10.The Zero Lower Bound in a two period economy

Transcription

1 .The Zero Lower Boud i a two period ecoomy Idex:. The Zero Lower Boud i a two period ecoomy.... Itroductio.... A two period closed ecoomy with moey.....osumptio.....the IS curve the Fisher equatio the omial IS curve the Zero lower boud ash i advace costrait two cases numerical example The case with flexible prices theory numerical example The case with sticky prices theory numerical example ommittig with high iflatio fiscal expasio....5 Discussio...4 Appedix...5 Review questios ad exercises...7 Review questios...7 Exercises...7

2 . Itroductio Whe prices are sticky, the real iterest rate will, i geeral, depart from its atural level. This gives the cetral bak a special role i stabilizig the ecoomy. I particular, whe the real iterest rate is too high, the cetral bak may use its policy tools to drive dow the iterest rates. A problem may arise, however, i that the omial iterest rate ca hardly go egative. Whe the omial iterest rate is zero, it does ot make sese for people to buy bods. Hece, people will hoard ay excess moey produced by the cetral bak, ad further moetary expasios will ot traslate ito higher bod prices ad lower iterest rates. I that case, the moetary policy will be impotet, uable to expad aggregate demad. This is the liquidity trap argumet, put forward by Keyes. This ote reviews this argumet i a two-period framework, followig a paper by Paul Krugma. Sectio 8. presets the basic model. I Sectio 8.3 we describe how the ZLB iterferes with moetary effectiveess i a world where prices are flexible. I Sectio 8.4, we aalyse the case with sticky prices, ad the aveues that ca be followed to get rid of the trap. I Sectio 8.5 we discuss how fiscal policy ca help drive the ecoomy out of the trap.. A two period closed ecoomy with moey.. osumptio osider a edowmet ecoomy closed to iteratioal trade. I this ecoomy, the atural levels of curret output ad future output are give, respectively by ad. As for the future, prices are assumed to be flexible ad full employmet output is always met. urret output, istead, may fall short the full employmet level i case prices are sticky. Households maximize their life-time utility fuctio subject to a life-time wealth costrait. I order to make the problem as simple as possible, let s assume that the rate of time preferece is equal to zero: U l l () The household life-time budget costrait is give by () r Krugma, P., 998: It s baaack! Japa s slump ad the retur of the liquidity trap, Brookigs Papers o Ecoomic Activity.

3 Where G G (3) r Equatio (3) already accouts for the govermet iter-temporal budget costrait. From this problem, oe obtais the usual Euler equatio r (4) Substitutig i (), the optimal cosumptio today will be (5).. The IS curve Sice the ecoomy is closed, private cosumptio each period has to be equal to curret output mius govermet cosumptio: G (6) G (7) Usig the Euler equatio, the real iterest rate i the closed ecoomy will be such that: G r (8) G Equatio (8) describes the IS curve. As we already kow, the real iterest rate may tur egative whe people perceive future cosumptio to be less tha curret cosumptio. I this model, this may occur if people aticipate future output to be low or, future govermet expeditures to be high, relative to the correspodig levels today...3 The Fisher equatio We ow erich the model with the omial dimesio. First, let s assume that the Fisher parity coditio holds. Thus, the omial iterest rate ad the real iterest rate will be related i the followig maer: I the appedix, it is show that i the coditios of the Ricardia Equivalece, the household iter-temporal budget costrait is ot affected by the presece of moey. 3

4 P i r (9) P..4 The omial IS curve Substitutig this i the Euler equatio (8), oe obtais a reicaratio of the IS curve, ow as a fuctio of the omial iterest rate: G i () P G P..5 The Zero lower boud I the followig, we assume that the omial iterest rate, i cotrast to the real iterest rate caot fall below zero 3. Hece, from (): P () P That is, curret cosumptio spedig caot exceed future cosumptio spedig, i omial terms. Whe coditio () holds i equality, the zero lower boud is reached. This is more likely whe people perceive: (a) Future prices to be lower tha curret prices (deflatio); (b) Future cosumptio to be less tha curret cosumptio (dark future). I both cases, it may happe that the equilibrium omial iterest rate falls below its feasible level...6 ash i advace costrait I this ecoomy, households face a cash-i-advace costrait. That is, private spedig each period caot exceed the quatity of moey held: P () P (3) Sice the household leaves oly two periods, there is o poit i carryig to period more moey tha the eeded to settle the desired trasactios. Hece, equatio () will hold i equality: P (3a) 3 You may argue that the omial iterest rates ca go below zero, as it happes, for istamce, today with the Euribor. The reaso is that baks caot hoard cash, because the Europea etral Bak is chargig a iterest rate o bak deposits, ad baks habe physical difficulties i parkig moey somewhere elese. So it is cheaper for baks to led, evem at a egative rate. Still, the observed egative rates are oly margially differet from zero (less tha oe percetage poit), ot costitutig a meaigful departure from the zero boud. 4

5 As for period, equatio (), it may hold i equality or ot, depedig o the opportuity cost of holdig moey: whe the omial iterest rate is positive, moey is domiated by bods as store of value, so the costrait holds i equality: it will ot make sese for households to hold moey i excess of their trasactio eeds 4. Whe however the omial iterest rate reaches a value of zero, the idividual will be idifferet betwee carryig wealth to the future i the form of moey or i the form of bods. Hece, at the zero lower boud, equatio () may hold i strict iequality...7 Two cases Sice the omial iterest rate caot fall below zero, the model does ot solve symmetrically. To fid out the equilibrium of the model, oe should cosider two differet cases: (a) whe the iterest rate is positive; (b), whe the iterest rate is egative. For the iterest rate to be positive, coditio () must hold i strict iequality, that is, P (usig 3a). I this case, coditio () holds i equality, that is, P. For the iterest rate to be zero, coditio () must hold i strict equality, that is, P. I this case, ay amout of moey holdigs satisfyig the cash i advace costrait will be feasible: P. Puttig all these pieces together, we get: for P G (4) for Equatio (4) is just aother way of statig what we saw i equatio (): curret cosumptio expeditures are bouded by the amout of omial cosumptio expeditures i the future. I the followig, we make use of equatio (4) for two distictive cases: the case with flexible prices (well fuctioig) ad the case with sticky prices (Keyesia)...8 Numerical example Before turig to the formal aalysis of the liquidity trap, let s cosider the followig umerical example: ; ; G ; G ; 8. The IS curve takes the followig form: 4 Note that with a positive iterest rate moey velocity becomes costat ad idepedet of the level of the iterest rate. This is a simplifyig assumptio, but ot relevat for our aalysis, as we are cocered with what happes whe the omial iterest rate is zero. 5

6 G 8 G r (8a) level. Usig (3a), the future price level will be: 8 P (3b) 8 I this case, the omial versio of the IS curve takes the followig form: G P 8 i (a) G P P Whether the ZLB is met or ot, it will deped o curret output ad the price As we show ext, as log as prices are flexible, the ecoomy will always be at full employmet, irrespectively as to whether the ZLB becomes bidig or ot. With sticky prices, however, the ZLB may determie a level of output that falls short the full employmet level. Our questio, the, is how to use the policy variables,, G, G to rescue the ecoomy out of the liquidity trap..3 The case with flexible prices.3. Theory I the case with flexible prices, the ecoomy will be always at full employmet. Hece, equatio (6) determies curret cosumptio as: G (6a) Whe curret output is at full employmet, the atural iterest rate holds: G r (5) G I this case, equatio (4) determies the price level, oly: P for for Thus, i case the iterest rate is positive (upper segmet) the quatitative theory holds, with prices beig proportioal to the amout of moey prited. If however the zero lower boud is met (lower segmet), pritig more moey will stop deliverig higher prices. This case is described with the help of figure. The figure describes what happes whe the moey supply icreases from some level to the critical 6

7 level. Below this critical level, aggregate demad expads together with the moey supply, ad the curret price level icreases proportioally to moey. Figure shows what happes to the omial iterest rate as curret moey supply expads from a level below the critical oe: it decreases. Whe the critical level is reached, the omial iterest rate meets the ZLB. After this poit, ay extra moey beig prited will be cashed by the household, without expadig further the aggregate demad. The price level will remai uchaged at its possible maximum without ay further implicatio. Figure : aximum feasible price level uder flexible prices Figure : Iterest rate ad moey supply Note however that i this case the zero lower boud does or rise ay policy problem: sice prices are flexible, the ecoomy will be at full employmet ad the atural real iterest rate is achieved. O this case, chages i moey oly affect the price level ad the omial iterest rate or othig, without impactig i ay circumstace i output or the real iterest rate. This is the classical dichotomy..3. Numerical example 7

8 To illustrate how the ZLB affects the ecoomy uder flexible prices, let s refer agai to our umerical example. I the well fuctioig case, curret output will be at atural level, implyig a egative atural real iterest rate: 8 r.8 (5a) form: Give the curret output, the omial IS will take the followig particular.8 i (b) P This IS curve is depicted i Figure 3. As show i the figure, the ZLB is met whe P. 8. At prices lower tha this, the iterest rate will be positive ad the cash i advace costrait will holds i equality, determiig the price level: P, for 8 P. Thus, for istace, whe 5, the price level will be P. 5. Give the price levels today, the iterest rate is determied i the IS curve ( i. 6 ), ad it will be exactly the eeded for the household to be i shape with the Euler equatio ( r. 8 ). Whe 8, the price level icreases to P. 8, ad the omial iterest rate becomes equal to zero, implyig agai r. 8. For ay moetary expasio above the critical level ( 8 ), the omial iterest rate will remai uchaged at the ZLB. Figure 4 describes the same story i the (,P) locus: As log as prices are lower tha the maximum allowed by the ZLB, moetary expasios will produce proportioal icreases i prices. After that limit is reached, the ecoomy will lie i the equilibrium (,.8), irrespectively of the moey supply. Beyod the critical level, people will hoard the moey i excess of their trasactio eeds, istead of tryig to buy bods. Because moey is hoarded, prices will be uaffected ad so will do the iterest rate. The moetary policy becomes irrelevat, but after all this is ot a problem because the ecoomy is at the full employmet. 8

9 Figure 3: ZLB uder flexible prices Figure 4: oetary expasio ad the ZLB uder flexible prices.4 The case with sticky prices.4. Theory The more iterestig case arises whe prices are sticky. The problem is how to achieve the atural real iterest rate i a world with sticky prices, ad where the cetral bak is committed with a certai moetary policy i the future. To solve the model with sticky prices, let s assume that: P (6) I this case, the omial IS curve becomes: i G Because the omial iterest rate caot fall below zero, the equivalet to coditio () is: (7) G 9

10 Equatio (7) is very importat, because it tell us the two policy aveues oe ca follow wheever curret output falls short full employmet output: to icrease govermet expeditures today or moey supply tomorrow. These two aveues will be further explored below. As before, the model solves differetly for the case i which the iterest rate is positive or zero. From (4), the curret level of output will be G for (4b) G for Graphically, the equilibrium of the model is illustrated i Figure 5. The Figure is similar to Figure, except that aggregate supply is ow horizotal. Departig from the upper segmet i (4b), as moey supply expads, the aggregate demad moves rightwards determiig a higher icome level. Alog this move, curret cosumptio expads ad the real iterest rate declies. Note that i this case (with sticky prices), a moetary expasio produces a real effect. Whe the critical level of moey supply is reached, however, a further expasio of moetary policy will ot deliver a aggregate demad expasio, so output will be stuck at its maximum feasible level G. From this poit, the oly way to expad output towards full employmet is usig fiscal policy or promisig to expad moey supply i the future. Figure 5: aximum feasible output level uder flexible prices.4. Numerical example Referrig to our umerical example, let s assume that curret prices are sticky at P. I this case, the IS curve becomes (Figure 6):

11 i 8 Sice the omial iterest rate caot fall below zero, this meas that the maximum output level at the reach of moetary policy will be Figure 6: ZLB uder sticky prices. Uder sticky prices, the cash i advace costrait determies the maximum amout of output the household ca buy. That is: G As before, as log as the iterest rate is positive, the cash-i-advace costrait () will hold i equality. Thus, for ay output level below, the moetary policy determies the output level: For istace, whe 5, the household ca oly cosume 5, ad aggregate output will be 7. I this case, the Euler equatio implies a omial 8 iterest rate (equal to the real iterest rate) equal to.6 r Whe 8, the household ca oly cosume 8, ad aggregate output will be. I this case, the Euler equatio implies a omial iterest 8 rate (equal to the real iterest rate) equal to. Thus, the ecoomy reaches 8 the ZLB. Whe the moey supply expads beyod the critical level ( 8 ), there will be o impact o cosumptio, eve if the cash i advace costrait is ot bidig. The reaso is that, with a zero real iterest rate is zero, the household will choose a flat cosumptio stream. Ad because cosumptio is oly 8, output

12 will be. Beyod the ZLB, the moetary policy becomes impotet to achieve further output expasios..4.3 ommittig with high iflatio The fact that a moetary expasio today is ot capable of drivig the ecoomy towards full employmet does ot mea that moetary policy is completely impotet. The moetary authority ca still drive the ecoomy towards full employmet if credibly committig with a higher price level i the future. To see this, suppose that the moetary authorities revised their price target i momet from P to P. 5. This will be achieved settig the moey target i period to be P. Give the sticky price today P, the IS curve will become: i 8P P Thus, the IS curve will shift rightwards, as depicted i Figure 7. Hece, departig from a depressed equilibrium at the ZLB ad 8, moetary authorities are able to drive the ecoomy to full employmet by implemetig a permaet moetary expasio, to. Figure 7: Permaet moetary expasio.4.4 Fiscal expasio A alterative aveue to get out of depressio is a expasio of govermet cosumptio. As we already kow, i a closed ecoomy, govermet cosumptio crowds out private cosumptio. Thus, a fiscal expasio today will make cosumptio scarcer today. As curret cosumptio gets scarcer, its relative price (the

13 real iterest rate) shall icrease. This mechaism ca drive the ecoomy out of the ZLB without the eed for the cetral bak to promise more iflatio i the future. To see this i terms of our model, let s cosider agai the equilibrium described by poit i Figure 3. Departig from this poit, suppose the govermet decided to aticipate period expeditures to period. That is, istead of a horizotal expediture path G ; G, the govermet implemeted G 3 ; G. Note that, as log as the iterest rate is zero, such chage will ot alter the preset value of govermet expeditures ad hece, private wealth. Sice the ecoomy is closed to iteratioal trade, curret ad future cosumptio will be, 9 ad 9. Thus, the equilibrium real iterest rate will be exactly zero: 9 r 9 Figure 5: oey-fiaced fiscal aticipatio I this case, it will be possible for moetary policy to drive the ecoomy exactly to full employmet without the eed to abdicate from price stability i the future. Uder sticky prices ( P ) ad a credible commitmet with price stability i the future ( P ), the IS curve will be as follows: i 9 The shift i the IS curve due to the fiscal expasio is depicted i Figure 8. The old IS curve crosses the ZLB at. The ew IS curve crosses the ZLB at the full employmet level. Hece, a fiscal expasio today fully fiaced with moey pritig will be able to drive the ecoomy towards its full employmet level. 3

14 A questio that arises is whether the cosumer will be better of: before the fiscal expasio, the ecoomy was depressed, ad the households cosumptio path was 8. After the expediture switchig, household s cosumptio icreased to 9. Defiitely, cosumers are better off, eve i a world where the govermet meets its iter-temporal budget costrait 5..5 Discussio Whe output is below its atural level, the cetral bak may use its policy tools to drive dow the moey market omial iterest rate ad by the, ifluece the log term real iterest rates. The effectiveess of this policy depeds however o a umber of factors. First, the cetral bak iflueces the moey market iterest rate, which is a omial variable, while ivestmet ad cosumptio deped o the real iterest rate. Hece, the effectiveess of moetary policy will deped critically o how much people expect the iflatio rate to be. Secod, the cetral bak iflueces the short term iterest rate, while the opportuity cost of ivestmet is the log term iterest rate. The impact of moetary policy is mediated through the term structure of iterest rates, whereby the log ru iterest rate correspods to the average of the future expected short term iterest rates. Hece, whe the cetral baks cuts the short term iterest today, the impact o the log term real iterest rate will deped o expectatios regardig future short term iterest rates 6. Third, whe the omial iterest rate reaches the zero lower boud, a further expasio of moetary policy today will ot be able to drive dow the iterest rate. There is however a aveue through which cetral bak could act to ifluece the real iterest rate, eve at the ZLB. As real iterest rates deped o expected iflatio, moetary policy ca be effective if the cetral baks commits with higher iflatio i the future. A permaet icrease i moey supply, by creatig iflatio expectatios, may lower the ex ate real iterest rate, eve is the omial iterest rate is stuck at the ZLB. A problem with creatig iflatio is that this would imply the cetral bak to get rid of its reputatio: as poited out by Paul Krugma, for moetary policy to be effective, the cetral bak should credibly promise to be irresposible. 5 It should be oted that this model does ot provide a good bechmark to evaluate the welfare effects of policies, because it does ot iclude leisure either a social value for govermet expeditures. 6 Durig the recet fiacial crisis, both the FED ad the EB aouced that the policy of keepig low moey market rates today would be exteded for a log period of time. The itetio was to help keep all the term structure of iterest rates dow ad thereby the log-term iterest rate. 4

15 I istitutioal setups i which this is ot feasible, the alterative aveue is fiscal policy. A fiscal expasio today may iduce the desire rise i the equilibrium real iterest rate, pavig the way for a successful expasio of moetary. If, i plus, such fiscal expasio is achieved through the aticipatio of future govermet expeditures, the output gai will be full appropriated by households. Appedix The model preseted above abstracts from how budget costraits are impacted with the presece of moey. This appedix fills the gap, showig how the hidde part of the model looks like. osider first the household budget costrait. The household is edowed with uits of output i period, ad uits of output i period. At the begiig of period, the household holds a iitial wealth i the form of bods ad moey. Next year wealth will be determied by last year wealth, icome, taxes ad cosumptio expeditures: i B P T B (a) This equatio takes ito accout that moey ad bods are omial, while disposable icome ad cosumptio expeditures are real. It also shows that moey, i cotrast to bods, pays o iterest. Sice idividuals eed moey to carry out trasactios, they are willig to hold a asset that is domiated as store of value. Dividig both terms by the price level oe obtais: b r b T P (a) Where b B P ad r i P P. t t t I a two-period ecoomy, b. However, the household still eeds to hold moey i period to make trasactios. Hece, the secod period budget costrait is: r b T (a3) P ombiig (a) ad (a3) together, we obtai the household budget costrait: (a4) r Where 5

16 T P r b T (a5) P r This equatio stresses the fact that a icrease i real moey balaces plays i the household budget costrait a role similar to that of taxatio. Govermet spedig is fiaced with taxes, debt emissio, or moey pritig. The dyamics of govermet debt i periods ad are as follows (ote that moey is a liability): D g i D PG g T (a6) g i D PG T (a7) ombiig these two, oe obtais the govermet itertemporal budget costrait: G T G P r d T (a8) r P r This equatio states the preset value of govermet expeditures shall be equal to the preset value of govermet reveues, which cosist i formal taxes ad seigiorage reveues. Replacig (a8) ad (a5), private wealth becomes: r b d G G (a9) r Whe the ecoomy is closed to capital flows, b d. Hece, private wealth becomes: G G (3) r This equatio reveals that private wealth is ot affected by moey holdigs, as it ot affected by the timig of taxatio. By holdig moey, households implicitly pay a tax correspodig to the iterest the govermet saves o moey-fiacig, as compared to bod-fiacig. Give the level of govermet expeditures, however, the iterest the govermet saves by pritig moey traslates ito lower taxes, today or i the future. Hece, the household wealth is uaffected by moey holdigs, just like it is ot affected by the timig of taxatio. Oly govermet spedig impacts o private wealth. 6

17 Review questios ad exercises Review questios.. ommet: I order to recover its moetary policy effectiveess, the cetral bak has to credibly commit with higher iflatio i the future. Exercises.. (Zero lower boud) osider a two-period edowmet ecoomy with o govermet, where the life-time utility fuctio of the represetative cosumer is give by U.. I this ecoomy the Fisher priciple holds istataeously, that is, i r P P, ad moey velocity is equal to oe. I this ecoomy, future potetial output is equal to oe ad so are prices ad moey, ie,. P. The curret period full employmet output is.. a) Use the Euler equatio ad the Fisher priciple to fid out a expressio relatig the omial iterest rate to curret ad future output ad prices. b) (Flexible Prices): Suppose that prices were fully flexible. Describe the equilibrium i this ecoomy (output, prices, omial iterest rate) whe: ; ;. I which case does the quatitative theory break out? Why? What happes whe more moey is prited? c) (Sticky Prices): Suppose prices i period are sticky at P. Departig. from ad, fid out how far close to full employmet ca moetary expasio drive the ecoomy? What happes whe the zero lower boud is reached? d) (ommitmet with high iflatio): Suppose the cetral bak commits with a higher price level i the future, that is P. Explai why such commitmet could chage the effectiveess of moetary policy today..3. (Sticky prices, ZLB) osider a closed ecoomy where full employmet output today ad i the future are, respectively ad, while govermet expediture are G ad G. Further assume that i this ecoomy, the prefereces of the represetative household are give by U l l. e) (Natural iterest rate). From the household optimizatio problem, ad usig the govermet budget costrait, fid out the expressio of the IS curve. Give the full employmet output, what will be the atural iterest rate i this ecoomy? Represet i a graph. f) (Uemploymet): assume that the real iterest was stuck at r. 6. How much would be curret output i this case? Represet i a graph. 7

18 g) (ZLB). Assume that i this ecoomy the Fisher priciple holds, curret prices are sticky at P, ad the cetral bak is credibly committed with price stability i the future, P. Further assume that households cosumptio is subject to a cash-i-advace costrait of the form. Departig from 5, how far could a moetary expasio today go i expadig output? h) (Policy optios) Departig from (o), explai how the authorities could use (p) the moetary policy; (p) the fiscal policy; to drive the ecoomy to full employmet. 8