Problem 1 / 25 Problem 2 / 10 Problem 3 / 15 Problem 4 / 30 Problem 5 / 20 TOTAL / 100

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1 Deparmen of Applied Eonomis Johns Hopkins Universiy Eonomis 60 Maroeonomi Theory and Poliy Miderm Exam Suggesed Soluions Professor Sanjay Chugh Summer 0 NAME: The Exam has a oal of five (5) problems and pages numbered one () hrough foureen (4). Eah problem s oal number of poins is shown below. Your soluions should onsis of some appropriae ombinaion of mahemaial analysis, graphial analysis, logial analysis, and eonomi inuiion, bu in no ase do soluions need o be exepionally long. Your soluions should ge sraigh o he poin soluions wih irrelevan disussions and derivaions will be penalized. You are o answer all quesions in he spaes provided. You may use one page (double-sided) of noes. You may no use a alulaor. Problem / 5 Problem / 0 Problem 3 / 5 Problem 4 / 30 Problem 5 / 0 TOTAL / 00

2 Problem : Two-Period Consumpion-Savings Framework (5 poins). Consider he woperiod eonomy (wih zero governmen spending and zero axaion), in whih he represenaive onsumer has no onrol over his real inome (y in period and y in period ). The lifeime uiliy funion of he represenaive onsumer is ( ) u, = ln + ln. The lifeime budge onsrain (in real erms) of he onsumer is, as usual, y + = y+ + ( + r) a0. + r + r Suppose he onsumer begins period wih zero ne asses (a 0 = 0), and as per he noaion in Chapers 3 and 4, r denoes he real ineres rae. For use below, i is onvenien o define he gross real ineres rae as R = +r (as a poin of erminology, r is he ne real ineres rae). a. (5 poins) Se up a lifeime Lagrangian formulaion for he represenaive onsumer s lifeime uiliy maximizaion problem. Define any new noaion you inrodue. y Soluion: The lifeime Lagrangian is ln + ln + λ y+ + (+ ra ) 0 + r + r, whih onains Lagrange muliplier λ.

3 Problem oninued b. (0 poins) Based on he Lagrangian from par a, ompue he firs-order ondiions wih respe o and. Then, use hese firs-order ondiions o derive he onsumpion-savings opimaliy ondiion for he given uiliy funion. NOTE: Your final expression of he onsumpion-savings opimaliy ondiion should be presened in erms of he raio. Furhermore, in obaining he represenaion of he onsumpion-savings opimaliy ondiion, you should express any (+r) erms ha appear as R insead (if you have no already done so). Thus, he final form of he ondiion o presen is =... in whih he righ hand side is for you o deermine. Your final expression may NOT inlude any Lagrange mulipliers in i. Clearly presen he imporan seps and logi of your analysis. Soluion: The firs-order ondiions wih respe o and are λ = 0 λ = 0 + r + r The seond expression ells us λ =. Insering his expression for he muliplier ino he firs + r expression gives =. Replaing (+r) by R, and muliplying boh sides by, he onsumpion-savings opimaliy ondiion in he requesed form is = R.

4 Problem b oninued (more work spae). (4 poins) Saring from your expression in par b (ha is, saring from he expression you obained ha has he form... = ), onsru he naural logarihm of he expression. In doing so, reall he following resuls regarding algebrai manipulaion of naural logs: for y any wo x > 0 and y > 0, ln(xy) = ln x + ln y, and ln( x ) = yln x. Your final expression here should be of he form ln =... Clearly presen he imporan seps and logi of your analysis. Soluion: Applying he naural logarihm o boh sides of he final expression in par b, we have = ln ln R. 3

5 Problem oninued Reall from basi miroeonomis ha he elasiiy of a variable x wih respe o anoher variable y is defined as he perenage hange in x indued by a one-peren hange in y. As you sudied in basi miroeonomis, elasiiies are espeially useful measures of he sensiiviy of one variable o anoher beause hey do no depend on he unis of measuremen of eiher variable. A onvenien mehod for ompuing an elasiiy (whih we will no prove here) is ha he elasiiy of one variable (say, x) wih respe o anoher variable (say, y) is equal o he firs derivaive of he naural log of x wih respe o he naural log of y. (Read his saemen very arefully.) d. (6 poins) Saring from your expression in par (ha is, saring from he expression you obained ha has he form ln =...), ompue he elasiiy of he raio wih respe o he gross real ineres rae R. The resuling expression is he elasiiy of onsumpion growh (beween period one and period wo) wih respe o he (gross) real ineres rae for he given uiliy funion. Clearly presen he imporan seps and logi of your analysis. Soluion: Being areful abou applying he mehod of ompuaion of an elasiiy desribed above, he elasiiy of onsumpion growh wih respe o he (gross) real ineres rae is ln( / ) =. ln R This follows immediaely from he final expression in par. 4

6 Problem : Governmen Budges and Governmen Asse Posiions (0 poins). Jus as we an analyze he eonomi behavior of onsumers over many ime periods, we an analyze he eonomi behavior of he governmen over many ime periods. Suppose ha a he beginning of period, he governmen has zero ne asses. Also assume ha he real ineres rae is always r = 0. The following able desribes he real quaniies of governmen spending and real ax revenue he governmen olles saring in period and for several periods hereafer. Period Real governmen expendiure (g) during he period Real ax olleions during he period Quaniy of ne governmen asses a he END of he period a. (6 poins) Complee he las olumn of he able based on he informaion given. Briefly explain he logi behind how you alulae hese values. Soluion: The algebra o ompue he lis of numerial values given in he las olumn above is o repeaedly apply he governmen flow budge onsrain b = g + ( + r) b, being areful o updae he ime period and he new bond posiion a he end of every ime period. b. (4 poins) Suppose insead he governmen ran a balaned budge every period (i.e., every period i olleed in axes exaly he amoun of is expendiures ha period). In his balaned-budge senario, wha would be he governmen s ne asses a he end of period +4? Briefly explain/jusify. Soluion: Again using he governmen flow budge onsrain b = g + ( + r) b, a balaned budge every period would mean g = 0 in every period. Doing he same proedure as in par a would learly give ha he governmen s ne asse posiion a he end of every ime period, inluding a he end of period +4, is exaly zero (given ha he governmen begins period wih zero ne asses). 5

7 Problem 3: A Conraion in Credi Availabiliy (5 poins). The graph below (on he nex page) shows he usual wo-period indifferene-urve/budge onsrain diagram, wih period- onsumpion ploed on he horizonal axis, period- onsumpion ploed on he verial axis, and he downward-sloping line represening, as usual, he onsumer s LBC. Throughou all of he analysis here, assume ha r = 0 always. Furhermore, here is no governmen, hene never any axes. Suppose ha he represenaive onsumer has lifeime uiliy funion u (, ) = ln+ ln, and ha he real inome of he onsumer in period and period is y = and y = 8. Finally, suppose ha he iniial quaniy of ne asses he onsumer has is a 0 = 0. EVERY onsumer in he eonomy is desribed by his uiliy funion and hese values of y, y, and a 0. a. (6 poins) If here are no problems in redi markes whasoever (so ha onsumers an borrow or save as muh or as lile as hey wan), ompue he numerial value of he opimal quaniy of period- onsumpion. (Noe: if you an solve his problem wihou seing up a Lagrangian, you are free o do so as long as you explain your logi.) Soluion: The onsumpion-savings opimaliy ondiion (given he naural-log uiliy funion) is given by / = +r = (he seond equaliy follows beause r = 0 here). Thus, a he opimal hoie, i is he ase ha =. Using his relaionship (and again using he fa ha r = 0 here), we an express he onsumer s LBC as + = y + y = 0, whih obviously implies he opimal hoie of period- onsumpion is = 0. Noe: alhough you were no asked o ompue i, you ould have ompued he implied value of he onsumer s asse posiion a he end of period one. Beause a 0 = 0, y =, and we jus ompued = 0, he asse posiion a he end of period one is a = y = (i.e., posiive ). b. (9 poins) Now suppose ha beause of problems in he finanial seor, no onsumers are allowed o be in deb a he end of period. Wih his finanial resriion in plae, ompue he numerial value of he opimal quaniy of period- onsumpion. ALSO, on he diagram on he nex page, qualiaively and learly skeh he opimal hoie wih his finanial resriion in plae (qualiaively skehed already for you is he opimal hoie if here are no problems in finanial markes). Your skeh should indiae boh he new opimal hoie and an appropriaely-drawn and labeled indifferene urve ha onains he new opimal hoie. (Noe: if you an solve his problem wihou seing up a Lagrangian, you are free o do so as long as you explain your logi.) Soluion: Beause in par a (ie, wihou any redi resriions), he represenaive onsumer was hoosing o NOT be in deb a he end of period (i.e., a > 0 under he opimal hoie in par a), he imposiion of he redi resriion, nohing hanges ompared o par a. Tha is, he opimal hoie of period- onsumpion is sill 0. Hene, in he diagram below, he opimal hoie in he presene of redi onsrains is exaly he same as he opimal hoie wihou redi onsrains. The general lesson o draw from his example and our analysis in lass is ha i is no neessarily he ase ha finanial marke problems mus and always spill over ino real eonomi aiviy (i.e., onsumpion in his ase). 6

8 Problem 3b oninued Opimal hoie if no redi-marke problems Consumer LBC 7

9 Problem 4: The Consumpion-Leisure Framework (30 poins). In his quesion, you will use he basi (one period) onsumpion-leisure framework o onsider some labor marke issues. Suppose he represenaive onsumer has he following uiliy funion over onsumpion and labor, A =, + φ + ul (,) ln n φ where, as usual, denoes onsumpion and n denoes he number of hours of labor he onsumer hooses o work. The onsans A and φ are ouside he onrol of he individual, bu eah is srily posiive. (As usual, ln( ) is he naural log funion.) Suppose he budge onsrain (expressed in real, raher han in nominal, erms) he individual faes is = ( ) w n, where is he labor ax rae, w is he real hourly wage rae, and n is he number of hours he individual works. Reall ha in one week here are 68 hours, hene n + l = 68 mus always be rue. The Lagrangian for his problem is A [ w ) n ], + φ + φ ln n + λ ( in whih λ denoes he Lagrange muliplier on he budge onsrain. a. (6 poins) Based on he given Lagrangian, ompue he represenaive onsumer s firsorder ondiions wih respe o onsumpion and wih respe o labor. Clearly presen he imporan seps and logi of your analysis. Soluion: The firs-order ondiions wih respe o onsumpion and labor are λ = 0 φ An + λ( ) w = 0 8

10 Problem 4 oninued b. (7 poins) Based on ONLY he firs-order ondiion wih respe o labor ompued in par a, qualiaively skeh wo hings in he diagram below. Firs, he general shape of he relaionship beween w and n (perfely verial, perfely horizonal, upward-sloping, downward-sloping, or impossible o ell). Seond, how hanges in affe he relaionship (shif i ouwards, shif i in inwards, or impossible o deermine). Briefly desribe he eonomis of how you obained your onlusions. (IMPORTANT NOTE: In his quesion, you are no o use he firs-order ondiion wih respe o onsumpion nor any oher ondiions.) Soluion: The firs-order ondiion wih respe o labor an be rearranged o (if we wan o pu φ An i in verial axis/horizonal axis form) w =. Given ha φ > 0, here is learly an λ ( ) upward sloping relaionship beween w and n. Ploing his below (and ignoring onvexiy/onaviy issues, whih are governed by he pariular magniude of φ ) gives an upward sloping relaionship holding A,, and λ onsan. This is he labor supply funion. Then, saring from his upward-sloping relaionship, a rise in he ax rae (holding A, λ, and n onsan) auses he enire funion o shif inwards. The laer effe is due o individuals working fewer hours when he ax rae rises, all else equal, due o he derease in heir afer-ax real wage. real wage labor 9

11 Problem 4 oninued. (4 poins) Now based on boh of he wo firs-order ondiions ompued in par a, onsru he onsumpion-leisure opimaliy ondiion. Clearly presen he imporan seps and logi of your analysis. Soluion: As usual, his requires eliminaing he Lagrange muliplier aross he wo expressions. The firs-order ondiion on onsumpion gives lambda = /. Insering his ino he firs-order ( )w ondiion on labor gives An φ =. Or, muliplying hrough by, he onsumpion-leisure opimaliy ondiion an be expressed as An φ = ( )w. / d. (7 poins) Based on boh he onsumpion-leisure opimaliy ondiion obained in par and on he budge onsrain, qualiaively skeh wo hings in he diagram below. Firs, he general shape of he relaionship beween w and n (perfely verial, perfely horizonal, upward-sloping, downward-sloping, or impossible o ell). Seond, how hanges in affe he relaionship (shif i ouwards, shif i in inwards, or impossible o deermine). Briefly desribe he eonomis of how you obained your onlusions. labor 0

12 Problem 4d oninued (more work spae) Soluion: The budge onsrain says ha = ( )wn. Subsiuing his ino he onsumpionleisure opimaliy ondiion from par, we have ( )wn An φ = ( )w. The (-)w erms on he lef-hand and righ-hand sides obviously anel, leaving n An φ =, or, ombing he powers in n, An +φ =. Ploing his in he spae above, we will have n = A whih learly does no depend on he real wage w a all. Hene, his is a verial line a he value A +φ. e. (6 poins) How do he onlusions in par d ompare wih hose in par b? Are hey broadly similar? Are hey very differen? Is i impossible o ompare hem? Desribe as muh as you an abou he eonomis when omparing he pair of diagrams. Soluion: Broadly, he differene beween par b and par d is ha par b is a miroeonomi analysis, while par d is a maroeonomi analysis. More preisely, par b is, inuiively, a purely slope argumen, raher han boh a slope and a level argumen in par d. The analysis in par b is anamoun o analyzing he effes of poliy on jus he labor marke (why? beause he analysis here reas onsumpion as a onsan). The analysis in par d insead is anamoun o analyzing joinly he effes of poliy on labor markes and goods markes. To he exen ha here are feedbak effes beween he wo markes, here is no reason o hink he answers from he analyses mus be he same. The laer is he basis for hinking of he analysis in par b as a miroeonomi analysis and he analysis in par d as a maroeonomi analysis. Wha his implies is ha one way (perhaps he mos imporan way) o undersand he differene beween miroeonomi analysis and maroeonomi analysis is ha he laer rouinely onsiders feedbak effes aross markes, whereas he former usually does no. +φ

13 Problem 5: Two Types of Sok (0 poins). Consider a variaion of he Chaper 8 infinieperiod sok-priing model. The variaion here is ha here are wo ypes of sok ha he DOW represenaive onsumer an buy: Dow sok and S&P sok. Denoe by a he represenaive onsumer s holdings of Dow sok a he beginning of period and by a he represenaive onsumer s holdings of S&P sok a he beginning of period. Likewise, le P and P denoe, respeively, he nominal prie of Dow and S&P sok in period, and DOW DOW D and D denoe, respeively, he per-share nominal dividend ha Dow and S&P sok pay in period. The period- budge onsrain of he represenaive onsumer is hus P + S a + S a = Y + ( S + D ) a + ( S + D ) a, DOW DOW DOW DOW DOW in whih all of he oher noaion is sandard: Y denoes nominal inome (over whih he onsumer has no onrol) in period, is real unis of onsumpion, and P is he nominal prie of eah uni of onsumpion. Also as usual, he lifeime uiliy of he onsumer saring from 3 period onwards is u ( ) + βu ( + ) + β u ( + ) + β u ( + 3) +..., where β (0,] is he usual measure of onsumer impaiene. The Lagrangian for he onsumer lifeime uiliy maximizaion problem, saring from he perspeive of he beginning of period, is u ( ) + βu ( ) + β u ( ) λ Y + S a + D a + S a + D a P S a S a DOW DOW DOW DOW DOW DOW + βλ Y + S a + D a + S a + D a P S a S a +... DOW DOW DOW DOW DOW DOW in whih λ denoes he Lagrange muliplier on he budge onsrain of period, λ + denoes he Lagrange muliplier on he budge onsrain of period +, and so on. a. (3 poins) Based on he given Lagrangian, ompue firs-order ondiions wih respe o a and a. NOTE: You do no need o ompue any oher firs-order ondiions. DOW Soluion: Taking FOCs wih respe o a and + + DOW a : ( + ) 0 ( + + ) λs + βλ S + D = DOW DOW DOW λs + βλ + S + D = 0

14 Problem 5 oninued b. (3 poins) Based on he firs-order ondiions obained in par a, re-express hem as period- sok-priing expressions for boh Dow sok and S&P sok. Your final expressions should DOW be of he form S =... and S =... Soluion: Rearranging he wo expressions obained in par a, we have βλ ( + + ) + S = S + D λ βλ S = S + D ( + + ) DOW + DOW DOW λ. (6 poins) Based on he sok-prie expressions obained in par b, is i he ase ha DOW S = S? If so, briefly explain why; if no, briefly explain why no; if i is impossible o ell, explain why. Soluion: No, given he informaion a hand hus far, here is no way o know if he wo sok pries are equal o eah oher. In pariular, you are hus far given no informaion on he dividends ha eah of hese wo differen asses pay. 3

15 Problem 5 oninued d. (8 poins Harder) Assume in his sub-quesion ha β =. Suppose he eonomy evenually reahes a seady sae. In his seady sae, Dow sok pay zero dividends bu S&P sok pay a posiive nominal dividend ha is always one-enh he nominal prie of a share of S&P sok. Tha is, in he seady sae, D = 0.S in every period. Furher suppose ha in he seady sae, he inflaion rae of onsumer goods pries beween one period and he nex is always 0 peren (i.e., π = 0.0 ). Compue numerially he seady-sae rae a whih he nominal prie of eah ype of sok grows every period (i.e., wha you are being asked o ompue is he inflaion or appreiaion raes of eah of he wo ypes of sok). Jusify your soluion wih any appropriae ombinaion of mahemaial, graphial, or qualiaive argumens. Also provide brief eonomi raionale/inuiion for your findings. Soluion: The firs poin o observe is ha we now need he firs-order ondiion on (and is period + analog): u'( ) λp = 0 (he period + analog is u'( + ) λ+ P+ = 0 ). Solving hese expressions for he mulipliers λ and λ +, and insering hem in he sok prie expressions obained in par b, gives βu '( + ) P S = ( S+ + D+ ) u'( ) P+ DOW βu'( + ) DOW DOW P S = ( S+ + D+ ) u '( ) P + Nex, in he seady sae, + = for all, by definiion. You are also old ha β = and P = = (be areful wih numeraors and denominaors). Imposing hese hree P+ + π +. ondiions gives S = ( S+ + D+ ). DOW DOW DOW S = ( S+ + D+ ). Finally, imposing he ondiion D = 0.S in every period gives. S =.S + and DOW DOW.S = S + in he seady sae. From hese wo seady-sae expressions, i is lear how Dow pries and S&P pries are hanging over ime: from he laer expression, learly S&P pries are no hanging over ime, while from he former expression, Dow pries are rising a a rae of 0 peren, he same as he rae of onsumer prie inflaion. The inuiion behind hese resuls is as follows. No maer whih we way we measure he real ineres rae (wheher using Dow reurns or S&P reurns), hey mus boh mus be equal o he onsumer s MRS. The Dow sok pays no dividend, hene is enire reurn mus ome hrough hanges in he prie of he sok iself i.e., here are apial gains on he Dow sok. In onras, beause S&P soks do pay a dividend, he required apial gains on S&P sok are lower. Wih he pariular numerial values given, he required apial gain on S&P sok urn ou o be zero. END OF EXAM 4