The Friedman Rule and the Zero Lower Bound

Transcription

1 Inernaional Journal of Economics and Finance; Vol. 5, No. 11; 2013 ISSN XE-ISSN Published by Canadian Cener of Science and Educaion The Friedman Rule and he Zero Lower Bound Sebasien Bue 1 & Udayan Roy 2 1 Deparmen of Business and Economics, School of Business and Informaion Sysems, York College, Unied Saes 2 Deparmen of Economics, College of Liberal Ars and Sciences, LIU Pos, Unied Saes Correspondence: Sebasien Bue, Deparmen of Business and Economics, School of Business and Informaion Sysems, York College, Guy R. Brewer Blvd, Jamaica, NY 11451, Unied Saes. Tel: sbue@york.cuny.edu Received: Sepember 2, 2013 Acceped: Ocober 5, 2013 Online Published: Ocober 26, 2013 doi: /ijef.v5n11p22 URL: hp://dx.doi.org/ /ijef.v5n11p22 Absrac We explain why cenral banks rarely implemen he Friedman rule by sudying he properies of a simple New Keynesian dynamic macroeconomic model ha is generalized o incorporae he zero lower bound on nominal ineres raes. We show ha wo long-run equilibria exis, one sable and he oher unsable, and we characerize he condiions under which he economy plunges ino a deflaion-induced depression following a conracionary demand shock. As long as he sum of he inflaion rae and he naural real ineres rae says posiive, he economy converges back o he long-run sable equilibrium, even when he zero lower bound is iniially binding. On he oher hand, a deflaionary spiral sars when he sum of he inflaion rae and he naural rae of ineres becomes negaive. The Friedman rule, which calls for a small dose of anicipaed deflaion, akes he sum of he inflaion rae and he naural real ineres rae uncomforably close o zero and hereby raises he probabiliy of he economy plunging ino a depression. Our heoreical resuls also shed ligh ono he differen correlaion values beween inflaion and economic growh in he daa. Keywords: Friedman rule, zero lower bound, inflaion, growh 1. Inroducion In his classic essay on he quaniy of money, Friedman (1969) proposed a simple rule o se nominal ineres raes in he long-run. In order for consumpion, savings, and labor supply decisions of hose who hold money no o be disored, he (privae) opporuniy cos of holding money should equal he social cos of creaing addiional money. Since in a fia moneary sysem cenral banks can prin money a virually no cos, he Friedman rule calls for seing nominal ineres raes o zero in he long-run. A large body of academic research has shown ha he Friedman rule holds in a variey of moneary economies wih monopolisic compeiion (Ireland, 1996), disoring axes (Chari e al., 1996) or search fricions (Lagos, 2010). In addiion, Cole & Kocherlakoa (1998) characerize he moneary policies ha will implemen zero raes in he one secor neoclassical growh model. They show ha he only resricion hese policies mus saisfy is ha asympoically money shrinks a a rae no greaer han he rae of discoun which implies a small amoun of deflaion in goods prices. While much ink has been spil abou he Friedman rule in academic circles, here is lile evidence ha cenral banks around he world have or are ready o adop a long-run arge of zero nominal ineres raes. In his paper, we explain why cenral banks rarely implemen he Friedman rule by revisiing he properies of a simple New Keynesian dynamic macroeconomic model ha is generalized o incorporae he zero lower bound on nominal ineres raes. Reduced-form New Keynesian models of shor erm economic flucuaions usually do no incorporae any resricions on nominal ineres raes and are buil around hree equaions: an IS curve (he oupu gap depends on he real ineres rae), a Phillips curve (inflaion depends on he oupu gap and las period s expecaion of his period s inflaion rae), and a Taylor rule wih a zero lower bound consrain. Wihou he zero lower bound on nominal ineres raes, he cyclical and long-run properies of New Keynesian models are well known. In he shor-run, oupu and real ineres raes are couner-cyclical and, in he absence of shocks, all variables converge oward a unique sable equilibrium where he economy is operaing a full capaciy and inflaion is equal o he 22

2 cenral bank s inflaion arge. The inroducion of he zero lower bound changes he model dynamics in he shor and long-run. Our economy has now wo long-run equilibria: (i) he sable one where nominal ineres raes are posiive and inflaion is equal o he cenral bank s arge and (ii) a new equilibrium wih zero nominal ineres raes ha can lead o a deflaionary spiral. Given he mulipliciy of equilibria, several quesions abou shor-erm dynamics arise. For a given se of parameer values and iniial condiions, o which of he wo equilibria does inflaion and oupu converge? Are here equilibrium pahs where deflaion does no lead o depression? And ulimaely, why do cenral banks rarely implemen he Friedman rule? We find ha answers o hese quesions depend on he sign of a simple expression, he sum of he inflaion rae and he naural rae of ineres. We show ha, as long as he sum of he inflaion rae and he naural real ineres rae says posiive, he economy converges back o he long-run sable equilibrium, even when he zero lower bound is iniially binding. On he oher hand, a deflaionary spiral sars when he sum of he inflaion rae and he naural rae of ineres becomes negaive. The Friedman rule, which calls for a small dose of anicipaed deflaion, akes he sum of he inflaion rae and he naural real ineres rae uncomforably close o zero and hereby raises he probabiliy of he economy plunging ino a depression. We believe ha he hrea of a policy-induced deflaionary spiral is a key facor accouning for policymakers relucance o adop any policy, including he Friedman rule, ha may cause deflaion. Our heoreical resuls also shed ligh ono he differen values for he correlaion beween inflaion and economic growh in he daa. There is a igh link beween deflaion and depression during he Grea Depression years and many believe ha deflaion conribued o he severe oupu losses in he U.S. and oher counries beween 1929 and 1932 (Bernanke & Carey, 1996; Eichengreen & Sachs, 1985). Ouside of he Grea Depression years, however, he empirical link beween deflaion and depression is much weaker (Akeson & Kehoe, 2004; Benhabib & Spiegel, 2009). Our model predicions ha (i) mild deflaion does no lead o a depression and (ii) deflaion and depression are closely linked when he economy is hi by a large demand shock help reconcile and explain he seemingly differen paerns for inflaion and growh in he daa. Noe ha our modeling choice of using a sysem of reduced-form equaions differs from he recen macroeconomic lieraure in wo imporan ways. Firs, agens choices are no maximizing a well-defined uiliy (objecive) funcion, implying ha he srucural equaions of he model, in paricular he IS and he Philips curves, are no grounded in microeconomic heory. In he New Keynesian models, he IS curve has consumpion (or oupu) growh depending on he ineres rae while he Phillips curve saes ha inflaion is a funcion of he real marginal cos (or oupu) and he expeced fuure inflaion rae. Second, we assume adapive expecaions in price seing insead of allowing prices o be se in a forward-looking manner. Our main poin however, ha implemening he Friedman rule is no recommended when he economy is in a liquidiy rap because of he possibiliy of deflaion-induced depression, remains valid as long as inflaion and oupu are posiively correlaed when he zero lower bound is binding (see Krugman (1998) for a fully specified model wih micro-foundaions where a deflaionary spiral can occur). The remainder of he paper is organized as follows. In Secions 2 and 3, we inroduce our model and characerize is long-run equilibria. We discuss he sabiliy of hese equilibria in Secion 4. We reinerpre empirical findings in ligh of our heoreical resuls in Secion 5. Finally, we offer concluding remarks in Secion The Model We exend a simple New Keynesian model of he business cycle by explicily incorporaing a non-negaiviy consrain on nominal ineres raes. The version of our model wihou he zero lower bound has replaced he IS-LM model as he mainsay of shor-run analysis in macroeconomics exbooks and is variously referred o as he dynamic AD-AS (or DAD-DAS) model in Mankiw (2010), he AS/AD model in Jones (2011), and he 3-equaion (IS-PC-MR) model in Carlin and Soskice (2006), where PC refers o he Phillips Curve and MR refers o he cenral bank s Moneary Policy Rule. Equilibrium in he marke for goods and services is given by Y α(r ϱ)+ε (1) where Ȳ denoes he naural or full-employmen level of oupu, r is he real ineres rae, ϱ is he naural real 23

3 ineres rae of ineres, α is a posiive parameer represening he responsiveness of aggregae expendiure o he real ineres rae, and ε represens boh demand shocks (say, from changes in privae-secor opimism or animal spiris ) and he governmen s fiscal policy sance. This equaion is essenially he well-known IS Curve. The key feaure of (1) is he negaive relaionship beween he real ineres rae r and oupu Y. When he real ineres rae increases, borrowing becomes more expensive and savings yields a higher reward. As a resul, firms engage in fewer invesmen projecs and consumers save more and spend less. These effecs reduce he demand for goods and services. Noe ha he economy operaes a full capaciy (Y ) when here are no demand shocks (ε 0) and when he real ineres rae is equal o he naural real ineres rae (r ϱ). The ex-ane real ineres rae in period is deermined by he Fisher equaion and is equal o he nominal ineres rae i minus he inflaion expeced in he nex period: r i E +1 (2) Inflaion in he curren period,, is deermined by a convenional Phillips curve augmened o include he role of expeced inflaion, E 1, and exogenous supply shocks, ν : wih φ is a posiive parameer. E 1 +φ(y Ȳ )+ν (3) The random variable ν capures all oher influences on inflaion oher han inflaion expecaions (which are capured in he erm E 1 ) and shor-run economic condiions (which are capured in he erm φ(y Ȳ )). For example, an increase in oil prices would mean a posiive value for ν since higher inpu prices migh force firms o raise he price of heir producs. Inflaion expecaions play a key role in boh he Fisher equaion (2) and he Phillips curve (3). To keep he model racable analyically and because he role of expecaions was analyzed in grea deail in Eggerson and Woodford (2003, 2004), we assume ha inflaion in he curren period is he bes forecas for inflaion in he nex period. Tha is, agens have adapive expecaions wih: E +1 (4) Finally, we complee he descripion of he model wih a rule for moneary policy. Dynamic New Keynesian models assume ha he cenral bank ses a arge for he nominal ineres rae, i, based on he inflaion gap and he oupu gap according o he Taylor rule (Taylor, 1993). Specifically, i is assumed ha i +ϱ+θ π ( π )+Θ Y (Y Ȳ ), where he cenral bank s policy parameers Θ π and Θ Y are non-negaive. We, however, wish o explicily incorporae he fac ha nominal ineres raes need o be non-negaive. Therefore, our moneary policy rule is: i max{0,π +ϱ+θ (π π π )+Θ Y (Y Ȳ )}. (5) 24

4 Alhough he cenral bank ses a arge for i, is rue influence on he economy works hrough he real ineres rae, r. When inflaion in period is equal o he cenral bank s arge (π π ) and oupu is a is naural level (Y ), he las wo erms in equaion (5) are equal o zero, implying ha he real rae is equal o he naural rae of ineres (r ϱ). As inflaion rises above he cenral bank s arge (π >π ) or oupu rises above is naural level (Y >Ȳ ), he Taylor rule ensures ha he real rae rises o bring down inflaion and/or oupu. Conversely, if inflaion falls below he cenral bank s arge (π <π ) or oupu falls below is naural level (Y <Ȳ ), he real ineres falls o provide a simulus o he economy. 3. Long-Run Equilibria For given values of he model s period- parameers (α, ϱ, φ, Θ π, Θ Y, π *, and Ȳ ), is period- shocks (ε and ν ), and he pre-deermined inflaion rae ( 1 ) for period 1, one can use he he model s five equaions o solve for is five period- endogenous variables (Y, r, i, E π, and π ). Once π is deermined, he process 1 can be repeaed for period +1, and so on and on. To sudy he model s dynamic properies algebraically as opposed o numerically we will simplify he model by assuming a consan arge inflaion rae: ha is, π * π* for all. Definiion 1. An equilibrium is a sequence for oupu, inflaion, nominal and real ineres raes {Y,π,E π,i,r } such ha he model s equaions (1) (5) are saisfied a all ime. 1 Definiion 2. A long-run equilibrium is any equilibrium sequence such ha, in he absence of shocks (ε ν 0 for all ), he inflaion rae is consan. I is hen sraighforward o idenify he wo long-run equilibria of he model. Firs, one can check ha if 1 π * and here are no shocks, hen +s π * for all subsequen periods. We will refer o his long-run equilibrium as he orhodox equilibrium. Real oupu is equal o poenial oupu in all periods, Y +s +s ; he real ineres raes is equal o he naural rae of ineres, r +s ϱ, and nominal ineres raes is consan and equal o i +s ϱ+π *. Second, one can check ha, if 1 ϱ and here are no shocks, hen +s ϱ is also an equilibrium. We refer 25

5 o his long-run equilibrium as he deflaionary equilibrium. Real oupu is equal o poenial oupu, Y +s +s, real ineres rae is equal o he naural rae of ineres, r +s ϱ, and nominal ineres raes are zero forever, i +s Sabiliy of Equilibrium The issue of sabiliy ineviably arises: How does he economy behave for arbirary values of 1? Will i converge o one of he wo long-run equilibria? If so, which one? To analyze he sabiliy properies of he DAD-DAS model, we need o solve for he model s endogenous variables for he case in which he zero lower bound on he nominal ineres is no binding and again for he case in which he zero lower bound is binding. 4.1 Zero Lower Bound is Non-Binding When he zero lower bound is no binding, he moneary policy rule (5) becomes i π +ϱ+θ (π π π )+Θ Y (Y Ȳ ). (6) Therefore, he economy is described by equaions (1) (4) and (6). Using hese equaions, i is sraighforward, if edious, o derive expressions for he model s endogenous variables in erms of he parameers of he model and he pre-deermined inflaion rae of he previous period. Specifically, i can be shown ha equilibrium oupu is Y + αθ (π π 1 ν )+ε 1+α(Θ φ+θ ), (7) π Y equilibrium inflaion is (1+αΘ ) (π +ν )+αθ φπ * Y 1 π +φε 1+α(Θ φ+θ ), (8) π Y and he equilibrium nominal ineres rae is 1 i 1 ( [(1 Y )( Y ) ) (1 ) * ( (1 ) ) ] 1 Y (9) As he Fisher equaion (2) and adapive expecaions (4) imply r i π, he equilibrium real ineres rae can be derived from equaions (8) and (9) above. I follows from (7) ha, in he usual case where he zero lower bound on he nominal ineres rae is no binding, oupu (Y ) is direcly relaed o full-employmen oupu (Ȳ ), he demand shock (ε ), and he cenral bank s arge inflaion (π * ), and inversely relaed o he previous period s inflaion ( 1 ), he inflaion shock (ν ), he responsiveness of inflaion o oupu in he Phillips Curve (φ), and he responsiveness of he nominal ineres rae o oupu in he moneary policy rule (Θ Y ). I follows from (8) ha inflaion ( ) is direcly relaed o 1, ν, 26

6 π *, and ε. Figure 1 graphs he dynamic mapping from 1 o for given values of all he parameers in equaion (8) and he shocks se a zero (ε ν 0). I can be checked ha his graph which we will call he inflaion mapping curve mus inersec he 45-degree line a 1 π *, because 1 π* implies π π π * 1. This represens he orhodox long-run equilibrium. Figure 1. Dynamics wihou he zero lower bound on he nominal ineres rae The dynamic mapping from 1 o when here are no shocks (ε ν 0) and he zero lower bound on he nominal ineres rae is non-binding is shown here. For any given value of 1, he inflaion mapping curve and he 45-degree line can be used o race inflaion in subsequen periods. The orhodox long-run equilibrium ( 1 π * ) can be seen o be sable. (TC "1 Dynamics wihou he zero lower bound on he nominal ineres rae The dynamic mapping from 1 o when here are no shocks (ε ν 0) and he zero lower bound on he nominal ineres rae is non-binding is shown here. For any given value of π, he inflaion mapping curve and 1 he 45-degree line can be used o race inflaion in subsequen periods. The orhodox long-run equilibrium (π π * ) can be seen o be sable. 1 Noe also ha he inflaion mapping curve mus be flaer han he 45-degree line as is slope is a posiive 27

7 fracion: ha is: / 1 (1+αΘ Y )/(1+αΘ Y +αθ π φ) (0,1) As is shown in Figure 1, his feaure of he inflaion mapping curve when he zero lower bound on he nominal ineres rae is no binding and here are no shocks implies ha he orhodox long-run equilibrium is sable. (The sabiliy of he deflaionary long-run equilibrium will be discussed below.) To guaranee he inuiive resul ha a posiive demand shock increases oupu in equaion (7), we assume ha 1 αφ>0 in his paper. I hen follows from (9) ha he nominal ineres rae, assuming i is no a he zero lower bound, is direcly relaed o ϱ and o all exogenous variables and shocks ha are direcly relaed o inflaion. Noe ha hese resuls prevail only when i, as given by (9), is non-negaive. I can be checked ha (9) implies where i > 0 if and only if -1 > πc -1, (10) π c 1 (1 αφ)θ π π * (1+αΘ π φ+αθ Y ) (Θ π φ+θ Y +φ)ε 1+Θ π +αθ Y ν (11) is he criical value for he previous period s inflaion rae such ha he equilibrium nominal ineres rae is given by equaion (9) when 1 π c 1, and is zero when 1 πc 1. The line segmen BC in Figure 2 graphs he dynamic mapping from 1 o in equaion (8) for 1 π c 1 when here are no shocks and he cenral bank has a consan arge inflaion (π * π* ). A quick comparison of Figures 1 and 2 emphasizes he fac ha he inroducion of he zero lower bound on he nominal ineres rae sharply resrics he domain over which equaion (8) is applicable. Noe, as before, ha he coefficien of π on he righ hand side of equaion (8) is a posiive fracion, which explains why BC is drawn 1 flaer han he 45-degree line. A 1 π *, BC inersecs he 45-degree line, indicaing he orhodox long-run equilibrium of secion. I can be checked from (11) ha, in he absence of shocks, π c 1 is a convex combinaion of π * and ϱ. Our assumpion ha π * +ϱ>0 hen implies π * >π c 1 > ϱ, as shown in Figure 2. 28

8 Figure 2. Sabiliy ABC represens he dynamic mapping from 1 o when here are no shocks (ε ν 0) AB represens an economy in which he zero lower bound on he nominal ineres rae is binding, and BC represens he same economy when he zero lower bound on he nominal ineres rae is non-binding. For any given value of 1, ABC and he 45-degree line can be used o race inflaion in subsequen periods. The orhodox long-run equilibrium ( 1 π * ) can be seen o be sable, whereas he deflaionary long-run equilibrium ( 1 ϱ) can be seen o be unsable. (TC "2 Sabiliy ABC represens he dynamic mapping from 1 o when here are no shocks (ε ν 0). AB represens an economy in which he zero lower bound on he nominal ineres rae is binding, and BC represens he same economy when he zero lower bound on he nominal ineres rae is non-binding. For any given value of 1, ABC and he 45-degree line can be used o race inflaion in subsequen periods. The orhodox long-run equilibrium ( 1 π * ) can be seen o be sable, whereas he deflaionary long-run equilibrium ( 1 ϱ) can be seen o be unsable. In he absence of shocks (ε ν 0), equaion (8) reduces o 1 αθ φ π 1+αΘ φ+αθ (π * π ). (12) π Y 1 Noe ha whenever he cenral bank s arge inflaion rae, π *, exceeds (respecively, is less han) he previous period s inflaion, 1, he curren period s inflaion rises (respecively, falls), while sill remaining less 29

9 (respecively, greaer) han he arge inflaion. In oher words, if here are no shocks and he arge inflaion is consan and π π c 1 1, he inflaion rae converges over ime o he cenral bank s arge inflaion rae. Under adapive expecaions and in he absence of shocks, he Phillips Curve yields φ (Y Ȳ ) 1 αθ φ π 1+αΘ φ+αθ (π * π π Y 1 ). The convergence of inflaion o arge inflaion herefore implies he convergence of equilibrium oupu o he full-employmen oupu. Therefore, according o he Taylor Rule (5 or 6), he nominal ineres rae se by he cenral bank will converge o i * π * +ϱ>0. And, by he Fisher equaion (2) he real ineres rae converges o ϱ. We have, herefore, esablished he following: Proposiion 1. When here are no shocks and π π c 1 1, he economy converges o he orhodox long-run equilibrium. 4.2 Zero Lower Bound Is Binding I follows from (10) ha, when π <π c 1 1, he zero lower bound on he nominal ineres rae becomes binding: ha is, i 0. (13) In his case, recall ha he economy is described by equaions (1) (4) and (13). Using hese equaions, i is sraighforward, as before, o derive expressions for he endogenous variables a ime in erms of he parameers of he model and he pre-deermined inflaion rae of he previous period. Specifically, and equilibrium inflaion is equilibrium oupu is Y αφ ( 1 +ν )+ α( 1 +ν +ϱ)+ε 1 αφ (14) φ 1 αφ (αϱ+ε ). (15) Noe ha he coefficien of 1 in (15) exceeds uniy. This is why he segmen AB in Figure 2, which shows he dynamic link beween 1 and for 1 π c 1 when here are no shocks, is seeper han he 45-degree line. Also, i can be checked ha equaing he righ hand sides of equaions (8) and (15) yields 1 π c 1, hereby confirming he coninuiy of he mapping of 1 o a 1 π c 1 in Figure 2. The conras beween equaion (7), when he zero lower bound is no binding, and equaion (14), when he zero lower bound is binding, is sriking. I follows from (14) ha now 1, ν, and φ are direcly relaed o oupu. Moreover, oupu is now direcly relaed o he responsiveness of aggregae demand o he real ineres rae (α) and o he long-run real ineres rae (ϱ), and unrelaed o all moneary policy ools (π *, Θ Y, and Θ ). The π 30

10 ineffeciveness of moneary policy follows from he fac ha he nominal ineres rae is now a he zero lower bound and moneary policy is, herefore, unable o influence he nominal ineres rae. I follows from equaion (15) ha inflaion is direcly relaed o 1, ν, φ, and ε, as i is in equaion (8), which describes inflaion when he zero lower is no binding. Now, however, α and ϱ have a direc effec on inflaion, and he moneary policy ools have no effec a all (for he reason discussed in he previous paragraph). Subracing 1 from boh sides of equaion (15) yields 1 αφ 1 αφ (π +ϱ). (16) 1 This confirms our earlier resul ha when here are no shocks and 1 ϱ, he economy is in he deflaionary long-run equilibrium of secion 3. However, equaion (16) also reveals he unsable naure of he deflaionary long-run equilibrium. If here are no shocks and if 1 < ϱ, equaion (16) implies ha inflaion keeps falling and, by equaion (14) drags oupu down wih i, indefiniely. This is he dreaded deflaionary spiral. On he oher hand, if here are no shocks and if π c 1 >π > ϱ, inflaion and oupu boh keep rising. In shor, we have esablished he following: 1 Proposiion 2. Assume here are no shocks. The deflaionary long-run equilibrium is unsable. As inflaion keeps rising when here are no shocks and π c 1 >π > ϱ, equaion (11) implies ha inflaion will, 1 in finie ime, exceed π c 1 and ha, herefore, he zero lower bound on he nominal ineres rae will swich from binding o non-binding. Therefore, proposiions 1 and 2 can be combined o yield he following: Proposiion 3. Assume here are no shocks. If 1 > ϱ, he economy converges o he orhodox long-run equilibrium. If 1 ϱ, he economy says in he deflaionary long-run equilibrium. If 1 < ϱ, he economy says in a deflaionary spiral. Proposiion 3, which is graphically illusraed in Figure 1, helps explain why cenral banks rarely implemen he Friedman rule as he opimal moneary policy. The Friedman rule, which calls for a small dose of anicipaed deflaion, makes he sum of inflaion and naural rae of ineres very close o zero, which makes he economy vulnerable o furher shocks and raises he probabiliy ha he economy plunges ino a depression. 5. Reinerpreaion of Empirical Findings In he inroducion, we cied he empirical findings of Akeson and Kehoe (2004), who show ha, while he deflaion-depression link is srong for he Grea Depression years, here is lile evidence supporing a saisical relaionship beween deflaion and depression ouside of he Grea Depression years. Here we briefly summarize heir empirical sraegy and revisi heir findings in ligh of our resuls. The auhors use he following simple specificaion o esimae he relaionship beween inflaion and growh: Δy i α+βπ i +ε i (17) where Δy i represens average annual growh for counry i during five-year period, π i represens average 31

11 annual inflaion for counry i during five-year period, and ε i represens an i.i.d. normal disurbance erm. The daa se consiss of hisorical daa on inflaion and oupu growh raes for a panel of 17 differen counries. Every daa series runs for more han a hundred years and ends in he year The counries, wih he year he daa sars given in parenhesis, are: Argenina (1885), Ausralia (1862), Brazil (1861), Canada (1870), Chile (1908), Denmark (1871), France (1820), Germany (1830), Ialy (1867), Japan (1885), he Neherlands (1900), Norway (1865), Porugal (1833), Spain (1849), Sweden (1861), he Unied Kingdom (1870), and he Unied Saes (1820). For all counries excep Ausralia and Denmark, he daa up o 1980 are aken from rolnick. The daa for Ausralia and Denmark up o 1980 are aken from backus. The daa from 1980 onwards are from he Inernaional Financial Saisics of he Inernaional Moneary Fund. For he Grea Depression years, he poin esimae for he slope β is equal o 0.4, implying ha, on average, a one percenage poin lower inflaion rae is associaed wih a drop in growh of.4 of a percenage poin, say, from 3.40 o 3.00 percen. When he enire ime period is considered, however, he value for β drops o 0.08, implying a much weaker link beween inflaion and growh beween 1820 and Finally, he poin esimae of β is slighly negaive for he years afer WWII. Our heoreical resuls in Proposiion 3 help explain he weak relaionship beween inflaion and oupu growh ouside of he Grea Depression years and he igh deflaion-depression link during he Grea Depression years. When inflaion falls below ϱ and a depression sars, our model clearly predics a igh link beween deflaion and depression (see Figure 2). However, he probabiliy of a large demand shock saring a depression is very small. Thus, many equilibrium pahs involve a mild case of deflaion bu no depression, implying a small correlaion beween inflaion and oupu growh even when he zero lower bound is binding. Using our model, we calculae he slope of he linear relaionship beween oupu growh and inflaion, when he zero lower bound is binding and when i is no. Tha is, we seek o find he parameer β such ha Δy γ+βπ +u (18) where Δy Y Y +1 Y and u is a disurbance erm. Assuming ha he naural oupu level is consan over ime and normalizing i o one, we use equaions (7) and (8) o calculae he slope β when he zero lower bound is no binding and equaions (14) and (15) when i is binding. When he zero lower bound is no binding, we find ha he value of β is approximaely equal o: β POS α 2 Θ 2 π φ (1+αΘ )(1+α(Θ φ+θ )) Y π Y (19) On he oher hand, when he zero lower bound is binding, he slope β is approximaely equal o: β ZLB α 2 φ 1 αφ (20) We calculae he slope β ZLB and β POS for he following parameer values. Table 1. Parameer values Ȳ Α ϱ Φ Θ π Θ Y * π The firs hree parameers (Ȳ,α,ϱ) appear in he aggregae demand equaion (1). A value of 100 for he naural level of oupu is convenien as flucuaions in Y Ȳ can be inerpreed as percenage deviaion of oupu from 32

12 is naural level. The parameer α1 implies ha a one percenage poin increase in he real ineres rae reduces he oupu gap by one percenage poin. The naural rae of ineres ϱ is equal o 2 percen, in line wih hisorical daa. The parameer φ appears in he Philips curve in equaion (3). The value of 0.25 implies ha inflaion rises by 0.25 percenage poin when oupu is one percen above is naural level. Finally, he las hree parameers affec he Taylor rule in equaion (5). The value of one half for he coefficiens Θ and Θ sems direcly from π Y Taylor (1993). For each percenage poin ha inflaion rises above he Fed s inflaion arge, he federal funds rae rises by 0.5 percen. For each percenage poin ha real GDP rises above he naural level of oupu, he federal funds rae rises by 0.5 percen. Since Θ is greaer han zero, our equaion for moneary policy follow π he Taylor principle: whenever inflaion increases, he cenral bank raises he nominal ineres rae by an even larger amoun. Finally, he inflaion arge π * is equal o 2 percen, reflecing he Fed s concerns of keeping inflaion under conrol, a policy ha sared hiry years ago under chairman Paul Volcker. Some ineresing remarks are in order. Firs, when uses he values of α1 and φ0.25, we find ha β ZLB 0.33 when he zero lower bound is binding which is very close o he 0.4 esimae in Akeson and Kehoe (2004) for he Grea Depression years, cerainly a period of ime when he zero lower bound was binding. Second, we find ha β POS if one ses Θ Θ 0.5 for he Taylor rule as suggesed by Taylor (1993). As a resul, we Y π have β ZLB >β POS, which is also in line wih he resuls from he empirical lieraure. For example, he esimaed slope for he Grea Depression years, when he zero lower bound is binding, is 0.4, while for he years excluding he Grea Depression bu including he wars, he slope is β Conclusion In his paper, we explained why cenral banks rarely implemen he Friedman rule by sudying he properies of a simple New Keynesian dynamic macroeconomic model ha is generalized o incorporae he zero lower bound on he nominal ineres rae. We showed ha he model has wo long-run equilibria, one sable and he oher unsable and demonsraed he exisence of a deflaionary spiral in which boh oupu and inflaion fall wihou bound. The Friedman rule, which calls for a small dose of anicipaed deflaion, akes he sum of he inflaion rae and he naural real ineres rae uncomforably close o zero and hereby raises he probabiliy of he economy plunging ino a depression. We believe ha he hrea of a policy-induced deflaionary spiral is a key facor accouning for policymakers relucance o adop any policy, including he Friedman rule, ha may cause deflaion. Our analysis which focuses on long-run sabiliy does no address he shor-erm limiaions ha high levels of public deb, which are refleced in a counry s credi raings, place on cenral bank s abiliy o raise ineres raes now and in he fuure. Cochrane (2008) argues ha risk premiums, which capure defaul probabiliies, are an imporan facor for undersanding moneary policy via he erm srucure of ineres raes. For example, Japan wih a sock of deb greaer han wo imes is gross domesic produc migh no be able o raise ineres raes in ime, when economic aciviy and inflaion grow a oo rapid a rae following recen expansionary policy. Sudying moneary policy in a model ha incorporaes he zero lower bound and he sock of public deb remains a promising opic for fuure research. 7. Appendix: Derivaion of Equilibrium Oucomes 7.1 Oupu and Inflaion When he Zero Lower Bound Is No Binding; Equaions (7) and (8) Our goal here is o derive expressions for oupu and inflaion as a funcion of he pre-deermined inflaion in period 1 and he model s deep" parameers, when he zero lower bound on he nominal ineres rae is no binding. 33

13 Our firs sep is o use he Fisher equaion (2) o replace he real ineres rae in equaion (1) by r i. We ge he expression for nominal ineres rae from he Taylor rule in equaion (5) and obain he following relaionship beween and Y : Y α(θ π ( π )+Θ Y (Y Ȳ ))+ε (21) Nex, we replace in he previous equaion by using he Phillips curve (3): Y α(θ π ( 1 +φ(y Ȳ )+ν π )+Θ Y (Y Ȳ ))+ε (22) Noe ha, in he previous expression, oupu is he only endogenous variable a ime. As a resul, we can solve for Y as a funcion of he pre-deermined inflaion in period 1 and he model parameers. Afer collecing he erms in Y, we obain: Y + αθ (π 1 ν )+ε 1+α(Θ φ+θ ), (23) π Y which is equaion (7). Noe ha he oupu gap can be obained from he previous expression: Y Ȳ αθ π (π 1 ν )+ε 1+α(Θ π φ+θ Y ) (24) To obain an expression for inflaion in period, we subsiue he oupu gap ino he Phillips curve (3) and use our assumpion ha agens have adapive expecaions. Tha is, E 1 1. Afer collecing erms in, we obain he following expression for in erms of he parameers of he model and 1 : 1 + αθ φ(π * 1 ν )+φε 1+α(Θ φ+θ ) +ν, (25) π Y which, when rearranged, yields equaion (8). 7.2 Nominal Ineres Rae When he Zero Lower Bound Is No Binding Equaion (9) The equilibrium ineres rae in (9) is obained by subsiuing he expression for oupu gap and inflaion ino he Taylor rule. Afer some edious algebra ha involve collecing erms and rearranging, we obain he following expression for ineres rae: i ϱ+ (1+Θ +αθ )(π +ν ) (1 αφ)θ π * π Y 1 π +(Θ Y +(1+Θ π )φ)ε, (26) 1+α (Θ φ+θ ) π Y which is equaion (9). To solve for he inflaion hreshold, se i 0 in he previous expression and solve for he value of 1. Afer 34

14 edious rearranging of erms, we obain: π c 1 (1 αφ)θ π π * (1+αΘ π φ+αθ Y )ϱ (Θ π φ+θ Y +φ)ε 1+Θ π +αθ Y ν, (27) which is equaion (11). 7.3 Oupu and Inflaion When he Zero Lower Bound Is Binding Equaions (14) and (15) Our goal here is o derive expressions for oupu and inflaion as a funcion of he pre-deermined inflaion in period 1 and he model s parameers, when he zero lower bound on he nominal ineres rae is binding. As before, our firs sep is o use he Fisher equaion (2) o replace he real ineres rae in equaion (1) by r i. However, since i 0, aggregae demand when he zero lower bound is binding is given by: Y +α( +ϱ)+ε (28) Nex, we replace in he previous equaion by using he Phillips curve (3): Y α( 1 +φ(y Ȳ )+ν +ϱ)+ε (29) We solve for Y as a funcion of he pre-deermined inflaion level 1 and he model parameers: α(π +ν +ϱ)+ε 1 Y Ȳ + 1 αφ which is equaion (14). Noe ha he oupu gap can be obained from he previous expression: (30) α(π +ν +ϱ)+ε 1 Y Ȳ 1 αφ. (31) To obain an expression for inflaion in period, we subsiue he oupu gap ino he Phillips curve (3) and use our assumpion ha agens have adapive expecaions. Tha is, E 1 1. Afer collecing erms in, we obain he following expression for in erms of he parameers of he model and 1 : Afer furher simplificaion, we ge 1 + αφ(π +ν +ϱ)+φε 1 1 αφ +ν. (32) 1 1 αφ ( 1 +ν )+ φ 1 αφ (αϱ+ε ). (33) which is equaion (15). References Akeson, A., & Kehoe, P. J. (2004). Deflaion and depression: Is here an empirical link? The American Economic Review, 94(2), hp://dx.doi.org/ / Backus, D. K., & Kehoe, P. J. (1992). Inernaional evidence on he hisorical properies of business cycles. The American Economic Review, 82(4), Benhabib, J., & Spiegel, M. M. (2009). Moderae inflaion and he deflaion-depression link. Journal of Money, Credi and Banking, 41(4), hp://dx.doi.org/ /j x Bernanke, B. S., & Carey, K. (1996). Nominal wage sickiness and aggregae supply in he grea depression. 35

15 Quarerly Journal of Economics, 111(3), hp://dx.doi.org/ / Carlin, W., & Soskice, D. (2006). Macroeconomics: Imperfecions, insiuions, and policies. Oxford Universiy Press. Chari, V. V., Chrisiano, L. J., & Kehoe, P. J. (1996). Opimaliy of he friedman rule in economies wih disoring axes. Journal of Moneary Economics, 37(2), hp://dx.doi.org/ / (96) Cochrane, J. H. (2008). On he need for a new approach o analyzing moneary policy. NBER Macroeconomics Annual, 23, hp://dx.doi.org/ / Cole, H. L., & Kocherlakoa, N. (1998). Zero nominal ineres raes: Why hey are good and how o ge hem? Federal Reserve Bank of Minneapolis Quarerly Review, 22(2), Eggersson, G. B,. & Woodford, M. (2004). Policy opions in a liquidiy rap. The American Economic Review, 94(2), hp://dx.doi.org/ / Eggersson, G. B., & Woodford, M. (2003). The zero bound on ineres raes and opimal moneary policy. Brookings Papers on Economic Aciviy, 1, Eichengreen, B., & Jeffrey, S. J. (1985). Exchange raes and economic recovery in he 1930s. Journal of Economic Hisory, 4(45), hp://dx.doi.org/ /s Friedman, M. (1969). The opimum quaniy of money. Chicago: Aldine. Ireland, P. N. (1996). The role of counercyclical moneary policy. The Journal of Poliical Economy, 104(4), hp://dx.doi.org/ / Jones, C. I. (2011). Macroeconomics (2nd ed.). Krugman, P. (1998). Is baack! Japan s slump and he reurn of he liquidiy rap. Brookings Papers on Economic Aciviy, 29(2), hp://dx.doi.org/ / Lagos, R. (2010). Some resuls on he opimaliy and implemenaion of he Friedman rule in he search heory of money. Journal of Economic Theory, 145(4), hp://dx.doi.org/ /j.je Mankiw, G. N. (2010). Macroeconomics worh (7h ed.). Rolnick, A. J., & Weber, W. E. (1997). Money, inflaion, and oupu under fia and commodiy sandards. Journal of Poliical Economy, 105(6), hp://dx.doi.org/ / Taylor, J. B. (1993). Discreion versus policy rules in pracice. Carnegie-Rocheser Conference Series on Public Policy, 39, hp://dx.doi.org/ / (93)90009-l Copyrighs Copyrigh for his aricle is reained by he auhor(s), wih firs publicaion righs graned o he journal. This is an open-access aricle disribued under he erms and condiions of he Creaive Commons Aribuion license (hp://creaivecommons.org/licenses/by/3.0/). 36