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1 Deparmen of Applied Economics Johns Hopkins Universiy Economics 602 Macroeconomic Theory and olicy Final Exam rofessor Sanjay Chugh Fall 2008 December 8, 2008 NAME: The Exam has a oal of four (4) quesions and pages numbered one () hrough eleven (). Each problem s oal number of poins is shown below. Your soluions should consis of some appropriae combinaion of mahemaical analysis, graphical analysis, logical analysis, and economic inuiion, bu in no case do soluions need o be excepionally long. Your soluions should ge sraigh o he poin soluions wih irrelevan discussions and derivaions will be penalized. You are o answer all quesions in he spaces provided. You may use wo pages (double-sided) of noes. You may use a calculaor. Quesion / 5 Quesion 2 / 5 Quesion 3 / 28 Quesion 4 / 42 TOTAL /00

2 . A Naional Service rogram (5 poins). Consider he following radical policy proposal: raher han axes being levied on individuals and he proceeds of hose axes being used by he governmen o fund various programs, suppose ha every individual pays no axes of any kind bu mus give en hours of his ime every week o naional service. Here you will analyze his naional service program in he conex of he (one-period) consumpion-leisure model we have sudied. Thus, here are now hree uses of he individual s ime: work, leisure, and naional service (he mandaory 0 hours). Assume he following: - Insiuing he naional service program has no effec on any prices or wages in he economy. - Any ime spen volunarily performing naional service beyond he required 0 hours is considered leisure. a. (8 poins) Using he noaion we developed in Chaper 2 (i.e., c o denoe consumpion, n o denoe hours of work per week, l o denoe hours of leisure per week, o denoe he nominal price of consumpion, and W o denoe he nominal hourly wage), consruc he represenaive agen s (weekly) budge consrain in his model wih a naional service program. Recall ha here are 68 hours in one calendar week. rovide brief economic jusificaion for your work. Soluion: The individual is required o give 0 hours per week o naional service. Thus he has (68-0)=58 hours per week lef o allocae o eiher labor or leisure. roceeding compleely analogously as in he sandard model, hen, we finally arrive a he budge consrain c + Wl = 58W. (Refer o Chaper 2 of he Reciaion Noes for he full algebraic derivaion: he derivaion here is idenical excep ha = 0 by assumpion and 68 is replaced by 58). b. (7 poins) Now recall he sandard consumpion-leisure model wih no naional service program and suppose ha boh he consumpion ax rae is zero and he labor ax rae is zero. How does he slope of he budge consrain in his economy compare wih he slope of he budge consrain in he economy wih he naional service program in par a? rovide brief economic inuiion. Soluion: Wih no axes and no naional service program, he budge consrain is c + Wl = 68W. Comparing his wih he budge consrain in par a above, we see ha he slopes of he wo budge consrains are idenical he wo consrains only differ in heir inerceps. The consrain in par a has inerceps which are smaller han he consrain here in par b. The inuiion is ha in he consumpion-leisure model he individual has ime o allocae beween working and no working. The naional service program akes away some of his ime bu oherwise has no effec on he real (inclusive of axes) wage W /.

3 2. Moneary olicy in he MIU Model (5 poins). In his quesion, you will analyze, using indifference curve/budge consrain diagrams, he implicaions of alernaive nominal ineres raes on he represenaive consumer s choices of consumpion and real money balances. Recall ha, wih an insananeous uiliy funcion uc (, M / ) (where, as usual, c denoes consumpion and M / denoes real money balances), he consumpion-money opimaliy condiion (which we derived in Chaper 4) can be expressed as um ( c, / ) M i =, uc ( c, / ) M + i where, again as usual, i is he nominal ineres rae, u c (.) denoes he marginal uiliy of consumpion, and u m(.) denoes he marginal uiliy of real money balances. a. (5 poins) Suppose he cenral bank is considering seing one of wo (and only wo) nominal ineres raes: i and i, wih i 2 > i. On he indifference map below, qualiaively (and 2 clearly) skech relevan budge lines and show he consumer s opimal choices of consumpion and real money under he wo alernaive policies. On he diagram below, noe he poin on he verical axis marked FIXED his denoes a poin ha mus lie on ANY budge consrain. Clearly label your diagram, including he slopes of he budge lines. Soluion: Examining he righ-hand-side of he above, i is clear ha he smaller is i, he flaer is he budge line. Saring from he FIXED poin, draw wo budge lines, wih he budge line wih slope i flaer han he budge line wih slope i2. On he flaer budge line, he consumer s opimal choice of money balances and consumpion is higher han on he seeper budge line. consumpion Real money 2

4 Quesion 2 coninued b. (5 poins) You are a policy adviser o he cenral bank, and any advice you give is based on he goal of maximizing he uiliy of he represenaive consumer. The cenral bank asks you o help i choose beween he wo nominal ineres raes 2 i and i (and only hese wo). Which nominal ineres rae would you recommend implemening? Briefly explain. Soluion: Again as is clear from he diagram, choosing he smaller value of i allows he represenaive consumer o aain a higher level of uiliy (a higher indifference curve), so i is preferred o i 2. c. (5 poins) Suppose insead he cenral bank is open o seing any nominal ineres rae, no jus eiher i or i 2. Wha would your policy recommendaion be? Briefly jusify your recommendaion, and also in he diagram in par (a) skech and clearly label a new budge line consisen wih your policy recommendaion. Soluion: Seing i = 0 (or, echnically speaking, very very very close o zero) would make he budge line compleely fla, and allow he consumer o obain he highes possible uiliy. Noe ha, because indifference curves are downward sloping, if i < 0, hen here would no be a poin of angency beween he budge line and an indifference curve here would no equilibrium. (Indeed, i = 0 is he lowes ha nominal ineres raes can ever go (somehing known as he zero-lower-bound on ineres raes were hey o go lower, a moneary economy (ie, one in which money is used as a medium of exchange) would no exis. A opic for a more advanced course in moneary economics.) 3

5 3. The Fiscal Theory of Exchange Raes (28 poins). In his quesion, you will use he fiscal heory of exchange raes o analyze some consequences of a fixed exchange rae sysem. The model is jus as we have sudied in class in paricular, consumpion is consan a c = in every period, real money demand is described by he funcion, M / = φ ( c, i), holds, and he foreign price level is equal o one in every period (i.e., * = in every period ). The domesic counry runs a fiscal defici of DEF = 5.5 (a negaive defici is a surplus ) every period, and here is no poliical will o ever change his defici. The real money demand funcion is given by φ ( ci, ) = c 0 i, and he exchange rae ha he counry is pegging (for as long as i can) is E = 2 unis of domesic currency per uni of foreign currency. Finally, he foreign real G ineres rae is r * = 0.0, he governmen sars period wih foreign reserves of B 0 = 22, and foreign reserves can never go below zero. a. (3 poins) As long as he fixed exchange rae is in place and is expeced o remain in place, wha is he numerical value of he domesic nominal ineres rae? Briefly jusify your answer. Soluion: Use he ineres pariy condiion. If he peg is expeced o remain in place, ha e means E+ = E, so he ineres pariy condiion ells us ha he domesic nominal ineres rae equals he foreign real ineres, so i = r* = 0.0 during his ime. b. (3 poins) As long as he fixed exchange rae is in place and is expeced o remain in place, wha is he numerical value of he domesic counry s BO surplus or BO defici? Briefly jusify your answer. G G M M Soluion: Recall he governmen budge consrain is B B = DEF. E Seignorage revenue is zero during his period, so he governmen budge consrain reduces G G o B B = DEF. The lef-hand side is he change in foreign reserves during period, which is our definiion of he balance of paymens. Tha is, a counry s BO during a paricular period equals he change in is foreign reserves during ha period. Wih DEF = 5.5, here is a BO surplus of 5.5 every period. c. (3 poins) Based on your answer in par b, is he floaing exchange rae higher han, lower han, or equal o E = 2? Briefly jusify your answer. Soluion: Because he counry is a running a BO surplus, i is accumulaing foreign reserves, meaning he domesic currency is being held weaker han a he floaing rae. The floaing rae is hus smaller han E = 2. 4

6 Quesion 3 coninued d. (4 poins) If markes/invesors for some reason never expec a change in he nominal exchange rae, how many periods will he fixed exchange rae las? Briefly jusify your answer. Soluion: Wih a BO surplus, he domesic cenral bank is accumulaing foreign reserves. There is no economic upper limi on how many foreign reserves a cenral bank can hold, so in principle, he fixed exchange rae could las forever. Suppose he governmen of he domesic counry announces in period T- ha in period T he nominal exchange rae will be E T =.9, and markes/invesors believe his announcemen. e. (6 poins) Wha is he numerical value of he nominal ineres rae in period T- (i.e., compue it )? Briefly jusify your answer, and provide economic inuiion for wha you find, including a brief economic explanaion for why it differs from r * if i does. Soluion: Use he ineres pariy in period T- o compue he nominal ineres rae: E + it = ( + r*) E e T T e Because markes/invesors believe he governmen s announcemen, E T =.9, in which he above ells us ha it = Thus he nominal ineres rae falls below he world ineres rae when markes expec an appreciaion of he currency in he near fuure (going from E = 2 o E =.9 is an appreciaion of he domesic currency). Because he domesic currency is expeced o become sronger in he near fuure, markes/invesors require/demand less ineres compensaion in order o hold bonds denominaed in he domesic currency. 5

7 Quesion 3 coninued f. (5 poins) Wha is he numerical value of seignorage revenue for he domesic governmen in period T-? Briefly jusify your answer, and provide economic inuiion for wha you find, including a brief economic explanaion for why seignorage revenue differs from zero if i does. Soluion: By definiion, seignorage revenue in period T- is given by SR M M T T 2 T =, T which in urn, because of he money demand funcion, can be expressed as SR Lci (, ) Lci (, ) T T T 2 T 2 T =. T Because = E in every period and he peg doesn change unil period T, he erms cancel in he above expression, meaning SRT = L( c, it ) L( c, it 2). We compued in par e ha it = We know from par a above ha it 2 = r* = 0.. Insering hese values in he given money demand funcion, SRT = 0( ) ( 0( + 0.)) = = Seignorage revenue is hus posiive in period T-. The reason for his is fundamenally he same as in par e above: he domesic currency is expeced o become sronger in period T, hence markes/invesors demand for i rises, which allows he domesic governmen o prin more of i and earn posiive seignorage revenue. g. (6 poins) How does he domesic counry s BO in period T- compare o is BO in period T-2? Does i rise, fall, or say he same? Explain precisely, including why. Soluion: In period T-2, because seignorage revenue was zero, he BO surplus was 5.5, as compued in par c above. In period T-, he fiscal surplus (i.e., a negaive DEF) is sill in place, bu he governmen also collecs posiive seignorage revenue, hence he governmen budge consrain shows us ha he BO surplus becomes even larger han i was in period T-2 (specifically, BO surplus = 0.55 (-5.5) = 6.05, bu you did no have o make his calculaion because i was no asked). 6

8 4. The Yield Curve (42 poins). An imporan indicaor of markes beliefs/expecaions abou he fuure pah of he macroeconomy is he yield curve, which, simply pu, describes he relaionship beween he mauriy lengh of a paricular bond (recall ha bonds come in various mauriy lenghs) and he per-year ineres rae on ha bond. A bond s yield is alernaive erminology for is (annual) ineres rae. A sample yield curve is shown in he following diagram: This diagram plos he yield curve for U.S. Treasury bonds ha exised in markes on February 9, 2005: as i shows, a 5-year Treasury bond on ha dae carried an ineres rae of abou 4 percen, a 0-year Treasury bond on ha dae carried an ineres rae of abou 4.4 percen, and a 30-year Treasury bond on ha dae carried an ineres rae of abou 4.52 percen. Recall from our sudy of bond markes ha prices of bonds and nominal ineres raes on bonds are negaively relaed o each oher. The yield curve is ypically discussed in erms of nominal ineres raes (as in he graph above). However, because of he inverse relaionship beween ineres raes on bonds and prices of bonds, he yield curve could equivalenly be discussed in erms of he prices of bonds. In his problem, you will use an enriched version of our infinie-period moneary framework from Chaper 4 o sudy how he yield curve is deermined. Specifically, raher han assuming he represenaive consumer has only one ype of bond (a one-period bond) he can purchase, we will assume he represenaive consumer has several ypes of bonds he can purchase a oneperiod bond, a wo-period bond, and, in he laer pars of he problem, a hree-period bond. Le s sar jus wih wo-period bonds. We will model he wo-period bond in he simples possible way: in period, he consumer purchases B TWO unis of wo-period bonds, each of, which has a marke price btwo and a face value of one (i.e., when he wo-period bond pays 7

9 off, i pays back one dollar). The defining feaure of a wo-period bond is ha i pays back is face value wo periods afer purchase (indeed, hence he erm wo-period bond ). The one-period bond is jus as we have discussed in class and in Chaper 4. Mahemaically, hen, suppose (jus as in Chaper 4) ha he represenaive consumer has a lifeime uiliy funcion saring from period M M 2 2 M M ln c ln ln β c+ + β ln + β ln c+ 2 + β ln + β ln c+ 3+ β ln..., and his period- budge consrain is given by c + B + B + M + S a = Y + M + B + B + ( S + D ) a. b b, TWO TWO TWO 2 (Based on his, you should know wha he period + and period +2 and period +3, ec. budge consrains look like). This budge consrain is idenical o ha in Chaper 4, excep of course for he erms regarding wo-period bonds. Noe carefully he iming on he righ hand side TWO in accordance wih he defining feaure of a wo-period bond, in period, i is B 2 ha pays back is face value. The res of he noaion is jus as in Chaper 4, including he fac ha he subjecive discoun facor (i.e., he measure of impaience) is β <. a. (5 poins) Qualiaively represen (using he axes below) he yield curve shown in he diagram above in erms of prices of bonds raher han ineres raes on bonds. Tha is, in he empy se of axes below, plo (qualiaively) on he verical axis he prices associaed wih he bonds of various mauriy lenghs show in he diagram above. Soluion: Wih mauriy lenghs ploed on he horizonal axis, he yield curve in erms of bond prices is downward-sloping. This follows simply because of he inverse relaionship beween bond prices and ineres raes. The yield curve shown above is in erms of ineres raes and is sricly increasing; hence he associaed yield curve in erms of prices mus be sricly decreasing. 8

10 Quesion 4 coninued b. (0 poins) Based on he uiliy funcion and budge consrain given above, se up an appropriae Lagrangian, and use i o derive he represenaive consumer s firs-order condiions wih respec o boh B and B (as usual, he analysis is being conduced from TWO he perspecive of he very beginning of period ). Define any auxiliary noaion ha you need in order o conduc your analysis. TWO Soluion: The only wo firs-order condiions ha you needed here are hose on B and B. Denoing by λ he Lagrange muliplier on he period- budge consrain and by λ + he Lagrange muliplier on he period-+ budge consrain, he wo firs-order condiions, respecively, are and λ + βλ + = b 0 btwo, 2 λ β λ =. Noe well he +2 ime subscrips in he second expression; his follows from he fac he a woperiod bond purchased in period does no repay is promised face value unil period +2. (Refer back o roblem Se 4 for an analogous sock-pricing model in which socks ook wo periods o pay off heir capial gains and dividends.) c. (0 poins) Using he wo firs-order condiions you obained in par b, consruc a relaionship beween he price of a wo-period bond and he price of a one-period bond. btwo, Your final relaionship should be of he form =..., and on he righ-hand-side of his expression should appear (poenially among oher hings), b. (Hin: in order o ge b ino his expression, you may have o muliply and/or divide your firs-order condiions by appropriaely-chosen variables.) b βλ + Soluion: From he firs expression above, we have, as usual ha =. From he second λ 2 btwo, β λ + 2 expression above, we analogously can obain =. We can rewrie his expression λ for he price of a wo-period bond as βλ βλ =, btwo, λ+ λ in which we have simply muliplied and divided he preceding expression by λ + (i.e., we have muliplied by one, always a valid mahemaical operaion). The final erm on he far righ-handside is nohing more han he price of a one-period bond, so we can wrie 9

11 Quesion 4c coninued (if you need more space) btwo, βλ + 2 b =, λ which saisfies he form of he relaionship you were asked o derive. We can acually boil his down furher, hough. Noe ha he price of one-period bond purchased in period +2 would b βλ + 2 be given by + =, which follows from opimizaion wih respec o period + oneperiod bond holdings. Using his expression in he period- price of a wo-period bond, we hus λ + obain + =, btwo, b b + which is a key idea in finance heory: he price of a muli-period asse (bond) is nohing more han he produc of he prices of wo consecuive one-period asses (bond). d. (5 poins) Suppose ha he opimal nominal expendiure on consumpion (c) is equal o in every period. Using his fac, is he price of a wo-period bond greaer han, smaller han, or equal o he price of a one-year bond? If i is impossible o ell, explain why; if you can ell, be as precise as you can be abou he relaionship beween he prices of he wo bonds. (Hin: you may need o invoke he consumer s firs-order condiion on consumpion) Soluion: Sar wih he relaionship βλ = derived above. If nominal consumpion btwo, + 2 λ + expendiures are consan (and equal o one) every period, his means ha λ = every period. (This conclusion follows from he fac ha he FOC wih respec o consumpion is /c λ = 0 in every period, which can be rearranged o λ = ). If he muliplier is one every period, we c immediaely have b Because β <, we conclude btwo, <. btwo, b = β. b e. (5 poins) Now suppose here is also a hree-period bond. A hree-period bond purchased in any given period does no repay is face value (also assumed o be ) unil hree periods afer i is purchased. The period- budge consrain, now including one-, wo-, and hree-period bonds, is given by c + B + B + B + M + S a = Y + M + B + B + B + ( S + D ) a, b b, TWO TWO b, THREE THREE TWO THREE 2 3 THREE, where B is he quaniy of hree-period bonds purchased in period and bthree is associaed price. Following he same logical seps as in pars b, c, and d above (and 0

12 Quesion 4e coninued coninuing o assume ha nominal expendiure on consumpion (c) is equal o one in period every period), is he price of a hree-year bond greaer han, smaller han, or equal o he price of a wo-year bond? If i is impossible o ell, explain why; if you can ell, be as precise as you can be abou he relaionship beween he prices of he wo bonds. (Noe: if you can answer his quesion wihou seing up a Lagrangian, you may do so.) Soluion: Exending he Lagrangian from above, he firs-order condiion wih respec o B is THREE λ + β λ + =, b, THREE which can be rearranged o yield muliplying by one, we can express his as 3 b, THREE + 3 λ 2 b, THREE λ+ 2 λ+ λ β λ =. Jus like in par c above, by cleverly β λ βλ βλ =, which, in exacly he same way as in par c, we can express in erms of chained one-period bond prices, βλ =. b, THREE + 3 b b + λ + 2 If he Lagrange muliplier λ is consan every period, we can conclude he price of a hreeperiod bond is smaller han he price of a wo-period bond (which in urn, from par c, is smaller han he price of a one-period bond). This again follows because β <. f. (5 poins) Suppose ha β = Using your conclusions from pars d and e, qualiaively plo a yield curve in erms of bond prices in he se of axes below (obviously, you can plo only hree differen mauriy lenghs here). Soluion: Based on he analyses in pars d and e, he price of bonds is clearly negaivelyrelaed o is mauriy lengh, hence he yield curve in erms of prices is sricly decreasing. This is jus as your skech of he empirical yield curve in par a. g. (5 poins) Wha is he single mos imporan reason (economically, ha is) for he shape of he yield curve you found in par f? (This requires only a brief, qualiaive/concepual response.)

13 Soluion: Re-examining our conclusions/analyses in pars d, e, and f, he sole reason we were able o reach he conclusions we reached in each of hose pars was he fac ha β <. Thus, he idea of impaience and is effecs on he macroeconomy rears is head again, his ime wih respec o bond prices of differen mauriies. The concepual idea is simple: because of impaience, he longer a bond purchaser mus wai o receive a given face value, he less he will be willing o pay for i oday (and his is refleced in bond marke prices hrough he bond demand funcion for differen mauriy bonds). END OF EXAM 2