All individuals have identical preferences. The representative consumer maximises

Transcription

1 MS Macro Hanou - Te Obsel-Rogo Moel Tis open economy moel inrouces a number o new eaures. We ave wo counries ineracing, so neier is small. Money is inrouce ino e uiliy uncion. PPP ols or consumer prices, bu no proucer prices (i.e. price o naional oupu). We ave a variey o goos, an consumers ol ES preerences beween em. We ormally log-linearise aroun a seay sae. Te moel incorporaes nominal rigiiy in a igly sylise, bu analyically simple way. None o ese aiions is concepually iicul: e main problem is keeping ouc wi e noaion. To elp, in some places I epar rom e original, bu ese eparures are explicily noe. Te bes reerence or is moel is aper 0 o Obsel & Rogo s exbook, Founaions o Inernaional Macroeconomics (Is E), an e page number reerences are o is. Te moel was irs publise in Excange rae ynamics reux, Journal o Poliical Economy (995) 03, I as since becomes a sanar saring poin or many open economy journal aricles. Moel Se Up Assumes perec oresig. Tere is a coninuum o iniviual monopolisic proucers/consumers, a racion n o wic are in e ome counry, wi e remaine in e overseas counry. Eac prouces a single iereniae goo, sol uner monopolisic compeiive coniions. Tere is no capial, bu labour supply is a coice variable. onsumers All iniviuals ave ienical preerences. Te represenaive consumer maximises s 2 U = [log + χ log( M / P ) κy / 2] s s s s s= β () β is e rae o ime preerence (using e paper s original noaion). is a consumpion bunle given by ( θ ) / θ θ /( θ ) = [ c( z) z] (2) 0 were c(z) is consumpion o goo z, an θ>. (Noe capial leers enoe aggregaion across goos, no iniviuals.) In is ES ype ormulaion, θ urns ou o be e elasiciy o eman. Te aggregae price inex base on (2) is

2 θ /( θ ) P = [ p( z) z] (3) 0 Tis inex solves e problem o coosing c(z) o minimise expeniure subjec o =. (O&R page 662.) Te oer wo erms in uiliy are real money balances, an a erm relecing isuiliy rom labour supply, base on reurns o scale o ½. (O&R page 662.) Noe a a all in κ implies an increase in prouciviy. Te only asse e consumer can ol is a real bon (enominae in erms o e consumer price bunle), wic pays a real rae o ineres r. Tese bons are enominae in erms o consumer prices. Tis is an imporan assumpion, or reasons noe uner rae below. Teir buge consrain is en P B + M = P + r ) B + M + ( + p y P Pτ (4) were r is e real ineres rae on bons el beween - an, p is e price o e iniviual s goo y, an τ are lump sum axes. (Noe e ierence beween e price o an iniviual goo p an e aggregae price P.) Proucion Te iniviual s eman or goo z is given by (O&R p664) p( z) θ c( z) = [ ] (5) P Deman or any paricular goo epens on aggregae eman an relaive prices. As all consumers are ienical, we can wrie a (worl) eman or goo z as y z p( z) P θ W ( ) = [ ] (6) were w = n + (-n), using e superscrips w or worl, or overseas an or ome. (O&R use * or overseas an noing or ome.) Trae We assume no barriers o rae, so e law o one price ols or any paricular goo i.e. p (z) = ep (z) were e is e ome currency price o oreign currency (+ is a epreciaion). Because preerences are equal or all consumers, we can also wrie e same expression or aggregae prices i.e. P = ep (O&R p663). Tis is PPP or e

3 consumpion bunle, no oupu. However i as a crucial consequence real ineres raes across e wo counries, eine in erms o consumer prices, mus be e same. Recall a uner real UIP, e real ineres rae ierenial equals e expece cange in e real excange rae. PPP implies a consan real excange rae, so real ineres rae ierenials mus be zero. As ere is no capial or governmen, we ave nb + (-n)b = 0 (7) i.e. e sum o omesic an overseas bons are zero. I B is posiive, is means a ome resiens ol asses issue by overseas resiens. Maximisaion Firs Orer oniions β ( r+ ) = + + (8) Te sanar Keynes/Ramsey rule wi log uiliy (ineremporal subsiuion elasiciy is uniy). M χ (9) / P = ( + i ) / i + + were i is e nominal ineres rae eine as +i =P + (+r + )/P. Noe money eman epens on consumpion, no oupu. y (+ θ ) / θ W ) / θ ϑ = ( θκ ( ) (0) Tis las equaion is in eec a labour supply equaion, given e eman curve. Tere is also a ransversaliy coniion (O&R p666). Seay Sae Denoe seay sae variables by an unerlining (O&R use overlining). From (7) we ave _ r = (-β)/β = δ () were δ is e rae o ime preerence. Te buge consrain implies (using e above) = δb + py/p (2) were we recall a B are bon olings (wic can be negaive). In general e equilibrium nee no be symmeric i B is no zero. I overseas resiens are borrowing rom omesic resiens, omesic consumpion will be iger,

4 an ey will prouce less (o enjoy more liesure), raising p /P. In e symmeric seay sae were B =0, we ave _ θ ( ) θκ / 2 y = (3) Tis is less an e welare maximising level o oupu because o monopoly power (O&R p668). Log-linearise aroun seay sae We now log-linearise aroun e symerical seay sae. (See O&R page o see ow is is one, in e conex o a ieren moel.) To avoi inroucing more noaion, rom now on e variable x will enoe (using e previous noaion) x / x, wic is (log x ) near equilibrium. (O&R use a bol on.) Aggregaing (3) or ome an overseas implies P = np + ( n)( e + p ) (4) P n( p e ) + ( n) p = (4) Te aggregae consumpion price in e ome counry is e sum o all ome prouce goos (assume ienical) an overseas goos. (In O&R noaion, P becomes p, an p is enoe as p().) We ave, as beore, aggregae PPP: e = P P (5) Te eman curves become y = ( P p ) + θ (6) w an a similar equaion replacing wi. A worl goos marke clearing coniion is = n + n) = ny + ( n) y = Y w ( (7) w Te labour/liesure rae-o becomes ( θ + ) y = θ + (8) w wi a similar equaion or overseas. Tus i ome consumpion rises relaive o e worl average, oupu alls. Te consumer Euler equaion becomes

5 + δ + δ = + r + (9) wi a similar equaion or overseas. (Noe r is equal across counries see above.) Money eman is M r P + δ + + P = P δ (20) wi, again, a similar equaion or overseas. A log-linearise version o e seay sae buge consrain (2) is = δ B + p + y P (2) were we exclue ime subscrips because is only ols in seay sae. (Deriving is, you irs iereniae (2), ivie roug by seay sae, an en noe a seay sae B is zero, so seay sae consumpion equals income. omparaive saics: lexible prices In is version o e moel, B is no an enogenous variable. (Tis is a consequence o assuming ininie lives.) Inee i is e only acor leaing o ierences beween e wo counries. We now analyse e impac o a cange in e isribuion o weal. I is cange is permanen, en e moel will immeiaely jump o a new equilibrium. We now a seay sae versions o equaions (4) o (20) an solve e moel. I is easies o o is by irs solving or e ierences beween counries, an en solving or e aggregae. Denoe x = x -x. Deman curves: y =θ ( e p ) (22) Labour/leisure: y θ + θ = (23) Buge consrain: = ( ) δ B + y ( e p ) (24) n (You nee o be sligly careul in eriving is las equaion.) Noe a a epreciaion as wo eecs on e buge consrain: income rises because

6 oupu rises (rom 22), bu ere is a negaive erms o rae eec. However, as θ>, e ormer mus ominae. We now ave ree equaions in ree unknowns, wi B as e only exogenous variable. ombining all ree gives ( n + θ ) δb 2θ = )( (25) an p = ( ) δb / 2θ n e p (26) A weal ranser (increase in B ) raises consumpion a ome relaive o overseas. Tis reuces relaive ome oupu (more leisure goes wi more consumpion), raising ome prices an improving ome erms o rae (e le an sie o 26). I is sraigorwar o sow (O&R p673) a worl oupu an consumpion are uncange by a cange in B. Tis solves or all real variables. We can en use e money eman equaions o eermine prices, an e nominal excange rae (O&R p673). In paricular we ave or eac counry P = M (28) as M is ixe, so seay sae inlaion is zero, an e = M (29) I is sraigorwar o sow a we ge a sanar money neuraliy resul, an (29) combine wi e ac a is inepenen o M, implies a canges in money lea o proporionae canges in e excange rae. omparaive saics: sicky prices Now suppose oupu prices are ixe or one perio, an consier a permanen unanicipae cange in e ome money supply. As prices are ully lexible by perio 2, en perio 2 will be e seay sae. By ixing prices in perio, we mus rop wo equaions: e oupu/labour supply equaions (8). Oupu now ollows eman. In aiion, B can now cange, because in perio we are no in a seay sae an so an iniviual in a counry can run a curren accoun eici: B + B = r B + p y / P (30) Log-linearising in perio gives B 2 y n) = ( e (3)

7 because B is prese, as are prices, an by using (4). Tus any canges in perio will lea o curren accoun canges a will aler e isribuion o worl weal, wic in urn will ave real seay sae eecs. Money will no longer be neural! Using (9) implies a = 2 Relaive consumpion canges in perio by e same as i canges in perio 2 = seay sae. Wy? Because all consumers are ienical, ey will respon in e same way o canges in real ineres raes, an real raes are equal or bo counries. Using (20) plus PPP gives M e = ( e )/δ (32) 2 e Using (29) or e 2, plus e lack o ynamics in consumpion, allows us o use e above o sow a 2 e M 2 2 = an (33) e = e 2 Te excange rae also jumps in perio o is seay sae value. We o no ge oversooing. Te reason or is is simple PPP implies a real ineres raes canno ier beween counries, so ere is no scope or canges in relaive real ineres raes inluencing e excange rae, as i oes in e Dornbusc moel. As prices are ixe in e irs perio, we ave rom (22) y θe = (34) B Using (3) an is overseas equivalen implies 2 ( e = n)( y ) (35) Bu using (25) allows us o relae e long run (=sor run) consumpion ierenial o is weal ranser. ombining all is gives δ ( + θ ) + 2θ θδ ( + θ ) + 2θ = (36) e M From UIP, we know a real (in erms o consumer price inlaion) ineres rae ierenials are equal o e expece cange in e real excange rae (eine in erms o consumer prices). Bu uner PPP, is real excange rae is consan, so real ineres rae ierenials are zero. Noe a aloug consumers benei rom an increase in money in perio, ey lose o an inlaion ax in perio 2.

8 An increase in ome money leas o a epreciaion, bu less an proporionae. Te reason or is is as ollows, As prices are iniially ixe, an oupu is below is perecly compeiive level, en iger money will raise oupu. As ome prouces more goos, i runs a curren accoun surplus. Tis raises is long run weal, raising consumpion also in e long run. From e lex price version, we know is appreciaes e excange rae, because iger consumpion goes wi more leisure, lower oupu an iger prices. As e iniial jump in e excange rae is also is long run level, en is real appreciaion parially oses e iniial epreciaion. In e long run, worl real variables o no cange as we noe above. In e sor run we can erive (O&R p 682) + δ w r = M (37) δ Te real ineres rae alls by an amoun epening on e size o e ome counry. Given e Euler equaions or consumpion, is means worl oupu an consumpion rise in perio. Is e non-neuraliy in is moel ineresing? In one sense i is a special case prouce by assuming ininie lives, because only en is weal ineerminae an subjec o yseresis eecs. However even wiou ininie lives, sor run Keynesian eecs coul lea o igly persisen consumpion eecs. I is also e case a e size o ese long-run non-neuraliies on oupu are secon orer (by an amoun equal o e rae o ineres) compare o e irs roun eecs. Noe a ere are wo ransmission mecanisms beween e wo perios in is moel. One is roug asse accumulaion, wereby socks in e irs perio inluence e seay sae in e secon perio. Te secon works in e reverse irecion (via raional expecaions), were e nominal excange rae in e secon perio eermines e excange rae in e irs.

9 We can use e moel o examine prouciviy canges by ecreasing e parameer κ. (O&R p696-) I e prouciviy increase is emporary (perio only), e eec is rivial, because oupu is eman eermine. All e prouciviy gain will go o increases in leisure: e same oupu will be prouce wi less labour. (Te κ parameer only eners e one equaion a is roppe because oupu is eman eermine.) A permanen increase in global prouciviy will raise global oupu, bu by no as muc, because consumers will ake some o is gain as iger leisure. Te more ineresing case is a permanen increase in ome prouciviy. In e long run, is will raise ome oupu, an ome s erms o rae (eir real excange rae) mus epreciae so a e exra goos can be sol. Home consumpion rises relaive o overseas consumpion in bo perios, wic given (33) implies a all in e, wic is an appreciaion. Te inuiion bein is appreciaion, wic occurs in bo e sor an e long run, can be seen rom consiering perio. A sor run increase in ome consumpion (anicipaing iger uure oupu) leas o an increase in money eman. As oupu prices are ixe, e only way is exra eman can be accommoae wi ixe nominal supply is i consumer prices all, implying an appreciaion. I is also relaively sraigorwar o a governmen o is moel (p700-). Noe a, unlike our earlier open economy moels, governmen spening is assume o be spli beween counries in e same way as privae consumpion. As a resul, an increase in only ome governmen spening will resul in iger eman or bo counries proucs. However only ome s axes rise, so relaive ome consumpion will all. Tere are a number o ineresing resuls o noe. Firs, iger spening as a long run posiive oupu eec. Tis is because iger axes lea consumers o subsiue work or leisure. In e sor run, as we ave alreay noe, a rise in ome governmen spening will prouce a nominal (an real: prices are ixe) epreciaion, wic will raise ome oupu (eman eermine) relaive o overseas. In e long run relaive ome oupu also rises, bu is is a supply sie eec, inuce by ome resiens giving up leisure because o iger axes.