Deparmen of Applied Economics Johns Hopkins Universiy Economics 60 acroeconomic Theory and Policy Final Exam Suggesed Soluions Professor Sanjay Chugh Spring 009 ay 4, 009 NAE: The Exam has a oal of four (4) problems and pages numbered one () hrough hireen (3) (followed by hree blank pages for any scrach work). 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 no use a calculaor. Problem / 5 Problem / 5 Problem 3 / 35 Problem 4 / 0 TOTAL / 00
Problem. oney and oney (0 poins). Consider an exended version of our infinie-period IU framework. In addiion o socks and nominal bonds, suppose here are wo forms of money: and. money (which we will denoe by ) and money (which we will denoe by ) boh direcly affec he represenaive consumer s uiliy. The period- uiliy funcion is assumed o be u c,, = ln + ln +κ ln P P P P c r, which, noe has hree argumens. The Greek leer kappa (κ) in he uiliy funcion is a number beween zero and one, 0 κ, over which he represenaive consumer has no conrol. The period- budge consrain of he consumer is Pc + + + B + S a = Y + + ( + i ) + ( + i ) B + ( S + D ) a, where i denoes he nominal ineres rae on bonds held beween period and + (and hence i on bonds held beween and ) and i denoes he nominal ineres rae on money held beween period and + (and hence i on money held beween and ). Thus, noe ha money poenially pays ineres, in conras o money, which pays zero ineres. As always, assume he represenaive consumer maximizes lifeime uiliy by opimally choosing consumpion and asses (i.e., in his case choosing all four asses opimally). a. (4 poins) Using he funcional form for uiliy given in his problem, wha is he marginal rae of subsiuion beween real money and real money? (Hin: You do no need o solve a Lagrangian o answer his all ha is required is using he uiliy funcion.) Explain he imporan seps in your argumen. Soluion: As always, he RS is simply he raio of marginal uiliies. The marginal uiliy / P funcion wih respec o is, and he marginal uiliy funcion wih respec o is / P κ / P (noe ha you had o use he chain rule o properly compue hese marginal uiliies). / P Consrucing he raio of hese and canceling erms, we have he RS is (i.e., his is he κ slope of any indifference curve over money and money).
Problem coninued b. (4 poins) A sudden, unexplained change in he value of κ would be inerpreable as which of he following: a preference shock, a echnology shock, or a moneary policy shock? Briefly explain. Soluion: The parameer κ, as we saw in par a above, affecs he RS beween wo of he argumens o he uiliy funcion ( money and money). A change in κ hus affecs he slope of he indifference curve, and hus is inerpreable as a preference shock. c. ( poins) Le φ ( c, i, i ) denoe he real money demand funcion for money. Noe he hree argumens o he funcion φ (.). Using he firs-order condiions of he represenaive consumer s Lagrangian, generae he funcion φ ( c, i, i ) (i.e., solve for real money demand as a funcion of c, i, and i ). Briefly explain (economically) why i appears in his money demand funcion. (Noe: you mus deermine yourself which are he relevan firs-order condiions needed o creae his money demand funcion draw on our approach from Chaper 4.) Soluion: Consruc he lifeime Lagrangian as usual, and compue he firs-order-condiions wih respec o c, B, and (he FOCs on and a urn ou o be irrelevan in his problem): λp = 0 c λ + βλ+ ( + i) = 0 κ / P λ + βλ+ ( + i ) = 0 / P Subsiue he FOC on consumpion and bonds ino he FOC on money o ge, afer several algebraic rearrangemens, κ c( + i) =. P i i Noe ha his is very similar o he usual money demand funcion obained when uiliy is logarihmic (in our usual model we implicily had κ = 0 and i = 0 ). The ineres rae i appears in his money demand funcion simply because here is an ineres benefi of holding his asse, as opposed o no ineres in money. As he above money demand funcion shows, he larger is i, he larger is money demand. Think of as a savings deposi agains which you can wrie checks, and money as cash. Cash earns you zero ineres, whereas a savings deposi earns you some posiive ineres; on he oher hand, cash is acceped everywhere, bu checks agains your savings deposi are no acceped everywhere (i.e., savings deposis are less liquid han cash).
Problem. oneary Neuraliy or Nonneuraliy? (5 poins). Suppose firms need o use only labor in order o produce oupu, and oupu in period is given by y = An, where A is a echnology shock and n is he number of hours of labor. Firms choose heir labor in a profi maximizing way every period. Suppose he represenaive consumer s period- uiliy funcion is given by ln( Bc ) + ln l+ ln, where, as always, l = n denoes leisure, B denoes a P preference shock, and / P is real money. Noe ha raher han supposing here are 68 unis of ime available, here we are supposing ha here is only one uni of ime available (hence, boh n and l will be fracions beween zero and one). Finally, assume he labor supply curve is always upward-sloping (i.e., i never bends backwards), and ha he labor ax rae is always zero. a. (7 poins) In he diagram below, show he effec on he consumer s opimal choice of consumpion and leisure in period due o a simulaneous rise in A and rise in B. Clearly label your diagram, and explain precisely any and all effecs perinen here. consumpion Soluion: Wih he represenaive firm maximizing profi, we know ha he real wage equals he marginal produc of labor. In his case (wih he linear producion echnology), he marginal produc of labor is simply A, which means he slope of he budge line in he above diagram (which is he real wage) is A. The rise in A means he budge line becomes seeper, pivoing around he fixed poin on he leisure axis. Nex, consider he RS beween consumpion and leisure. We know he RS beween consumpion and leisure (i.e., he slope of he indifference curve) is ul / u c. Wih he given uiliy funcion, ul = / l and uc = B/( Bc) = / c. Noe ha he B erm drops ou of he marginal uiliy of consumpion here (you had o use he chain rule o fully differeniae). Thus, he RS is unaffeced by he preference shifer in his case, meaning he indifference map is unaffeced. 3
Problem coninued b. (7 poins) Suppose insead of he preference and TFP shocks in par a, here were a posiive money shock (and no oher shocks) a he end of period (as we described in class i.e., he acual urns ou o be differen han he planned ). Suppose boh nominal prices and nominal wages are compleely flexible. In he diagram below, skech he effec on he consumer s opimal choice of consumpion and leisure in period due o he surprise exra quaniy of money in he economy. Clearly label your diagram, and explain precisely any and all effecs perinen here. (Hin: The consumpion-money opimaliy condiion is relevan for he analysis here.) c. consumpion iniial opimal choice leisure Soluion: Recall he consumpion-money opimaliy condiion wih log-log uiliy (derived Pc i in class and on Problem Se 0): = (again, noe ha he preference shifer B does + i no appear). A surprise rise in in he RBC view is me wih an adjusmen in jus (all) prices, since he choice regarding consumpion represens an opimal choice. Thus P rises. Bu in he above diagram, he slope is W/P (he nominal wage divided by he nominal price level). In order o leave he planned opimal consumpion choice unaffeced, he slope of he budge line mus remain he same, which requires ha W rises by he same percenage as he rise in P. Graphically, hen, nohing changes: he budge line is unaffeced, and opimal choice remains unchanged. 4
Problem coninued c. (7 poins) Consider again he same money shock from par b. Suppose he nominal wage is compleely rigid (i.e., i can never change), bu nominal goods prices are compleely flexible. In he diagram below, skech he effecs on he consumer s opimal choice of consumpion and leisure in period due o he surprise exra quaniy of money in he economy. Clearly label your diagram, and explain precisely any and all effecs perinen here. (Hin: The consumpion-money opimaliy condiion is relevan for he analysis here.) consumpion Soluion: The logic proceeds as in par c unil he discussion abou he nominal wage. In order o leave he consumpion choice unchanged, we saw in par c ha W had o rise by he same percenage as P. Bu here W is sicky, so he budge line mus roae somehow. Wih a rise in P and W unchanged, he real wage is lower, so he budge line pivos downward, meaning opimal consumpion is lower here. Thus, in his case expansionary moneary policy ends up causing consumpion o fall because of he sickiness in nominal wages. d. (4 poins) The RBC view is capured by which scenario, par b or par c? The New Keynesian view is capured by which scenario, par b or par c? Soluion: The RBC view is par b (all prices are flexible), he NK view is par c (some prices in his case wages are sicky). 5
Problem 3. Consumpion, Savings, and Financing Consrains. (35 poins) Because he consumpion expendiures ha occur laer in an individual s lifeime are ypically on biggericke iems (e.g., cars, refrigeraors, ec.) and hus more expensive, an individual ofen has o begin planning for his/her fuure consumpion expendiures well in advance of heir acual purchase and use. In conras, consumpion expendiures during he earlier years of an individual s life are predominanly on smaller iems (e.g., movies, food, enerainmen, ec.) and hus migh no require as much advance planning. We can analyze his idea in a wo-period represenaive consumer framework. As always, suppose he represenaive consumer has uiliy funcion of period- consumpion and period- consumpion given by uc (, c ). Naurally, period is he early sage of an individual s economic life, and period is he laer sage of an individual s economic life. Suppose he financial asses ha individuals have a heir disposal are socks, jus as we sudied in Chaper 8. The represenaive consumer begins period wih zero sock holdings (i.e., a 0 = 0). The period- and period- budge consrains of he represenaive consumer are hus Pc + Sa = Y Pc + S a = Y + ( S + D ) a in which he res of he noaion is as always: P denoes he per-uni nominal price of a consumpion good (in a given ime period), Y denoes he nominal income he consumer earns (in a given ime period), S denoes he nominal price of each share of sock (in a given ime period), and D denoes he per-uni nominal dividend each share of sock pays (in a given ime period). Because period- goods are weighed more owards big-icke iems, consumers ypically have o borrow o purchase hem. This means ha asymmeric informaion issues may be a facor in lenders being willing o exend credi o consumers for heir period- purchases. Suppose he financing consrain (aka credi consrain) ha has evolved in markes o deal wih hese informaion issues is Pc = Sa. The inerpreaion of his consrain is ha he marke value of asses accumulaed during period of he individual s life forms he basis for period- consumpion. Finally, jus as in class, define he nominal ineres rae on sock as S + D i + =, S + i define he real ineres rae on sock as + r = + π framework, we know a = 0. (OVER), and, because his is a wo-period 6
Problem 3 coninued a. (6 poins) Formulae he sequenial Lagrangian for he represenaive consumer s uiliy maximizaion problem, saring, as usual, from he perspecive of he beginning of period. (Several hins and noes are useful here: i) Do no subsiue he financing consrain direcly ino any of he oher consrains; i will be mos informaive o conduc he analysis wih he financing consrain as a separae consrain; ii) Use he muliplier μ (he Greek leer mu ) for he financing consrain; iii) Be exremely careful abou your seup of he Lagrangian here, because i is he basis for almos all of he analysis ha follows!) Soluion: The sequenial Lagrangian is [ ] uc (, c) + λ[ Y Pc Sa] + λ [ Y+ ( S + D) a Pc] + μ Sa Pc b. (8 poins) Based on he Lagrangian in par a, compue he firs-order condiions wih respec o c, c, and a. Soluion: The FOCs are u( c, c) λp = 0 u( c, c) λp μp = 0 λs + λ ( S + D ) + μs = 0 For use in he nex pars of he problem, noe ha he firs equaion can be rearranged o u( c, c) u( c, c) λ = ; he second equaion can be rearranged o λ = μ ; and he hird P P S + D equaion can be wrien as λ = λ + μ, or using he definiion of he ineres rae on S sock, ( i ) λ = λ + + μ. 7
Problem 3 coninued c. (4 poins) If here were no financing consrain on consumers purchases of period- consumpion, he value of he Lagrange muliplier μ could be hough of as being equal o wha numerical value? Be as precise as possible, and briefly explain. (Noe: You can answer his par even if you were unable o ge all he way hrough par a and par b.) Soluion: If financing condiions did no maer a all for consumpion purchases, we could hink of he value of he Lagrange muliplier on he financing consrain as being exacly equal o zero. S + D + i d. (0 poins) Using he definiions + i = and + r =, rearrange S + π he firs-order condiions you obained in par b above o derive he consumpion-savings u( c, c) opimaliy condiion. Your final expression should be of he form =..., where he u( c, c) ellipsis on he righ hand side indicae erms ha you mus deermine. Clearly explain/show he seps in your logic/derivaion. (Noe: I is fine if he final expression conains he muliplier μ in i. This derivaion requires a few algebraic seps, bu he logic of he derivaion is exacly he same as our iniial sudy of he wo-period framework.) Soluion: Insering he expressions for λ and λ obained in par b above ino he hird expression obained in par b above, we have u( c, c) u( c, c)( + i ) = μ( + i ) + μ. P P On he righ hand side, we can cancel μ, leaving us wih u( c, c) u( c, c)( + i ) = μi. P P On boh sides of his expression, muliply by P and divide by u( c, c ) o ge u( c, c) P( + i ) μpi =. u( c, c) P u( c, c) Using he Fisher relaion, we could also express his as u( c, c) μpi = + r u ( c, c ) u ( c, c ) Eiher of he laer wo expressions are wrien in he requesed form. Noe ha if μ = 0, his is simply RS across ime periods is equal o he gross real ineres rae (where he ineres rae is ha measured according o sock-marke reurns). 8
Problem 3 coninued e. (7 poins Harder) Based on he consumpion-savings opimaliy condiion you derived in par d above, does he financing consrain Pc = Sa maer for consumer s consumpion and savings decisions over ime? In oher words, does he sandard consumpion-savings opimaliy condiion sudied in Chaper 3 and 4 ge alered by he presence of his financing consrain? If so, explain he economic inuiion behind why; if no, explain he economic inuiion behind why no. (Hin: A diagrammaic explanaion, alhough no required, may be useful. In any case, here are likely several differen ways o usefully describe he economic effecs here.) Soluion: Clearly, if μ is differen from zero, he sandard consumpion-savings radeoff is affeced. To gain inuiion for how/why, we can re-express he condiion in par d as u ( c, c ) + μpi u ( c, c ) = + r. Here, he lef-hand-side is a sor of generalized RS (noe ha he erm μ Pi has unis of uils because μ has unis of uils/dollar). If μ = 0, hen clearly we have he usual Chaper 3 and 4 opimaliy condiion. Imagine drawing he corresponding indifference curve/lifeime budge consrain diagram of his oucome. Then, saring from he unconsrained opimal choice, if he value of μ rises above (falls below) zero, hen, holding consan he real reurn on sock (i.e., holding consan he slope of he LBC), he indifference curve passing hrough he unconsrained allocaion becomes seeper (flaer). The new opimal choice (i.e., he one aking ino accoun he binding financing consrain) hen feaures more (less) c (and hus less (more) c ) compared o he unconsrained opimal choice. Thus, in his case, consumers can be forced o consume eiher more or less consumpion across ime periods due o he financing consrain. This seems a odds wih our discussion of credi consrains earlier in he semeser. The difference arises because here we are considering an always-binding financing consrain (i.e., i always holds wih sric equaliy), whereas in our earlier sudy we were considering a credi consrain ha only affeced consumpion-savings oucomes if consumers waned o borrow during period bu no oherwise. 9
Problem 4. Proporional Taxes and Ricardian Equivalence? (0 poins) (Harder) Consider a modified version of he wo-period framework wih governmen sudied in Chaper 7. By governmen here we will mean jus he fiscal auhoriy; suppose here is no moneary auhoriy a all. The governmen and he represenaive consumer each live for boh periods of he economy, and suppose here are never any credi consrains on he consumer. The governmen does no have access o lump-sum axes, only proporional consumpion axes. However (his is differen from our baseline framework), he consumpion axes he governmen collecs in a given period are no resriced o be levied on consumpion from only in ha period. To be more precise, suppose ha oal consumpion ax revenues he governmen collecs in period are based only on period- consumpion (because here was no period zero, say). However, oal consumpion ax revenues he governmen collecs in period are based on boh period- consumpion and period- consumpion. Tha is, a porion of he revenue colleced in period is based on period- consumpion, and he remaining porion of he revenue colleced in period is based on period- consumpion. Denoe by τ, he ax rae on period- consumpion ha is levied in period ; denoe by τ, he ax rae on period- consumpion ha is levied in period ; and by τ, he ax rae on period- consumpion ha is levied in period. There is no τ,3 (which would represen he ax rae on period- consumpion ha is levied in period 3) because he economy does no exis in period 3. Wih his noaion, he governmen s period- and period- budge consrains in real erms are: g + b = ( + r) b + τ c 0, g + b = ( + r) b + τ c + τ c,, The represenaive consumer s period- and period- budge consrains in real erms are: ( + τ ) c + a = ( + r) a + y, 0 τ c + ( + τ ) c + a = ( + r) a + y,, For simpliciy, suppose he governmen and consumer each begin period wih zero asses. As usual, you can hink of all he ax raes as being numbers beween zero and one (bu hey need no be so resriced). The remainder of he noaion is as in class. Noe carefully how he ax raes τ,, τ,, and τ, appear in hese budge consrains. (OVER) 0
Problem 4 coninued a. (5 poins) Consruc he governmen s LBC, showing imporan seps. Provide brief economic inerpreaion. Soluion: The way o consruc he governmen LBC is he same as always: firs solve he period- budge consrain for b (in which we use he usual condiion b = 0), hen inser his soluion ino he period- consrain. Rearranging erms a bi, we arrive a g τ,c τ,c g+ = τ,c+ + + r + r + r, which saes, as does any governmen LBC, ha he presen-value of lifeime governmen spending equals he presen-value of lifeime governmen ax collecions. Here, hose lifeime ax collecions involve axes on period- consumpion ha are colleced in period (he τ, c erm) as well axes on period- consumpion ha are colleced in period (he τ,c erm), which mus be discouned because i is colleced in period. b. (5 poins) The essence of he way we defined Ricardian Equivalence in class was: An economy exhibis Ricardian Equivalence if, holding fixed is sequence of governmen spending, a change in he iming of lump-sum axes (and also assuming no credi consrains and ha consumers planning horizons are he same as he governmen s planning horizon) has no effec on consumpion or naional savings. In he analysis a hand here, suppose he governmen keeps is sequence of g and g unchanged, bu decides o cu he ax rae in period on period- consumpion ha is, i lowers he ax rae τ,. Is i possible for his economy o exhibi Ricardian Equivalence even hough in he framework in his problem axes are NOT lump-sum? If so, carefully show how/why and provide brief economic inerpreaion. If no, precisely explain why no. (Hin: Derive he consumer s LBC and focus your aenion on he slope of he consumer s LBC.) Soluion: To begin, consruc he LBC of he consumer in he usual manner. Solve he period budge consrain for a (using he No-Ponzi condiion a = 0) o ge τ, + τ, a = c+ c y. Inser his ino he period- budge consrain doing so and + r + r + r rearranging a bi gives us τ, + τ, y ( + τ,) c + c + c = y + + r + r + r. Le s combine he erms involving c o ge
Solve his for c : τ, + τ, y + τ, + c c y r + = +. + + r + r τ, + τ, + r ( + r) y y c = + ( + r) c+ +, + τ, + τ, + τ, and, as you re old, focus on he slope of he LBC, i.e., he coefficien in fron of c. Before we proceed, le s ouline he res of he argumen. Ricardian Equivalence is a saemen abou how changes in axes do (or do no) lead o changes in naional savings, which, recall, is na s = y g c in period. As we ve discussed, he issue hen boils down o deermining wheher a change in axes affecs period- consumpion (because y and g are assumed o be unchanging in period ). In he (simpler) framework wih proporional consumpion axes we sudied in class, we had (implicily) ha τ, = 0, which means ha period- consumpion decisions did no lead o ax paymens in period. Thus, in ha framework, a fall in τ, would mean ha τ, mus rise in order for he governmen LBC o hold he fall in τ, and he rise in τ, necessarily means he slope of he LBC in ha framework became flaer, which hen in general would mean ha he opimal choice of c would change, which would hus mean ha naional savings changes, hence Ricardian Equivalence does no hold. However, here τ, can be nonzero. I is possible ha, when τ, falls, τ, could rise by an amoun ha leaves he overall coefficien in fron of c, which is afer all he slope of he LBC, unchanged. If he LBC doesn change, hen he opimal choice of consumpion doesn change, hence naional savings doesn change, hence Ricardian Equivalence holds even hough axes are no lump-sum here and none of he oher reasons for he failure of Ricardian Equivalence are presen eiher. Allowing for he governmen o ax pas choices (ha is, allowing he governmen o ax in period consumpion which occurs in period ) hrough non-lump-sum axes urns ou o resusciae he Ricardian Equivalence Theorem here. A bi more specifically, wha s happening in his framework is ha he governmen has a richer ax srucure available o i compared o he proporional-ax framework we sudied in class he governmen is no resriced here o ax only conemporaneous choices in a given period. The axes paid in period according o he rae τ, were incurred in period ; i s jus ha hey are no paid unil period. I urns ou ha i does no maer exacly when he axes incurred in period are paid hey can be paid a any ime, and he iming of hose ax collecions don affec real economic aciviy.
Problem 4 coninued This resul is only a recenly-undersood one in heoreical macreconomics, even hough he idea of Ricardian Equivalence has been a benchmark resul in macroeconomic heory for he pas 30 years. END OF EXA See arco Basseo and Narayana Kocherlakoa, On he Irrelevance of Governmen Deb When Taxes are Disorionary, Journal of oneary Economics, Vol. 5, 004, p. 99-304, for he firs exposiion of his resul, a very readable aricle, one which in fac is developed using he simple wo-period model. 3