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1 Deparmen of Appled Economcs Johns Hopkns Unversy Economcs 60 Macroeconomc Theory and Polcy Fnal Exam Suggesed Soluons Professor Sanjay Chugh Fall 009 NAME: The Exam has a oal of fve (5) problems and pages numbered one () hrough welve (). Each problem s oal number of pons s shown below. Your soluons should conss of some approprae combnaon of mahemacal analyss, graphcal analyss, logcal analyss, and economc nuon, bu n no case do soluons need o be exceponally long. Your soluons should ge sragh o he pon soluons wh rrelevan dscussons and dervaons wll be penalzed. You are o answer all quesons n he spaces provded. You may use wo pages (double-sded) of noes. You may no use a calculaor. Problem / 5 Problem / 5 Problem 3 / 5 Problem 4 / 0 Problem 5 / 5 TOTAL / 00
2 Problem : Consumpon and Savngs n he Two-Perod Economy (5 pons). Consder a wo-perod economy (wh no governmen), n whch he represenave consumer has no conrol over hs ncome. The lfeme uly funcon of he represenave consumer s u( c, c) = lnc+ lnc, where ln sands for he naural logarhm. We wll work here n purely real erms: suppose he consumer s presen dscouned value of ALL lfeme REAL ncome s 6. Suppose ha he real neres rae beween perod and perod s zero (.e., r = 0), and also suppose he consumer begns perod wh zero ne asses. a. (7 pons) Se up he lfeme Lagrangan formulaon of he consumer s problem, n order o answer he followng: ) s possble o numercally compue he consumer s opmal choce of consumpon n perod? If so, compue ; f no, explan why no. ) s possble o numercally compue he consumer s opmal choce of consumpon n perod? If so, compue ; f no, explan why no. ) s possble o numercally compue he consumer s real asse poson a he end of perod? If so, compue ; f no, explan why no. Soluon: We know ha wh zero nal asses, he LBC of he consumer s c y c+ = y+, + r + r where he noaon s sandard from class. The Lagrangan s hus y c uc (, c) + λ y+ c, + r + r where λ of course s he Lagrange mulpler (noe here s only one mulpler snce hs s he lfeme formulaon of he problem no he sequenal formulaon of he problem). The frsorder condons wh respec o c and c (whch are he objecs of choce) are, as usual: u( c, c) λ = 0 λ u( c, c) = 0 + r (And of course he FOC wh respec o he mulpler jus gves back he LBC.) Also as usual, hese FOCs can be combned o gve he consumpon-savngs opmaly condon, u( c, c) = + r. Wh he gven uly funcon, he margnal uly funcons are u = / c and u ( c, c ) u = / c, so he consumpon-savngs opmaly condon n hs case becomes c / c = + r. Ths can be rearranged o gve c = ( + r ) c, whch we can hen nser n he LBC o y gve c+ c = y+ (no, ha s no a ypo, s c+ c afer he subsuon ). + r y In hs problem, you are gven neher y nor y. Insead, wha you are gven s y + = 6. + r * Thus, we have ha he opmal quany of perod- consumpon s c = 3 (whch solves par ). * We can compue c because you are gven he neres rae r (whch you would need n order o * use he expresson c = ( + r) c compued above; pluggng n, we have c = 3 (Ths solves par
3 ). To compue he asse poson a he end of perod, we would need o compue snce we don know y, we canno compue hs eher (whch solves par ). y c, bu * b. (8 pons) To demonsrae how mporan he concep of he real neres rae s n macroeconomcs, an nerpreaon of (n addon o he several dfferen nerpreaons we have already dscussed n class) s ha reflecs he rae of consumpon growh beween wo consecuve perods. Usng he consumpon-savngs opmaly condon for he gven uly funcon, brefly descrbe/dscuss (ramblng essays wll no be rewarded) wheher he real neres rae s posvely relaed o, negavely relaed o, or no a all relaed o he rae of consumpon growh beween perod one and perod wo. For your reference, c he defnon of he rae of consumpon growh rae beween perod and perod s c (compleely analogous o how we defned n class he rae of growh of prces beween perod and perod ). (Noe: No mahemacs are especally requred for hs problem; also noe hs par can be fully compleed even f you were unable o ge all he way hrough par a). Soluon: u The famlar consumpon-savngs opmaly condon s r u = +. As we jus saw above, for / c he gven uly funcon, hs becomes r / c = +, or, rewrng, c r c = +. The lef-hand-sde of hs expresson obvously measures he consumpon growh rae beween perod and perod. Tha s, f c = 00 and c = 03, clearly he consumpon growh rae s 3 percen beween perod and perod. Whch would mean ha r = If he real neres rae were nsead larger, clearly he lef-hand-sde, c /c, would be larger as well. Thus, he hgher s he real neres rae, he hgher s he consumpon growh rae beween perods he real neres rae and he consumpon growh rae are posvely relaed o each oher. Ths s hus ye anoher way o hnk abou he real neres rae. The wo oher ways we dscussed n class of hnkng nuvely abou he real neres rae s ha measures he prce of curren (perod-) consumpon n erms of fuure (perod-) consumpon; and as reflecng he fundamenal degree of (human) mpaence of ndvduals n he economy. All of hese varous (and ulmaely ner-relaed) ways of hnkng abou he real neres underlne s fundamenal mporance n macroeconomc heory. Noe ha smply argung/explanng here ha a rse n he real neres rae leads o a fall n perod- consumpon does no address he queson he queson s abou he rae of change of consumpon beween perod and perod, no abou he level of consumpon n perod by self.
4 Problem : The Keynesan-RBC-New Keynesan Evoluon (5 pons). Here you wll brefly analyze aspecs of he evoluon n macroeconomc heory over he pas 5 years. Address each of he followng n no more han hree bref phrases/senences each. a. (5 pons) Descrbe brefly wha he Lucas crque s and how/why led o he demse of (old) Keynesan models. Soluon: The old Keynesan models were large esmaed sysems of equaons, and he esmaed coeffcens could no (because hey were jus based on hsorcal observaons) ake no accoun how behavor mgh change f polcy changed. In he 970 s, hs led o he downfall of such models as polcy-makers red more and more o explo hese relaonshps, bu he coeffcens began o vary a lo (for some reason ) wh polcy, evenually causng he professon (hrough he Lucas crque) o undersand ha such models really were no all ha useful for polcy advce afer all. b. (5 pons) In wrng down uly funcons and producon funcons for use n RBC-syle macro models, he assumed funcons are ypcally esmaed usng daa (.e., a common assumpon s he logarhmc uly funcon we have ofen used, based on some sascal evdence ha s conssen wh observed mcroeconomc and macroeconomc evdence). Is hs pracce subjec o a Lucas-ype crque? Brefly explan why or why no? Soluon: Yes, seems ha hs pracce s also subjec o a Lucas-ype crque he parameers/coeffcens n he uly and producon funcons, for example, could n prncple be dependen on polcy. If hey are, and polcy changes n parcular way ha, say, changes consumers uly funcons, hen he same pfalls facng he old Keynesan models could arse. So far, seems we have no wnessed hs aspec of he Lucas crque. c. (5 pons) Brefly defne and descrbe he neuraly vs. nonneuraly debae surroundng moneary polcy oday. Whch ype of shock does hs debae concern? Soluon: The RBC vew holds ha money shocks do no affec real varables (.e., consumpon or GDP) n he economy (neuraly), whle he New Keynesan vew holds ha hey do (nonneuraly) because prces ake me o adjus (are scky ). 3
5 Problem 3: Opmal Tax Polcy (5 pons). Consder our sac (.e., one perod) consumpon-lesure framework from Chaper. In hs problem, you wll use hs framework as a bass for offerng gudance regardng opmal (.e., he bes ) labor ncome ax polcy. Recall he basc consumpon-lesure opmaly condon ul (,) c l = ( w ), u (,) c l c n whch all of he noaon s as n Chaper : denoes he labor ncome ax rae, w denoes he real wage, c denoes consumpon, l denoes lesure, u c denoes he margnal uly of consumpon, and u l denoes he margnal uly of lesure. Suppose ha frms are monopolscally compeve (raher perfecly compeve). I can be shown n hs case ha when frms are makng her prof-maxmzng choce regardng labor hrng, he followng condon s rue: mpn = w( monpol). Here, mpn denoes he margnal produc of labor and monpol s a measure of he degree of monopoly power ha frms weld. For example, f monpol = 0, hen frms weld no monopoly power whasoever, n whch case we are back o our perfecly-compeve framework of frm prof maxmzaon from Chaper 6. If nsead monpol > 0, hen frms do weld some monopoly power. (Noes: The varable monpol can never be less han zero. You also do no need o be concerned here wh how he above expresson s derved jus ake as gven. Furher, noe ha here are no fnancng consran ssues here whasoever.) Suppose he followng:. The only goal polcy makers have n choosng a labor ax rae s o ensure ha he perfecly-compeve oucome n labor markes s aaned.. Any monopoly power ha frms have canno be drecly elmnaed by polcy makers. Tha s, f monpol > 0, he governmen canno do anyhng abou ha; all he governmen can do s choose a ax rae. Based on all of he above, derve a relaonshp beween he opmal (.e., n he sense ha aans he goal of polcymakers descrbed n pon # above) labor ncome ax rae and he degree of frms monopoly power. Carefully explan your logc and any mahemacal dervaons nvolved. (OVER) 4
6 Problem 3 connued Soluon: Noe for use below ha we can express he second equaon above as mpn w =. monpol The frs logcal sep n he argumen s he observaon ha here s no monopoly power f markes are perfecly compeve (by defnon, obvously), n whch case monpol = 0. In hs case, we have ha mpn = w. Pung hs concluson ogeher wh he consumpon-lesure opmaly condon gves us ul (,) c l = ( mpn ). uc(,) c l By he basc heory of perfec compeon, you had o hen recognze (mplcly or explcly) ha any ax rae dfferen from zero creaes a wedge (.e., a deadwegh loss) n he labor marke. Hence snce perfec compeon means zero deadwegh losses, he opmal ax polcy n he case of monpol = 0 s = 0. Ths mples ha n perfec compeon, mus be ha ul (,) c l = mpn. u (,) c l c Now le s generalze he argumen for he case of monpol > 0. Inserng he expresson mpn w = no he consumpon-lesure opmaly condon, we have monpol ul (,) c l ( mpn ) =. u (,) c l monpol c ul (,) c l The goal of ax polcy now s o pck a so ha he perfec-compeon oucome = mpn uc(,) c l s acheved despe he fac ha monpol > 0. Examnng he prevous condon, s clear ha seng = monpol acheves he perfec-compeon oucome. Thus, he opmal labor-ncome ax rae s = monpol. 5
7 Problem 4: Fnancng Consrans and Labor Demand (0 pons). In our class dscusson abou he way n whch fnancng consrans affec frms prof maxmzaon decsons, we focused on he effecs on frms physcal capal nvesmen. In realy, mos frms spend wce as much on her wage coss (.e., her labor coss) han on her physcal nvesmen coss. (Tha s, for mos frms, roughly wo-hrds of her oal coss are wages and salares, whle roughly one-hrd of her oal coss are devoed o mprovng or expandng her physcal capal.) For many frms, paymen of wages mus be made before he recep of revenues whn any gven perod. (For example, magne a frm ha has o pay s employees o buld a compuer; he revenues from he sale of hs compuer ypcally don arrve for many weeks or monhs laer because of delays n he shppng process, he real process, ec.) For hs reason, frms ypcally need o borrow o pay for her ongong wage coss. Bu, because of asymmerc nformaon problems, lenders ypcally requre ha he frm pu up some fnancal collaeral o secure loans for hs purpose. Here, you wll analyze he consequences of fnancng consrans on frms wage paymens usng a varaon of he acceleraor framework we suded n class. For smplcy, suppose ha he represenave frm, whch operaes n a wo-perod economy, mus borrow n order o fnance only perod- wage coss; for some unspecfed reason, suppose ha perod- wage coss are no subjec o a fnancng consran. As n our sudy of he acceleraor framework n class, he represenave frm s wo-perod dscouned prof funcon s Pf ( k, n) + Pk + ( S+ D) a0 Pwn Pk Sa Pf( k, n) Pk ( S + D) a Pw n Pk 3 Sa and suppose now he fnancng consran ha s relevan for frm prof-maxmzaon s Pwn + = Sa. (The presen-dscouned-value appears on he lef-hand-sde because we are conducng he analyss, as always, from he perspecve of he begnnng of perod.) The noaon s as always: P denoes he nomnal prce of he oupu he frm produces and sells; S denoes he nomnal prce of sock; D denoes he nomnal dvdend pad by each un of sock; n denoes he quany of labor he frm hres; w s he real wage; a 0, a, and a are, respecvely, he frm s holdngs of sock a he end of perod 0, perod, and perod ; k, k, and k 3 are, respecvely, he frm s ownershp of physcal capal a he end of perod 0, perod, and perod ; denoes he nomnal neres rae beween perod and perod ; and he producon funcon s denoed by f(.). Also as usual, subscrps on varables denoe he me perod of reference for ha varable. Fnally, because hs s a wo-perod framework, we know a = 0 and k 3 = 0. (OVER) The commercal paper marke, abou whch much has been dscussed n he news meda n he pas year, s one ype of channel for such frm fnancng needs. 6
8 Problem 4 connued The Lagrangan for he frm s prof maxmzaon problem s hus Pf ( k, n) + Pk + ( S+ D) a0 Pwn Pk Sa P f( k, n ) Pk ( S + D ) a Pw n Pk S a Pwn + λ Sa + 3 n whch λ denoes he Lagrange mulpler on he fnancng consran. a. (4 pons) Based on he Lagrangan above, compue he frs-order condons wh respec o k and a. Soluon: The frs-order condons are smply: Pf( k, n) P + + S + D S+ + λs = 0 + k P + + = 0 b. (4 pons) Based on he Lagrangan above, compue he frs-order condons wh respec o n and n. Soluon: The frs-order condons are smply: Pf ( k, n) Pw = 0 n Pf n( k, n) Pw λpw = Noe for reference below ha he second equaon here can be expressed as f ( n k, ) ( ) n = w + λ. 7
9 Problem 4 connued Suppose ha a he begnnng of perod, he real reurn on STOCK, r STOCK, all of a sudden falls below r, he real reurn on rskless ( safe ) asses. Suppose ha before hs shock occurred (.e., n perod zero ), was he case ha r = r STOCK. c. (4 pons) Below s a graph of he nvesmen (capal) marke n perod. Does he adverse shock o r STOCK shf eher he nvesmen demand and/or he savngs supply funcon? If so, explan how, n wha drecon, and why. Soluon: The nvesmen demand funcon s unaffeced by he fnancng consran (see he FOC on k above), hence exogenous changes n rstock have no effec on capal demand/nvesmen demand. Furhermore, because nohng s sad abou wheher fnancng frcons mpnge on he savngs supply sde of he economy (.e., on consumers consumponsavngs decsons), here s no bass for asserng any shf of he savngs funcon. Hence, here s no drec effec on he marke for physcal capal. d. (4 pons) Below s a graph of he labor marke n perod. Does he adverse shock o r STOCK shf eher he labor demand and/or he labor supply funcon? If so, explan how, n wha drecon, and why. Soluon: No shf n labor supply because, as above, no saemens are made abou wheher fnancng frcons affec consumers behavor (whch s wha would be requred for a shf of he labor supply funcon). The fnancng consran does NOT affec perod- wage paymens, hence a fall n rstock has no drec effec on he perod- labor demand funcon: here s no shf n he perod- labor demand funcon. 8
10 Problem 4 connued e. (4 pons) Below s a graph of he labor marke n perod. Does he adverse shock o r STOCK shf eher he labor demand and/or he labor supply funcon? If so, explan how, n wha drecon, and why. Soluon: No shf n labor supply because, once agan, no saemens are made abou wheher fnancng frcons affec consumers behavor (whch s wha would be requred for a shf of he labor supply funcon. A fall n rstock wll cause a RISE n λ, hence for a gven level of w, he effecve margnal produc of perod- labor FALLS, hence he labor demand curve shfs nwards. Ths effec arses because λ drecly appears n he perod- FOC on labor above. Supply Demand Perod Labor Marke n 9
11 Problem 5: The Cash-n-Advance Framework (5 pons) (Harder). We suded he MIU framework and assered ha was a smple way of nroducng money no our basc dynamc economy seup. An alernave formulaon by whch o nroduce money s o assume ha some subse of consumpon goods are cash goods and he res are cred goods. Smply pu, cash goods are consumpon goods ha mus be purchased usng cash (money), whle cred goods do no requre money for purchase (hey may be purchased on cred ). Denoe by c cash good consumpon by he represenave consumer n perod, and by c cred good consumpon n perod. Suppose ha n every perod, he consumer s uly funcon s gven by uc (, c ). In every perod, he consumer faces wo consrans: he flow budge consran b Pc + Pc + M + P B + Sa = Y + M + B + ( S + D) a and he cash-n-advance consran, Pc = M, whch s he requremen ha all nomnal expendures on cash goods requre money. Noe ha cash goods and cred goods have he same nomnal prce P. Also noe ha as n our MIU model, here are hree asses: money, nomnal bonds, and sock. The res of he noaon s compleely dencal o wha we have suded. Here, n perod, he consumer chooses c, c, M, B, and a. Fnally, also as usual, suppose he represenave consumer has an mpaence facor β <. a. (5 pons) Se up an approprae Lagrangan for hs problem. (Hn : Do no use he second consran o subsue ou any varables ha s, use each consran above as a dsnc consran. Hn : Because n each me perod here are hus wo consrans ha he represenave consumer mus respec, how many unque Lagrange mulplers mus be nroduced no he problem for each me perod?) Soluon: The Lagrangan s uc (, c ) + βuc (, c ) + β uc (, c ) λ φ b Y M B ( S D) a Pc Pc M P B Sa [ M Pc ] + βλ Y + M + B + ( S + D ) a P c P c M P B S a + βφ+ [ M + P+ c+ ] +... b where λ s, as usual he Lagrange mulpler on he me- flow budge consran, and φ s he mulpler on he me- cash-n-advance consran (recall ha all consrans are ncluded n he Lagrangan wh a mulpler). 0
12 Problem 5 connued b. (0 pons) Based on he Lagrangan you formulaed n par a, and recallng he relaonshp b P = beween he prce of bonds and he nomnal neres rae, derve he consumer s + opmaly condon beween cash goods and cred goods, showng very carefully he mporan seps. (Noe ha you mus deermne for yourself whch frs-order-condons are he ones you mus compue,) Your fnal expresson should be of he form u( c, c) =... u ( c, c ) and on he rgh-hand-sde he only varable ha should appear s he nomnal neres rae. ). The frs-order condons wh respec o c, c, a, u( c, c) λp φp = 0 u( c, c) λp = 0 λs + βλ+ ( S+ + D+ ) = 0 λ + φ + βλ = 0 λ + = 0 + b P βλ+ M, and B are, respecvely, Combnng he frs wo expressons gves u ( c, c ) u ( c, c ) = φp. Combnng he las wo expressons gves φ = λ, from whch clearly follows ha + φ P = λ P. + Subsue usng hs for φ P n he mmedaely-prevously-derved expresson o ge u( c, c) u( c, c) = λp. + Nex, use he FOC on cred good consumpon, λ P = u( c, c) o wre he prevous lne as u( c, c) u( c, c) = u( c, c). + Re-arrangng a b more, we have ha he opmaly condon s u( c, c) = +, u( c, c) + whch s n he requesed form. Ths s he opmaly condon beween cash goods and cred goods n hs cash-n-advance (CIA) framework.
13 Problem 5 connued c. (0 pons) As we ve alluded o n class, n consrucng economc frameworks or heores, here are ofen many ways o represen he same dea. The MIU and cash-n-advance frameworks are wo alernave ways of modelng money. Recallng he seup of he MIU framework we suded and examnng he cash-n-advance framework here, brefly descrbe he seps you would need o go hrough o ransform one framework no he oher. Tha s, how could you conver one framework no he oher? (Hn: examne he uly funcons and consrans of he framework here and our MIU framework.) The analyss here s no purely mahemacal, bu raher requres you o draw comparsons beween he wo analycal frameworks. Soluon: In he cash-n-advance (CIA) model here, we have he CIA consran Pc = M, M whch we can obvously wre as c =, whch of course s real balances. We could hen P M subsue hs drecly no he uly funcon and wre he uly funcon as u, c. If P we now nerpre cash goods o be all consumpon, hen hs uly funcon s smply an MIU funcon. Havng made hs subsuon and re-nerpreaon, cash goods have been dropped from he model here, so we need o ake a FOC wh respec o any longer, n whch case we re back o he MIU model. Ths s jus an nsance of, as menoned, deas/models/heores beng somehow somorphc o each oher. END OF EXAM
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