Deparmen of Applied Economics Johns Hopkins Universiy Economics 602 Macroeconomic Theory and Policy Prolem Se 9 Professor Sanjay Chugh Spring 2012 1. Sock, Bonds, Bills, and he Financial Acceleraor. In his prolem, you will sudy an enriched version of he acceleraor framework we sudied in class. As in our asic analysis, we coninue o use he wo-period heory of firm profi maximizaion as our vehicle for sudying he effecs of financial-marke developmens on macroeconomic aciviy. However, raher han supposing i is jus sock ha is he financial asse a firms disposal for faciliaing physical capial purchases, we will now suppose ha oh sock and onds are a firms disposal for faciliaing physical capial purchases. Before descriing more precisely he analysis you are o conduc, a deeper undersanding of ond markes is required. In normal economic condiions, (i.e, in or near a seady sae, in he sense we firs discussed in Chaper 8), i is usually sufficien o hink of all onds of various mauriy lenghs in a highly simplified way: y supposing ha hey are all simply one-period face-value = 1 onds wih he same nominal ineres rae. Recall, in fac, ha his is how our asic discussion of moneary policy proceeded. In unusual (i.e., far away from seady sae) financial marke condiions, however, i can ecome imporan o disinguish eween differen ypes of onds and hence differen ypes of nominal ineres raes on hose onds. You may have seen discussion in he press aou cenral anks, such as he U.S. Federal Reserve, considering wheher or no o egin uying onds as a way of conducing policy. Viewed hrough he sandard lens of how o undersand open-marke operaions, his discussion makes no sense ecause in he sandard view, cenral anks already do uy (and sell) onds as he mechanism y which hey conduc open-marke operaions! A difference ha ecomes imporan o undersand during unusual financial marke condiions is ha open-marke operaions are conduced using he shores-mauriy onds ha he Treasury sells, of duraion one monh or shorer. In he lingo of finance, his ype of ond is called a Treasury ill. The erm Treasury ond is usually used o refer o longer-mauriy Treasury securiies hose ha have mauriies of one, wo, five, or more years. These longermauriy Treasury onds have ypically no een asses ha he Federal Reserve uys and sells as regular pracice; uying such longer-mauriy onds is/has no een he usual way of conducing moneary policy. In he ensuing analysis, par of he goal will e o undersand/explain why policy-makers are currenly considering his opion. Before eginning his analysis, hough, here is more o undersand. PS 9 Sanjay K. Chugh 1
(coninued) In privae-marke orrower/lender relaionships, longer-mauriy Treasury onds ( onds ) are ypically allowed o e used jus like socks in financing firms physical capial purchases. 1 We can capure his idea y enriching he financing consrain in our financial acceleraor framework o read: P ( k k ) = R S a + R P B. S B 1 2 1 1 1 1 1 The lef hand side of his richer financing consrain is he same as he lef hand side of he financing consrain we considered in our asic heory (and he noaion is idenical, as well refer o your noes for he noaional definiions). The righ hand side of he financing consrain is richer han in our asic heory, however. The marke value of sock, S 1 a 1, sill affecs how much physical invesmen firms can do, scaled y he governmen regulaion R S. In addiion, now he marke value of a firm s ond-holdings (which, again, means long-mauriy governmen onds) also affecs how much physical invesmen firms can do, scaled y he governmen regulaion R B. The noaion here is ha B 1 is a firm s holdings of nominal onds ( long-mauriy ) a he end of period 1, and P 1 is he nominal price of ha ond during period 1. Noe ha R B and R S need no e equal o each oher. In he conex of he wo-period framework, he firm s wo-period discouned profi funcion now reads: Pf( k, n) + Pk + ( S + D) a + B Pwn Pk Sa P B 1 1 1 1 1 1 1 0 0 1 1 1 1 2 1 1 1 1 Pf 2 ( k2, n2) Pk 2 2 ( S2 + D2) a1 B1 Pwn 2 2 2 Pk 2 3 Sa 2 2 PB 2 2 + + + + 1+ i 1+ i 1+ i 1+ i 1+ i 1+ i 1+ i 1+ i The new noaion compared o our sudy of he asic acceleraor mechanism is he following: B 0 is he firm s holdings of nominal onds (which have face value = 1) a he sar of period one, B 1 is he firm s holdings of nominal onds (which have face value = 1) a he end of period one, and B 2 is he firm s holdings of nominal onds (which have face value = 1) a he end of period wo. Noe ha period-2 profis are eing discouned y he nominal ineres rae i: in his prolem, we will consider i o e he Treasury ill ineres rae (as opposed o he Treasury ond ineres rae). The Treasury-ill ineres rae is he one he Federal Reserve usually (i.e., in normal imes ) conrols. We can define he nominal ineres rae on Treasury onds as i BOND 1 1 = 1 P1 = P 1+ i 1 BOND Thus, noe ha i BOND and i need no equal each oher. 1 Whereas, for various insiuional and regulaory reasons, very shor-erm Treasury asses ( T-ills ) are ypically no allowed o e used in financing firms physical capial purchases. PS 9 Sanjay K. Chugh 2
(coninued) The res of he noaion aove is jus as in our sudy of he asic financial acceleraor framework. Finally, ecause he economy ends a he end of period 2, we can conclude (as usual) ha k 3 = 0, a 2 = 0, and B 2 = 0. Wih his ackground in place, you are o analyze a numer of issues. a. Using λ as your noaion for he Lagrange muliplier on he financing consrain, consruc he Lagrangian for he represenaive firm s (wo-period) profi-maximizaion prolem.. Based on his Lagrangian, compue he firs-order condiion wih respec o nominal ond holdings a he end of period 1 (i.e., compue he FOC wih respec o B 1 ). (Noe: This FOC is criical for much of he analysis ha follows, so you should make sure ha your work here is asoluely correc.) c. Recall ha in his enriched version of he acceleraor framework, he nominal ineres rae on Treasury ills, i, and he nominal ineres rae on Treasury onds, i BOND, are poenially differen from each oher. If financing consrains do NOT a all affec firms invesmen in physical capial, how does i BOND compare o i? Specifically, is i BOND equal o i, is i BOND smaller han i, is i BOND larger han i, or is i impossile o deermine? Be as horough in your analysis and conclusions as possile. Your analysis here should e ased on he FOC on B 1 compued in par aove. (Hin: if financing consrains don maer, wha is he value of he Lagrange muliplier λ?) d. If financing consrains DO affec firms invesmen in physical capial, how does i BOND compare o i? Specifically, is i BOND equal o i, is i BOND smaller han i, is i BOND larger han i, or is i impossile o deermine? Furhermore, if possile, use your soluion here as a asis for jusifying wheher or no i is appropriae in normal economic condiions o consider oh Treasury ills and Treasury onds as he same asse. Be as horough in your analysis and conclusions as possile. Once again, your analysis here should e ased on he FOC on B 1 compued in par aove. (Noe: he governmen regulaory variales R S and R B are oh sricly posiive ha is, neiher can e zero or less han zero). The aove analysis was framed in erms of nominal ineres raes; he remainder of he analysis is framed in erms of real ineres raes. PS 9 Sanjay K. Chugh 3
(coninued) e. By compuing he firs-order condiion on firms sock-holdings a he end of period 1, a 1, and following exacly he same algera as presened in class, we can express he Lagrange muliplier λ as STOCK r r 1 λ = 1+ r R S. (1.1) Use he firs-order condiion on B 1 you compued in par aove o derive an analogous expression for λ excep in erms of he real ineres rae on onds (i.e., r BOND ) and R B (raher han R S ). (Hin: Use he FOC on B 1 you compued in par aove and follow a very similar se of algeraic manipulaions as we followed in class.) f. Compare he expression you jus derived in par e wih expression (1.1). Suppose r = r STOCK. If his is he case, is r BOND equal o r, is r BOND smaller han r, is r BOND larger han r, or is i impossile o deermine? Furhermore, in his case, does he financing consrain affec firms physical invesmen decisions? Briefly jusify your conclusions and provide rief explanaion. g. Through lae 2008, suppose ha r = r STOCK was a reasonale descripion of he U.S. economy for he preceding 20+ years. In lae 2008, r STOCK fell dramaically elow r, which, as we sudied in class, would cause he financial acceleraor effec o egin. Suppose governmen policy-makers, oh fiscal policy-makers and moneary policy-makers, decide ha hey need o inervene in order o ry o choke off he acceleraor effec. Furhermore, suppose ha here is no way o change eiher R S or R B (ecause of coordinaion delays amongs various governmen agencies, perhaps). Using all of your preceding analysis as well as drawing on wha we sudied in class, explain why uying onds (which, again, means long-mauriy onds in he sense descried aove) migh e a sound sraegy o pursue. (Noe: The analysis here is largely no mahemaical. Raher, wha is required is an careful logical progression of hough ha explains why uying onds migh e a good idea.) PS 9 Sanjay K. Chugh 4
2. The Yield Curve. An imporan indicaor of markes eliefs/expecaions aou he fuure pah of he macroeconomy is he yield curve, which, simply pu, descries he relaionship eween he mauriy lengh of a paricular ond (recall ha onds come in various mauriy lenghs) and he per-year ineres rae on ha ond. A ond s yield is alernaive erminology for is ineres rae. A sample yield curve is shown in he following diagram: This diagram plos he yield curve for U.S. Treasury onds ha exised in markes on Feruary 9, 2005: as i shows, a 5-year Treasury ond on ha dae carried an ineres rae of aou 4 percen, a 10-year Treasury ond on ha dae carried an ineres rae of aou 4.4 percen, and a 30-year Treasury ond on ha dae carried an ineres rae of aou 4.52 percen. Recall from our sudy of ond markes ha prices of onds and nominal ineres raes on onds are negaively relaed o each oher. The yield curve is ypically discussed in erms of nominal ineres raes (as in he graph aove). However, ecause of he inverse relaionship eween ineres raes on onds and prices of onds, he yield curve could equivalenly e discussed in erms of he prices of onds. In his prolem, you will use an enriched version of our infinie-period moneary framework from Chaper 14 o sudy how he yield curve is deermined. Specifically, raher han assuming he represenaive consumer has only one ype of ond (a one-period ond) he can purchase, we will assume he represenaive consumer has several ypes of onds he can purchase a oneperiod ond, a wo-period ond, and, in he laer pars of he prolem, a hree-period ond. Le s sar jus wih wo-period onds. We will model he wo-period ond in he simples possile way: in period, he consumer purchases B TWO unis of wo-period onds, each of, which has a marke price P TWO and a face value of one (i.e., when he wo-period ond pays PS 9 Sanjay K. Chugh 5
off, i pays ack one dollar). The defining feaure of a wo-period ond is ha i pays ack is face value wo periods afer purchase (indeed, hence he erm wo-period ond ). The one-period ond is jus as we have discussed in class and in Chaper 14. Mahemaically, hen, suppose (jus as in Chaper 14) ha he represenaive consumer has a lifeime uiliy funcion saring from period M M 1 2 2 M 2 3 3 M + + + ln c ln ln 3 + + β c+ 1+ β ln + β ln c+ 2 + β ln + β ln c+ 3+ β ln..., P P+ 1 P+ 2 P+ 3 and his period- udge consrain is given y Pc + P B + P B + M + S a = Y + M + B + B + ( S + D ) a., TWO TWO TWO 1 1 2 1 (Based on his, you should know wha he period +1 and period +2 and period +3, ec. udge consrains look like). This udge consrain is idenical o ha in Chaper 14, excep of course for he erms regarding wo-period onds. Noe carefully he iming on he righ hand side TWO in accordance wih he defining feaure of a wo-period ond, in period, i is B 2 ha pays ack is face value. The res of he noaion is jus as in Chaper 14, including he fac ha he sujecive discoun facor (i.e., he measure of impaience) is β < 1. a. Qualiaively represen he yield curve shown in he diagram aove in erms of prices of onds raher han ineres raes on onds. Tha is, wih he same mauriy lenghs on he horizonal axis, plo (qualiaively) on he verical axis he prices associaed wih hese onds.. Based on he uiliy funcion and udge consrain given aove, se up an appropriae Lagrangian in order o derive he represenaive consumer s firs-order condiions wih respec o oh B and B (as usual, he analysis is eing conduced from he TWO perspecive of he very eginning of period ). Define any auxiliary noaion ha you need in order o conduc your analysis. c. Using he wo firs-order condiions you oained in par, consruc a relaionship eween he price of a wo-period ond and he price of a one-period ond. Your final TWO, relaionship should e of he form P =..., and on he righ-hand-side of his expression should appear (poenially among oher hings), P. (Hin: in order o ge P ino his expression, you may have o muliply and/or divide your firs-order condiions y appropriaely-chosen variales.) d. Suppose ha he opimal nominal expendiure on consumpion (Pc) is equal o 1 in every period. Using his fac, is he price of a wo-period ond greaer han, smaller han, or equal o he price of a one-year ond? If i is impossile o ell, explain why; if you can ell, e as precise as you can e aou he relaionship eween he prices of he wo onds. (Hin: you may need o invoke he consumer s firs-order condiion on consumpion) e. Now suppose here is also a hree-period ond. A hree-period ond purchased in any given period does no repay is face value (also assumed o e 1) unil hree periods afer PS 9 Sanjay K. Chugh 6
i is purchased. The period- udge consrain, now including one-, wo-, and hreeperiod onds, is given y Pc + P B + P B + P B + M + S a = Y + M + B + B + B + ( S + D ) a,, TWO TWO, THREE THREE TWO THREE 1 1 2 3 1 THREE, where B is he quaniy of hree-period onds purchased in period and P THREE is associaed price. Following he same logical seps as in pars, c, and d aove (and coninuing o assume ha nominal expendiure on consumpion (Pc) is equal o one in period every period), is he price of a hree-year ond greaer han, smaller han, or equal o he price of a wo-year ond? If i is impossile o ell, explain why; if you can ell, e as precise as you can e aou he relaionship eween he prices of he wo onds. (Hin: you may need o invoke he consumer s firs-order condiion on consumpion) f. Suppose ha β = 0.95. Using your conclusions from pars d and e, plo a yield curve in erms of ond prices (oviously, you can plo only hree differen mauriy lenghs here). g. Wha is he single mos imporan reason (economically, ha is) for he shape of he yield curve you found in par f? (This requires only a rief, qualiaive/concepual response.) PS 9 Sanjay K. Chugh 7