Lecture Note on the Real Exchange Rate

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1 Lecure Noe on he Real Exchange Rae Barry W. Ickes Fall Inroducion The real exchange rae is he criical variable (along wih he rae of ineres) in deermining he capial accoun. As we shall see, his is because he real exchange rae is he relaive price of goods across counries. Hence, changes in he real exchange rae affec he compeiiveness of raded goods. The nominal exchange rae, S, refers o he dollar price of foreign exchange. 1 As wih mos variables in economics we disinguish beween he nominal and real values. The real exchange rae measures he cos of foreign goods relaive o domesic goods. I gives a measure of compeiiveness, and i is a useful variable for explaining rade behavior and naional income. One of he grea puzzles in inernaional macroeconomics is why he real exchange rae is so volaile. Consider Þgure 1 which shows he real exchange rae since You can see ha he real exchange rae is no only volaile, i does no appear o move around some equilibrium level. There are long swings. Given ha his is a relaive price one would suspec ha such large changes would have big welfare implicaions on he economy. This is a second surprise i appears ha his is no he case eiher. These are wo poins o hink abou. 0.2 DeÞniion We can deþne he real exchange rae, Q, by Q = SP P 1 IuseanS o denoe spo exchange rae. (1) 1

2 Figure 1: Real Exchange Rae where P is he price level in he foreign counry. An appreciaion of he real exchange rae indicaes ha he foreign price (in dollars) of a bundle of goods has risen relaive o he domesic price. If he real exchange rae appreciaes i means ha he real value of he dollar has depreciaed; ha is, he purchasing power of he dollar has fallen in relaive erms. Noice ha o deþneherealexchangeraeweneedospecifyheprice levels. If he baskes of goods in he domesic and foreign counries were he same his would be sraighforward; in pracice, hey are no. We ypically use some broad measure of he price level, such as he GDP deßaor or he CPI. I should be noed ha his means ha P will place a relaively heavy weigh on goods produced and consumed domesically, while P will likewise place a relaively heavier weigh on goods produced in he foreign counry. 2 We will soon see he imporance of his. Wha causes changes in Q? Two speciþc causes are worh discussing here. 1. A change in world relaive demand for US goods. Suppose ha preferences shifed so ha oal world spending on US goods increased. This could be due o shifs in privae demand owards US goods, or an increase in US governmen spending which is concenraed on US goods. A curren exchange raes his would cause an excess demand for US goods. To resore equilibrium he relaive price of US goods 2 This is especially rue because of non-raded goods, which we shall discuss shorly. 2

3 mus rise relaive o foreign goods; hence, Q mus fall, and he dollar has appreciaed in real erms. In oher words, he purchasing power of he dollar has increased relaive o foreign goods. Q d y 0 d y 1 Q 0 Q 1 y Figure 2: An Increase in he Demand for Domesic Goods 2. A change in relaive oupu supply. Suppose ha here is a relaive echnological shock ha increases he efficiency of US oupu relaive o foreign oupu. Wih given socks of capial and labor US oupu rises. Hence, a unchanged world demand here is an excess supply of US oupu. Why? This posiive supply shock raises US income (wealh), bu no all of he increase in income is spen on domesic goods. Some will be spen on foreign goods. Hence, he increase in he demand for US goods will be less han he supply. To resore equilibrium he relaive price of US goods mus fall; in oher words, Q mus rise, and he dollar mus fall in real erms. This real depreciaion of he dollar (or real appreciaion of he foreign currency, say he DM) means ha he purchasing power of he foreign currency has increased. Thus relaive produciviy growh causes he real exchange rae o appreciae and he real value of he currency o depreciae. We will reurn o he opic of real exchange rae movemens afer we ake a deour o discuss exchange rae deerminaion when he real exchange rae is consan. Why is he real exchange rae so imporan for hinking abou he curren accoun? Because i is he relaive price of foreign goods in erms of domesic 3

4 Q Q 1 s y 0 s y 1 d y 0 d y 1 Q 2 Q 0 y Figure 3: A change in relaive supply goods, changes in his variable will impac ne expors, and hence, he curren accoun. If a curren accoun deþci is o be reversed an appreciaion of he real exchange rae may be one of he mechanisms of adjusmen A Special Case: Purchasing Power Pariy An ineresing case o consider is he special case where he real exchange rae is consan over ime. Suppose ha he baske of goods ha were produced in he US and Germany were idenical, and ha all goods were radeable. In ha case, ne of ransporaion coss we would have he law of one price: arbirage would insure ha he dollar prices of he various goods would be idenical across counries. This yields a heory of exchange rae deerminaion known as PPP. Noice ha we could use he deþniion of he real exchange rae o wrie: S = Q P (2) P Now suppose ha he real exchange rae is consan over ime; hence, Q = Q for all ime. Then S = QP (3) P and i follows ha any changes in naional price levels resuls in a movemen of he exchange rae. PPP hus deermines he exchange rae by he movemens in relaive price levels. If US inßaion is higher han foreign inßaion 4

5 heexchangeraewillappreciaeandhedollarwilldepreciaerelaiveo he foreign currency. I will ake more dollars o purchase a DM. This is inuiive: he nominal exchange rae is he relaive price of currencies, and inßaion is he measure of he decrease in purchasing power of a currency. If he dollar is losing purchasing power faser han a DM, hen he DM should gain in value relaive o he dollar. This can be seen more clearly, perhaps, by aking logs of boh sides of (3): s = q + p p (4) wherewehaveusedlower-caseleersoreferohelogofavariable. 3 Now supposeweakeþrs differences of (4), i.e., s s s 1 : s = p p. (5) Expression (5) says ha he percenage change in he nominal exchange rae is equal o he difference beween he inßaion raes in he domesic andheforeigncounry. 4 When price levels are changing very rapidly hese movemens can dwarf all oher facors, and hen PPP provides a raher effecive heory of exchange rae movemens. A simple example of his heory is provided by he Big Mac index. The Big Mac is essenially he same good in every counry. Hence, we can compare he dollar price of Big Mac s across counries. Where he currency appears over-valued we should expec he exchange rae o appreciae, and vice versa. This does surprisingly well. See Þgure. There are several reasons why PPP does no hold in he shor run. Noice, ha PPP is a heory of exchange rae deerminaion based on goods ßows. I is ied o rade (i is no he only heory of his kind), and i ignores capial ßows. Exchange raes can also ßucuae because of expecaions of fuure changes, hough even hese mus be based on somehing. We have alked abou curren accouns. Of course, he need o Þnance deþcis can lead o differen raes of inßaion, and so back o PPP. So i is no rivial o dismiss i. Here are some imporan issues. 3 If you dees aking logs see equaion 20 where we derived he same expression wihou aking logs. 4 Recall ha X X 1 X 1 = X X 1 1 is he percenage change in X, and hus 1+g = X X 1, where x is he percenage growh rae. Now if we ake logs of his expression, for small g, i follws ha log(1 + g) g x x 1 x. 5

6 Firs, ariffs and ransporaion coss creae a band in which prices can ßucuae before arbirage becomes proþable. Second, permanen shifs in he erms of rade can cause Q o change, if counries differ in he composiion of oupu. An oil shock (posiive) will have a differen effec on an energy producer and a producer of energy-inensive producs. The laer counry will experience a relaive decline in he world demand for is goods, so is currency will experience a real depreciaion. Third, if prices are sicky in he shor run he law of one price, by deþniion, does no hold. Then i follows ha movemens in nominal exchange raes will also affec he real exchange rae. This may hold in he shor run, bu over longer periods of ime prices do adjus and PPP is more likely o hold. Fourh, he presence of non-raded goods, is probably he mos imporan facor. Think of haircus versus whea. Even if raded goods are idenical across counries and obey he law of one price, non-raded goods do no. Shifs in he relaive price of raded and non-raded goods can cause PPP o fail. This is raher easy o see. Le us wrie he price index of he domesic counry as P = Pn α P 1 α, where P is he price of raded goods, and α is he share of non-raded goods in he domesic price index. Now we can wrie he real exchange rae as: Q = S = S " # Pn α P (1 α ) Pn α P 1 α = S ³ P α n µ P P P ³ P n P α ³ P α n P P ³ α P n P P ³ bu if we assume ha PPP holds for raded goods, i follows ha S P P = 1, so Q = ³ P n P α ³ P n P α 6. (6)

8 measure he rue differences in purchasing power. This is because purchasing power of a currency differs depending on he shares of raded and non-raded goods. To see his, recall ha from he deþniion of he real exchange rae, Q = SP P,wecanwrie: q = s + p p hence, p = s + p q (8) which implies ha counries wih lower q will have higher price levels compared wih prices elsewhere, since he foreign price level measured in unis of domesic currency is jus s + p. If produciviy growh is rapid in he US relaive o he foreign counry, our price level will be higher, when measured in common currency unis. One way o hink abou his is simply ha PPP values exchange raes according o he relaive price of raded goods. Bu in LDC s he price of non-raded goods is lower. When hese are included, he price level in he advanced counry is higher. Tha is essenially wha is implied by The Role of Produciviy Growh Here is a sraighforward way o see why a counry ha has rapid growh in produciviy experiences real exchange rae appreciaion. The key noion is wha happens o wages. Sar again wih he deþniion of he price level as P = Pn α P 1 α,and use aserisks for he foreign counry. The law of one price implies ha for radable goods we have P = ep (9) Now proþ maximizaion means ha wages equal marginal producs. So we will have w = MPL and w P P = MPL (10a) or P = MPL w. Then i follows ha or MPL w ew w = e MPL w = MPL MPL 8 (11)

11 also due o recovery from he depreciaed exchange raes ha resuled from he iniial collapse of heir currencies and capial ßigh Ineres Differenials and he Real Exchange Rae If PPP holds, hen ineres differenials are a funcion of differences in expeced inßaion. The law of one price suggess ha he spo exchange rae is deermined by relaive price levels: e = P US (15) P E Expression (15) is a heory of exchange rae deerminaion purchasing power pariy based on he assumpion ha all goods are radeable. 6 Hence, i assumes ha real exchange raes are consan. I is no a bad assumpion for he long run, bu i may be problemaic for he shor run. If each counry produced one and he same good, and if ranspor coss and naional prejudices did no exis, hen arbirage would clearly bring abou (15). Of course counries produce many goods, and no all are radeable. I is noneheless worhwhile o see is implicaions. From (15) we can wrie: e e 1 = P US, P E, P US, 1 P E, 1 = P US, P US, 1 P E, P E, 1 (16) Now deþne inßaion as π = P US, P US, 1 1. Sowecanwrie(16) as: e e 1 = 1 + π us 1 = 1 + π us 1 + π E (17) e π E 1 + π E 1 + π E = π US π E (18) 1 + π E Now i is clear ha π US π E =(π US π E )(1 + π E π E ),soicanwrie(18) as: (1 + π E )(π US π E ) π E (π US π E ) =(π US π E ) π E (π US π E ) (19) 1 + π E 1 + π E 1 + π E 6 Suppose ha he baske of goods ha were produced in he US and Germany were idenical, and ha all goods were radeable. In ha case, ne of ransporaion coss we would have he law of one price: arbirage would insure ha he dollar prices of he various goods would be idenical across counries. This yields a heory of exchange rae deerminaion known as PPP. 11

12 Bu if inßaion raes are raher low he difference beween hem is likely o be low, and he produc of his difference and he inßaion rae is likely o be even lower. Hence, for low inßaion raes he las erm on he righ hand side of 19 0, which means ha we have he approximaion: e e 1 e 1 = π US π E (20) which is called relaive purchasing power pariy. Expression (20) says ha he percenage change in he nominal exchange rae is equal o he difference beween he inßaion raes in he domesic and he foreign counry. 7 When price levels are changing very rapidly hese movemens can dwarf all oher facors, and hen PPP provides a raher effecive heory of exchange rae movemens. A grea advanage of his expression is ha i holds even if absolue PPP does no. Now recall ha from UIPC we have: e e +1 e e = i US i E (21) So if expeced inßaion differences correspond wih acual inßaion differences, i follows ha he ineres differenial will be equal o he difference in expeced inßaion raes. Moreover, as marke paricipans undersand expression (20) i follows ha expeced inßaion will be equal o he growh rae of he exchange rae. Hence, i US i E = π e US π e E (22) where π e US is he expeced US inßaion rae. Thus expression (22) says ha he ineres differenial will be equal o he difference in expeced inßaion raes. Noice wha expression (22) implies. If agens expec higher US inßaion relaive o Europe i follows ha US ineres raes mus rise relaive o European raes. Hence, according o expression (22) he real reurn on US asses relaive o Europe will be unchanged. This is called he Fisher effec. The Fisher effec is usually wrien as r = i π e. Movemens in expeced 7 Recall ha X X 1 X 1 = X X X 1, X 1 1 is he percenage change in X, and hus 1+g = where x is he percenage growh rae. Now if we ake logs of his expression, for small g, i follws ha log(1 + g) g x x 1 x. 12

13 inßaion leave real ineres raes unchanged. Bu his is also wha is implied by expression (22). Suppose hen ha money growh in he US rises relaive o Europe. In he long run we would expec ha inßaion would rise by an equal amoun. And so would expeced inßaion. Hence we would expec he ineres rae in he US o rise relaive o Europe. This also implies ha he exchange rae musappreciae ahesameraeasheineresdifferenial. In he long run oupu and real reurns are unchanged. The only effec of he rise in money growh is on he nominal quaniies. Of course, we know in he shor run here will be impacs. Bu we can also see ha in he long run only nominal quaniies are effeced. A simple example of his heory is provided by he Big Mac index. The Big Mac is essenially he same good in every counry. Hence, we can compare he dollar price of Big Mac s across counries. Where he currency appears over-valued we should expec he exchange rae o appreciae, and vice versa. This does surprisingly well hough i is no perfec. Noice ha even wih hebigmac,however,price differences persis. No even all he Big Mac coss are really radeable. The assumpion of PPP is equivalen o assuming ha he real exchange rae is consan. Ye, we know i is no. So how is he heory modiþed? Recall ha he real exchange rae is deþned as Q = e P E. We are ineresed in an P US expression for he expeced growh rae of he real exchange rae, Qe Q 1 Q 1. Suppose ha inßaion was expeced o equal in he US and euroland. Then clearly we would have Qe Q 1 Q 1 = ee e 1 e 1.Ofcourseexpecedinßaion raes are no equal, however, so how is he expression alered? Suppose ha he exchange rae was no expeced o change. Then clearly we would have Q e Q 1 Q 1 = π e E π e US. If US inßaionishigherhanineurolandhereal exchange rae depreciaes. Pu hese wo facors ogeher and i is clear ha: Q e Q 1 Q 1 The expression is inuiive. 8 Now recall he ineres pariy condiion, ee e 1 e 1 = ee e 1 (π e US e πe E ). (23) 1 = i US i E. Using his o 8 To prove his ake logs of he expression for he real exchange rae expression: log Q =loge +logp E log P US 13

14 replace he expeced rae of depreciaion in (23), we obain: i US i E = Qe Q 1 +(π e US Q πe E ) (24) 1 which implies ha ineres differenials depend on expeced movemens in he real exchange rae in addiion o differences in expeced inßaion. When he real exchange rae is no expeced o change we have he same expression as before. Bu in general, nominal ineres differenials are explained by movemens in he real exchange rae as well. Of course you can hink of he laer as capuring all he reasons for exchange rae movemens oher han differenial price levels. This now les us derive an expression for real ineres pariy. While nominal reurns are hose ha are he acual componens of exchanges, i is real reurns, or raher expeced real reurns ha govern decisionmaking. Tha is, you may earn a nominal ineres rae of 10% from an asse, bu wheher you choose o hold i or no depends on how he real reurn relaes o he opporuniy cos of he funds. So an invesor hinking of where o hold her wealh mus consider he expeced real reurns. We know ha nominal ineres differenials are relaed o expeced changes in he real exchange rae and expeced inßaion. We obain he real ineres pariy condiion by Þrs using he Fisher equaion,i = r + π e. I follows ha he expeced real rae of ineres in he US, rus, e = i US, π e US,, and likewise for Euroland, re, e = i E, π e E,. Using hese in expression (24) yields he real ineres pariy condiion: rus, e re, e = Qe Q 1 (25) Q 1 which says ha expeced real ineres rae differenials are equal o expeced changes in he real exchange rae. Why should (25) hold? Suppose ha people expec he real exchange rae o appreciae, say because Euroland produciviy growh in radeables Now differeniae boh sides wih respec o ime and we obain: µ µ µ µ 1 dq 1 de 1 dp E = Q d e d + 1 dp US P E d P US. d Bu hese expressions are jus he (coninuous ime) growh raes of he real and nominal exchange rae and of he price levels in euroland and he US, respecively. So as he period shrinks, we obain he expression in he ex. 14

15 is expeced o be higher han in Euroland non-radeables, and also higher han in he US. Thus people expec he real value of he dollar o depreciae relaive o he euro. To compensae for his he real reurn on dollar asses mus exceed hose of Euroland asses. Does his mean ha here are proþ opporuniies ha are no being arbiraged away? No. The differences in he real raes do no reßec differen reurns on he same asse. I reßecs differen reurns on wo bundles of goods. The absence of arbirage opporuniies is guaraneed by ineres pariy, since any invesor ha compares relaive reurns has a unique consumpion baske. When I compare he rae of reurn on holding dollars or euros, he real reurn is compued by subracing my expeced rae of inßaion, whaever consumpion baske is relevan for me. Bu he expeced real differenial on he lef-hand side of (25) is comparing wo expeced inßaion raes ha reßec wo differen consumpion baskes. Noice ha if all agens were idenical PPP would hold and we would no have real ineres differenials. Bu because people in differen counries consume differen baskes of goods, here is no way for hem o arbirage away any difference. Why is his ineresing? Recall when we spoke of he Feldsein-Horioka puzzle we commened ha a direc way o es for capial marke inegraion would be o look a wheher excess reurns were arbiraged away. So ha would sugges looking a real ineres differenials in differen counries. If capial markes are inegraed hese should go o zero. Bu as we have jus seen, his is only rue if PPP holds. If i does no hold, hen real ineres differenials should be expeced o persis. Suppose, for example, ha US savings was so low ha people expeced he real exchange rae o appreciae in he fuure. 9 Then real reurns on US asses would have o exceed hose in he res of he world, according o (25). 9 Why do hey expec Q o appreciae? We have o pay back he debs we have incurred, so curren accoun deþcis mus become surpluses. So he relaive price of our goods mus fall relaive o he res of he world o allow us o expor more and impor less. 15

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