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# Advanced International Economics ECON 872

UW

GPA 3.6

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This 17 page Class Notes was uploaded by April Jerde on Thursday September 17, 2015. The Class Notes belongs to ECON 872 at University of Wisconsin - Madison taught by Charles Engel in Fall. Since its upload, it has received 44 views. For similar materials see /class/205143/econ-872-university-of-wisconsin-madison in Economcs at University of Wisconsin - Madison.

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Date Created: 09/17/15

We will investigate models of nominal exchange rates that share two properties longrun purchasing power parity and uncovered interest parity Long run purchasing power parity states that the real exchange rate is stationary The real exchange rate is de ned as qt 2 S p pt By stationary we mean covariance stationary or integrated of order 0 10 Such a stationary random variable has an unconditional mean and it is expected to converge to its conditional mean liggwtqak a Where a E q There is a large industry that tests for the null that real exchange rates among major countries have a unit root that is 1l against the alternative of 10 The literature is inconclusive Perhaps it is fair to say there is a consensus that these real exchange rates are stationary but very persistent Uncovered interest parity The return on a oneperiod safe domestic nominal investment is it A rstorder approximation to the expected return on a foreign investment is it Etstl St A foreign investment pays off both from the foreign interest rate and the expected appreciation of the foreign currency The foreign investment is risky because there is exchange rate uncertainty But uncovered interest parity is the theory that this risk is not priced it it Et tl St We Will return to empirical tests of this theory later There seems to be some evidence of violations of uncovered interest parity at horizons of a year or less But we Will assume it holds for now Now rewrite uncovered interest parity as it it Etstl St39 From each side of the equation subtract home relative to foreign expected in ation Etpt1 pt Et pt1 pt We get I 7i thHl qt where I E it Et pt1 p and r E 1 Etp1 19 These are the ex ante real interest rates at home and abroad We will rst explore the implications of real interest parity We can write qt thtl 7i 7i But qtl Et1qt2 ri1 n1 SO thtl tht2 Etri1 n1 Substituting above we have qt I 1 Etl1 n1 tht2 Continue this into the future q I 1EII1 n1gt Etltrtk1 nku thak Take the limit as k gt oo and we have qt n I Etl1I1 That is the deviation of the real exchange rate from its unconditional mean is equal to minus the in nite sum of current and expected future home foreign real interest differentials This relationship would hold in a purely neoclassical exible price model But exibleprice monetary models tend not to put much emphasis on the determination of the real exchange rate Monetary models of the exchange rate With sticky nominal prices in contrast emphasize that monetary policy determines real interest rates in the short run That is monetary policy determines it and i But in sticky price models a change in the nominal interest rate for example a drop in it represents a decline in the real rate of interest That is it I Etpt1 pt A monetary easing means it falls this is common usage in the real world But if monetary policy is easing then expected in ation Etpt1 pt actually rises How can it fall When EMU1 pt rises Only if 1 falls and falls more than the increase in EMU1 pt The famous Dombusch overshooting model of the exchange rate displays this mechanism It adds two elements to real interest parity First there is a money demand equation described by mt 2 pt xll39t There is an analogous equation in the foreign country m p ll39 These give us that relative interest rates are determined by relative real money supplies itl39 mtptmfp Dornbusch assumed that any money supply changes were permanent That is the money supply follows a random walk mt mt l I gtxlt gtxlt gtxlt mt mt1 111 If prices were exible and the real interest rate were constant we would have mt at lit pt llEtl3tl i t When the money supply follows a random walk the solution to this difference equation is simply m 9 In the foreign country m 9 Then Dombusch also assumed an ad hoc price adjustment scheme He assumes that in ation re ects partial adjustment of the price towards its longrun value pr1 pt In the foreign country pt1pt 26l3t pt Put these together With our previous results and we have Etpm pt Emil pf 1490 if Then the relative real interest rate is given by nI11lt9itif In the next period I1 Iquot 2 1 149W 11 so Etri1 it1 111 39 Now we can derive 1 1 Etlt1 EEtmtl pt1 Zmt Etpt1 1 1 Zmt pt Etpt1 pt Z lt 161t16lt Therefore EM 191 1 149Etl1 411 1 MXI exit 1 We can derive similarly EM 1k 1 MXl 971 1 Hence q an IEtn1n1m 1h911 91 92 z t i 1 149 1 16 6 It 1t 26 mt pt mt pt Since pt and p are predetermined they are not affected by any shocks to the money supply So the effect of a shock in the home money supply is to cause times the 116 the real exchange rate to use a home real deprec1ation by increase in money The real exchange rate is given by qt 2 st p pt Since pt and p are 116 predeterm1ned the nominal exchange rate also uses by 16 In the long run What happens to the nominal exchange rate In the long run prices adjust fully so pt ultimately rises oneforone With an increase in m The real exchange rate goes to its unconditional mean in the long run so the exchange rate is given by st pt 19 That is in the long run the nominal exchange rate goes up oneforone With the increase in mt The shortrun depreciation 1 1 9 is greater than the longrun depreciation 1 That is there is overshooting Intuitively why does this happen Uncovered interest parity tells us it 1 Etst1 st An increase in mt lowers it But uncovered interest parity tells us that this means E s t H1 S then must fall Since E s t H1 r1ses we must have s rise by even more That is when it we need an expectation of an appreciation of the currency a drop in Etst1 st That means the immediate depreciation increase in st must be greater than the increase in the expected future value of the exchange rate Etsm This model is useful in explaining the high volatility of exchange rates We can write a solution for the nominal exchange rate 1 x16 1 a t mt mt pt pt S We can derive a more general expression for the nominal exchange rate in the following way mt pt mt pt ll39t l39t Rewrite this as mt mt qt St lEtStl St39 Now rewrite this as m m qt1LEs 5 ttl39 11 1 Iterate forward to get 1 gtxlt X k St 11mt mt qt mEtmtl mtl qt1 j Etmtk mtk qtk 11 1 k1 E tstk1 Now we assume k1 11m Hm 11 1 St Zmmt mr qt Es t tk1 0 So we can write 1 Et mt1 mt1 qt1 11 That is the exchange rate is the expected present discounted value of xt 1 1 J St Etxtj 110 11 where xt mt mt qt Note that x is an 11 random variable but it is not a random walk even if m m is a random walk In fact we saw that if m m is a random walk then 116 9 AR1 random variable This in turn implies that q is an ARl So x is the sum of a pure random walk mt m and an AR1 qt qt 2 it i But BIG1 i1l 9it i indicating that it i is an The EngelWest theorem says that nonetheless S will nearly be a random walk when is near to one Note we can solve for st from the in nite sum j 00 SF A 12 110 11 J J 110 11 1 1 mt mt qt 111 11 9j 11 m m 1 1 11 9qt 1 gt1lt gt1lt mt mtmt pt mt pt 119 1 2116mt mt pt pt J39 Etch Now consider the effect of expected in ation on exchange rates From above mt p ltmf p gt zltz z gt We can rewrite this as m m p st pfst 10 I1Ept1ptEtp 11pf 3 mt m qt St 10 If METm E7511 Where We 7ft1 pm pt and 721 711 p We can rewrite as St mt m qt 1Et7zt Et7z1 Suppose expected hometoforeign in ation Ert1 Et7r1 increases Imagine that the current real interest differential is unaffected and all future real interest differentials are unaffected so that qt does not change The increase in home 1 relative to foreign in ation causes the home currency to depreciate st increases Intuitively holding real interest rates constant this change reduces the demand for home relative to foreign money causing the home depreciation Now consider a model in which monetary policy reacts to current conditions In particular the policymaker uses the nominal interest rate as the policy instrument and the instrument rule targets expected in ation and the output gap it pEt7zH1 yfzt and it pEt7Z39t1 7ft 0 gt Ly gt 0 Assume that a home depreciation increases domestic output relative to foreign output Let Vt v be other factors that affect homeforeign output 7 7t Vt Vt 5qt Taking home relative to foreign interest rates we get it i pEt7ztl 7z17Vt V 5qt Using uncovered interest parity and the definition of the real exchange rate 57 1 Er St1 l 5y 1 5y t pt Ii 0 9 7 9 E 7 7r v v l l l l l l 1 Note that an increase in Elm1 7r1 leads to a home appreciation Why 5 at 2 pt pt LEt tl 7 7r v v l57 l57 H1 157t I Now de ne 2 E We can solve the previous equation forward to get StZZ EtZtj39 F0 15y Although we have not fully speci ed the model it is likely that pt p is an 11 random variable Since the other variables in z are likely stationary it follows that z is 11 But there is nothing that says 2 is a random walk The EngelWest theorem says that nonetheless S will nearly be a random walk when is near to one

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