MSc Financial Economics: International Finance. Bubbles in the Foreign Exchange Market. Anne Sibert. Revised Spring Contents

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1 MSc Financial Economic: International Finance Bubble in the Foreign Exchange Market Anne Sibert Revied Spring 203 Content Introduction The Mone Market Solving for the Path of the Exchange Rate Introduction During the Bretton Wood era between the end of World War II and Augut 97, the world wa on a xed exchange rate tem and there wa little international capital mobilit. The primar determinant of the balance of pament wa the trade balance. The exchange rate wa viewed a the relative price of home and foreign good. The equilibrium value of the exchange rate wa one that led to trade balance. In a world where capital i mobile acro indutrialied countrie, thi view i not appropriate and the exchange rate i better viewed a the relative price of di erent currencie or the relative price of nancial aet denominated in di erent currencie. The model in thi lecture i an example of the former approach. It i particularl ueful for looking at how anticipated change in the current and future value of the mone uppl a ect the current exchange rate. The model i of a mall open econom with a ingle good. The home currenc i held onl b dometic reident and home reident do not hold foreign currenc. Both reident of the home countr and reident of the ret of the world hold home-currenc and foreign-currenc bond. Output i aumed to be exogenou. Time occur in dicrete interval. Equilibrium in the mone market i given b 2. The Mone Market

2 MSc Financial Economic: International Finance Bubble in the Foreign Exchange Market 2 M t =P t = Y I t ; > 0 () where M t i the mone uppl, P t i the home-currenc price of the good, Y i (contant) output or income and I t i one plu the home-currenc interet rate. The above equation i imilar to the LM curve of undergraduate macroeconomic coure. The righthand ide i the demand for real balance, which i aumed to be increaing in income and decreaing in the nominal interet rate. The parameter i referred to a the emi-elaticit of mone demand with repect to the interet rate and it meaure how enitive mone demand i to the interet rate. The higher i, the greater the enitivit; a goe to zero, mone demand become inentive to the interet rate. The particular functional form i choen becaue it i epeciall tractable. Take the logarithm of both ide of equation () and let mall letter denote the logarithm of capital letter o that, for example, x = ln X. We have m t p t = i t : (2) 2.. Purchaing power parit. Aume that purchaing power parit hold. Let E t be the exchange rate, or the price of the foreign currenc in term of the home currenc and let P t be the foreign currenc price of the ingle good. The aumption of purchaing power parit here mean that it cot the ame amount to bu the good whether one bu it directl, paing P t unit of home currenc, or whether one bu Pt unit of the foreign currenc with E t Pt unit of the home currenc and bu the good with the foreign currenc. Thu, P t = E t P t : (3) A the countr i mall, we take the foreign currenc price of the good a given or exogenou. To ave on notation we will aume that it i contant. We can then pick the unit in which the foreign good i meaured o that the foreign currenc price equal one. Then, taking logarithm of both ide of the above equation we have

3 MSc Financial Economic: International Finance Bubble in the Foreign Exchange Market 3 p t = e t : (4) 2.2. Uncovered interet parit. The fundamental aumption of the monetar approach i that interetbearing nancial aet denominated in di erent currencie are perfect ubtitute. Thi mean that invetor care onl about the expected return on aet and not about how rik the are. Let I t be one plu the interet rate on foreign-currenc denominated aet. The mall-countr aumption mean that thi variable i alo exogenou. We will further aume that it i a contant. An invetor can take one unit of home currenc and invet it in home-currenc bond. At the end of the period he will have I t unit of home currenc. Or, he can convert one unit of home currenc into =E t unit of foreign currenc and then invet the foreign currenc in foreign-currenc bond. At the end of the period he will have I =E t unit of foreign currenc, which he can then trade back for E t I =E t unit of home currenc. Thu, for invetor to be willing to hold both home-currenc bond and foreign-currenc bond, on average I t mut be equal to E t I =E t. Thi condition i called uncovered interet parit Perfect foreight. We aume that there i no uncertaint in the model. Market participant know the entire hitor and the entire future path of the mone uppl. Thu, the can olve for the path of the exchange rate. Hence, we aume that market participant can correctl predict the future exchange rate. Thi aumption i called perfect foreight. In thi cae uncovered interet parit implie that I t = E t I =E t : (5) Taking logarithim of both ide ield i t = i e t e t : (6) Subtituting equation (4) and (6) into equation (2) ield m t e t = (e t e t ): (7)

4 MSc Financial Economic: International Finance Bubble in the Foreign Exchange Market 4 Thi ummarie the model. It i a rt-order linear di erence equation with a contant coe cient. We could rewrite thi a e t = a cm t be t ; (8) where the value of a; b and c can be found from equation (7) The equilibrium i not table. Tpicall in mathematic book, equation uch a equation (8) would be olved backward to nd e t : Thi would work a follow. B equation (8), the time-(t ) exchange rate depend on the time-t mone uppl and the time-t exchange rate. Alo b equation (8), the time-t exchange rate depend on the time-t mone uppl and on the time-(t ) exchange rate. Hence, the time-t exchange rate can be expreed a a function of the time-t mone uppl, the time-(t ) mone uppl and the time-(t ) exchange rate. The time-(t ) exchange rate depend on the time-(t 2) mone uppl and the time-(t 2) exchange rate. Thu, the time-t exchange rate can be expreed a a function of the time-t, the time-(t ) and the time-(t 2) mone upplie and the time-(t 2) exchange rate. Continuing in thi fahion, the time-t exchange rate depend on the mone uppl for period zero through t and on e 0. The equation i the dicrete-time analogue of a di erential equation. Recall that there are a continuum of a olution to a rt-order di erential equation and the wa we tpicall pick out one of them i to a that we know the value of the olution a ome point. Solving the above equation backward and uing ome hitorical value to pin down the olution i the analogue to thi. Conider a imple example of equation (8) where m t i contant and n a cm. Then we can graph the equation. It will turn out that everthing intereting depend on the ign and magnitude of the variable b: The gure below depict the cae of < b:

5 e t e t = e t e t = n be t MSc Financial Economic: International Finance Bubble in the Foreign Exchange Market 5 e e 2 e e 0 et I have graphed the time-t value of the exchange rate on the horizontal axi and the time-(t ) value on the vertical axi. The two upward loping line are e t = n be t and e t = e t. The aumption that b > enure that the lope of the former exceed the lope of the latter. The point where the two curve interect i the tationar point. There i equilibrium and the exchange rate i contant. Thu, if the econom i ever at thi point, the exchange rate doe not change. Imagine we tarted o at ome initial value e 0. Then we can nd e on the vertical axi b uing the curve e = n be 0 :Once we have e on the vertical axi we can nd e on the horizontal axi b uing the curve e = e 0. Repeating the proce, it i clear from the picture that the exchange rate goe o monotonicall to minu in nit. If we had choen an e 0 greater than the tationar point, then we would have found that the exchange rate goe o to plu in nit. Onl when we tart at the interection of the two curve doe thi not happen. 2 Looking at equation (7), we ee that n in equation (8) mut be ( ) = >. Our model of the exchange rate i not table! The above reult i tpical in model of nancial aet; it turn out to be enible. Conider equation (7) again. The current exchange rate depend on the current tate of the econom, ummaried b the mone uppl, and the expected depreciation of the exchange rate. People have perfect foreight; hence, the The olution i monotonic if it i alwa increaing or alwa decreaing. 2 Other cae are 0 < b < ; which i monotonic and table (the exchange rate goe to the tationar point), < b < 0, which i table and ha ocillation and b < which ha exploive ocillation. Tr drawing the picture to ee for ourelf.

6 MSc Financial Economic: International Finance Bubble in the Foreign Exchange Market 6 expected future exchange rate i the actual future exchange rate. Thu, the current exchange rate depend on the current tate of the econom and next period exchange rate. Similarl, next period exchange rate depend on next-period tate of the econom and the two-period-ahead exchange rate. Thu, the current exchange rate depend on the current tate of the econom, next-period tate of the econom and the two-period-ahead exchange rate. Continuuing with thi logic, the current exchange rate depend on the current and all future tate of the econom and the exchange rate in nitel far out into the future. The graphical olution ued a di erent approach. With our graph, we tried to nd the current exchange rate, given the pat exchange rate and the current tate of the econom. Thi wa how people in the cience would tpicall proceed and how mathematic book tend to view the problem. If we think about the intuition, there i omething odd about thi. Firt, a the above intuition ugget, it doe not eem enible to view the current exchange rate a depending on the pat. Second, in the graphical example, the mone uppl i contant. So, wh hould the exchange rate var over time? Doe it not eem more reaonable that it would be contant a well? What if intead of tring to olve the equation backward a we did with the graphical approach we olve it forward, a the thought experiment in the previou paragraph ugget. In the graphical olution, we aw that the path of the exchange rate depended on the initial value of the exchange rate. Relating thi to the math, rt-order di erence equation have an in nite number of olution, jut a rt-order di erential equation do. When we olve a di erence or di erential equation backward we pick out a particular olution b aing that we know it value at ome point. Tpicall, we would a we know what the value i at zero. Thi i called a boundar condition. When we olve the model forward we will till have an in nite number of olution. But, now we cannot pick out a path b aing we know the initial value. From the thought experiment, the current exchange rate i going to depend on the current and all future value of the mone uppl and on the exchange rate in nitel far out into the future. Thu, our boundar condition i going to be a condition on thi in nitel far in the future exchange rate. We will aume that it i unreaonable for the exchange rate to go to plu or minu in nit when it i not warranted b the fundamental. It will turn on that there i onl one path that ati e thi. In the above picture, thi amount to aing that the onl reaonable equilibrium when the mone uppl i contant i the tead-tate equilibrium where the exchange rate i contant a well. In the next ection, I how how to

7 MSc Financial Economic: International Finance Bubble in the Foreign Exchange Market 7 olve the rt-order linear di erence equation forward and how to pick out the olution we want. 3. Solving for the Path of the Exchange Rate In thi ection I conider two method for olving for the path of the exchange rate. The rt i b brute force; the econd b uing lag operator. 3.. Solving b brute force. Conider equation (7) again: m t e t = (e t e t ): (9) rate The rt tep in olving forward i to olve for the current exchange rate in term of next-period exchange e t = m t e t: (0) Equation (0) mut hold for ever t; hence it hold at t : e t = m t e t2: () Subtitute equation () into equation (0): e t = m t mt e t2 : (2) Gather term e t = m t m t 2 e t2: (3) Do thi again. B equation (0) e t2 = m t2 e t3: (4)

8 MSc Financial Economic: International Finance Bubble in the Foreign Exchange Market 8 Subtitute equation (4) into equation (3): e t = m t m t 2 mt2 e t3 : (5) Gather term e t = m t 2 m t m t2# # 2 3 e t3: (6) We now notice a pattern; it we keep repeating the proce we would get e t = m t m t ::: ::: T # T m tt # T e tt : (7) To ee thi, note that when we went out to the exchange rate e t3 ; we multiplied thi exchange rate b a term raied to the power 3. Thu, when we go out to the exchange rate e tt, we multipl thi exchange rate b a term raied to the power T. When we went out to the exchange rate e t3 ; we included mone upplie and contant income out to the period t 3 = t 2:Thu, when we go out to the exchange rate e tt, we include mone upplie and contant income out to the period t T : Rewriting equation (7) uing the ummation notation ield e t = TX m t TX T e tt : (8)

9 MSc Financial Economic: International Finance Bubble in the Foreign Exchange Market 9 Now uppoe that we repeat thi an in nite number of time. That i, we let T go to in nit. 3 e t = X m t X lim T! T e tt : (9) Recall (or memorie!) the rule that X x = i jxj < : (20) x Thu we have X = = = : (2) Subtituting equation (2) into equation (9) ield e t = X m t T lim e tt : (22) T! The lat term goe to zero unle the exchange rate goe to plu or minu in nit. Thu, our boundar condition i that the lat term mut equal zero. Thu our olution i e t = X m t : (23) Thi ha intuitive appeal. The exchange rate depend on the current and all future value of the mone uppl. The importance of a future value of the mone uppl in determing the preent exchange rate depend on how far out in the future it i. The further in the future, the le e ect it ha on the current exchange rate. If we allow for bubble then it can be hown that e t = X m t t c ; (24) 3 Technicall, thi require that the mone uppl doe not grow o fat that the limit of the um involving the mone uppl fail to exit.

10 MSc Financial Economic: International Finance Bubble in the Foreign Exchange Market 0 where c i an contant. Suppoe that the mone uppl i contant then we can how that e t = m t c (25) ati e the equilibrium condition. Subtitute thi into m e t = (e t e t ) t t # t m m c = m c m c, t t # t c = c c, = which i true

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