Economics 602 Macroeconomic Theory and Policy Problem Set 9 Suggested Solutions Professor Sanjay Chugh Summer 2010

Transcription

1 Deparmen of Appled Economcs Johns Hopkns Unversy Economcs 60 Macroeconomc Theory and olcy rolem Se 9 Suggesed Soluons rofessor Sanjay Chugh Summer 00. Sock, onds, lls, and he Fnancal Acceleraor. In hs prolem, you wll sudy an enrched verson of he acceleraor framework we suded n class. As n our asc analyss, we connue o use he wo-perod heory of frm prof maxmzaon as our vehcle for sudyng he effecs of fnancal-marke developmens on macroeconomc acvy. However, raher han supposng s jus sock ha s he fnancal asse a frms dsposal for faclang physcal capal purchases, we wll now suppose ha oh sock and onds are a frms dsposal for faclang physcal capal purchases. efore descrng more precsely he analyss you are o conduc, a deeper undersandng of ond markes s requred. In normal economc condons, (.e, n or near a seady sae, n he sense we frs dscussed n Chaper 8), s usually suffcen o hnk of all onds of varous maury lenghs n a hghly smplfed way: y supposng ha hey are all smply one-perod face-value = onds wh he same nomnal neres rae. Recall, n fac, ha hs s how our asc dscusson of moneary polcy proceeded. In unusual (.e., far away from seady sae) fnancal marke condons, however, can ecome mporan o dsngush eween dfferen ypes of onds and hence dfferen ypes of nomnal neres raes on hose onds. You may have seen dscusson n he press aou cenral anks, such as he U.S. Federal Reserve, consderng wheher or no o egn uyng onds as a way of conducng polcy. Vewed hrough he sandard lens of how o undersand open-marke operaons, hs dscusson makes no sense ecause n he sandard vew, cenral anks already do uy (and sell) onds as he mechansm y whch hey conduc open-marke operaons! A dfference ha ecomes mporan o undersand durng unusual fnancal marke condons s ha open-marke operaons are conduced usng he shores-maury onds ha he Treasury sells, of duraon one monh or shorer. In he lngo of fnance, hs ype of ond s called a Treasury ll. The erm Treasury ond s usually used o refer o longer-maury Treasury secures hose ha have maures of one, wo, fve, or more years. These longermaury Treasury onds have ypcally no een asses ha he Federal Reserve uys and sells as regular pracce; uyng such longer-maury onds s/has no een he usual way of conducng moneary polcy. In he ensung analyss, par of he goal wll e o undersand/explan why polcy-makers are currenly consderng hs opon. efore egnnng hs analyss, hough, here s more o undersand.

2 (connued) In prvae-marke orrower/lender relaonshps, longer-maury Treasury onds ( onds ) are ypcally allowed o e used jus lke socks n fnancng frms physcal capal purchases. We can capure hs dea y enrchng he fnancng consran n our fnancal acceleraor framework o read: ( k k ) = R S a + R. S The lef hand sde of hs rcher fnancng consran s he same as he lef hand sde of he fnancng consran we consdered n our asc heory (and he noaon s dencal, as well refer o your noes for he noaonal defnons). The rgh hand sde of he fnancng consran s rcher han n our asc heory, however. The marke value of sock, S a, sll affecs how much physcal nvesmen frms can do, scaled y he governmen regulaon R S. In addon, now he marke value of a frm s ond-holdngs (whch, agan, means long-maury governmen onds) also affecs how much physcal nvesmen frms can do, scaled y he governmen regulaon R. The noaon here s ha s a frm s holdngs of nomnal onds ( long-maury ) a he end of perod, and s he nomnal prce of ha ond durng perod. Noe ha R and R S need no e equal o each oher. In he conex of he wo-perod framework, he frm s wo-perod dscouned prof funcon now reads: f( k, n) + k + ( S + D) a + wn k Sa 0 0 f ( k, n) k ( S + D) a wn k 3 Sa The new noaon compared o our sudy of he asc acceleraor mechansm s he followng: 0 s he frm s holdngs of nomnal onds (whch have face value = ) a he sar of perod one, s he frm s holdngs of nomnal onds (whch have face value = ) a he end of perod one, and s he frm s holdngs of nomnal onds (whch have face value = ) a he end of perod wo. Noe ha perod- profs are eng dscouned y he nomnal neres rae : n hs prolem, we wll consder o e he Treasury ll neres rae (as opposed o he Treasury ond neres rae). The Treasury-ll neres rae s he one he Federal Reserve usually (.e., n normal mes ) conrols. We can defne he nomnal neres rae on Treasury onds as = = + Thus, noe ha and need no equal each oher. Whereas, for varous nsuonal and regulaory reasons, very shor-erm Treasury asses ( T-lls ) are ypcally no allowed o e used n fnancng frms physcal capal purchases.

3 (connued) The res of he noaon aove s jus as n our sudy of he asc fnancal acceleraor framework. Fnally, ecause he economy ends a he end of perod, we can conclude (as usual) ha k 3 = 0, a = 0, and = 0. Wh hs ackground n place, you are o analyze a numer of ssues. a. Usng λ as your noaon for he Lagrange mulpler on he fnancng consran, consruc he Lagrangan for he represenave frm s (wo-perod) prof-maxmzaon prolem. Soluon: The Lagrangan, whch y now should e exremely sraghforward o consruc, s f( k, n) + k + ( S + D) a + wn k Sa 0 0 f ( k, n) k ( S + D) a wn k 3 Sa S + λ RSa + R k ( k). ased on hs Lagrangan, compue he frs-order condon wh respec o nomnal ond holdngs a he end of perod (.e., compue he FOC wh respec o ). (Noe: Ths FOC s crcal for much of he analyss ha follows, so you should make sure ha your work here s asoluely correc.) Soluon: ased on Lagrangan aove, he FOC wh respec o s + + λr = 0. + c. Recall ha n hs enrched verson of he acceleraor framework, he nomnal neres rae on Treasury lls,, and he nomnal neres rae on Treasury onds,, are poenally dfferen from each oher. If fnancng consrans do NOT a all affec frms nvesmen n physcal capal, how does compare o? Specfcally, s equal o, s smaller han, s larger han, or s mpossle o deermne? e as horough n your analyss and conclusons as possle (.e., ell us as much aou hs ssue as you can!). Your analyss here should e ased on he FOC on compued n par aove. (Hn: f fnancng consrans don maer, wha s he value of he Lagrange mulpler λ?) Soluon: As dscussed n deal n class, fnancng consrans are sad o no maer (n he conex of he acceleraor framework) when he value of he Lagrange mulpler s zero, λ = 0. Inserng hs value for he mulpler n he FOC derved n par, we have ha =. + Keepng n mnd ha n hs prolem we are dsngushng eween and, hs las expresson can e wren as 3

4 = + +, from whch s ovous ha =. Thus, n normal economc condons (.e., when λ = 0), he nomnal neres raes on Treasury lls and Treasury onds are exacly equal. Ths analycal resul n fac jusfes he usual pracce of reang all onds as he same n normal economc condons, her neres raes are (roughly) equalzed. (Indeed, f we nroduced even longer maury onds no our framework wo-perod onds, hree-perod onds, fve-perod onds, ec. we would e led o same concluson, ha all of her neres raes are equal o each oher, provded ha fnancng consrans don affec macroeconomc oucomes alhough mpaence nroduces anoher cavea no hs, u we have gnored mpaence ssues n hs prolem.) d. If fnancng consrans DO affec frms nvesmen n physcal capal, how does compare o? Specfcally, s equal o, s smaller han, s larger han, or s mpossle o deermne? Furhermore, f possle, use your soluon here as a ass for jusfyng wheher or no s approprae n normal economc condons o consder oh Treasury lls and Treasury onds as he same asse. e as horough n your analyss and conclusons as possle. Once agan, your analyss here should e ased on he FOC on compued n par aove. (Noe: he governmen regulaory varales R S and R are oh srcly posve ha s, neher can e zero or less han zero). Soluon: From he FOC on compued n par, and now whou mposng λ = 0, we can perform he followng algerac rearrangemens: + + λr = 0 + λ = + R λ = + R λ = + + R ( ) λ = R λ = + R Ths fnal expresson shows ha, f fnancng consrans maer (whch means ha λ 0 ), hen clearly. Whou knowng more aou how fnancal marke condons are affecng nvesmen ehavor ha s, wheher fnancng condons are gh or loose (whch would govern he sgn of he mulpler λ), s mpossle o say anyhng more aou how he T-ll neres rae and he T-ond neres rae compared o each oher. u, regardless, 4

5 s clear ha s no approprae o consder he wo neres raes as eng dencal n hs case. The aove analyss was framed n erms of nomnal neres raes; he remander of he analyss s framed n erms of real neres raes. e. y compung he frs-order condon on frms sock-holdngs a he end of perod, a, and followng exacly he same algera as presened n class, we can express he Lagrange mulpler λ as STOCK r r λ = r + R S. (.) Use he frs-order condon on you compued n par aove o derve an analogous expresson for λ excep n erms of he real neres rae on onds (.e., r ) and R (raher han R S ). (Hn: Use he FOC on you compued n par aove and follow a very smlar se of algerac manpulaons as we followed n class.) Soluon: ased on he dervaons n par d aove, hs sep smply requres applyng he Fsher relaon a couple of mes. Specfcally, le s sar agan wh he fnal condon oaned n par d aove λ = + R and rewre as ( ) λ = R Nex, mulply and dvde he erm nsde square rackes y (+π) (whch of course smply means we re mulplyng y one), whch gves y he Fsher relaon, we can express hs as + + π π λ = R + π or, fnally, r ( r ) λ = r R r r λ =, + r R, 5

6 whch s ovously smlar o he ype of condon we derved n class regardng sock fnancng. f. Compare he expresson you jus derved n par e wh expresson (.). Suppose r = r STOCK. If hs s he case, s r equal o r, s r smaller han r, s r larger han r, or s mpossle o deermne? Furhermore, n hs case, does he fnancng consran affec frms physcal nvesmen decsons? refly jusfy your conclusons and provde ref explanaon. Soluon: The expresson derved n par e and expresson (.) oh feaure λ on he lef hand sde. We can hus ovously se hem equal o each oher, gvng us STOCK r r r r S r = R r + + R Alhough you dd no have o perform he nex algerac sep (.e., you could conduc he ensung logcal analyss ased jus on hs las expresson), we can mulply hs enre expresson hrough y +r and hen also mulply he enre expresson y R S, whch would gve us S STOCK R r r = ( r r ), R whch makes ovous wha he consequence of r = r STOCK s. If r = r STOCK, ovously mus also e ha r = r (ecause you are old ha R S canno e zero). Thus, f he reurns on socks ( rsky asses ) are equal o he reurn on physcal asses (r), he reurn on onds ( safe fnancal asses ) are also equal o he reurn on physcal asses (r). Ths s essenally jus a saemen of he Fsher relaon recall from Chaper 4 ha one way o undersand/nerpre he Fsher equaon s ha says he reurns on safe and rsky asses are equal o each oher (he no-arrage relaonshp). Here, he underlyng vew s ha he reurns on all ypes of onds are rskless ( safe ), jus as s he reurns on physcal capal. g. Through lae 008, suppose ha r = r STOCK was a reasonale descrpon of he U.S. economy for he precedng 0+ years. In lae 008, r STOCK fell dramacally elow r, whch, as we suded n class, would cause he fnancal acceleraor effec o egn. Suppose governmen polcy-makers, oh fscal polcy-makers and moneary polcy-makers, decde ha hey need o nervene n order o ry o choke off he acceleraor effec. Furhermore, suppose ha here s no way o change eher R S or R (ecause of coordnaon delays amongs varous governmen agences, perhaps). Usng all of your precedng analyss as well as drawng on wha we suded n class, explan why uyng onds (whch, agan, means long-maury onds n he sense descred aove) mgh e a sound sraegy o pursue. (Noe: The analyss here s largely no mahemacal. Raher, wha s requred s an careful logcal progresson of hough ha explans why uyng onds mgh e a good dea.) Soluon: As we dscussed n class, one way of offseng he feedack effecs of a declne n fnancal marke reurns s o relax fnancal marke regulaons ncreasng R S and/or R n hs case. The reason hs may e helpful s ha, all else equal, would serve o lower λ (examne he condons derved aove), whch s, analycally, where he prolems can e raced o (e, he fac ha fnancng consrans maer ). From he fnancng consran self, s ovous ha rasng R S. 6

7 and/or R ncreases he effecve marke value of frms collaeralzeale fnancal asses (recall he asc nformaon asymmery prolems ha underle hs fnancng consran): ( k k ) = R S a + R. S Tha s, rasng R S and/or R ncreases he rgh hand sde of he fnancng consran. u f ha s nfeasle for nsuonal or polcal oher reasons, anoher polcy nervenon ha has he same effec s o ry o rase any of he oher componens of he rgh hand sde of he fnancng consran: ncludng governmen effors o ry o rase he prce of onds y drecly uyng hem n markes (.e., he ncreased demand for onds n ond markes should, all else equal, rase he prce of onds). Ths s eyond he scope of hs queson, u hs ype of analyss can shed lgh on a hos of polcy proposals and programs ha are eng/have een dscussed he pas year: many of hem share he road goal of ryng o rase he effecve: marke value of he prvae secor s collaeralzeale fnancal asses. Ths could e acheved y some comnaon of drec governmen purchases of a varey of fnancal asses (socks, onds), smply gvng frms more asses (.e., drecly gvng hem more a and/or more ), allowng new ypes of fnancal asses o e used for collaeral purposes (.e., addng a hrd asse o he rgh hand sde of he fnancng consran, a fourh asse o he rgh hand sde of he fnancng consran, ec.): roadly speakng, s all aou rasng he rgh hand sde of he fnancng consran aove! 7

8 . The Yeld Curve. An mporan ndcaor of markes elefs/expecaons aou he fuure pah of he macroeconomy s he yeld curve, whch, smply pu, descres he relaonshp eween he maury lengh of a parcular ond (recall ha onds come n varous maury lenghs) and he per-year neres rae on ha ond. A ond s yeld s alernave ermnology for s neres rae. A sample yeld curve s shown n he followng dagram: Ths dagram plos he yeld curve for U.S. Treasury onds ha exsed n markes on Feruary 9, 005: as shows, a 5-year Treasury ond on ha dae carred an neres rae of aou 4 percen, a 0-year Treasury ond on ha dae carred an neres rae of aou 4.4 percen, and a 30-year Treasury ond on ha dae carred an neres rae of aou 4.5 percen. Recall from our sudy of ond markes ha prces of onds and nomnal neres raes on onds are negavely relaed o each oher. The yeld curve s ypcally dscussed n erms of nomnal neres raes (as n he graph aove). However, ecause of he nverse relaonshp eween neres raes on onds and prces of onds, he yeld curve could equvalenly e dscussed n erms of he prces of onds. In hs prolem, you wll use an enrched verson of our nfne-perod moneary framework from Chaper 4 o sudy how he yeld curve s deermned. Specfcally, raher han assumng he represenave consumer has only one ype of ond (a one-perod ond) he can purchase, we wll assume he represenave consumer has several ypes of onds he can purchase a oneperod ond, a wo-perod ond, and, n he laer pars of he prolem, a hree-perod ond. Le s sar jus wh wo-perod onds. We wll model he wo-perod ond n he smples possle way: n perod, he consumer purchases TWO uns of wo-perod onds, each of, whch has a marke prce TWO and a face value of one (.e., when he wo-perod ond pays 8

9 off, pays ack one dollar). The defnng feaure of a wo-perod ond s ha pays ack s face value wo perods afer purchase (ndeed, hence he erm wo-perod ond ). The one-perod ond s jus as we have dscussed n class and n Chaper 4. Mahemacally, hen, suppose (jus as n Chaper 4) ha he represenave consumer has a lfeme uly funcon sarng from perod M M M 3 3 M ln c ln ln β c+ + β ln + β ln c+ + β ln + β ln c+ 3+ β ln..., and hs perod- udge consran s gven y c M + S a = Y + M ( S + D ) a., TWO TWO TWO (ased on hs, you should know wha he perod + and perod + and perod +3, ec. udge consrans look lke). Ths udge consran s dencal o ha n Chaper 4, excep of course for he erms regardng wo-perod onds. Noe carefully he mng on he rgh hand sde TWO n accordance wh he defnng feaure of a wo-perod ond, n perod, s ha pays ack s face value. The res of he noaon s jus as n Chaper 4, ncludng he fac ha he sujecve dscoun facor (.e., he measure of mpaence) s β <. a. Qualavely represen he yeld curve shown n he dagram aove n erms of prces of onds raher han neres raes on onds. Tha s, wh he same maury lenghs on he horzonal axs, plo (qualavely) on he vercal axs he prces assocaed wh hese onds. Soluon: Wh maury lenghs ploed on he horzonal axs, he yeld curve n erms of ond prces s downward-slopng. Ths follows smply ecause of he nverse relaonshp eween ond prces and neres raes. The yeld curve shown aove s n erms of neres raes and s srcly ncreasng; hence he assocaed yeld curve n erms of prces mus e srcly decreasng.. ased on he uly funcon and udge consran gven aove, se up an approprae Lagrangan n order o derve he represenave consumer s frs-order condons wh respec o oh and (as usual, he analyss s eng conduced from he TWO perspecve of he very egnnng of perod ). Defne any auxlary noaon ha you need n order o conduc your analyss. TWO Soluon: The only wo frs-order condons ha you needed here are hose on and. Denong y λ he Lagrange mulpler on he perod- udge consran and y λ + he Lagrange mulpler on he perod-+ udge consran, he wo frs-order condons, respecvely, are and λ + βλ + = 0 λ + β λ + =. TWO, 0 9

10 Noe well he + me suscrps n he second expresson; hs follows from he fac he a woperod ond purchased n perod does no repay s promsed face value unl perod +. (Refer ack o rolem Se 4 for an analogous sock-prcng model n whch socks ook wo perods o pay off her capal gans and dvdends.) c. Usng he wo frs-order condons you oaned n par, consruc a relaonshp eween he prce of a wo-perod ond and he prce of a one-perod ond. Your fnal TWO, relaonshp should e of he form =..., and on he rgh-hand-sde of hs expresson should appear (poenally among oher hngs),. (Hn: n order o ge no hs expresson, you may have o mulply and/or dvde your frs-order condons y appropraely-chosen varales.) βλ + Soluon: From he frs expresson aove, we have, as usual ha =. From he second λ TWO, β λ + expresson aove, we analogously can oan =. We can rewre hs expresson λ for he prce of a wo-perod ond as βλ βλ =, TWO, + + λ+ λ n whch we have smply mulpled and dvded he precedng expresson y λ + (.e., we have mulpled y one, always a vald mahemacal operaon). The fnal erm on he far rgh-handsde s nohng more han he prce of a one-perod ond, so we can wre βλ =, TWO, + λ + whch sasfes he form of he relaonshp you were asked o derve. We can acually ol hs down furher, hough. Noe ha he prce of one-perod ond purchased n perod + would βλ + e gven y + =, whch follows from opmzaon wh respec o perod + oneperod ond holdngs. Usng hs expresson n he perod- prce of a wo-perod ond, we hus λ + oan =, TWO, + whch s a key dea n fnance heory: he prce of a mul-perod asse (ond) s nohng more han he produc of he prces of wo consecuve one-perod asses (ond). d. Suppose ha he opmal nomnal expendure on consumpon (c) s equal o n every perod. Usng hs fac, s he prce of a wo-perod ond greaer han, smaller han, 0

11 or equal o he prce of a one-year ond? If s mpossle o ell, explan why; f you can ell, e as precse as you can e aou he relaonshp eween he prces of he wo onds. (Hn: you may need o nvoke he consumer s frs-order condon on consumpon) Soluon: Sar wh he relaonshp βλ = derved aove. If nomnal consumpon TWO, + λ + expendures are consan (and equal o one) every perod, hs means ha λ = every perod. (Ths concluson follows from he fac ha he FOC wh respec o consumpon s /c λ = 0 n every perod, whch can e rearranged o λ = ). If he mulpler s one every perod, we c mmedaely have ecause β <, we conclude TWO, <. TWO, = β. e. Now suppose here s also a hree-perod ond. A hree-perod ond purchased n any gven perod does no repay s face value (also assumed o e ) unl hree perods afer s purchased. The perod- udge consran, now ncludng one-, wo-, and hreeperod onds, s gven y c M + S a = Y + M ( S + D ) a,, TWO TWO, THREE THREE TWO THREE 3 THREE, where s he quany of hree-perod onds purchased n perod and THREE s assocaed prce. Followng he same logcal seps as n pars, c, and d aove (and connung o assume ha nomnal expendure on consumpon (c) s equal o one n perod every perod), s he prce of a hree-year ond greaer han, smaller han, or equal o he prce of a wo-year ond? If s mpossle o ell, explan why; f you can ell, e as precse as you can e aou he relaonshp eween he prces of he wo onds. (Hn: you may need o nvoke he consumer s frs-order condon on consumpon). Soluon: Exendng he Lagrangan from aove, he frs-order condon wh respec o s THREE λ + β λ + =,, THREE whch can e rearranged o yeld mulplyng y one, we can express hs as 3 THREE, + 3 λ β λ =. Jus lke n par c aove, y cleverly THREE, λ+ λ+ λ β λ βλ βλ =,

12 whch, n exacly he same way as n par c, we can express n erms of chaned one-perod ond prces, βλ =. THREE, λ + If he Lagrange mulpler λ s consan every perod, we can conclude he prce of a hreeperod ond s smaller han he prce of a wo-perod ond (whch n urn, from par c, s smaller han he prce of a one-perod ond). Ths agan follows ecause β <. f. Suppose ha β = Usng your conclusons from pars d and e, plo a yeld curve n erms of ond prces (ovously, you can plo only hree dfferen maury lenghs here). Soluon: ased on he analyses n pars d and e, he prce of onds s clearly negavelyrelaed o s maury lengh, hence he yeld curve n erms of prces s srcly decreasng. Ths s jus as your skech of he emprcal yeld curve n par a. g. Wha s he sngle mos mporan reason (economcally, ha s) for he shape of he yeld curve you found n par f? (Ths requres only a ref, qualave/concepual response.) Soluon: Re-examnng our conclusons/analyses n pars d, e, and f, he sole reason we were ale o reach he conclusons we reached n each of hose pars was he fac ha β <. Thus, he dea of mpaence and s effecs on he macroeconomy rears s head agan, hs me wh respec o ond prces of dfferen maures. The concepual dea s smple: ecause of mpaence, he longer a ond purchaser mus wa o receve a gven face value, he less he wll e wllng o pay for oday (and hs s refleced n ond marke prces hrough he ond demand funcon for dfferen maury onds).