STATE UNIVERSITY OF NEW YORK AT ALBANY Department of Economics. Ph. D. Comprehensive Examination: Macroeconomics Spring, Partial Answer Key

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1 STATE UNIVERSITY OF NEW YORK AT ALBANY Deparmen of Economcs Ph. D. Comprehensve Examnaon: Macroeconomcs Sprng, 200 Paral Answer Key Par I. Please answer any 2 of he followng 3 quesons.. (From McCallum, 989.) Recall ha producers solve: max K 0,L 0 Π = Y R K W L, Y = Z L α K α, 0 <α<, (PRF) whle he represenave household solves he followng problem: ³X max E 0 {C,K +,L } =0 β [ln (C )+χln ( L )], =0 0 <β<, χ > 0, s.. C + K + = R K + W L + Π, (CA) µ lm E β J K +J+ =0, (NPG) J C +J along wh he oher usual boundary and non-negavy consrans. Noe ha capal deprecaes a a rae of 00 percen. Fnally, producvy follows an AR() process n logs: z ln (Z )=( φ) z + φz + ε, 0 φ<, (TS) where {ε } s an exogenous saonary marngale dfference sequence. (a) The frs order condons for prof maxmzaon are αz L α K α = W, (PM) ( α) Z L α K α = R, whle he frs order condons for uly maxmzaon are W = χ, C L C = βe µ C + R +.

2 Usng equaon (PM) o elmnae wages, he labor allocaon condon becomes αz L α C K α = χ, (LL) L whle he Euler equaon becomes C = βe µ C + ( α) Z + L α +K α + (EE) The capal accumulaon equaon can be wren as K + = Y C, (CA) wh Y gven by equaon (PRF). Ths consran can eher be derved drecly as a resource consran, or found by combnng he consumer s budge consran wh he defnon of profs. (b) We are conjecurng ha L = L, C = Π 0 Z L α K α K + = Π 20 Z L α K α = Π 0 Z L α K α = Π 20 Z L α K α Π Z K α, (CRULE) Π 2 Z K α. (KRULE) Imposng hese conjecures on equaon (EE) yelds µ Π Z K α = βe Π Z + K+ α ( α) Z + L α K+ α µ α = βe L α Π K µ + α = βe Π Π 2 Z K α L α, so ha and µ α =βe L α Π 2 Π 2 =( α) βl α. Repeang he exercse wh equaon (CA) yelds so ha Π 2 Z K α = Z L α K α Π Z K α, Π = L α Π 2 = [ ( α) β] L α. 2

3 Imposng he conjecure on equaon (LL) and nserng he equaon for Π shows ha so ha χ L = = L L = Π Z K α αz L α K α αl α, [ ( α) β] L α α χ [ ( α) β], and α L = α + χ [ ( α) β], hus confrmng our conjecure. (c) Loggng boh sdes of equaon (KRULE) yelds z = k + ( α) k π 2. Inserng hs resul no equaon (TS) shows ha whch reduces o k + ( α) k π 2 = φ [k ( α) k π 2 ]+ε, k + =(φ + α) k φ ( α) k +( φ) π 2 + ε. Combnng equaons (CRULE) and (KRULE) shows ha k + = π 2 π + c. Inserng hs no our expresson for logged capal yelds c =(φ + α) c φ ( α) c 2 + π 3 + ε. Smlarly, follows from equaons (KRULE) and (PRF) k + = π 2 α ln (L)+y = ln (( α) β)+y, so ha y =(φ + α) y φ ( α) y 2 + π 4 + ε. I can also be shown ha π 3 = ( φ)(απ ( α) π 2 ), π 4 = ( φ) α 2 ln (L) ( α) π 2. (d) Le s consder he economy s response o a echnology shock. Le b denoe logged devaons from seady sae values. 3

4 . If φ =0.9 and α =0.7, our answers o par (c) become bz = 0.9bz + ε, b k+ =.2 b k 0.27 b k + ε, by =.2by 0.27by 2 + ε, whch generaes he followng mpulse response funcons ε bz b k by Snce he mpulse response for oupu dffers from he mpulse response for oal producvy, hs economy clearly has a propagaon mechansm. In parcular, capal accumulaon amplfes he effec of he shock mproved echnology no only ncreases oupu drecly, bu nduces agens o accumulae more producve capal, whch ncreases oupu even furher. 3. Noe ha he mpulse response funcon for oupu has a hump shape, whch s n fac he shape observed n he daa. The reason for he hump n hs model s ha capal accumulaon occurs wh a lag. In parcular, whle echnology s slghly lower n perod 2 han n perod, he capal sock s consderably hgher. The ne effec s ha oupu s hgher n perod Were we o use a more realsc deprecaon rae, he relave change n he capal sock nduced by a echnology shock would be much lower. Wh a small deprecaon rae, nvesmen comprses a small fracon of capal, so ha even large changes n nvesmen have relavely lle effec on he capal sock. Ths would effecvely elmnae he propagaon effecs of capal accumulaon. Insead, he mpulse response funcon for oupu would have he same shape as he mpulse response funcon for echnology. Ths s a well-known feaure of he baselne real busness cycle model. 2. Recall ha he preferences of he represenave consumer are X β [u (c )+v( )], =0 0 <β<. The consumer has one un of me n each perod o allocae beween work and lesure,.lengb p + denoe one-perod bonds purchased n perod, he budge consran of he consumer s c + b p + = w ( ) τ +(+r ) b p where w s he real wage and r sherealneresraeonabondpurchasednperod. The represenave frm has access o a producon echnology y = z n 4

5 where y s oupu, n s labor, and z deermnes aggregae labor producvy. The governmen purchases g uns of oupu n perod- and hrows hem no he ocean. The governmen s budge consran s g +(+r ) b g = τ + b g + where b g + denoes one-perod bonds ssued n perod and τ denoes lump sum axes. (a) The socal planner s problem s wren as follows: max {c, } X β [u (c )+v( )] =0 s.. c + g = z ( ), gven {g }. Subsung for c n he socal welfare funcon yelds he equvalen problem: X max β {u [z ( ) g ]+v( )}. { } =0 Dfferenang wh respec o and seng he resul equal o zero gves he followng frs-order condon o he planner s problem: u 0 [z ( ) g ]( z )+v 0 ( )=0. Ths condon mplcly defnes he opmal polcy = (z,g ). (b) Snce here are no mssng markes or dsorons, he socal planner s opmum corresponds o he compeve equlbrum n hs economy. The uly maxmzaon problem solved by he represenave agen s wren: max {c,,b p +} X β [u (c )+v( )] =0 s.. c + b p + = w ( ) τ +(+r ) b p gven {w r } =0, {τ r} =0,andbp =0. Leng λ denoe he Lagrange mulpler on he perod- budge consran, he frs-order condons o hs problem are wren as follows: u 0 (c ) λ =0 v 0 ( ) λ w =0 λ +(+r + ) λ + =0. Elmnang he Lagrange mulpler from hese expressons yelds he followng: u 0 ( ) v 0 (c ) = w () βu 0 (c + ) u 0 (c ) = +r +. (2) 5

6 Snce g = g and z = z for all, he condon n par (a) mples ha lesure s also consan: = = (z, g) for all. Gven ha lesure s consan, oupu s also consan: y = z ( ) = zn for all. Subsequenly, goods marke clearng mples ha consumpon s also consan: c = y g for all. Imposng c = c + = c n (2) mples β = +r + r + = β for all. (c) The represenave frm solves he followng problem n each perod: max n z n w n max n (z w ) n gven z and w.zeroprof mples w = z. A emporary ncrease n z ncreases he prce of lesure nducng he consumer o subsue consumpon for lesure by workng harder (subsuon effec). The ncome effec also ncreases consumpon. (Addve separably ensures ha consumpon s a normal good. Ths s verfed by dong he comparave sacs on ().) Therefore, z T = z >z T = z mples c T >c T. From (2) we know ha r T > β. Also, z T + = z<z T = z mples c T + <c T. From (2) we know ha r T + < β. In all oher perods he neres rae equals. β (d) Usng he same logc as n par (c), consumpon s consan over he perods =0,, 2,...,T and = T,T +,T +2,... wh consumpon n he laer perods greaer han durng he earler perods. From (2) we deduce ha r T > β. In all oher perods he neres rae equals. β (e) For a emporary ncrease n producvy, he worker mus consume more emporarly (here s no sorage or nvesmen n hs economy). Snce he consumer wshes o smooh consumpon, he neres rae mus rse before ncome ncreases o preven equlbrum borrowng. Smlarly, he neres rae mus fall o preven he consumer from accumulang asses when producvy reurns o normal. For a permanen ncrease, he neres rae need only rse o preven borrowng agans he hgher fuure ncome. 6

7 3. We are consderng a varan of he Lucas ree model where here are wo ypes of rees: apple rees, whch wll be ndexed by a ; and banana rees, whch wll be ndexed by b. (a) The dvdend processes for he wo rees are gven by (d a d) = φ d a d + ε a, d b d = φ d b d + ε b, wh 0 φ<, d>0, andε b = αε a,where <α<0. Recursvely subsung (or usng lag operaors) shows ha d b d X X = φ j ε b j = α φ j ε a j = α (d a d). j=0 Le d = d a + d b denoe aggregae dvdends. I mmedaely follows ha d = d a + d b =2d +(d a d)+ d b d =2d +(+α)(d a d), so ha (d 2d) = (+α) φ d a d + ε a = φ (d 2d)+(+α) ε a. mn λ 0 max ln c + λ c 0, s a +,sb +,y( ) j=0 The sae of he economy can be characerzed by aggregae dvdends, d.(snce d a and d b are one-o-one ransformaons of d, hey would work as well.) To see ha d exhauss he sae space, frs recall ha he consumer cares only abou oal consumpon, and noe ha d gves he economy s aggregae consumpon endowmen. Snce all consumers are dencal, n equlbrum aggregae resources wll be suffcen o characerze consumer behavor. Fnally, here s no nformaon beyond d ha s necessary for predcng fuure resources. I follows ha d summarzes he sae of he aggregae economy. (b) We le q (d 0,d) be he kernel used o prce one-sep-ahead conngen clams. Wrng he consumer s problem as a Lagrangean, we ge V (x,d )= x c X Z p s + y (d + ) q (d +,d) dd + Z +β V X {a,b} {a,b} p + (d + )+d + s + + y (d + ),d + f (d +,d ) dd +, where f (, ) gves he condonal densy of d +. The FOC for an neror soluon are: /c = λ, Z V [ +] λ p = β p x + (d + )+d + f (d+,d ) dd +, {a, b}, + λ q (d 0,d ) = β V (x + (d 0 ),d 0 ) f (d 0,d ), d 0. x + 7

8 Noe ha we pck y ( ) as f we were pckng a scalar for each possble realzaon of d + (value of d 0 ). Snce (followng Benvense-Schenkman), V [] = λ, x he Euler equaons are µ, {a, b}, p = βe p c c + + d + + c q (d +,d )=β c + (d + ) f (d +,d ), d +. (c) To acheve equlbrum,we mpose c (d )=d, s + =, {a, b}, andy (d + )= 0, d +. Then he equlbrum sock prce and prcng kernel follow µ p = d βe, (EE) q (d +,d )=β d + p + + d + µ d d + f (d +,d ), d +, (EE 0 ) where d + denoes a parcular realzaon of nex perod s dvdends. Snce 0 <β<, makes sense o solve equaon (EE) forward: E µ βl d p d = + βe, p d = βl βe d + µ Ã d + X + b = E d + wh he bubble erm b obeyng E (b + )=β b. j= β j d +j d +j! + b, We now requre ha lm E β J p +J d+j = 0,. (TVC) J (TVC) wll be sasfed only f b =0and he prce of a sock s Ã! X p = p (d )=d E β j d +j. d +j (d) I follows from arbrage argumens ha f an asse pays w (d) uns of consumpon goods when d + = d, sprces Z p w = w (d + ) q (d +,d ) dd +. When he asse s a rsk-free dscoun bond, w (d) =.Imposngequaon(EE 0 ), follows ha he prce of a rsk-free bond, R (d ) s gven by Z Z µ µ R d (d )= q (d +,d ) dd + = β f (d +,d ) dd + = βd E. d + d + 8 j=

9 (e) Defnehereurnonsocks,R,by R = p + + d +, {a, b}, p and defne he equy premum, e,bye = E (R) R.Usehedefnon of R o rewre equaon (EE) µ d = βe R d µ + µ d = βe E R + Cov β d,r d + d + Inserng he resul from par (d) shows ha: µ = R E R Cov β d,r, d µ + E R = R R Cov β d,r, d + = R + e, e Cov (/d +,R). E (/d + ) Informally, we would expec e a o be bgger han e b,whchsosayhawe would expec nvesors o demand a hgher expeced rae of reurn on apple rees. Recall ha nnovaons o apples and bananas move n oppose drecons, wh nnovaons o apples beng he larger of he wo. Ths means ha mes of hgh oupu/consumpon are mes when dvdends on apples are hgh, bu dvdends on bananas are low. Ths suggess ha he reurn on apple socks wll be srongly procyclcal, whle he reurn on banana socks wll be weakly procyclcal or even counercyclcal. If consumpon and apple sock reurns move n he same drecon, hen he margnal uly of consumpon and apple sock reurns move n oppose drecons, so ha Cov (/d +,R a ) < 0 and e a > 0. If apple socks yeld he hghes reurns when he margnal uly of consumpon s low, households wll no purchase hem unless hey are compensaed wh a hgher expeced reurn. On he oher hand, f banana socks yeld he hghes reurns when aggregae consumpon s low, households wll be wllng o accep lower yelds. The Capal Asse Prcng Model says ha asses whose reurns move srongly n he same drecon as he reurns of he marke porfolo (have a large posve bea ) n hs case, apple socks wll n general have o offer hgher reurns han asses whose reurns move n he oppose drecon n hs case, bananas. Fnally, noe ha as α goes o, he varance of aggregae oupu/consumpon goes o zero. The equy prema go o zero accordngly when here s no consumpon uncerany, he paern of an asse s payoffs acrosshesaespaces rrelevan. 9

10 Par II. Please answer any 3 of he followng 5 quesons. 4. We are consderng several ssues of naonal ncome accounng. (a) An auomoble s reaed as a consumpon good. The prce of he auomoble s par of naonal produc durng he year of producon, and s couned as consumpon. Suppose, for example, ha he prce of he auomoble s $0,000. Then naonal produc and consumpon s $0,000 hgher durng he year of producon. (b) Alernavely, he auomoble could be reaed as an nvesmen good (housng s reaed hs way). The prce of he auomoble s par of he naonal produc durng he year of producon and s couned as nvesmen. Each year afer, deprecaon s calculaed and reduces ne naonal produc and ne nvesmen. Also, each year he value of he servces of he auomoble o he consumer s calculaed and s par of naonal produc and consumpon. For he $0,000 auomoble, suppose ha he auomoble lass 0 years, and he deprecaon s $,000 per year. Also, suppose ha he value of he servces of he auomoble o he consumer s $,500 per year. The frs year he addon o ne naonal produc s $0, 000 $, $, 500 = $0, 500. In he remanng nne years, he addon o ne naonal produc s $, $, 500 = $500. Added over he full 0 years, he oal effec on ne naonal produc s $, = $5, 000, whch s he oal value of he servces of he auomoble o he consumer. The deprecaon offses he nal cos exacly. (c) For an nvesmen good bough by a busness, he value of he good o he busness s mplc n he value of s produc, so no separae measure of he value of he servces of he nvesmen good s necessary. Consder sofware cosng $500 ha lass fve years; deprecaon s $00 per year. The accounng change convers he sofware from an nermedae good o an nvesmen good. For he frs year, gross naonal produc and gross nvesmen rse by $500; hese gross quanes are unaffeced nlaeryears. Forhefrs year, ne naonal produc and ne nvesmen rse by $500 $00 = $400. In he remanng four years, ne naonal produc and ne nvesmen fall by $00. Added over he full fve years, he oal effec on ne naonal produc and ne nvesmen s zero; he deprecaon offses he nal cos exacly. 5. Brhdays and Weddngs. We are consderng a world wh N dencal agens. Each ndvdual solves max ln (e + e 0 + αe)+(e e ), e [0,E] where e s ndvdual s expendures on he socal even, e s he average expendure of oher ndvduals n he economy, and E>s ndvdual s resources. (a) When e 0 =and α =, so ha ndvduals vew hese evens as compeons, he frs-order condon for he consumer s problem s e + e =, 0

11 so ha e = e, whchcanholdforanye [0,E]. Equlbra wh hgher expendures are no welfare-enhancng, however. In fac, mmedaely follows from u (e,e)=ln(e e +)+(E e ) ha hgher values of e reduce welfare. In Cooper and John s (988) ermnology we have sraegc complemenares and negave spllovers. When oher people spend more on socal evens, your opmal expendure goes up, presumably because your socal saus depends on your relave expendures hs s he sraegc complemenary. On he oher hand, expendures by oher people make your expendures less producve, presumably because you now need more spendng o mach he socal norm hs s he negave spllover. (b) If 0 <e 0 < and α =, so ha ndvduals vew hese expendures as subsuable across evens, he frs order condon s e + e 0 + e =, e = e e 0. In a symmerc equlbrum, e = e, and he equlbrum expendure s e =( e 0 ) /2. Noe ha he socal planner solves max ln (2e + e 0)+(E e), [0,E] he soluon for whch s e =(2 e 0 ) /2. Because ndvdual agens do no ake accoun of he way n whch her acons benef oher agens, hey spend oo lle. In par (a) we had negave spllovers and sraegc complemenares, whch could lead o oo much expendure on socal evens. Here we have he mrror oppose, posve spllovers and negave sraegc complemenares (sraegc subsuables), whch leads o oo lle expendure. 6. We are consderng a T -perod economy wh a sngle agen whose preferences are E 0 TX β [ln c +ln( n )] = where c s perod- consumpon and n s he fracon of he perod worked by he agen n perod. The agen owns a echnology ha convers me no oupu accordng o he formula y = z n α, 0 <α<, where z > 0 s an..d. random varable wh posve suppor. There s a governmen ha mposes an ncome ax τ n each perod, and hrows he proceeds no he ocean. (a) Solve he followng problem faced by he household/frm: max {n } E 0 TX β [ln ( τ ) z n α +ln( n )] =

12 gven {τ r } =0. Noe ha bond marke equlbrum (boh prvae borrowng and publc borrowng s zero) has been mposed as a shorcu. The agen s frs-order condon gves he (sae-ndependen) decson rule for employmen: α =0 n = α n n +α. Perod- governmen revenue s herefore µ α α z τ. +α The graph s lnear wh slope z α +α α > 0. (b) From par (a), he governmen mus choose τ such ha µ α α z τ = G τ = +α z µ α α G. +α Noe ha he governmen observes z. (c) The populaon momen (condonal on z )sgvennpar(a): z α α +α > 0. Snce he ax rae s chosen by he governmen o manan consan revenue, he economercan ges a (very precse!) esmae of zero nsead. He used an nconssen esmaor by wrongly assumng ha τ was an exogenous varable. Revse and resubm. 7. Consder he compuaon of he Solow resdual measure of dsemboded echnologcal change, for an economy wh capal and labor as npus. (a) In Solow s sandard formulaon, he conrbuon of an npu o growh s he ncome share for he npu mulpled by he growh rae of he npu. To see hs, noe ha he ncrease n oupu Y caused by an ncrease n labor L s mp L L, so he ncrease n oupu growh Y/Y caused by he ncrease n labor s mp L L/Y. For an economy n compeve general equlbrum, he real wage s he margnal produc of labor, w = mp L. Hence he ncrease n oupu growh caused by he ncrease n labor s mp L µ L Y µ L = w Y µ µ wl L = Y L = ncome share npu growh. (b) We assume ha: he labor force s 0, he real wage s 0; prof (capal ncome) s half of naonal ncome; and labor ncome s half. I follows ha labor ncome s 0 0 = 00, so capal ncome s also 00; and naonal ncome s 200. Durng he year, labor rses by 2%, capal rses by 8%, and real oupu rses by 20%. The conrbuon of labor growh o oupu growh s hus /2.2 =.06, and he 2

13 conrbuon of capal growh o oupu growh s /2.8 =.09. Consequenly he oal conrbuon of npu growh o oupu growh s =.5. Snce oal oupu growh s.20, herefore he resdual measure of echnologcal change s.20.5 =.05. Oupu rses faser han eher npu, so he Solow resdual s necessarly posve. (c) Alernavely, suppose ha he economy s no perfecly compeve, bu here s sgnfcan monopoly. Wh monopoly, he oupu prce s a markup over npu cos. Then he conrbuon of an npu o growh s he cos share for he npu mulpled by he growh rae of he npu. To see hs, recall ha oal cos c s labor cos plus capal cos, c = wl + rk; Cos mnmzaon by he frm mples ha he margnal cos equals he npu prce dvded by he margnal produc, mc = w mp L = r mp K. Here mc s he margnal cos, mp L s he margnal produc of labor, and mp K s he margnal produc of capal. Wh consan reurns o scale, he average cos ac equals he margnal cos mc. Then he ncrease n oupu growh caused by he ncrease n labor s µ L mp L = w µ L = w µ L Y mc Y ac Y µ µ wl L = acy L = cos share npu growh. (d) Usng he resul n (c), we recompue he Solow resdual assumng ha half of he prof s he acual cos of capal, whereas he oher half s a monopoly markup over oal npu cos. The labor cos s 00, and he capal cos s 50 (half of prof),so he cos shares are 2/3 and /3. The conrbuon of labor growh o oupu growh s 2/3.2 =.08, and he conrbuon of capal growh o oupu growh s /3.8 =.06. Consequenly he oal conrbuon of npu growh o oupu growh s =.4. Snce oal oupu growh s.20, hereforehe resdual measure of echnologcal change s.20.4 =.06. Because he wegh assgned o he faser growng npu (capal) has declned, he compued rae of echnologcal change s hgher. 8. The neoclasscal growh model s largely slen on hs ssue snce he balanced growh rae of per-capa consumpon s deermned exogenously by summng he growh raes of oal facor producvy and populaon. The saemen of he queson s evocave of a smple endogenous growh model wh human capal (and nondecreasng reurns) ala Lucas. In erms of hs framework, a subsdy o educaon causes ndvduals o reallocae her me away from producon and oward educaonal acves. Alhough he subsdy ncreases he growh rae of per capa consumpon, decreases 3

14 he lfeme wealh of he represenave consumer, and hence he level of per-capa consumpon, by roughly he presen value of resources requred o fund he program. One mus compare he lfeme welfare of he represenave agen generaed by he wo compeng pahs of per-capa consumpon. (Graphng he log of he wo me pahs yelds posvely-sloped sragh lnes ha nersec a some fuure dae.) If one assumes ha educaonal acves are no valued for her own sake (and herefore do no ener he momenary uly funcon), hen he answer depends prmarly on wo feaures of preferences: he dscoun facor and he curvaure of he momenary uly funcon over consumpon. A large (small) dscoun facor, reflecng abundan paence, favors larger (smaller) subsdes. A large (small) degree of curvaure means ha he agen s relavely noleran of growh n hs or her consumpon profle mplyng a preference for smaller (larger) subsdes. 4