Problem 1 / 25 Problem 2 / 25 Problem 3 / 11 Problem 4 / 15 Problem 5 / 24 TOTAL / 100

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1 Deparmen of Economics Universiy of Maryland Economics 35 Inermediae Macroeconomic Analysis Miderm Exam Suggesed Soluions Professor Sanjay Chugh Fall 008 NAME: The Exam has a oal of five (5) problems 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 one page (double-sided) of noes. You may no use a calculaor. Problem / 5 Problem / 5 Problem 3 / Problem 4 / 5 Problem 5 / 4 TOTAL / 00

2 Problem : Consumpion and Savings in he Two-Period Economy (5 poins). Consider a wo-period economy (wih no governmen), in which he represenaive consumer has no conrol over his income. The lifeime uiliy funcion of he represenaive consumer is u( c, c) = lnc+ lnc, where ln sands for he naural logarihm. We will work here in purely real erms: suppose he consumer s presen discouned value of ALL lifeime REAL income is 6. Also suppose he consumer begins period wih zero ne asses. a. (7 poins) Se up he lifeime Lagrangian formulaion of he consumer s problem, in order o answer he following: i) is i possible o numerically compue he consumer s opimal choice of consumpion in period? If so, compue i; if no, explain why no. ii) is i possible o numerically compue he consumer s opimal choice of consumpion in period? If so, compue i; if no, explain why no. iii) is i possible o numerically compue he consumer s real asse posiion a he end of period? If so, compue i; if no, explain why no. Soluion: We know ha wih zero iniial asses, he LBC of he consumer is c y c+ = y+, + r + r where he noaion is sandard from class. The Lagrangian is hus y c uc (, c) + λ y+ c, + r + r where λ of course is he Lagrange muliplier (noe here s only one muliplier since his is he lifeime formulaion of he problem no he sequenial formulaion of he problem). The firsorder condiions wih respec o c and c (which are he objecs of choice) are, as usual: u( c, c) λ = 0 λ u( c, c) = 0 + r (And of course he FOC wih respec o he muliplier jus gives back he LBC.) Also as usual, hese FOCs can be combined o give he consumpion-savings opimaliy condiion, u( c, c) = + r. Wih he given uiliy funcion, he marginal uiliy funcions are u = / c and u ( c, c ) u = / c, so he consumpion-savings opimaliy condiion in his case becomes c / c = + r. This can be rearranged o give c = ( + r ) c, which we can hen inser in he LBC o y give c+ c = y+ (no, ha s no a ypo, i s c+ c afer he subsiuion ). + r y In his problem, you are given neiher y nor y. Insead, wha you are given is y + = 6. + r * Thus, we have ha he opimal quaniy of period- consumpion is c = 3 (which solves par i). * We can no compue c, however, because we are no given he ineres rae r (which you would need in order o use he expression c = ( + r ) c compued above. (This solves par ii). * To compue he asse posiion a he end of period, we would need o compue y c, bu since we don know y, we canno compue his eiher (which solves par iii).

3 Problem coninued b. (8 poins) To demonsrae how imporan he concep of he real ineres rae is in macroeconomics, an inerpreaion of i (in addiion o he couple of differen inerpreaions we have already discussed in class) is ha i reflecs he rae of consumpion growh beween wo consecuive periods. Using he consumpion-savings opimaliy condiion for he given uiliy funcion, briefly describe/discuss (rambling essays will no be rewarded) wheher he real ineres rae is posiively relaed o or negaively relaed o he rae of consumpion growh beween period one and period wo. For your reference, he definiion of he rae of c consumpion growh rae beween period and period is (compleely analogous o how c we defined in class he rae of growh of prices beween period and period ). (Noe: No mahemaics are especially required for his problem; also noe his par can be fully compleed even if you were unable o ge all he way hrough par a). Soluion: u The familiar consumpion-savings opimaliy condiion is r u = +. As we jus saw above, for / c he given uiliy funcion, his becomes r / c = +, or, rewriing, c r c = +. The lef-hand-side of his expression obviously measures he consumpion growh rae beween period and period. Tha is, if c = 00 and c = 03, clearly he consumpion growh rae is 3 percen beween period and period. Which would mean ha r = If he real ineres rae were insead larger, clearly he lef-hand-side, c /c, would be larger as well. Thus, he higher is he real ineres rae, he higher is he consumpion growh rae beween periods he real ineres rae and he consumpion growh rae are posiively relaed o each oher. This is hus ye anoher way o hink abou he real ineres rae. The wo oher ways we discussed in class of hinking inuiively abou he real ineres rae is ha i measures he price of curren (period-) consumpion in erms of fuure (period-) consumpion; and as reflecing he fundamenal degree of (human) impaience of individuals in he economy. All of hese various (and ulimaely iner-relaed) ways of hinking abou he real ineres underline is fundamenal imporance in macroeconomic heory. Noe ha simply arguing/explaining here ha a rise in he real ineres rae leads o a fall in period- consumpion does no address he quesion he quesion is abou he rae of change of consumpion beween period and period, no abou he level of consumpion in period by iself.

4 Problem : European and U.S. Consumpion-Leisure Choices (5 poins). Europeans work fewer hours han Americans. There are likely very many possible reasons for his, and indeed in realiy his fac arises from a combinaion of many reasons. In his quesion, you will consider wo reasons using he simple (one-period) consumpion-leisure model. a. (3 poins) Suppose ha boh he uiliy funcions and pre-ax real wages W / P of American and European individuals are idenical. However, he labor income ax rae in Europe is higher han in America. In a single carefully-labeled indifference-curve/budge consrain diagram (wih consumpion on he verical axis and leisure on he horizonal axis), show how i can be he case ha Europeans work fewer hours han Americans. Provide any explanaion of your diagram ha is needed. Soluion: If Europeans work fewer hours han Americans, hen Europeans have more leisure ime han Americans, simply because (in our weekly model) n+ l = 68. Europeans and Americans have idenical uiliy funcions, which means ha heir indifference maps are idenical. This means ha he difference in hours worked mus arise compleely from differences in heir budge consrains. Wih a higher labor income ax in Europe, he budge consrain of he European consumer is less seep han he budge consrain of he American, as he diagram below shows (because he slope of he budge consrain is ( W ) / P, and you are given ha W / P is he same in he wo counries). The diagram shows ha he European opimally chooses more leisure (hence less labor) and less consumpion han he American. Here, he difference beween Europeans and Americans is solely in he relaive prices (embodied by he slope of he budge consrain) hey face. (For full credi here, you had o somehow make clear ha he indifference maps of he represenaive European and he represenaive American are idenical.) consumpion American s budge consrain Opimal choice of American Opimal choice of European European s budge consrain 68 leisure 3

5 Problem coninued. b. ( poins) Suppose ha boh he pre-ax real wages W / P and he labor ax raes imposed on American and European individuals are idenical. However, he uiliy funcion AMER EUR u (,) c l of Americans differs from ha of Europeans u ( c, l ). In a single carefullylabeled indifference-curve/budge consrain diagram (wih consumpion on he verical axis and leisure on he horizonal axis), show how i can be he case ha Europeans work fewer hours han Americans. Provide any explanaion of your diagram ha is needed. Soluion: In his case, he budge consrains of he European consumer and American consumer are idenical, so he difference in hours worked mus arise compleely from differences in heir uiliy funcions. Graphically, his means ha he wo ypes of consumers have differen indifference maps (i.e., a differen se of indifference curves). In he diagram below, he budge line is he common budge line of he European and he American. The solid indifference curves are he American s, while he dashed indifference curves are he European s. Wih seeper indifference curves, he European s opimal choice along he same budge line mus occur a a poin ha feaures more leisure (hence less labor) and less consumpion han he American s opimal choice. Here, he difference beween Europeans and Americans is solely in heir preferences. 68(-)W/P opimal choice of American opimal choice of European 68 leisure 4

6 Problem 3: Governmen Budges and Governmen Asse Posiions ( poins). Jus as we can analyze he economic behavior of consumers over many ime periods, we can analyze he economic 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 describes he real quaniies of governmen spending and real ax revenue he governmen collecs saring in period and for several periods hereafer. Period Real governmen expendiure (g) during he period Real ax collecions during he period Quaniy of ne governmen asses a he END of he period a. (7 poins) Complee he las column of he able based on he informaion given. Briefly explain he logic behind how you calculae hese values. Soluion: If his were he wo-period model, we could compue he governmen asse posiion a he end of period, say, one, by rearranging he period- governmen budge consrain: b = g+ b0 -- in his expression we have used he assumpion ha r = 0. Furhermore (and again wih r = 0), we can compue he governmen asse posiion a he end of period wo as: b = g + b (In he simple wo-period model, we assumed b = 0, bu if we wan o exend pas wo periods, we of course would no make his assumpion.) Direcly exending his logic o an infinie-period seing, hen, he governmen s asse posiion a he end of any paricular period is given by: b = g + b. Successively applying his rule hen gives rise o he ne asse posiions presened in he able above. b. (4 poins) Suppose insead he governmen ran a balanced budge every period (i.e., every period i colleced in axes exacly he amoun of is expendiures ha period). In his balanced-budge scenario, wha would be he governmen s ne asses a he end of period +4? Briefly explain/jusify. Soluion: A balanced budge means g equals ax collecions every period. If his were rue in he above able, and applying he logic of par a above, he governmen ne asses a he end of every period would always be zero; hus a he end of period +4 hey are zero as well. 5

7 Problem 4: A Conracion in Credi Availabiliy (5 poins). The graph below shows our usual wo-period indifference-curve/budge consrain diagram, wih period- consumpion ploed on he horizonal axis, period- consumpion ploed on he verical axis, and he downward-sloping line represening, as always, he consumer s LBC. Throughou all of he analysis here, assume ha r = 0 always. Furhermore, here is no governmen, hence never any axes. Suppose ha he represenaive consumer has lifeime uiliy funcion uc (, c) = lnc+ lnc, and ha he real income of he consumer in period and period is y = and y = 8. Finally, suppose ha he iniial amoun of ne asses he consumer has is a 0 = 0. EVERY consumer in he economy is described by his uiliy funcion and hese values of y, y, and a 0. a. (6 poins) If here are no problems in credi markes whasoever (so ha consumers can borrow or save as much or as lile as hey wan), compue he numerical value of he opimal quaniy of period- consumpion. (Noe: if you can solve his problem wihou seing up a Lagrangian, you are free o do so as long as you explain your logic.) Soluion: The consumpion-savings opimaliy condiion (given he naural-log uiliy funcion) is given by c /c = +r = (he second equaliy follows because r = 0 here). Thus, a he opimal choice, i is he case ha c = c. Using his relaionship (and again using he fac ha r = 0 here), we can express he consumer s LBC as c + c = y + y = 0, which obviously implies he opimal choice of period- consumpion is c = 0. Noe: alhough you were no asked o compue i, you could have compued he implied value of he consumer s asse posiion a he end of period one. Because a 0 = 0, y =, and we jus compued c = 0, he asse posiion a he end of period one is a = y c = (i.e., posiive ). b. (9 poins) Now suppose ha because of problems in he financial secor, no consumers are allowed o be in deb a he end of period. Wih his credi resricion in place, compue he numerical value of he opimal quaniy of period- consumpion. ALSO, on he diagram on he nex page, qualiaively and clearly skech he opimal choice wih his credi resricion in place (qualiaively skeched already for you is he opimal choice if here are no problems in credi markes). Your skech should indicae boh he new opimal choice and an appropriaely-drawn and labeled indifference curve ha conains he new opimal choice. (Noe: if you can solve his problem wihou seing up a Lagrangian, you are free o do so as long as you explain your logic.) Soluion: Because in par a (ie, wihou any credi resricions), he represenaive consumer was choosing o NOT be in deb a he end of period (ie, a > 0 under he opimal choice in par a), he imposiion of he credi resricion, nohing changes compared o par a. Tha is, he opimal choice of period- consumpion is sill 0. Hence, in he diagram below, he opimal choice in he presence of credi consrains is exacly he same as he opimal choice wihou credi consrains. The general lesson o draw from his example and our analysis in class is ha i is no necessarily he case ha financial marke problems mus and always spill over ino real economic aciviy (i.e., consumpion in his case). 6

8 Problem 4b coninued c Opimal choice if no credi-marke problems Consumer LBC c 7

9 Problem 5: Two Types of Sock (3 poins). Consider a variaion of our usual infinie-period sock-pricing model. The variaion here is ha here are wo disinc ypes of sock (raher han jus one) ha he represenaive consumer can buy: Dow sock and S&P sock. Denoe by a he represenaive consumer s holdings of Dow sock a he beginning of period and by a he represenaive consumer s holdings of S&P sock a he beginning of period. Likewise, le S and S denoe, respecively, he nominal price of Dow and S&P sock in period, and D and D denoe, respecively, he per-share nominal dividend ha Dow and S&P sock pay in period. The period- budge consrain of he represenaive consumer is hus Pc + S a + S a = Y + ( S + D ) a + ( S + D ) a, in which all of he oher noaion is sandard: Y denoes nominal income (over which he consumer has no conrol) in period, c is real unis of consumpion, and P is he nominal price of each uni of consumpion. Also as usual, he lifeime uiliy of he consumer saring from 3 period onwards is uc ( ) + βuc ( + ) + β uc ( + ) + β uc ( + 3) +..., where β (0,] is he usual measure of consumer impaience. The sequenial Lagrangian for his problem is uc ( ) + βuc ( ) + β uc ( ) λ Y + S a + D a + S a + D a Pc S a S a + βλ Y + S a + D a + S a + D a P c S a S a a. (8 poins) Based on he Lagrangian presened above, compue he firs-order condiions wih respec o boh a and a. Soluion: Taking FOCs wih respec o c, a, and a and combining hese FOCs as usual yields wo (very similar) sock-pricing equaions. Noe you could have sopped simply a he FOCs on he wo ypes of sock, as direcly asked in he quesion. βu'( c+ ) P S = ( S+ + D+ ) u'( c) P βu'( c ) P S = ( S + + D + ) P + u'( c) + + 8

10 Problem 5a coninued (if you need more space) b. (7 poins) Based on he expressions you obained in par a above, deermine wheher i is he case ha S = S? If so, briefly explain why; if no, briefly explain why no; if i s no possible o ell, explain why no. Soluion: No, i is no possible o ell wheher or no S = S simply because you are hus far given no informaion on he dividends ha each of hese wo differen asses pay. 9

11 Problem 5 coninued c. (8 poins Harder) Assume here for simpliciy ha β =. Suppose he economy evenually reaches a seady-sae. In his seady sae, Dow sock coninue o pay zero dividends bu S&P sock pay a nominal dividend ha is always one-enh he nominal price of a share of S&P sock. Tha is, in he seady sae, D = 0.S. Furher suppose ha in he seady-sae, he inflaion rae of consumer goods prices beween one period and he nex is always 0 percen (i.e., π = 0.0 ). Compue numerically he seady-sae rae a which he nominal price of each ype of sock grows every period (i.e., wha you re being asked o compue is he inflaion or appreciaion raes of each of he wo ypes of sock). Jusify your answer wih any appropriae combinaion of mahemaical, graphical, or qualiaive argumens. Also provide brief economic raionale/inuiion for your findings. Soluion: Recall ha we can express hings in erms of he MRS beween period- and period+ consumpion. Doing his using each ype of sock, we have and u'( c ) ( S D ) P u'( c ) S P β = u'( c ) ( S D ) P =. u'( c ) S P β The reason ha we have wo alernaive ways of expressing he consumpion-savings opimaliy condiion here is simply because we are considering wo alernaive asses. Wih he assumpion P ha β =, D + = 0, D+ = 0.S+, and he given informaion = =, we can P + π. wrie he above wo expressions as + + and u'( c) S = u'( c ) S. + u'( c).s+ =. u'( c+ ) S. Nex, recall ha a seady sae means ha consumpion becomes consan from one period o he nex. If consumpion is consan from one period o he nex, clearly marginal uiliy of consumpion becomes consan from one period o he nex as well, meaning he lef-hand-side of he las wo expressions equals one in seady sae: + and S = S. + 0

12 .S =. S. + From hese wo seady-sae expressions, i is clear how Dow prices and S&P prices are changing over ime: from he laer expression, clearly S&P prices are no changing over ime, while from he former expression, Dow prices are rising a a rae of 0 percen, he same as he rae of consumer price inflaion. The inuiion behind hese resuls is as follows. No maer which we way we measure he real ineres rae (wheher using Dow reurns or S&P reurns), hey mus boh mus be equal o he consumer s MRS. The Dow sock pays no dividend, hence is enire reurn mus come hrough changes in he price of he sock iself i.e., here are capial gains on he Dow sock. In conras, because S&P socks do pay a dividend, he required capial gains on S&P sock are lower. Wih he paricular numerical values given, he required capial gain on S&P sock urn ou o be zero.