Macroeconomic Stabilization When The Natural Real Interest Rate Is Falling

Transcription

1 Ciy Universiy of New York (CUNY) CUNY Academic Works Publicaions and Research Guman Communiy College Fall 2015 Macroeconomic Sabilizaion When The Naural Real Ineres Rae Is Falling Sebasien Bue CUNY Guman Communiy College Udayan Roy Long Island Universiy - C W Pos Campus How does access o his work benefi you? Le us know! Follow his and addiional works a: hp://academicworks.cuny.edu/nc_pubs Par of he Macroeconomics Commons Recommended Ciaion Bue, Sebasien and Roy, Udayan, "Macroeconomic Sabilizaion When The Naural Real Ineres Rae Is Falling" (2015). CUNY Academic Works. hp://academicworks.cuny.edu/nc_pubs/9 This Aricle is brough o you for free and open access by he Guman Communiy College a CUNY Academic Works. I has been acceped for inclusion in Publicaions and Research by an auhorized adminisraor of CUNY Academic Works. For more informaion, please conac AcademicWorks@cuny.edu.

2 MACROECONOMIC STABILIZATION WHEN THE NATURAL REAL INTEREST RATE IS FALLING Shor ile for running header: Teaching Secular Sagnaion Sebasien Bue and Udayan Roy Absrac: The auhors modify he Dynamic Aggregae Demand Dynamic Aggregae Supply model in Mankiw s widely-used inermediae macroeconomics exbook o discuss moneary policy when he naural real ineres rae is falling over ime. Their resuls highligh a new role for he cenral bank s inflaion arge as a ool of macroeconomic sabilizaion. They show ha even when he zero lower bound is no binding, a pruden cenral bank will need o mach every decrease in he naural real ineres rae wih an equal increase in he arge rae of inflaion in order o sabilize he risk of he economy falling ino a deflaionary spiral, which is an acue case of simulaneously falling oupu and inflaion in which he economy s self-correcing forces are inacive. Keywords: inermediae macroeconomics, naural ineres rae, secular sagnaion, zero lower bound JEL codes: E12, E52, E58 Sebasien Bue ( sebasien.bue@guman.cuny.edu) is an associae professor of economics a he Ciy Universiy of New York, Guman Communiy College. Udayan Roy ( udayan.roy@liu.edu) is a professor of economics a Long Island Universiy. Bue is he corresponding auhor.

3 Recen discussions of secular sagnaion have drawn aenion o he seady decrease in he naural real ineres rae in he Unied Saes since he 1980s. 1 In his aricle, we invesigae how a naion s cenral bank should respond o decreases in he naural real ineres rae when he zero lower bound on nominal ineres raes (ZLB) is poenially binding. In paricular, if oupu is a he full-employmen level and inflaion is a he cenral bank s inflaion arge, should he cenral bank feel free o ignore decreases in he naural real ineres rae? Our answer is no. We argue ha decreases in he naural real ineres rae increase an economy s vulnerabiliy o he deflaionary spiral, which is an acue case of simulaneously falling oupu and inflaion in which he economy s self-correcing forces become inacive. We also show ha a cenral bank can neuralize his danger by simply raising is inflaion arge. Inermediae macroeconomics exbooks do no discuss wha policy makers should do in response o changes in he naural real ineres rae. 2 We hope o show ha a sandard exbook model and sandard graphical echniques can be used o presen his new aspec of macroeconomic sabilizaion o sudens. We work ou he comparaive saic effecs of a fall in he naural real ineres rae in he dynamic Aggregae Demand Aggregae Supply (DAD DAS) model of shor-run macroeconomic dynamics in Mankiw (2013, Chaper 15), a widely-used inermediae macroeconomics exbook, modified, as in Bue and Roy (2014), o formally include he zero lower bound on he nominal ineres rae. In Mankiw s DAD DAS model, equilibrium oupu and inflaion are deermined a he inersecion of wo curves: a negaively-sloped dynamic aggregae demand curve and a posiively-sloped dynamic aggregae supply curve. And, irrespecive of he equilibrium levels of inflaion and oupu a a given dae, he economy converges over ime o he model s unique long-run equilibrium, in which oupu is a he full-employmen level and inflaion is a he 2

4 cenral bank s arge rae. However, when he explici requiremen ha he nominal ineres rae se by he cenral bank mus be non-negaive is added, Bue and Roy (2014) show ha he familiar negaively-sloped aggregae demand curve becomes a kinked curve wih a negaively-sloped segmen (when he ZLB is nonbinding) and a posiively-sloped segmen (when he ZLB is binding). 3 And, his leads o wo (raher han one) long-run equilibria: (1) he sable equilibrium discussed above and (2) an unsable equilibrium a which he ZLB is binding and he slighes shock can se off a deflaionary spiral. 4 The equilibrium inflaion rae a any given dae urns ou o be he crucial deerminan of he economy s subsequen desiny. If inflaion a a given dae is higher han he negaive of he naural real ineres rae, hen all s well: he economy converges o he sable long-run equilibrium. If, on he oher hand, inflaion a a given dae drops below he negaive of he naural real ineres rae, he economy eners a deflaionary spiral. So, inflaion s danger level is equal o he negaive of he naural real ineres rae; inflaion mus be kep above his danger level a all coss. Now consider an economy ha is a he sable long-run equilibrium, wih inflaion equal o he cenral bank s arge inflaion rae and higher han he danger level. An unfavorable demand shock (or a favorable inflaion shock) could reduce inflaion from he cenral bank s arge inflaion rae o below he danger level, hereby iniiaing a deflaionary spiral. This is why he job of he cenral bank in our model is no longer resriced o he sandard one of achieving full employmen and low inflaion. The cenral bank mus also do wha i can o reduce he risk of a shock-induced deflaionary spiral. And his is why a decrease in he naural real ineres rae makes macroeconomic sabilizaion harder. When he naural real ineres rae decreases, he negaive of i increases, raising he danger level of inflaion closer o he cenral bank s arge 3

5 rae. Consequenly, he possibiliy ha a shock would cause inflaion o fall from he cenral bank s arge rae (in he sable equilibrium) o below he danger level becomes more likely. 5 Luckily, he very naure of he danger poins o a way of neuralizing i. If he risk of a deflaionary spiral has increased because he danger level of inflaion has increased and comes closer o he arge rae, an obvious soluion is o raise he arge rae oo. We show ha any increase in he economy s vulnerabiliy o he deflaionary spiral can be neuralized by an increase in he cenral bank s inflaion arge. 6 If he naural real ineres rae decreases by a cerain amoun and if he cenral bank responds by increasing he arge inflaion by he same amoun, hen he gap beween he inflaion arge and he negaive of he naural real ineres rae would remain unaffeced and herefore he chance ha a shock of a given size would precipiae a deflaionary spiral would say unchanged. 7 While i is fairly obvious ha when he zero lower bound is binding, a cenral bank ha wishes o mainain full employmen would need o increase is arge inflaion rae in locksep wih decreases in he naural real ineres rae, he analysis oulined above shows ha even when he zero lower bound is no binding, a pruden cenral bank will need o mach every decrease in he naural real ineres rae wih an equal increase in he arge inflaion rae in order o keep he economy s vulnerabiliy o he deflaionary spiral from increasing. 8 Inermediae macroeconomics exbooks do no address his issue. We hope o show ha a sandard exbook model and sandard graphical echniques can be used o presen his new aspec of macroeconomic sabilizaion o sudens. Noe ha our focus on he naural real ineres rae is relaed o he curren research and debae on secular sagnaion (e.g., Eggersson and Mehrora 2014). In an address o he Naional Associaion for Business Economics ha is ofen cied in discussions of secular sagnaion, Larry 4

6 Summers refered o changes in he srucure of he economy ha have led o a significan shif in he naural balance beween savings and invesmen, causing a decline in he equilibrium or normal real rae of ineres ha is associaed wih full employmen (Summers 2014a). 9 However, economiss do no agree on how o define secular sagnaion, wha causes i, or wheher i exiss a all (Eichengreen 2014). 10 Our aricle shows how macroeconomic sabilizaion is affeced by secular sagnaion as defined by Summers. The res of he aricle is organized as follows. The nex secion reviews hisorical daa abou real ineres raes and he oupu gap for he Unied Saes in he las hiry years and proposes a simple way o esimae he naural real ineres rae for use in inermediae macroeconomics courses. In he following secion, we presen he DAD DAS model of Mankiw (2013) modified o include he ZLB consrain. We derive he kinked demand curve and characerize he model s long-run equilibrium and sudy heir sabiliy. In subsequen wo secions, we show how decreases in he naural real ineres rae increases he economy s vulnerabiliy o a deflaionary spiral and how raising he cenral bank s arge inflaion can insulae he economy from his danger. We conclude by discussing some issues no formally modeled in his aricle. MOTIVATING EVIDENCE A salien feaure of bond markes, boh in he Unied Saes and abroad, is ha real ineres raes and he naural real ineres rae have been in a downward rend for he pas hiry years (King and Low 2014; Barsky, Jusiniano, and Melosi 2014; Laubach and Williams 2003). 11 While real ineres raes can be easily inferred from inflaion and nominal yield daa, he naural real ineres rae is a heoreical consruc ha is no direcly observable and needs o be esimaed. 12 Here we propose a model, simple enough o be included in inermediae macroeconomics courses, o 5

7 esimae he naural real ineres rae. We run a linear regression beween real ineres raes, r, and he oupu gap, Y 13 : r = β + β 1 + ε (1) 0 Y where he disurbance erm ε is independen and idenically disribued (i.i.d.) over ime. We inerpre he esimae of he inercep β 0 as an esimae of he naural real ineres rae, because β 0 esimaes he real ineres rae when oupu gap is zero. We presen esimaes of β 0 and β 1 for each decade beween 1981 and 2013 in able 1. While our model is much simpler han Laubach and Williams (2003), we oo find ha he naural real ineres rae has been falling since he 1980s and has become negaive afer he financial crisis. 14 [Inser able 1 abou here.] In he nex secion, we presen a heoreical macroeconomic model of he business cycle where he ZLB is poenially binding. Laer, we use his model o analyze he impac of a decline in he naural ineres rae on inflaion and oupu. A MODEL OF MACROECONOMIC DYNAMICS As we saw in he previous secion, he naural real ineres rae has been decreasing in he Unied Saes since he 1980s. We wish o show ha a decrease in he naural real ineres rae has imporan implicaions for macroeconomic sabilizaion, and ha hese implicaions can be easily discussed in undergraduae inermediae macroeconomics courses. To ha end, we analyze in he nex secion he comparaive saic effecs of a decrease in he naural real ineres rae using a modified version of Mankiw s DAD DAS model ha incorporaes he zero lower bound on he 6

8 nominal ineres rae. Tha modified DAD DAS model discussed in Bue and Roy (2014) is summarized in his secion. The Kinked DAD Curve The DAD curve of he DAD DAS model is shown in figure 1. Aggregae demand is inversely relaed o inflaion when he ZLB is nonbinding, bu direcly relaed o inflaion when he ZLB is binding. [Inser figure 1 abou here] To undersand he kinked shape of he DAD curve, we need o look a he equaions ha drive Mankiw s DAD DAS model. Goods marke equilibrium in period is given by Y = Y α ( r ρ) + ε. (2) Here Y denoes he naural or long-run level of oupu, r is he real ineres rae, ρ is he naural or long-run real ineres rae, α is a posiive parameer represening he responsiveness of aggregae expendiure o he real ineres rae and ε represens demand shocks. Under fiscal simulus (an increase in governmen expendiure or a decrease in axes), ε is posiive; under fiscal auseriy, ε is negaive. 15 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 for he nex period: r = i Eπ. 1 (3) + The expeced inflaion in he above equaion is assumed o follow adapive expecaions: Eπ + = π. (4) 1 The nominal ineres rae in he Fisher equaion (3) is assumed o be se by he cenral bank according o is moneary policy rule. This rule is i = π + ρ+ θπ ( π π ) + θ ( Y Y ), Y 7

9 where π is he cenral bank s inflaion arge, and he parameers θ π and θ Y are non-negaive. Bu he nominal ineres rae mus obey he zero lower bound (ZLB): ha is, i needs o be non-negaive. Therefore, he generalized moneary policy rule is: i = max{0, π + ρ+ θπ ( π π ) + θ ( Y Y )}. (5) Y Figure 1 shows he border ha separaes he ( Y, π )-oucomes for which he ZLB is no binding from he ( Y, π )-oucomes for which he ZLB is binding. Algebraically, he moneary policy rule (5) implies ha he ZLB border saisfies π + ρ+ θπ ( π π ) + θ ( Y Y ) = 0. (6) Above his border, he ZLB is no binding and he nominal ineres rae se by he cenral bank is posiive ( i = π + ρ+ θπ ( π π ) + θ ( Y Y ) > 0 ). On he border, he cenral bank chooses a Y zero ineres rae, bu does so willingly, and no because i waned a negaive rae bu could no choose i. Below he border, he ZLB is binding ( i = 0). Using equaions (2) hrough (5), i is sraighforward o show, following Bue and Roy (2014), ha Y Y = Y 1 αθ π ( π π ) + + αθ + Y 1 1 αθ Y ε (7) when he ZLB is nonbinding, and Y = Y + α ( π + ρ) + ε. (8) when he ZLB is binding. Equaion (7) is graphed in figure 1 as he negaively-sloped segmen of he DAD curve above he ZLB border, which is where he ZLB is no binding. And equaion (8) is graphed as he posiively-sloped segmen of he DAD curve below he ZLB border, which is where he ZLB 8

10 is binding. In his way, he inroducion of he zero lower bound on he nominal ineres rae yields a kinked DAD curve. The posiively-sloped segmen is mean o capure he idea ha falling inflaion is a special nighmare a he zero lower bound. As i = 0, any decline in curren inflaion ( π ) means an increase in he curren real ineres rae ( r = i Eπ 1 = i π = 0 π = π ). The + rising real ineres rae reduces aggregae demand and oupu ( Y ), as he familiar IS curve (2) dicaes. 16 The kinked DAD curve in figure 1 has been drawn assuming ha he demand shock is absen ( ε = 0). Consequenly, π =π and Y = Y saisfy equaion (7). This is poin O in he figure. Similarly, i is sraighforward o check ha π = ρ and Y = Y saisfy equaion (8). This is poin D in he figure. Oucomes O and D will play an imporan role in our discussion of he model s equilibrium below. The DAS Curve Coming now o aggregae supply, inflaion, π, is deermined in Mankiw s DAD DAS model by a convenional Phillips Curve augmened o include he role of expeced inflaion, E π 1, and an exogenous inflaion shock, ν : π = E 1 + ν (9) π + φ ( Y Y ) where φ is a posiive parameer. When adapive expecaions (4) is subsiued ino he Phillips Curve (9), we ge Mankiw s dynamic aggregae supply or DAS curve: π π + φ ( Y Y ) + ν. (10) = 1 9

11 I follows ha he slope of he DAS curve is d π / dy = φ > 0. Noe also ha Y = Y and π = π + ν saisfies equaion (10). Figure 2 shows hree posiively-sloped DAS curves 1 DAS O, DAS R, and DAS for hree differen predeermined values of π 1 namely, D π, π R, and ρ, respecively and zero inflaion shock ( ν = 0). I follows ha he heighs of hese hree DAS curves a he full-employmen oupu mus also be π, π R, and ρ, respecively, as shown. Specifically, poin O, a which Y = Y and π =π, mus be on DAS O, which is he DAS curve when π 1 =π. And poin D, a which Y = Y and π = ρ, mus be on DAS D, which is he DAS curve when π = 1 ρ. In shor, when here are no shocks, he heigh of he DAS curve a full employmen is necessarily equal o he previous period s inflaion rae. [Inser figure 2 abou here] Following Bue and Roy (2014), we make he echnical assumpion ha φ, he slope of he DAS curve, is smaller han 1/ α, he slope of he posiively-sloped segmen of he DAD curve (wih ZLB binding): 1/ α > φ. Shor-Run and Long-Run Equilibrium A shor-run equilibrium a any period is graphically represened by he inersecion of he DAD and DAS curves for period. Figure 2 shows hree shor-run equilibria a O, R, and D for he same DAD curve and hree differen DAS curves. Le us consider hese hree equilibria one by one. Unless oherwise specified, we assume ha (a) here are no demand or inflaion shocks ( ε ν = 0 for all ), (b) he full-employmen oupu is consan ( Y = Y = ), and (c) he parameers of he model (α, φ, ρ, θ π, θ Y, π, and Y ) are consan. We do his o focus on he dynamic forces of change ha are inernal or endogenous o he economy (as opposed o 10

12 change caused by shocks and parameer changes). As we saw in our subsecion on he kinked DAD curve, under hese assumpions, he DAD curve does no shif over ime. So, le he DAD curve in figure 2 be he economy s DAD curve in all periods. Suppose he economy is in shor-run equilibrium in period 1 a R in figure Therefore, π = π. Recall from our subsecion on he DAS curve ha under our no-shocks 1 R and consan-parameers assumpions, he heigh of he DAS curve a he full-employmen oupu, Y, is equal o he pre-deermined rae of he previous period s inflaion. Therefore, he DAS curve in period would have o be DAS R and he shor-run equilibrium in period would herefore have o be a R wih inflaion a π = π r > π R = π 1. This example illusraes he inernal dynamics of he DAD DAS model whereby change can occur in his case from R a 1 o R a even hough here are no shocks or parameer changes. (The reader can check ha he shor-run equilibrium a + 1 will be somewhere beween R and O on he DAD curve and ha he economy converges o O over ime.) Nex, suppose π 1 =π, which is he cenral bank s inflaion arge. Therefore, he DAS curve in period would have o be DAS O. Therefore, he shor-run equilibrium in period would have o be a O because, as we saw in our subsecions on he kinked DAD curve and he DAS curve, poin O lies on boh DAD and DAS. Consequenly, π = π = π 1. In oher O words, O is a long-run equilibrium, which is a shor-run equilibrium ha repeas forever, as long as here are no shocks or parameer changes. Following Bue and Roy (2014), we will refer o O as he orhodox long-run equilibrium. I can be checked ha D oo is a long-run equilibrium. Following Bue and Roy (2014), we will refer o D as he deflaionary long-run equilibrium. 11

13 Now ha we have seen a shor-run equilibrium, R, and wo long-run equilibria, O and D, we can sae he sabiliy resuls esablished by Bue and Roy (2014). They show ha if π ρ, hen in subsequen periods inflaion and oupu will converge o > 1 π and Y, respecively. Tha is, if inflaion in some period exceeds he negaive of he naural real ineres rae, here s nohing o worry abou as long as here are no shocks and no parameer changes: he economy will converge o he orhodox long-run equilibrium a O. On he oher hand, if π < 1 ρ, hen in subsequen periods, inflaion and oupu will decrease indefiniely, moving souhwes along he DAD curve away fron D. This is he deflaionary spiral, an especially undesirable oucome. 18 Having discussed he model s equilibrium and sabiliy properies, we nex explain how a fall in he naural real ineres rae which is Summers definiion of secular sagnaion affecs inflaion and oupu in he shor and long runs. THE EFFECTS OF A DECREASE IN THE NATURAL RATE As he naural real ineres rae ρ does no appear in equaion (10), i is clear ha changes in ρ canno shif he DAS curve. Similarly, ρ does no appear in equaion (7), implying ha changes in ρ canno shif he negaively-sloped segmen of he DAD curve, which applies when he ZLB is no binding. We herefore have he following lemma: Lemma 1: When he ZLB is no binding in equilibrium (shor-run or long-run), a decline in he naural real ineres rae has no effec on he DAD and DAS curves. Therefore, he shor-run equilibrium values of oupu and inflaion are unaffeced. This resul, when coupled wih he relevan moneary policy rule i = π + ρ+ θπ ( π π ) + θ ( Y Y ), implies ha, for any decrease (respecively, increase) in Y he naural rae, ρ, he nominal ineres rae decreases (respecively, increases) by he same 12

14 percenage-poin amoun. Subsiuing adapive expecaions (4) ino he Fisher equaion (3) hen yields he same resul for he real ineres rae. In an informal sense, i is his full adjusmen of he wo ineres raes, i and r, o changes in ρ ha makes i unnecessary for eiher oupu or inflaion o adjus. There are limis, however, o he adjusmen of he nominal ineres rae o a falling naural rae. Alhough ρ can decrease indefiniely, he nominal ineres rae canno: i canno fall below zero. Once i has been driven down o zero by repeaed decreases in ρ, he economy will reach he zero lower bound (ZLB). For he ZLB case, noe ha he naural real ineres rae ρ does appear in he equaions for he ZLB border (6) and he posiively-sloped segmen of he DAD curve (8). I is sraighforward o check ha a decrease in ρ shifs boh he ZLB border (6) and he posiively-sloped segmen of he DAD curve upward, as shown by he dashed lines in figure 3. I is also clear from equaion (8) ha any decrease (respecively, increase) in ρ leads o an equal upward (respecively, downward) shif in he rising segmen of he kinked DAD curve. The effecs of hese shifs on shor-run equilibrium are shown in figure 3. The economy is iniially a Z, bu hen moves o D. We herefore have he following lemma: [Inser figure 3 abou here] Lemma 2: When he zero lower bound on he nominal ineres rae is binding in equilibrium (shor-run or long-run), a decrease in he naural real ineres rae ( ρ ) leads o decreases in boh oupu and inflaion in he shor run. 13

15 To sum up, we have shown ha in he shor run, a fall in he naural real ineres rae has no impac on inflaion and oupu when he ZLB is no binding, bu leads o declines in inflaion and oupu when he ZLB is binding. Nex, we show ha a decrease in he naural real ineres rae brings he (unsable) deflaionary long-run equilibrium closer o he (sable) orhodox long-run equilibrium, and hereby i increases he likelihood ha a demand and/or inflaion shock migh push he economy from he (sable) orhodox long-run equilibrium ino a deflaionary spiral. Proposiion 1: A decrease in he naural real ineres rae makes an economy more vulnerable o a deflaionary spiral in he sense ha he minimum size, in absolue value, ha a demand or inflaion shock would have o be in order o be big enough o iniiae a deflaionary spiral becomes smaller when he naural real ineres rae decreases. As in figure 1, le he DAD curve in figure 4 iniially be OKD. Noe ha he DAD and DAS curves inersec a O, indicaing ha he economy is iniially a he orhodox long-run equilibrium wih Y = Y and π =π. Now, consider a decrease in he naural or long-run real ineres rae from ρ 1 o ρ. As in figure 3, he DAD curve shifs from OKD o O K D. The deflaionary long-run 2 <ρ 1 equilibrium shifs from D o D, coming closer o O, he orhodox long-run equilibrium. From he equaions (7) and (8), for he negaively-sloped and posiively-sloped segmens of he kinked DAD curve, and from equaion (6) for he ZLB border, i is sraighforward o check ha an adverse demand shock a decrease in ε shifs he kinked DAD curve o he lef and leaves he ZLB border unaffeced. 19 So, when ρ =ρ1, if here is a big enough negaive demand shock, he DAD curve could shif jus o he lef of DAD 2 in figure 4, and hereby 14

16 precipiae a deflaionary spiral by reducing he inflaion rae below ρ1. 20 On he oher hand, when ρ =ρ2, he demand shock would only need o be big enough o shif he DAD curve o he lef of DAD 3. In oher words, a fall in he naural real ineres rae makes he economy more vulnerable o a deflaionary spiral because i reduces he size of he adverse demand shock ha is jus big enough o sar a deflaionary spiral. [Inser figure 4 abou here] We now consider he effec of an inflaion shock in figure 5. Recall from he secion on he DAS curve ha a negaive inflaion shock ( ν < 0) leads o a verically downward shif of he DAS curve. When ρ =ρ1, a big enough negaive (ha is, favorable) inflaion shock could ake he DAS curve o somewhere below DAS 2, which would reduce inflaion below ρ1, and hereby iniiae a deflaionary spiral. Bu when ρ = ρ2 < ρ1, a smaller inflaion shock would be able o iniiae a deflaionary spiral, because he DAS curve would have o be pushed down only somewhere below DAS 3. [Inser figure 5 abou here] Finally, consider an exreme scenario where he naural real ineres rae decreases o such an exen ha π = ρ. Then he wo long-run equilibria collapse ino he same long-run equilibrium. In figure 6, his case is represened by he DAD curve AOE 2. Here O is sill he long-run equilibrium in he sense ha an economy a O remains a O in he absence of shocks, parameer changes, and policy changes. However, his long-run equilibrium is neiher sable nor unsable. If π 1 > π = ρ2, hen he economy will converge o O (in he absence of any furher shocks, parameer changes, and policy changes). However, if π 1 < π = ρ2 hen a deflaionary spiral will ake he economy away from O., 15

17 [Inser figure 6 abou here] If ρ falls furher o ρ 3, hen ρ 3 >π and he DAD curve becomes AK E 3 3 in figure 6. In his case, here are no long-run equilibria. For all values of π 1, he economy will be in a deflaionary spiral (in he absence of any furher shocks, parameer changes, and policy changes). As we will show in he nex secion, he good news is ha his exreme case can be avoided by seadily raising he cenral bank s inflaion arge ( π ) whenever he naural real ineres rae falls, hereby ensuring ha ρ would never cach up o π. RAISING THE INFLATION TARGET TO NEUTRALIZE DECREASES IN THE NATURAL RATE We have jus seen in figures 4 and 5 ha a decline in he naural real ineres rae brings he (unsable) deflaionary long-run equilibrium closer o he (sable) orhodox long-run equilibrium, hereby increasing he danger ha a demand or inflaion shock migh push he economy from he orhodox long-run equilibrium ino a deflaionary spiral. An obvious soluion is o move he orhodox long-run equilibrium in response o every move of he deflaionary long-run equilibrium in such a way ha a consan disance is mainained beween he wo. While a decrease in ρ raises he posiively-sloped segmen of he DAD curve (8) by he same percenage-poin amoun and leaves he negaively-sloped segmen (7) unaffeced, i can be easily checked ha an increase in π raises he negaively-sloped segmen by he same percenage-poin amoun and leaves he posiively-sloped segmen unaffeced. Therefore, as shown in figure 7, if ρ decreases by percenage poins and, simulaneously, π increases by percenage poins, hen he DAD curve will shif from DAD (or, OKD ) o DAD 2 (or, O K D ). Therefore, while he deflaionary long-run equilibrium will move from D o D, he 16

18 orhodox long-run equilibrium will move from O o O. In his way, an increase in he cenral bank s inflaion arge insulaes he economy from he increased risk of a deflaionary spiral associaed wih a decrease in he naural real ineres rae. [Inser figure 7 abou here] We hus have he following proposiion: Proposiion 2: Alhough a decrease in he naural real ineres rae increases an economy s vulnerabiliy o a deflaionary spiral, he hrea would be neuralized if he cenral bank raises is arge inflaion rae by he same percenage-poin amoun as he decrease in he naural real ineres rae. To see he irony in his resul, consider an economy ha is safely ensconced a he orhodox long-run equilibrium wih Y = Y and π =π, and hen imagine seady decreases in ρ. As we saw in lemma 1, oupu and inflaion would be unaffeced and no danger would be apparen. Neverheless, he probabiliy ha a sudden shock would push he economy ino a deflaionary spiral would be rising all he while, in an invisible, suberranean way. Therefore, our analysis suggess ha decreases in he naural real ineres rae should be couneraced wih maching increases in he cenral bank s inflaion arge even when oupu is a full employmen and inflaion saisfies he cenral bank s inflaion arge. Anoher noable poin is ha a negaive naural real ineres rae makes i possible for an economy o be in a deflaionary spiral even when here is no deflaion! The naural real ineres rae could very well drop o negaive levels, as indeed our esimaes in our secion on moivaing evidence sugges i did in he pos-2009 Unied Saes. Therefore, ρ, he criical rae of inflaion a which a deflaionary spiral is riggered, could be posiive. In such a siuaion, inflaion could be posiive, and ye be low enough o rigger a deflaionary spiral! 17

19 As he esimaes in Laubach and Williams (2003) and our own esimaes in his aricle sugges, he naural real ineres rae has been falling in he Unied Saes since he 1980s. Bu nobody saw any reason for worry a he ime because oupu and inflaion were doing fine. Our lemma 1 explains why decreases in he naural real ineres rae did no affec he economy back hen: because he zero lower bound on he nominal ineres rae was no binding before December Bu our proposiion 1 argues ha he decrease in he naural real ineres rae back in he pre-december 2008 period was making he economy more and more vulnerable o a deflaionary spiral even hough here was no visible impac on oupu and inflaion a he ime. And, as our proposiion 2 argues, maching increases in he Fed s inflaion arge should have been implemened in he placid period before Pu anoher way, cenral banks should quesion he sraegy of keeping inflaion sable over he long run and consider insead a sraegy of keeping he nominal ineres rae sable over he long run. To see why, recall ha Y = Y and π =π in he sable orhodox long-run equilibrium. Then, from (2), he real ineres rae in his equilibrium is r = ρ, and, by (5), he nominal ineres rae is i = π = ρ+ π. In line wih our argumen ha decreases in ρ r + should be mached by equal increases in π, i follows ha ρ+ π should be kep consan. In oher words, insead of keeping inflaion sable over he long run, cenral banks should consider keeping he nominal ineres rae sable over he long run. These are new ideas in inermediae macroeconomics ha follow from a sandard model in a sandard exbook. And hese ideas can be augh o inermediae macroeconomics sudens using sandard graphical and algebraic echniques. Finally, le us briefly consider how an economy ha is in he orhodox long-run equilibrium would adjus over ime if and when he cenral bank raises is inflaion arge

20 Suppose a ime he DAD curve is OKD in figure 7, he DAS is DAS O, and he economy is herefore a he (sable) orhodox long-run equilibrium a O. Therefore, a, oupu is Y = Y and inflaion is π =π. A ime + 1, he DAS curve would sill be DAS O. (Recall from our subsecion on he DAS curve ha he heigh of he DAS curve a he full-employmen oupu is he previous period s inflaion, assuming no shocks.) However, he DAD curve a + 1 would be DAD 2, following he simulaneous decrease in ρ by percenage poins and an equal increase in π. Therefore, he shor-run equilibrium a + 1 would be a R, wih Y + 1 > Y = Y and π > π = π + 1. In subsequen periods, i can be shown, by following he reasoning by which we showed he movemen of he economy from R o R in figure 2 in our subsecion on he DAS curve, ha he economy will over ime move from R owards O, he new (sable) orhodox long-run equilibrium, along O K. In shor, when he cenral bank raises is inflaion arge, he economy will go immediaely from O o R and, hen gradually, from R o O. Oupu will rise above and hen reurn o he full-employmen level. Inflaion will rise gradually from π o π +. As for he ineres raes, i can be shown ha he real ineres rae will fall from r = ρ in he iniial orhodox long-run equilibrium, by more han percenage poins, o r + 1 < ρ and hen seadily increase o he new sable level of ρ. The nominal ineres rae will also fall from i = ρ+ π in he iniial orhodox long-run equilibrium and hen seadily increase o reurn o he unchanged level of ( ρ ) ( ) = + π + π + ρ. CONCLUSION 19

21 We have argued ha changes in he naural real ineres rae have imporan implicaions for he conduc of macroeconomic sabilizaion. We have made our argumen using he DAD DAS model in Mankiw (2013, chaper 15), modified o incorporae he zero lower bound on nominal ineres raes. One key prescripion for moneary policy ha emerges from our analysis is ha a pruden cenral bank should raise is arge inflaion rae when he naural real ineres rae decreases, irrespecive of wheher he ZLB is binding or no. A higher arge inflaion insulaes he economy from adverse demand shocks and favorable inflaion shocks ha could rigger a deflaionary spiral. We end by discussing wo issues ha have been menioned in recen debaes, bu are no discussed in our aricle. Firs, Summers (2013) has argued ha aemps o figh negaive real ineres raes by raising he arge inflaion will lead o asse price bubbles and relaed financial insabiliy. Krugman (2013) and Kocherlakoa (2014) have argued ha a separae se of policies called macroprudenial policies aimed direcly a he regulaion of financial markes are appropriae ways of addressing Summers concerns. Finally, boh Summers (2013) and Krugman (2013) have argued in favor of prolonged fiscal simulus as a response o secular sagnaion. While aware ha such fiscal simulus would require governmen borrowing and increasing levels of governmen deb, boh have argued ha, when real ineres raes are lower han he growh rae of real GDP, prolonged governmen borrowing may be possible wihou any increase in he deb-o-gdp raio, and should herefore be considered safe. While we accep he imporance of he issues summarized in he las wo paragraphs, we were unable o address hem formally in he model ha we have used in his aricle. Our goal hroughou has been o ake a opic ha is a he cener of curren macroeconomic debae how 20

22 o conduc moneary policy when real raes are decreasing and show ha a formal analysis of i can be made accessible o undergraduaes wihou requiring hem o learn a whole new model. 21

23 NOTES 1 See Summers (2014a) and he collecion of papers in Teulings and Baldwin (2014). 2 Curren ediions of prominen inermediae macroeconomics exbooks (e.g., Mankiw 2013; Blanchard and Johnson 2013; Jones 2011; and Mishkin 2011) all discuss he ZLB, and all make he poin ha expansionary fiscal policy works a he ZLB whereas expansionary moneary policy (a leas of he convenional kind) does no. However, hese exbooks do no explain he implicaions for macroeconomic sabilizaion of changes in he naural real ineres rae. Carlin and Soskice (2015, Secion 3.3.3) provide a deailed accoun of he deflaionary spiral. 3 The posiively-sloped segmen of he DAD curve capures he idea ha falling inflaion is a special nighmare a he ZLB. As nominal ineres raes canno be reduced any furher, any decline in inflaion means an increase in he real ineres rae, which in urn reduces aggregae demand and oupu. 4 Several cenral banks, including he Swiss Naional Bank, he European Cenral Bank, and he Danish Naional Bank have recenly adoped a negaive nominal ineres rae policy and as a resul, he zero lower bound is no a firm lower bound. Negaive nominal ineres raes are no an issue for our analysis, however, because he only assumpion needed for our heoreical resuls o carry hrough is ha nominal ineres raes are bounded from below. The floor value, wheher posiive or negaive, is inconsequenial. The negaive nominal ineres raes charged by banks reflec he cos of sorage and, because of compeiive pressure; his cos is unlikely o become much greaer han 1 percen of deposi, hereby creaing a floor on nominal ineres raes (Cecchei and Schoenholz 2014). 22

24 5 In he exreme scenario where he naural real ineres rae has fallen ino negaive erriory, he economy could experience a deflaionary spiral even hough curren inflaion is posiive (and presumably low). 6 Fiscal simulus also can reverse a deflaionary spiral, as shown in Bue and Roy (2014). Bu ax cus and/or increases in governmen purchases necessarily require increases in governmen borrowing, which may no always be an available opion, especially in a weak economy and especially if he governmen has already piled up so large a deb ha privae lenders would be leery of lending i more. Therefore, here is a need o avoid geing ino a deflaionary spiral in he firs place, and o avoid a deflaionary spiral omorrow, i s necessary o ensure ha oday s inflaion says above he negaive of he naural real ineres rae, which is he danger level. 7 Chadha and Perlman (2014) who analyze he Gibson Paradox noe ha, in he presence of an uncerain naural rae, he need o sabilize he banking secor s reserve raio can lead o persisen deviaions of he marke rae of ineres from is naural level and consequenly long-run swings in he price level. 8 Noe ha real ineres raes have been declining in he Unied Saes since he 1980s wih seady decreases in he esimaed naural real ineres rae. Bu cenral banks have no raised heir inflaion arges during ha period. (Even he Bank of Japan s recen move in his direcion was o increase is inflaion arge from 2 percen o a mere 3 percen.) This unwillingness needs o be re-examined in he ligh of our aricle. 9 In his address o he Naional Associaion for Business Economics and elsewhere, Summers (2014a) cauions ha even hough he zero lower is no echnically binding, low nominal and real ineres raes undermine financial sabiliy in various ways. The financial sabiliy channel is no presen in our aricle. 23

25 10 Eichengreen (2014) emphasizes four differen causes of secular sagnaion in his review essay: slower growh of echnological progress (Gordon 2014), sagnan aggregae demand (Summers 2014b; Krugman 2014), he failure of counries like he Unied Saes o inves in infrasrucure, educaion and raining, and finally arophy of skills caused by long-erm unemploymen and forgone on-he-job raining (Crafs 1989; Gordon and Krenn 2010). We believe ha i would be very hard, le alone desirable, o wrie an aricle, which encompasses all aspecs of secular sagnaion, so we focus on analyzing only one aspec of secular sagnaion. 11 A recen Inernaional Moneary Fund (IMF) repor (IMF 2014) cies hree main reasons for he decline in real raes since he mid-1980s: a) higher saving raes in emerging marke economies, b) greaer demand for safe asses reflecing he rapid reserve accumulaion of emerging marke economies as well as increased riskiness of equiy relaive o bonds, and c) a sharp and persisen decline in invesmen raes in advanced economies since he global financial crisis. All hree facors lead o greaer saving propensiies and lower invesmen propensiies. 12 Knu Wicksell (1898/1936, 102) offers he following definiion for he naural real ineres rae: There is a cerain rae of ineres on loans which is neural in respec o commodiy prices, and ends neiher o raise nor o lower hem. This is necessarily he same as he rae of ineres which would be deermined by supply and demand if no use were made of money and all lending were effeced in he form of real capial goods. More recenly, Kocherlakoa (2014) refers o he naural real ineres rae as he mandae-consisen real ineres rae. 13 Laubach and Williams (2003) use maximum likelihood o esimae changes in naural real ineres rae over ime, while he equilibrium resuls in Barsky, Jusiniano, and Melosi (2014) are derived from solving a dynamic uiliy-maximizaion problem. We believe ha he echnical and criical hinking skills needed o fully undersand hese wo modeling echniques are beyond he 24

26 skill se ha sudens possess when hey ake heir inermediae macroeconomics courses a mos colleges. 14 There are poenial economeric issues wih esimaing changes in he naural rae using equaion (1), such as co-inegraion of he variables, which could affec our esimaes for he naural rae of ineres. For he sake of space, however, and because our aricle proposes an innovaion for inermediae macroeconomics courses, we do no discuss economeric issues relaed o esimaing changes in he naural real ineres rae here. Raher, we leave his discussion for an upper elecive course on economerics or ime series analysis. 15 Noe ha an increase in he real ineres rae leads o a decrease in aggregae demand, as in he sandard IS curve. For he graphical analysis in he res of he aricle, we will make he simplifying assumpion: for all. 16 The negaive feedback loop beween oupu and inflaion is he mechanism ha leads o a deflaion-induced depression, as previously explained by Fisher (1933) and Krugman (1998). In normal imes, when nominal ineres raes are posiive, he cenral bank can afford o cu ineres raes following a negaive demand shock o provide shor-run simulus o he economy. When he zero lower bound is binding, however, cuing raes is no feasible and real ineres raes spike up as a resul of lower inflaion. Higher real ineres raes in urn depress he economy furher, which pu furher pressure on real raes, which depress he economy furher, and so on and so forh. 17 The DAS curve hrough R has no been drawn for simpliciy. 18 To see he logic behind he unsable naure of he deflaionary long-run equilibrium, D, and he deflaionary spiral, see pages 46 and 47 of Bue and Roy (2014). We saw above in figure 2 how an economy saring a R moves o R in he nex period and furher owards O in 25

27 subsequen periods. I is sraighforward o see he workings of he deflaionary spiral by repeaing ha analysis, bu saring a D insead of R. Bue and Roy (2014) emphasize he need o keep inflaion above ρ, and discuss how fiscal and moneary policy can be used (a) o sop inflaion from falling below ρ and hereby precipiaing a deflaionary spiral, and (b) o raise inflaion above ρ afer i has already fallen below ha level, hereby ending he deflaionary spiral. They show ha he only way ou of a deflaionary spiral once i has begun is fiscal simulus. Now, fiscal simulus usually involves a ax cu or an increase in governmen purchases or boh, and his usually requires an increase in governmen borrowing. And such borrowing, especially in condiions of economic weakness, may no be possible, especially for a governmen ha has already borrowed a lo and is, herefore, reaed warily by privae lenders. Tha is why i is crucial ha π < 1 ρ be avoided a all coss. 19 See figure 2 of Bue and Roy (2014) for a more deailed explanaion. 20 Recall our discussion of he sabiliy resul in Bue and Roy (2014): if inflaion falls below he negaive of he naural real ineres rae, he economy will hereafer be in a deflaionary spiral (if here are no furher shocks or parameer changes), wih oupu and inflaion falling repeaedly. 21 A deailed discussion of his dynamic adjusmen is provided in pages of Mankiw (2013). 26

28 REFERENCES Barsky, R., A. Jusiniano, and L. Melosi The naural rae of ineres and is usefulness for moneary policy. American Economic Review: Papers and Proceedings 104(5): Blanchard, O., and D. R. Johnson Macroeconomics. 6h ed. Upper Saddle River, NJ: Prenice Hall. Bue, S., and U. Roy A simple reamen of he liquidiy rap for inermediae macroeconomics courses. Journal of Economic Educaion 45(1): Carlin, W., and D. Soskice Macroeconomics: Insiuions, insabiliy, and he financial sysem. Oxford, UK: Oxford Universiy Press. Cecchei, S., and K. L. Schoenholz Money, banking, and financial markes. 4h ed. New York: McGraw-Hill. Chadha, J. S., and M. Perlman Was he Gibson paradox for real? A Wicksellian sudy of he relaionship beween ineres raes and prices. Financial Hisory Review 21(2): Crafs, N Long erm unemploymen and he wage equaion in Briain, Economica 56: Eggersson, G. B., and N. R. Mehrora A model of secular sagnaion. NBER Working Paper No Cambridge, MA: Naional Bureau of Economic Research. Eichengreen, B Secular sagnaion: A review of he issues. In Secular sagnaion: Facs, causes, and cures, ed. C. Teulings and R. Baldwin, London, UK: Cenre for Economic Policy Research, VoxEU.org. Fisher, I The deb-deflaion heory of grea depressions. Economerica 1(4): Gordon, R. J The demise of U.S. economic growh: Resaemen, rebual, and reflecions. NBER Working Paper No Cambridge, MA: Naional Bureau of Economic Research 27

29 Gordon, R. J., and R. Krenn The end of he grea depression : Policy Conribuions and Fiscal Mulipliers. NBER Working Paper No Cambridge, MA: Naional Bureau of Economic Research. Inernaional Moneary Fund (IMF) Perspecives on global real ineres raes. World economic oulook: Recovery srenghens, remains uneven, chaper 3. Washingon, DC: IMF. hp:// Jones, C. I Macroeconomics. 2nd ed. New York: W.W. Noron. King, M., and D. Low Measuring he world real ineres rae. NBER Working Paper No Cambridge, MA: Naional Bureau of Economic Research. Kocherlakoa, N Low real ineres raes. Speech a he Ninh Annual Finance Conference, Carroll School of Managemen, Boson College, Boson, Massachuses, 5 June hp:// Krugman, P I s baack! Japan s slump and he reurn of he liquidiy rap. Brookings Papers on Economic Aciviy 1998(2): Washingon, DC: The Brookings Insiuion Bubbles, regulaion, and secular sagnaion. The New York Times, November 16. hp://krugman.blogs.nyimes.com/2013/11/16/secular-sagnaion-coalmines-bubbles-and-larr y-summers/ Four observaions on secular sagnaion. In Secular sagnaion: Facs, causes, and cures, ed. C. Teulings and R. Baldwin, London, UK: Cenre for Economic Policy Research, VoxEU.org. Laubach, T., and J. C. Williams Measuring he naural rae of ineres. Review of Economics and Saisics 85(4): Mankiw, G. N Macroeconomics. 8h ed. Duffield, UK: Worh Publishers. 28

30 Mishkin, F.S Macroeconomics: Policy and pracice. 1s ed. Upper Saddle River, NJ: Prenice Hall. Summers, L. H Uniled speech. IMF foureenh annual research conference in honor of Sanley Fischer, Washingon, DC, November 8. hp://larrysummers.com/imf-foureenh-annual-research-conference-in-honor-of-sanley-fisc her a. Reflecions on he new secular sagnaion hypohesis. In Secular sagnaion: Facs, causes, and cures, ed. C. Teulings and R. Baldwin, London, UK: Cenre for Economic Policy Research, VoxEU.org b. U.S. economic prospecs: Secular sagnaion, hyseresis, and he zero lower bound. Business Economics 49: Teulings, C., and R. Baldwin Inroducion. In Secular sagnaion: Facs, causes, and cures, ed. C. Teulings and R. Baldwin, London, UK: Cenre for Economic Policy Research, VoxEU.org. Wicksell, K. 1898/1936. Ineres and prices. Trans. R. F. Kahn. London: Macmillan. 29

31 TABLE 1: The Naural Real Ineres Rae, Quarerly daa (sd error in parenhesis) Decade N (0.0041) ( ) (0.0013) ( ) (0.0030) ( ) (0.0104) ( ) All: (0.0021) ( )

32 FIGURE 1: The Kinked DAD Curve is shown here. I is assumed ha ε = 0 and Y = Y for all. The negaively-sloped segmen of he DAD curve saisfies equaion (7), he posiively-sloped segmen saisfies equaion (8), and he ZLB border saisfies equaion (6). 31

33 FIGURE 2: Poins O and D are long-run equilibria, while poin R is a shor-run equilibrium. Boh shocks are assumed zero. The DAS curves all saisfy equaion (10) bu for differen levels of he previous period s inflaion. Noe ha he heigh of a DAS curve a Y, he full-employmen oupu, is equal o he previous period s inflaion rae. 32

34 FIGURE 3: The effec of a fall in he naural rae ( ρ ) on oupu and inflaion is shown here. I is assumed ha ε = 0 and Y = Y for all. A falling ρ has no effec on he orhodox long-run equilibrium, O, bu moves he deflaionary long-run equilibrium from D o D, hereby bringing he deflaionary equilibrium closer o he orhodox equilibrium. The shor-run equilibrium moves from Z o D, implying decreases in boh oupu and inflaion. Had he naural real ineres rae fallen even slighly below ρ, a deflaionary spiral would have been iniiaed. 33

35 FIGURE 4: The Naural Real Ineres Rae and he Deflaionary Spiral Demand Shock: A fall in he naural real ineres rae makes he economy more vulnerable o a deflaionary spiral. I is assumed ha ν = 0 and Y = Y for all. When he naural real ineres rae decreases, he adverse demand shock ha can ip he economy ino a deflaionary spiral becomes smaller. When ρ =ρ1, a deflaionary spiral can be iniiaed by he DAD curve moving o he lef of DAD 2. On he oher hand when ρ = ρ2 < ρ1, he DAD curve would need o move only o he lef of DAD 3. 34

36 FIGURE 5: The Naural Real Ineres Rae and he Deflaionary Spiral Inflaion Shock: A fall in he naural real ineres rae makes he economy more vulnerable o a deflaionary spiral. I is assumed ha ε = 0 and Y = Y for all. When he naural real ineres rae decreases from ρ 1 o ρ 2, he deflaionary long-run equilibrium moves from D o D. When ρ =ρ 1, a favorable inflaion shock can iniiae a deflaionary spiral by moving he DAS curve o below DAS 2. On he oher hand when ρ = ρ2 < ρ1, he DAS curve would need o drop only o below DAS 3. 35

37 FIGURE 6: If ρ falls o π or below, he economy has eiher one neiher-sable-nor-unsable long-run equilibrium or no long-run equilibrium. I is assumed ha ε = ν = 0 and Y = Y for all. As he naural real ineres rae falls from ρ 0 o ρ <ρ 1 0 o ρ 2 o ρ 3, he DAD curve shifs from AKE o AK 1 E 1 o AOE 2 o AK 3 E 3. For each of he firs wo, here are wo long-run equilibria: O which is sable and he oher unsable. For AOE 2, O is he only long-run equilibrium, and i is neiher sable nor unsable. When he DAD curve is AK 3 E 3, here is no long-run equilibrium; he deflaionary spiral is he only possible oucome. 36

38 FIGURE 7: An effecive way o neuralize a fall in he naural real ineres rae is o raise he cenral bank s inflaion arge. I is assumed ha ε = ν = 0, Y = Y for all, and > 0. When he naural real ineres rae decreases from ρ o ρ, he deflaionary long-run equilibrium moves from D o D, coming closer o he sable long-run equilibrium, O, and increasing he chance ha a shock would push he economy ino a deflaionary spiral. However, an increase in he cenral bank s arge inflaion from π o π + mainains he disance beween he wo long-run equilibria and herefore does no allow he economy o become more vulnerable o a deflaionary spiral when he naural real ineres rae decreases. 37