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# ch8ans - Foundations of International Macroeconomics1...

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Foundations of International Macroeconomics 1 Workbook 2 Maurice Obstfeld, Kenneth Rogo f , and Gita Gopinath Chapter 8 Solutions 1 . First think about the continuous-time case. At time t =0thema rke t t = T level from time T onward. In that case, we can show that the exchange rate is indeterminate ,sotha tth egov e rnm en t spo l icyi sno tcoh e r en t( in that it does not tie down a unique market equilibrium). Let perfect-foresight equilibrium be described by the continuous-time Cagan model [eq. (70) on p. 559 of the book], m t = e t η o e t . Let e a T be an arbitrary time- T exchange rate and suppose the market rmly expects that rate to prevail. Then the preceding Cagan equation, coupled with the terminal condition e T = e a T , shows that the exchange rate path e t = 1 η Z T t exp[( t s ) / η ] m s d s +exp[( t T ) / η ] e a T will equilibrate markets for t [0 ,T ]. T when con- fronted with this exchange rate path. The authority has no choice, in view of its vow of a constant exchange rate from T on, but to set the fundamental m T = e a T and to hold m t = e a T for all t>T .W h y

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interval [ T, authority must validate (ratify) any market expectation whatsoever. Any initial exchange rate can be an equilibrium because each is conditioned on a di f erent expectation of what the (constant) money supply path will be from date T onward. The situation is subtly di f erent in discrete time ,inwh ichcasethemode l is m t = e t η ( e t +1 e t ) . The basic reason discrete time makes a di f erence is that now, if the market rmly foresees a date T rate of e a T ,thereare two distinct ways the author- ity can ful ll its promise of a constant exchange rate from date T on (two alternatives which collapse to one in continuous time). First, the authority could still set m t = e a T for t = T and for t>T , as in the continuous-time setting±in which case the exchange rate again is not uniquely determined. Alternatively, if the authority can commit not to adjust m T fully to validate e a T , but instead only to set m t = e a T for t strictly greater than T only, we again get an exchange rate path constant at e a T starting on date T .Inth i s second case, however, the exchange rate is uniquely determined. To xideas, suppose the authority can commit to a xed value m T for date T .I nt
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ch8ans - Foundations of International Macroeconomics1...

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