eparmen of Applied Economics Johns Hopkins Universiy Economics 602 Macroeconomic Theory and Policy Miderm Exam Suggesed Soluions Professor Sanjay hugh Fall 2008 NAME: The Exam has a oal of five (5) problems and pages numbered one () hrough welve (2) (wih he las hree pages blank). 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 / 25 Problem 2 / 20 Problem 3 / 0 Problem 4 / 5 Problem 5 / 30 TOTAL / 00
Problem : onsumpion and Savings in he Two-Period Economy (25 poins). onsider 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, c2) = lnc+ lnc2, 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 26. Suppose ha he real ineres rae beween period and period 2 is zero (i.e., r = 0), and 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 2? 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 LB of he consumer is c2 y2 c+ = y+, + r + r where he noaion is sandard from class. The Lagrangian is hus y2 c2 uc (, c2) + 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 2 (which are he objecs of choice) are, as usual: u( c, c2) λ = 0 λ u2( c, c2) = 0 + r (And of course he FO wih respec o he muliplier jus gives back he LB.) Also as usual, hese FOs can be combined o give he consumpion-savings opimaliy condiion, u( c, c2) = + r. Wih he given uiliy funcion, he marginal uiliy funcions are u = / c and u ( c, c ) 2 2 u2 = / c2, so he consumpion-savings opimaliy condiion in his case becomes c 2 / c = + r. This can be rearranged o give c 2 = ( + r ) c, which we can hen inser in he LB o y2 give c+ c = y+ (no, ha s no a ypo, i s c+ c afer he subsiuion ). + r y2 In his problem, you are given neiher y nor y 2. Insead, wha you are given is y + = 26. + r * Thus, we have ha he opimal quaniy of period- consumpion is c = 3 (which solves par i). We can compue c * 2, because we are given he ineres rae r -- using r = 0 in he expression c2 = ( + r) c obained above, we have c 2 = c = 3. (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 (which solves par iii).
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, negaively relaed o, or no a all relaed o he rae of consumpion growh beween period one and period wo. For your reference, he definiion c of he rae of consumpion growh rae beween period and period 2 is 2 c (compleely analogous o how we defined in class he rae of growh of prices beween period and period 2). (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 2 / c he given uiliy funcion, his becomes r / c = +, or, rewriing, 2 c2 r c = +. The lef-hand-side of his expression obviously measures he consumpion growh rae beween period and period 2. Tha is, if c = 00 and c 2 = 03, clearly he consumpion growh rae is 3 percen beween period and period 2. Which would mean ha r = 0.03. If he real ineres rae were insead larger, clearly he lef-hand-side, c 2 /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-2) 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 2, no abou he level of consumpion in period by iself. 2
Problem 2: European and U.S. onsumpion-leisure hoices (20 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. (0 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
Problem 2 coninued. b. (0 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
Problem 3: Governmen Budges and Governmen Asse Posiions (0 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 0 Real ax collecions during he period 2 Quaniy of ne governmen asses a he EN of he period + 8 4 +2 5 0 +3 0 0 +4 8 2 a. (6 poins) omplee 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: b2 = 2 g2 + b (In he simple wo-period model, we assumed b 2 = 0, bu if we wan o exend pas wo periods, we of course would no make his assumpion.) irecly 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
Problem 4: A onracion in redi 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-2 consumpion ploed on he verical axis, and he downward-sloping line represening, as always, he consumer s LB. 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 (, c2) = lnc+ lnc2, and ha he real income of he consumer in period and period 2 is y = 2 and y 2 = 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 2, 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 2 /c = +r = (he second equaliy follows because r = 0 here). Thus, a he opimal choice, i is he case ha c = c 2. Using his relaionship (and again using he fac ha r = 0 here), we can express he consumer s LB as c + c = y + y 2 = 20, 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 = 2, and we jus compued c = 0, he asse posiion a he end of period one is a = y c = 2 (i.e., posiive 2). 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
Problem 4b coninued c 2 Opimal choice if no credi-marke problems onsumer LB c 7
Problem 5: Effecs of Tax Policy on Sock Prices (30 poins). onsider our infinie-period model wih socks as he only asse. Socks held a he beginning of period pay a nominal dividend a he very beginning of period. Suppose ha dividend paymens are subjec o a proporional ax rae in period, where is a number beween zero and one. For example, if = 0.20, hen 20 percen of all dividends received by he represenaive consumer in period mus be paid o he governmen (we ll disregard here any issues relaed o wha he governmen does wih hose revenues). a. (5 poins) Se up he period- flow budge consrain, briefly explaining how he dividend ax eners he expression. Soluion: In period, he flow budge consrain reads Pc + S a = Y + S a + ( ) a, in which he erm on he far-righ-side of he equaion says ha he consumer only keeps he fracion of dividend paymens afer paying he dividend ax. b. (0 poins) Using he flow budge consrain you se up above, show algebraically (i.e., using a Lagrangian) how he nominal sock price in period, denoed as usual by S, depends on he dividend ax when he represenaive consumer is maximizing lifeime uiliy from period onwards. Also, he dividend ax rae in WHIH period affecs he period- sock price? Provide brief economic inerpreaion/logic. Soluion: We only need he period- and and period-(+) erms of he Lagrangian (since here is no habi persisence and he holding period is jus one period here), which are: uc ( ) + λ Y + Sa + ( ) a Pc Sa + βuc ( + ) + βλ + Y+ + S+ a + ( + ) + a P+ c+ S+ a + +... where as usual λ is he muliplier on he period- budge consrain and λ + is he muliplier on he period + budge consrain. The firs-order condiion wih respec o a (which is one of he objecs of choice in period ) is (afer a sligh algebraic rearrangemen) βλ + ( ( S ) = S+ + + + ), λ which shows ha i is he period-(+) dividend ax rae ha affecs he period- sock price. This should make sense because i s he period-+ reurn ha maers for he period- sock price i.e., when markes (represened by our represenaive consumer) deermine prices in he curren period, he fuure afer-ax reurn is wha maers. As he above expression shows, he lower is + (all else equal), he higher is S. Thus, if i s announced ha here will be a lowering of he dividend ax rae in he fuure, ha migh be expeced o boos sock prices in he presen. Noe ha here you did no even need o compue he FO wih respec o c (nor proceed o consruc he pricing kernel), since ha wasn essenial o he argumen. 8
Problem 5b coninued (if you need more space) c. (5 poins Harder) Suppose in addiion o he dividend ax described above, here is also a proporional ax on consumpion (a sales ax). The consumpion ax rae in period is. Suppose ha rises, bu all oher ax raes (including hose beyond period ) remain unchanged. Show algebraically (i.e., using a Lagrangian) how his one-ime consumpionax hike policy change affecs he period- sock price, assuming all else is equal? (i.e., his is a usual ceeris paribus exercise) Also provide brief economic inerpreaion for your finding. Soluion: The flow budge consrain here modifies o ( + ) Pc + Sa = Y + Sa + ( ) a, and he Lagrangian modifies o uc ( ) + λ Y + Sa + ( ) a ( + ) Pc Sa + βuc ( + ) + βλ + Y+ + S+ a + ( + ) + a ( + + ) P+ c+ S+ a + +... The firs-order condiion wih respec o a is unchanged from above, bu here you did need o explicily consider he FO wih respec o c (and proceed o consruc he kernel) in order o see he effec of he consumpion ax on sock prices. The FO wih respec o consumpion in u'( c period is '( ) ( ) u c λ + ) P = 0, which can be solved for he muliplier, λ =. ( + ) P Similarly, in period + we d have λ u'( c ) + + =. Insering hese expressions for he ( + + ) P+ mulipliers ino he sock price equaion, β u'( c+ ) ( + ) P S = ( S ( ) + + + + ). u'( c ) ( + ) P + + If he period- sales ax rises (and all else remains unchanged), he above expression shows ha S rises. The economics behind his is ha if he period- price of consumpion (inclusive of axes) rises, consumers will desire less period- consumpion and subsiue ino period + consumpion. oing so means increased demand for asses (socks) in period as a means by which o ransfer resources from he presen (period ) o he fuure (period +). The increased demand for socks in period bids up he price S. 9