h = N (, h f df Households receive wage income, ( h N( h, aggregae profis from home firms, Γ They pay lump-sum axes each period, T Each household can rade in a complee marke in coningen claims (arbirarily denominaed in he home currency The budge consrain is given by: + + (A5 PC ( h + Z( ( h, = ( h N ( h +Γ T + ( h,, + Ω + where (, h represens household h s payoffs on sae-coningen claims for sae + + Z( is he price of a claim ha pays one dollar in sae, condiional on sae occurring a ime In his equaion, P is he exac price index for consumpion, given by: ν / ( ν / ( ν / ν / (A6 P = k PH PF, k = ( ( ν / ( ν / P H is he Home-currency price of he Home aggregae good and P F is he Home currency price of he Foreign aggregae good Equaion (A6 follows from cos minimizaion Also, from cos minimizaion, P H and P F are he usual CES aggregaes over prices of individual varieies, f: (A7 ( ξ ξ P = P ( f df, and ( P P ( f df H H F = F ξ ξ Foreign households have analogous preferences and face an analogous budge consrain Because all Home households are idenical, we can drop he index for he household and use he fac ha aggregae per capia consumpion of each good is equal o
he consumpion of each good by each household The firs-order condiions for consumpion are given by: ν (A8 PC H H = PC, ν (A9 PC F F = PC, ξ PH ( f (A CH( f = CH P and PF ( f CF ( f = C H PF + σ + (A β C( / C( ( P / P = Z ( ( + In equaion (A, we explicily use an index for he sae a ime for he purpose of clariy Z + ( is he normalized price of he sae coningen claim Tha is, i is + defined as Z( divided by he probabiliy of sae ξ F, + condiional on sae + Noe ha he sum of Z( across all possible saes a ime + mus equal /, where denoes he gross nominal yield on a one-period non-sae-coningen bond Therefore, aking a probabiliy-weighed sum across all saes of equaion (A, we have he familiar Euler equaion: + σ (A β Ε ( C( / C( ( P / P+ = Analogous equaions hold for Foreign households Since coningen claims are (arbirarily denominaed in Home currency, he firs-order condiion for Foreign households ha is analogous o equaion (A is: + σ + (A β C ( / C ( ( E P / E P = Z ( ( + + Here, E refers o he home currency price of foreign currency exchange rae As noed above, we will assume a his sage ha labor inpu of all households is he same, so N = N( h Ab Firms Each Home good, Y ( f is made according o a producion funcion ha is linear in he labor inpu These are given by: (A4 Y( f = AN ( f Noe ha he produciviy shock, N ( f (A5 A, is common o all firms in he Home counry is a CES composie of individual home-counry household labor, given by: η η η η N( f = N( h, f dh, e offer an apology o he reader here e wan o sick o CGG s noaion, who use for he log of he nominal exchange rae Consisency requires us o use rae, so we have used he disinc bu similar noaion E o refer o he level of he nominal exchange Ε o be he condiional expecaion operaor e
where he echnology parameer, η, is sochasic and common o all Home firms Profis are given by: (A6 Γ ( f = PH ( f CH ( f + E PH ( f CH ( f ( τ N ( f In his equaion, PH ( f is he home-currency price of he good when i is sold in he Home counry P H ( f is he foreign-currency price of he good when i is sold in he Foreign counry C ( f is aggregae sales of he good in he home counry: H H (A7 C ( f = C ( h, f dh H CH ( f is defined analogously I follows ha Y( f = CH( f + CH( f There are analogous equaions for Y ( f, wih he foreign produciviy shock given by A, he foreign echnology parameer shock given by η, and foreign subsidy given by τ Ac Equilibrium Goods marke clearing condiions in he Home and Foreign counry are given by: ν PC ν P C ν ( ν / ν ν / (A8 Y = CH + CH = + = k S C + S C PH P ( H, ν PC ν P C ν ( ν / ν ν / (A9 Y = CF + CF = + = k ( S C + S C PF P F e have used S and S o represen he price of impored o locally-produced goods in he Home and Foreign counries, respecively: (A S = P / P, F = H / H (A S P PF Equaions (A and (A give us he familiar condiion ha arises in openeconomy models wih a complee se of sae-coningen claims when PPP does no hold: (A σ C EP EPH / ( / ( S ν S ν C P PH = = Toal employmen is deermined by oupu in each indusry: (A N N( f df A = = Y( f df = A ( CHVH + CHV where ξ PH ( f (A4 VH df, and V PH A Price and age Seing H ξ PH ( f d P f H Households are monopolisic suppliers of heir unique form of labor services Household h faces demand for is labor services given by: η ( h (A5 N( h = N, where H
(A6 ( η ( h dh η = The firs-order condiion for household h s choice of labor supply is given by: ( h σ φ (A7 = ( + μ ( C( h ( N( h, where μ P η The opimal wage se by he household is a ime-varying mark-up over he marginal disuiliy of work (expressed in consumpion unis Because all households are idenical, we have = ( h and N = N( h Since all households are idenical, we have fro m equaion (A7: σ φ ( v/ (A8 / P = ( + μ C N S H e adop he following noaion For any variable K : K is he value under flexible prices K is he value of variables under globally efficien allocaions In oher words, his is he value for variables if prices were flexible, and opimal subsidies o monopolisic suppliers of labor and monopolisic producers of goods were in place This includes a ime-varying subsidy o suppliers of labor o offse he ime-varying mark-up in wages in equaion (A7 Flexible Prices Home firms maximize profis given by equaion (A6, subec o he demand curve ( A They opimally se prices as a mark-up over marginal cos: (A9 P ( ( ( ( P H f = E P H f = τ + μ / A, where P μ ξ hen opimal subsidies are in place: (A PH( f = EPH( f = / A From (A7, (A9, and (A, i is apparen ha he opimal subsidy saisfies P (A ( τ( + μ ( + μ = Noe from (A9 ha all flexible price firms are idenical and se he same price Because he demand funcions of Foreign residens have he same elasiciy of demand for Home goods as Home residens, firms se he same price for sale abroad: (A EP = P H H and EP H = PH From (A, using (A8, we have: (A P H = E P H = /(( +μ A and P /(( F = E P F = + μ A e can conclude: (A4 S = S Because P H ( f is idenical for all firms, (A collapses o (A5 Y = AN PCP 4
e assume a sandard Calvo pricing echnology A given firm may rese is prices wih probabiliy θ each period e assume ha when he firm reses is price, i will be able o rese is prices for sales in boh markes e assume he PCP firm ses a single price in is own currency, so he law of one price holds The firm s obecive is o maximize is value, which is equal o he value a saeconingen prices of is enire sream of dividends Given equaion (A, i is apparen ha he firm ha selecs is price a ime, chooses is rese price, P ( f, o maximize (A6 Ε θ Q, + PH ( f( CH+ ( f + CH+ ( f ( τ + N+ ( f, = subec o he sequence of demand curves given by equaion (A and he corresponding Foreign demand equaion for Home goods In his equaion, we define ( σ (A7 Q, + β C+ / C ( P / P + The soluion for he opimal price for he Home firm for sale in he Home counry is given by: (A8 P H ξ θ Q, + ( τ + PH+ ( CH+ CH+ / = ξ θ Q, + PH+ ( CH+ CH+ = Ε + ξ ( z = ξ Ε + H A + Under he Calvo price seing mechanism, a fracion θ of prices remain unchanged from he previous period From equaion (A7, we can wrie: (A9 ξ ξ /( ξ H = θ( H + ( θ( H P P P, LCP The same environmen as he PCP case holds, wih he sole excepion ha he firm ses is price for expor in he imporer s currency raher han is own currency when i is allowed o rese prices The Home firm, for example, ses P ( f in Foreign currency The firm ha can rese is price a ime chooses is rese prices, P ( f, o maximize H (A4 Ε θ Q, + PH ( z CH+ ( f + E PH CH+ ( f ( τ + N+ ( f = The soluion for (A4 P H P ( z H is given by: Ε ξ ( z = ξ Ε e find for expor prices, ξ θ Q, + ( τ + PH+ CH+ / A + = ξ θ Q, + PH+ CH+ = H P ( f H and 5
(A4 P H Ε ξ ( z = ξ Ε ξ θ Q, + ( τ + ( PH+ CH+ / A+ = ξ θ Q, + E+ ( PH+ CH+ = Equaion (A9 holds in he LCP case as well However, he law of one price does no hold For expor prices, we have: ξ ξ /( ξ (A4 PH = θ( PH + ( θ( PH Subsidies As in CGG, we will assume ha subsidies o monopoliss are no se a heir opimal level excep in seady-sae Tha is, insead of he efficien subsidy given in equaion (A, we have: (A44 ( ( P τ + μ ( + μ = Here, μ is he seady-sae level of μ e have dropped he ime subscrip on he subsidy rae τ because i is no ime-varying 6
Appendix B Log-linearized Model In his secion, we presen log-linear approximaions o he models presened above In his secion, we presen all of he equaions of he log-linearized model, bu we separae ou hose ha are used in he derivaion of he loss funcion (which do no involve price seing or wage seing and hose ha are no Equaions used for derivaion of loss funcions e define he log of he currency misalignmen as he average of he difference beween Foreign and Home prices: (B m ( e + ph ph + e + pf p F In he flexible-price and PCP models, m = e also define he expor premium as he average by which consumer prices of impored goods exceeds he average of locally produced goods: (B z ( ph + pf pf ph In all hree models, o a firs order, ln( VH = ln( VH = ln( VF = ln( VF = Tha allows us o approximae equaion (A and is foreign counerpar as: (B n = y a, and (B4 n = y a The marke-clearing condiions, (A8 and (A9 are approximaed as: ν ν ν ν ν ν (B5 y = c + c + s s, ν ν ν ν ν ν (B6 y = c + c s + s The condiion arising from complee markes ha equaes he marginal uiliy of nominal wealh for Home and Foreign households, equaion (A, is given by: ν ν (B7 σc σc = m + s s e define relaive and world log oupu by: (B8 y ( y y (B9 y ( y + y For use laer, i is helpful o use equaions (B5-(B7 o express, c, s, in erms of and y and he price deviaions, m and z (B ν ν( ν c = y + y + m c y 7
ν ν( ν (B c = y + y m, where σν ( ν + ( ν e can furher simplify hese by defining: (B c ( c c (B c ( c + c Then ν ν( ν (B4 c = y + m (B5 c = y And, solving for he erms of rade, we find: σ ( ν (B6 s = y + z m, σ ( ν (B7 s = y + z + m Under a globally efficien allocaion, he marginal rae of subsiuion beween leisure and aggregae consumpion should equal he marginal produc of labor imes he price of oupu relaive o consumpion prices To see he derivaion more cleanly, we inser he shadow real wages in he efficien allocaion, w p H and w p F ino equaions (B8 and (B9 below So, he efficien allocaion would be achieved in a model wih flexible wages and opimal subsidies These equaions hen can be undersood inuiively by looking a he wage seing equaions below ((B-(B, and (B4-(B5 assuming he opimal subsidy is in place Bu, o emphasize, hey do no depend on a paricular model of wage seing, and are us he sandard efficiency condiion equaing he marginal rae of subsiuion beween leisure and aggregae consumpion o he marginal rae of ransformaion σ (B8 a = w ph = + φ y + ( σ + φ y φa, (B9 a w p σ = F = + φ y + ( σ + φ y φa Equaions of wage and price seing The real Home and Foreign produc wages, from equaion (A8, are given by: ν (B w ph = σ c + φn + s + μ, ν (B w pf = σ c + φn + s + μ e can express w ph, and w pf in erms of y and y and he exogenous disurbances,, a, μ, and μ : a 8
(B σ ( ν ν w ph = + φ y + ( σ + φ y + m + z φa + μ, (B σ ( ν ν w pf = + φ y + ( σ + φ y m + z φa + μ, Flexible Prices e can solve for he values of all he real variables under flexible prices by using equaions (B, (B4, (B, (B, (B6, (B and (B, as well as he price-seing condiions, from (A: (B4 w p H = a, (B5 w p F = a PCP Log-linearizaion of equaions (A8 and (A9 gives us he familiar New Keynesian Phillips curve for an open economy: (B6 π H = δ( w ph a + βε π H +, where δ = ( θ( βθ / θ e can rewrie his equaion using (B and (B8 as: (B7 σ πh = δ + φ y + ( σ + φ y + βε π H + + u where u = δμ Similarly for foreign producer-price inflaion, we have: (B8 σ πf = δ + φ y + ( σ + φ y + βε π F + + u LCP Equaion (B6 holds in he LCP model as well Bu in he LCP model, he law of one price deviaion is no zero e have: (B9 σ ( H ( ν ν π = δ + φ y + σ + φ y + m + z + βε π H + + (B σ ( ν ν πf = δ + φ y + ( σ + φ y m + z + βε π F + + u In addiion, from (A4 and (A4, we derive: (B π H = δ( w ph e a + βε πh+ = δ( w ph m z a + βε πh+ e can rewrie his as (B σ H ( + ν ν π = δ + φ y + σ + φ y m z + βε π H + + u Similarly, we can derive: (B σ + ν ν πf = δ + φ y + ( σ + φ y + m z + βε π F + + Consider equaions (B9-(B If z = in hese equaions, henπ π = π π This in urn implies z + = By inducion, if he iniial F H F H 9
condiion z = holds, i follows ha z = in all periods in he LCP model, or, in oher words, s = s Tha is, he relaive price of Foreign o Home goods is he same in boh counries e emphasize ha his is rue for a firs-order approximaion in he LCP model So we can simplify equaions (B9-(B: σ ( ν (B4 πh = δ + φ y + ( σ + φ y + m + βε π H + + u, σ ( ν (B5 πf = δ + φ y + ( σ + φ y m + βε π F + + u σ + ν (B6 πh = δ + φ y + ( σ + φ y m + βε π H + + u σ + ν (B7 πf = δ + φ y + ( σ + φ y + m + βε π F + + u elaionship o CGG s Phillips Curve The Phillips curve in CGG s PCP model has Home inflaion depending only on he Home oupu gap Our model should be equivalen o heirs when here is no home bias in preferences, bu equaion (B7 has π H depending on boh y and y This will no reduce o a funcion only of y excep in he case of σ = However, i can be seen ha in fac (B7 is equivalen o CGG s equaion when one recognizes ha CGG s definiion of he oupu gap differs from he one used here CGG CGG CGG define he oupu gap as he difference beween y and wha I will call y y is he efficien level of Home oupu when he Foreign oupu level is aken as given Tha conrass o our definiion in which y is he globally efficien oupu level CGG s definiion is convenien because heir analysis focuses on non-cooperaive moneary policy, while he definiion used here is more convenien because of he focus on cooperaive moneary policy Bu his is a maer of convenience: algebraically he equaions are he same To see his, noe ha equaion (B6 is he same as in CGG hen here are no deviaions from he law of one price, equaion (B can be wrien as: σ w ph = + φ y + ( σ + φ y φ a + μ (B8 σ y y y + y φ = + + ( σ + φ φ a + μ hen here is no home bias in preferences, his simplifies o: σ + σ (B9 w ph = + φ y + y φa + μ Then, CGG define: (B4 CGG σ ( φ a y = + y σ + φ
So we can wrie using (B9 and (B4: σ + CGG (B4 w ph a = + φ ( y y + μ Subsiuing his expression ino equaion (B6 gives us CGG s version of he Phillips curve, in which inflaion depends only on he Home oupu gap under heir definiions
Appendix C CI erivaion of elfare Funcion in Clarida-Gali-Gerler model wih Home Bias in preferences The obec is o rewrie he welfare funcion, which is defined in erms of home and foreign consumpion and labor effor ino erms of he squared oupu gap and squared inflaion e derive he oin welfare funcion of home and foreign households, since we will be examining cooperaive moneary policy Mos of he derivaion requires only s -order approximaions of he equaions of he model, bu in a few places, nd -order approximaions are needed If he + o a o indicae ha here are nd - approximaion is s -order, I ll use he noaion ( order and higher erms lef ou, and if he approximaion is nd -order, I will use + o( a (a is noaion for he log of he produciviy shock From equaion (A, he period uiliy of he planner is given by: σ σ + φ +φ (C υ ( C + C ( N + N σ + φ Take a second-order log approximaion around he non-sochasic seady sae e assume allocaions are efficien in seady sae, so we have C σ C σ φ N + + φ = = = N The fac ha C σ + φ = N follows from he fac ha in σ φ seady sae, C = N from marke clearing and symmery, and C = N from he condiion ha he marginal rae of subsiuion beween leisure and consumpion equals one in an efficien non-sochasic seady sae e ge: σ σ σ σ υ = C + C ( c + c + C (( c + ( c (C σ + φ σ + φ σ C ( n + n C (( n + ( n + o( a Since we can equivalenly maximize an affine ransformaion of (C, i is convenien o simplify ha equaion o ge: (C σ ( + φ υ = c + c n n + c + c ( n + n + o( a Uiliy is maximized when consumpion and employmen ake on heir efficien values: max σ + φ (C4 υ = c + c n n + ( c + c ( n + n + o( a In general, his maximum may no be aainable because of disorions e can wrie x = x + x, where x x x So, we have:
(C5 σ + φ υ = c + c n n + ( c + c ( n + n σ + φ + c + c n n + ( c + c + cc + c c ( n + n + nn + n n + o( a or, max σ + φ υ υ = c + c n n + ( c + c ( n + n (C6 + ( σ ( cc + cc ( + φ ( nn + nn + o( a e can rewrie his as: max υ υ = c n + ( σ (( c + ( c ( + φ (( n + ( n (C7 + ( σ ( c c + c c ( + φ ( n n + n n + o( a The obec is o wrie (C7 as a funcion of squared oupu gaps and squared inflaion if possible e need a second-order approximaion of c n Bu because he res of he erms are squares and producs, he s -order approximaions ha have already been derived will be sufficien ecalling ha m = in he PCP model, we can wrie equaions (B4-(B5 as: ν (C8 c = y + o( a, (C9 c = y + o( a I follows from (C8 and (C9 ha: ν (C c = y + o( a, (C c = y + o( a, ν (C c = y + o( a, (C c = y + o( a Nex, we can easily derive: n = y + o a, and (C4 ( (C5 n = y + o( a These follow as in (B-(B4 because n y a o( a in he Foreign counry e need expressions for n and = + and n = y a (and similarly n e have, using (B8-(B9:
σ a = + φ y φ a + o ( ( a, a = ( σ + φ y φa + o a Using a = y n and a = y n, we can wrie hese as σ ( ν (C6 n = y + o( a, and + φ σ (C7 n = y + o( a + φ Turning aenion back o he loss funcion in equaion (C7, we focus firs on he erms ( ( ( σ ( c ( ( ( ( ( ( ( ( + c + φ n + n + σ c c + c c + φ n n + n n These involve only squares and cross-producs of c, c, c, c, n, n, n, and n e can subsiue from equaions (C-(C7 ino his expression I is useful provide a few lines of algebra since i is a bi messy: ( σ ( c + ( c ( + φ ( n + ( n (C8 ( ( ( c c c c φ ( n n n n + ( σ + ( + + v ( σ ( v = ( σ ( + φ ( y ( σ + φ( y + ν( ν y y Now reurn o he c n erm in equaion (C7 and do a nd -order approximaion Sar wih equaion (A8, dropping he affec he approximaion, and noing ha in he PCP model, S k erm because i will no = S : ν ( ν/ ν ν / (C9 Y = S C + S C Then use equaion (A, bu using he fac ha S = S and here are no deviaions from he law of one price: ν σ (C C = CS Subsiue in o ge: ν ν ν (C Y = S C + S Solve for : ν ν + σ C C ν ν ν ν ν + σ (C C = Y S + S Then we ge his nd -order approximaion: 4
ν ν ν ν σ ν ( ν (C c = y + s s + o a σ σ Symmerically, ( ( ν ν ν ν σ = + ν + ( ν + (C4 c y s s o a σ σ Averaging hese wo equaions, we ge: ν ν (C5 ( σ c = y ν s + o( a σ Now we can ake a s -order approximaion for o subsiue ou for From equaion (B6, seing m = and z =, we have: 4σ (C6 s = ( y + o( a Subsiuing ino equaion (C6, we can wrie: ν( ν ( ν ( σ (C7 c = y ( y + o( a Evaluaing (C7 a flexible prices, we have: ν( ν ( ν ( σ (C8 c = y ( y + o( a I follows from he fac ha c = c c ha ν( ν ( ν ( σ (C9 c = y (( y + y y + o( a See secion C below for he second-order approximaions for n and n : ξ n = y + σ + o a (C p ( H ξ (C n = y + σ p + o( a F Adding hese wo equaions ogeher gives us: ξ ξ (C n = y + σ p ( H + σ pf + o a Subsiue expressions (C9-(C along wih (C8 ino he loss funcion (C7: max ( ν ( σ ξ = ( y + y y ph + pf υ υ ν( ν ( ( σ σ v ( σ ( v (C + ( σ ( + φ ( y ( σ + φ( y + ν( ν y y σ ξ = + φ ( y ( σ + φ( y ( σ p H + σ pf s s 5
This expression reduces o CGG s when here is no home bias ( ν = To see his from heir expression a he op of p 9, muliply heir uiliy by (since hey ake average uiliy, and se heir γ equal o ½ (so heir counry sizes are equal C erivaion of elfare Funcion under LCP wih Home Bias in Preferences The second-order approximaion o welfare in erms of logs of consumpion and employmen of course does no change, so equaion (C6 sill holds As before, we break down he derivaion ino wo pars e use firs-order approximaions o srucural equaions o derive an approximaion o he quadraic erm ( σ ( c + ( c ( + φ ( n + ( n + ( σ c c + c c ( + φ n n + n n Then we use second order approximaions o he srucural equaions o derive an expression for c n ( ( ( ( The quadraic erm involves squares and cross-producs of c, c, c, c, n, n, n, and n Expressions (C-(C sill gives us firs-order approximaions for c and c ; equaions (C4-(C5 are firs-order approximaions for n and n ; and, (C6-(C7 are firs-order approximaions for n and n Bu we need o use equaions (B4-(B5 and (C-(C o derive: ν ν( ν (C4 c = y + m + o( a, c = y + o a (C5 ( ih hese equaions, we can follow he derivaion as in equaion (C8 Afer edious algebra, we arrive a he same resul, wih he addiion of he erms ( σν ( ν ( σ ν( ν( ν m and my Noe ha he las erm involves 4 oupu levels, no oupu gaps Tha is, we have: (C6 ( ( ( σ ( c + ( c ( + φ ( n + ( n + ( σ ( c c + c c ( + φ ( n n + n n v ( σ ( v = ( σ ( + φ ( y ( σ + φ( y + ν( ν y y ( σν ( ν ( σν ( ν( ν + m + my 4 The derivaion of c n is similar o he PCP model However, one edious aspec of he derivaion is ha we canno make use of he equaliy S = S ha holds under PCP and flexible prices e wrie ou he equilibrium condiions for home oupu, and is foreign equivalen, from equaions (A8 and (A9: 6
ν ( ν / ν ν / (C7 Y = S C + ( S C, ν ( ν / ν ν / (C8 Y = ( S C + S C e direcly ake second-order approximaions of hese equaions around he efficien non-sochasic seady sae: ν ν ν ν ν ν y + y = c + c + s s (C9 ν ν ν ν ν ν + c + c + s + s ( ν ν ν ν + sc sc + o a ν ν ν ν ν ν y + y = c + c + s s ν ν ν ν ν ν (C4 + c + c + s + s ν ν ν ν + sc sc + o( a Averaging (C9 and (C4, we find: (C4 ( ( ( ν ν y ( ( ( ( ( + y + y = c + c + c + s + s + o( a 6 Nex, we can use equaions (B4-(B7 o ge approximaions for (, (,, and s These equaions are linear approximaions for, c, s, and, bu since we are looking o approximae he squares of hese variables, ha is sufficien ih some algebra, we find: (C4 ν( ν ( ν ( σ ν( ν ( σ ν( ν( ν = ( c y y m m y 8 ν( ν z + o( a 8 Noe ha if se m = and z = in (C4, we would arrive a he second-order approximaion for c from he PCP model Then following he derivaions as in he PCP model derivaion of (C9, we can wrie: c c s c s 7
(C4 ν( ν ( ν ( σ ν( ν = (( + c y y y y m 8 ( σν ( ν( ν ν( ν my z ++ o a ( 8 As shown in secion B, we can make he following second-order approximaion: ξ ν ν ν ν (C44 n = y + σ p ( H σ ph σ pf σ pf o a + + + + e hen can subsiue (C4, and (C44, along wih (C6 ino he loss funcion (C7 e find: max ( ν ( σ ν( ν ν ν = y ν ( ν ( ( y + y y m 4 ( σν ( ν( ν ν( ν my z 4 ξ ν ν ν ν y σ ph + σ ph + σ pf + σ pf (C45 + + + ν ( σ ( φ ( y ( σ φ( ( σν ( ν ( σν ( ν( ν m m y y ν( ν + + 4 σ ν( ν ν( ν = + φ ( y ( ( y m z σ + φ 4 4 ξ ν ν ν ν σ p ( H σ ph σ pf σ pf o a + + + + C erivaions of Price ispersion Terms in Loss Funcions In he PCP case, we can wrie PH ( f (C46 AN = A N( f df = Y df YV P =, H ξ PH ( f where V df Taking logs, we can wrie: PH (C47 a + n = y + v pˆ H ( f e have v ln e ξ df, ( where we define (C48 pˆ H( f ph( f ph Following Gali (8, we noe ( ξ ˆ ( ( (C49 p H f ξ e ( ξ pˆ ( ˆ H f ph ( f ( = + + +o a ξ ( σ( ν + y y 8
By he definiion of he price index P H, we have (C5 ξ pˆ ( ˆ H f df = ph ( f df + o( a e also have ξ (C5 pˆ H ( f ξ e = ξ pˆ ( ˆ ( ( H f + ph f + o a I follows, using (C5: (C5 e ( ξ pˆ H ( f df = Hence, from (C49, ( ( ˆ ξ p H ( f ξ ξ e df = ξ pˆ ( ˆ ( ˆ ( H f df + p H f df + o a = + p H f df + o a Noe he following relaionship: (C5 p ˆ ( f df ( ( ( ( ( H = p f H Ε p f df f H + o a = var( p H + o ( a Using our noaion for variances, σ var( p, and aking he log of (C5 we arrive a ξ (C54 v σ p o( a = + H ph Subsiuing his ino equaion (C47, and recalling ha y = n + a, we arrive a equaion (C The derivaion of (C for he Foreign counry proceeds idenically For he LCP model, we will make use of he following second-order approximaion o he equaion Y = CH + C H : ν (C55 ν ν ν ( y = ch + ch + ch + chch + ch + o( a In he LCP model, we can wrie: (C56 H ξ ξ P ( H f PH ( f = ( = H H + = H H + H H P, H PH AN A N f df C df C df C V C V where he definiions of V H and V H are analogous o ha of V in he PCP model Taking a second-order log approximaion o (C56, we have: ν ν a + n = ( ch + vh + ( ch + vh (C57 ν ν + ( ( ch + vh + ( ch + vh ( ch + vh + ( ch + vh + o( a e can follow he same seps as in he PCP model o conclude: ξ v = σ + o a (C58 H p ( H ξ (C59 vh = σ p + o( a H 9
find: (C6 Subsiuing hese expressions ino (C57 and cancelling higher order erms, we ν ξ ν ξ ν ν a + n = ( ch + σ p ( ( H + ch + σ ph + ch + chch + ch + o a Then using equaion (C55, we can wrie: ξ (C6 a n y ν ν ξ σ p ( H σ + = + ph o a + + Keeping in mind ha y = n + a, we can wrie: ξ (C6 n y ν ν ξ σ p ( H σ = + + ph + o a Following analogous seps for he Foreign counry, ξ (C6 n y ν ν ξ σ = + p ( F σ pf o a + + Adding (C6 and (C6 gives us equaion (C44 Finally, o derive he loss funcions for policymakers, we noe ha he loss funcion is he presen expeced discouned value of he period loss funcions derived here (equaion (C for he PCP model and (C45 for he LCP model Tha is, he policymaker seeks o minimize Ε β ( u u max + + = Following oodford (, chaper 6, we can see ha, in he PCP model, if prices are adused according o he Calvo price mechanism given by equaion (A9 for P H ha (C64 θ βσ ph + = βπh+ = ( βθ( θ = Analogous relaionships hold for PF in he PCP model, and for P H, PF, PF, and P H in he LCP model e can hen subsiue his relaionship ino he presen value loss max funcion, Ε β ( u+ u+, o derive he loss funcions of he wo models presened = in he ex The ex wries he loss funcion as: σ ν( ν Ψ + φ ( y ( ( y m + σ + φ + 4 (C65 ξ ν ν ν ν + ( πh + ( πf + ( πf + ( πh δ ν ν e can use he fac ha if π H + πf = π, hen (
ν ν ν ν ( πh + ( πf = π + ( πf π H This follows because for any a, x, and y, if ax + ( a y = z, hen ax + ( a y = z + a( a( x y Likewise, ν ν ν ν ( πf + ( πh = ( π + ( πh π F I follows ha ν ν ν ν ( πh + ( πf + ( πf + ( πh ν ν ν ν = π + ( πf πh + ( π + ( πh π F (C66 ν( ν = π + ( π + ( s s ν( ν = ( π + ( π + ( s s The second equaliy follows because, under Calvo pricing, o a firs order πf πh = πf πh = s s The hird equaliy follows because ( π π + ( π π π + ( π ( π + ( π = = Then subsiue (C66 ino (C65 o 4 ge he simplified loss funcion presened in he paper
Appendix The model is closed wih equaions for moneary policy This appendix solves he model algebraically when here are no cos push shocks and labor supply elasiciy parameer, φ, is se o zero These soluions can be used o derive he impulse response funcions in Figure of he paper e also assume for simpliciy ha he foreign produciviy shock is zero (Since he model is symmeric, he soluion for he response o foreign produciviy shocks is sraighforward e assume he Home produciviy shock follows he A process given by: a = ρa + ε, E ε = PCP model ih no mark-up shocks, he Phillips curves, (B7 and (B8 simplify o: ( πh = δ y + σy + βεπh + ( π F = δ y + σy + βεπ F + The opimal argeing rules can be wrien as: ( y y + ξπ H = (4 y y + ξπ F = e will assume y = y = I follows immediaely from hese equaions ha under he opimal policy, y = y = π H = πf = From equaions (B8 and (B9 we have σ (5 a = y + σ y σ (6 a = y + σ y I follows ha : + (7 y = a + a σ σ + (8 y = a + a σ σ Since y = y =, hese wo equaions solve for acual Home and Foreign oupu Since s = y, we have s =, or s = s = a Since q = ( ν s, we have q =, which implies q = ( ν a ν ν ν Then π = πh + ( e e + πf = ( e e Assuming ph, = pf, =, we have ph = pf =, so s = e + pf ph = e ν ν Therefore, π = ( a a = (( ρ a + ε
To calculae impulse responses for he exchange rae, we have, seing a =, k Ee = Es = ρ ε for k > For impulse responses for consumer price inflaion, we + k + k ν k have E π + k = ρ ( ρ ε for k > LCP Model Under LCP, we can wrie he Phillips curves as: δ (9 π = q + βεπ + ( π = δσ y + βεπ + The argeing rules are: ( q q + σξπ = ( y y + ξπ = Assuming a ime period - all variables are a heir efficien levels, hese equaions immediaely imply ha π = π =, which imply π = π = Also, y = and q = ( ν Since q = ( ν y + ν( ν m =, we have m = y Then, because ν( ν σ ( ν s = y m, we find ν ( ν ( y = s Noe also ha since y =, we have y = y The deviaion of he relaive price of impors from he efficien level is given by s = s a The evoluion of s in urn is deermined by he expecaional difference equaion: (4 s s = δ s + βε( s+ s + δa This equaion has he soluion (5 s = γs + γ a, where + β + δ (( + β + δ 4β δ γ =, and γ = β + β ( γ ρ + δ However, recognizing ha δ ( θ( βθ / θ, we see ha hese soluions simplify o: ( θ ( βθ γ = θ and γ = βρθ I follows ha s = θ s + ( γ a + θa To ge impulse responses, wih some algebra, we can show, seing a = and s =, k ( γ + θ k+ γ (6 Es + k= ρ θ ε ρ θ ρ θ
I follows immediaely from ( ha ν( ν k ( γ + θ k+ γ (7 Ey + k = ρ θ ε ρ θ ρ θ From he Phillips curve (B9, we have σ ( ν (8 πh = δ + φ y + m + βε π H+ ( ν Subsiuing in he relaionships m = y and (, we ge: ν( ν δ ( ν (9 πh = s + βεπ H + The forward soluion o his equaion is given by: δ ( ν ( πh = ( s + βε s + + β βεs + + Seing a = and s =, we can ge impulse responses from his equaion: ( δ( ν ρ ( γ + θ θ γ ε βρ ρ θ βθ ρ θ k k+ E π H + k = 4