Optimal Discretionary Monetary Policy in a Micro-Founded Model with a Zero Lower Bound on Nominal Interest Rate

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1 Opimal Discreionary Moneary Policy in a Micro-Founded Model wih a Zero Lower Bound on Nominal Ineres Rae Phuong V. Ngo a, a Deparmen of Economics, Cleveland Sae Universiy, 2121 Euclid Avenue, Cleveland, OH Absrac This paper invesigaes opimal discreionary moneary policy under he zero lower bound on he nominal ineres rae ZLB) in he case of a disored seady sae due o monopoly and axaion. Solving a fully nonlinear micro-founded FNL) model using a global mehod, I find ha he cenral bank in a more disored economy would cu he ineres rae less aggressively under a paricularly adverse demand shock. This occurs because inflaion and nominal ineres raes are higher on average, making he ZLB less likely o bind and causing he economy o escape from he ZLB sooner. However, he social planner would choose he opimal inflaion rae of approximaely zero. The resul emerges because he uncondiional benefi of avoiding he ZLB is no big enough o offse he cos of higher relaive price dispersion when inflaion is significanly posiive. In addiion, I show ha he convenional linear-quadraic LQ) mehod is inaccurae in he case of a sufficienly disored seady sae. JEL classificaion: C61, E31, E32, E52. Keywords: opimal discreionary moneary policy, ZLB, disored seady sae, opimal inflaion rae, Calvo price adjusmens, nonlinear mehod Corresponding auhor. Tel address: p.ngo@csuohio.edu Phuong V. Ngo) Preprin submied o Elsevier April 24, 2014

2 1. Inroducion The focus of researchers concerned wih opimal moneary policy under he ZLB has been he case of a non-disored seady sae, where he overall economic disorion due o monopoly and axaion is assumed o be zero. Specifically, governmen subsidies exis o fully offse monopolisic disorion so ha he seady sae oupu is no disored from is socially efficien level. Hence, we can simplify a fully nonlinear micro-founded problem of opimal discreionary moneary policy using he LQ approach developed by Woodford 2001, 2003) and can avoid compuaional difficuly. This paper aims a filling he hole in he ZLB lieraure by invesigaing opimal discreionary moneary policy under he ZLB in he case of a disored seady sae due o posiive overall economic disorion. To his end, I solve a FNL micro-founded model using a global mehod. Also, I use he LQ mehod o simplify he FNL model, which I solve using he same mehod. I hen provide a comparison beween he FNL and LQ models. Sudying he case of a disored seady sae brings he ZLB lieraure closer o realiy. McGraan 1994) repors ha labor income axes range from 10 40%, while Diewer and Fox 2008) esimae ha monopolisic markups in some main indusries range from 11 44%. As a resul, he overall economic disorion ranges from 20 60%. This informaion migh influence privae expecaions, which, in urn, would affec opimal policy before, during and afer he ZLB period. 1 1 I is well-known in he lieraure of inflaion bias ha he greaer he overall economic disorion, he higher he average inflaion under discreion and, as a resul, he higher he nominal ineres rae, see Woodford 2003). Wheher in realiy we observe he kind of inflaion bias ha emerges in he model under discreionary policy in he case of a disored seady sae is sill debaable and beyond he scope of his paper. 2

3 Solving FNL models also helps us o answer he quesion saed in Adam and Billi 2007): o wha exen does he full nonlineariy affec he opimal moneary policy under discreion in he presence of he ZLB? In addiion, we can sudy he role of relaive price dispersion as an endogenous sae variable, which is eliminaed in he LQ framework due o he linear approximaion. 2 I obain four ses of main findings. Firs, under a paricularly adverse shock driving he economy near he ZLB, he cenral bank in an economy wih a larger economic disorion would cu he ineres rae less aggressively. The inuiion is simply ha, in his economy, inflaion and nominal ineres rae are higher on average. When he nominal ineres rae is near he ZLB, a paricularly adverse demand shock migh have occurred. Given he mean-revering naure of shocks, he condiional probabiliy ha anoher adverse shock occurs and pushes he economy ino he liquidiy rap wih binding ZLB is very small. Furhermore, even when his shock occurs and he ZLB binds, he oupu losses and reducion in inflaion are smaller han hey would be in an economy wih a smaller economic disorion. Therefore, downward pressure on he condiional expeced inflaion is smaller and he cenral bank cus he nominal ineres rae less aggressively. Second, wih a larger overall economic disorion, inflaion and ineres raes are higher on average, resuling in a smaller probabiliy of reaching he ZLB. However, he social planner would choose he opimal inflaion rae of approximaely zero, corresponding o very small overall economic disorion. This occurs because he uncondiional benefi from avoiding he ZLB is no big enough o offse he cos of higher relaive price dispersion when inflaion is high. In sum, he uncondiional 2 Alvarez e al. 2011) find ha relaive price variance is significanly posiive when inflaion is high, while Zandweghe and Wolman 2010) show ha iniial relaive price dispersion could affec moneary policy. So sudying he role of relaive price dispersion is ineresing. 3

4 expeced welfare is maximal when he average long-run inflaion is around zero. Third, when he iniial relaive price disorion is greaer han he seady sae value, he cenral bank ends o pursue a higher nominal ineres rae, making he ZLB less likely o bind. The inuiion is ha he relaive price dispersion is an inefficiency wedge: when he relaive price dispersion is high, he cenral bank would like o reduce i by ighening he moneary policy and, as a resul, lowers he fronloading behavior by firms in seing heir prices, leading o a smaller curren relaive price dispersion. This resul is ineresing and can no be found using he LQ mehod because he change in relaive price dispersion is always zero. Finally, he FNL model and he LQ model produce differen resuls if here is a paricularly adverse shock ha makes he ZLB binding. When he ZLB binds, he cenral bank canno sabilize oupu and he price level, making he relaive price dispersion disan from he seady sae. While he impac of he relaive price dispersion as an endogenous sae variable in he FNL model is significan, i is always zero and has no role in he LQ model due o he firs order approximaion. However, he difference beween he FNL and LQ model is no significan in he case of a non-disored seady sae. When he overall economic disorion is large, he wo mehods produce very differen resuls, especially when he ZLB binds. The approximaed inflaion and ineres rae in he LQ model are subsanially smaller han he rue values derived using he FNL model. Consequenly, given he ZLB binds in boh models, he oupu losses in he FNL model are significanly smaller han hose in he LQ model. In addiion, he ineres rae cu in he FNL model is less aggressive under a shock driving he ineres rae near he ZLB. The relaed lieraure on opimal moneary policy under he ZLB was inspired by seminal work by Krugman 1998), which exensively discusses causes and conse- 4

5 quences of he ZLB in a series of simple wo-period perfec-foresigh models. Since hen, exensive research relaed o he ZLB has been implemened, including Eggersson and Woodford 2003), Jung e al. 2005), Adam and Billi 2006), Nakov 2008), Levin e al. 2010), Bodensein e al. 2010), Eggersson and Krugman 2010), and Werning 2011). The common feaure of hese papers is ha hey focus on he case of non-disored seady sae and use he LQ mehod. The papers closes o mine are Adam and Billi 2007), and Anderson e al. 2010). Like my paper, Adam and Billi 2007) use a global mehod o solve an opimal discreionary moneary policy problem ha allows for an occasionally-binding ZLB. However, because hey use he LQ approach, he only nonlineariy in heir paper is he ZLB. This paper exends he work of Adam and Billi 2007) by considering a fully nonlinear model. In addiion, his paper sudies he implicaions of posiive overall economic disorion on discreionary policy and opimal inflaion rae in he presence of he ZLB. Anderson e al. 2010) invesigae he size of inflaionary biases under discreion in he presence of overall economic disorion using nonlinear mehods. However, in heir model, he nominal ineres rae can be adjused freely because he ZLB is no imposed. Hence, he average long-run inflaion is he same as he deerminisic seady sae inflaion. This paper exends heir work by considering he ZLB, a very imporan consrain faced by policymakers. There are hree recen working papers sudying he ZLB using fully nonlinear mehods. Nakaa 2011) sudies opimal fiscal and moneary policy in a nonlinear sicky price model of he Roemberg-ype insead of he Calvo-ype as in my model. I choose o use he Calvo-ype price adjusmens so ha I can examine relaive price dispersion as an endogenous variable and compare my resuls direcly o he resuls in he previous lieraure. I sudy he role of economic disorion, while Nakaa 5

6 2011) focuses on fiscal policy. Fernandez-Villaverde e al. 2012) sudy he ZLB in a fully nonlinear model using he collocaion mehod associaed wih Smolyak nodes. Judd e al. 2011) solve a fully nonlinear New Keynesian model wih he ZLB using a cluser-grid algorihm. The moneary policy in hese wo papers is a Taylor rule. They do no consider opimal discreionary moneary policy. The remainder of his paper is organized as follows. Secion 2 presens he srucure of he economy, and Secion 3 describes he discreionary moneary policy problem faced by a cenral bank and explains briefly soluion mehods. I calibrae key parameers and repor main resuls in Secion 4, and conclude in Secion 5. Numerical algorihms and mahemaical derivaions are presened in Appendices. 2. Model The economic srucure in his paper presens wo key New Keynesian feaures, such as in Roemberg and Woodford 1997) and Yun 1996). Specifically, inermediae goods producers are monopolisic compeiors. In addiion, hey rese heir prices infrequenly à la Calvo 1983) Household The represenaive household maximizes his oal expeced discouned flow uiliies: { C 1 γ χn1+η ) max E 1 γ η subjec o he budge consrain: j=1 { j 1 ) )}} C 1 γ χn1+η +j β +k 1 γ +j 1 + η k=0 ) 1 + i 1 C + B = 1 τ w ) w N + B π D i)di + T 1)

7 where C, N are composie consumpion and oal labor; B, D, T denoe real bonds, dividends, and lump sum ransfers; i, π are he ne nominal ineres rae and he inflaion rae, respecively; w is he real wage; τ w is he labor income ax rae; γ, η, χ are he risk aversion parameer, he inverse elasiciy of labor wih respec o wages, and he seady sae labor deermining parameer; β is he sochasic subjecive discoun facor or preference ha follows an AR1) process wih a seady sae value β : ln β +1 ) = 1 ρβ ) ln β ) + ρ β ln β ) + ε β,+1, where β is given. 2) The opimal choices of he household mus saisfy he following condiions: [ ) γ ) ] C i E β = 1 3) C 1 + π +1 χn η C γ = 1 τ w ) w 4) The firs condiion shows he marginal iner-emporal rade-off beween oday s and omorrow s consumpion. The second condiion is he marginal rade-off beween working and consuming. The sochasic preference is a reduced form of more realisic forces ha can drive he nominal ineres rae o he ZLB. From he Euler equaion, an increase in he discoun facor causes he nominal ineres rae o fall, given privae expecaions and households desire o smooh heir consumpion. 3 3 The convenional echnology shock is no able o cause he ZLB o bind realisically. The reason is ha we need a very big posiive echnology shock o generae massive savings ha can drive he nominal ineres rae o he ZLB. We did no observe his ype of shock before he onse of he las 7

8 To produce he composie consumpion goods, C, he household buys and aggregaes a variey of inermediae goods using a CES echnology. His cos-minimizaion problem is given below. min P i) C i) di s.. C = 0 C i) ɛ 1 ɛ ) ɛ ɛ 1 di 5) where C i) is he amoun of inermediae goods i [0, 1] and ε is he elasiciy of subsiuion among inermediae goods. The opimal condiion gives rise o he demand for he inermediae goods i, C i), and he aggregae price level, P, below : ) ɛ P i) C i) = C 6) P P = ) 1 P i) 1 ɛ 1 ɛ di 7) crisis, see Amano and Shukayev 2012). Guerrieri and Lorenzoni 2011) model deb limi and household heerogeneiy in labor produciviy. They show ha an exogenous decline in he deb limi acs as an increase in he subjecive discoun facor in my represenaive agen model. The decline in he deb limi causes fuure consumpion o be more volaile because wih a lower deb limi households will be less able o insure heir consumpion agains risks. Therefore, he savers will save more and he borrowers will borrow less due o precauionary savings. Eggersson and Krugman 2010) also model household deb limi and deleveraging as a key facor o drive he nominal ineres rae o he ZLB. In heir model, an iniial shock o he deb limi causes borrowers o deleverage by cuing back heir consumpion, resuling in a decrease in he price level. This deflaion pus more pressure on he real deb he borrowers have o pay back now, leading o furher deleveraging and a sharper decline in he nominal ineres rae. Ngo 2013) exends Eggersson and Krugman 2010) by endogenizing he deb limi. He sudies he ineracion beween he ZLB and he endogenous deb limi in explaining he collapse of he housing marke and he Grea Recession. Hall 2011) models excessive capial sock and a sharp decline in capial uilizaion as he reason for he nominal ineres rae o be pinned a he ZLB. Curdia and Woodford 2009) model a shock o he wedge beween deposi and lending raes as a driving force. 8

9 2.2. Inermediae goods producers There is a mass one of inermediae goods producers ha are monopolisic compeiors. In each period, a firm keeps is previous price wih probabiliy θ and reses is price wih probabiliy 1 θ). Given is price P i) and demand Y i), he firm i chooses labor ha min {w N i)} s.. Y i) = N i) 8) Le ϕ i) be he Lagrange muliplier wih respec o he producion. The firs order condiion gives he same marginal cos o all firms, ϕ : ϕ = ϕ i) = w 9) Max P i) Ei Whenever a firm has a chance o rese is price, i chooses he new price o solve: { [P ] i) ϕ Y i) + P { j 1 ) C+j ) γ [ ] }} θ j P i) β +k ϕ +j Y +j i) j=1 subjec o is demand in equaion 6). k=0 C P +j 10) The opimal relaive price, P i)/p, is he same for all firms ha are able o rese heir prices oday: P i) P = p = ε ) E {C γ Y ϕ + E { C γ Y + { j 1 ) θ j β +k j=1 k=0 { j 1 ) θ j β +k j=1 k=0 C γ +j C γ +j P+j P ) ε Y+j ϕ +j ) P+j Y+j P }} }} 11) 9

10 Wih some manipulaion, we can rewrie he opimal pricing rule as below: where S, F are wrien in he following recursive forms: p = S F 12) S = ) ε [ ] C γ Y ϕ ε 1 + θe β Π ε +1S +1 13) [ ] F = C γ Y + θe β Π +1F +1 14) and Π = 1 + π) is gross inflaion. Combining 13) wih 4) and 9), we obain: S = χc N η 1 Φ) + θe [ ] β Π ε +1S +1 15) where Φ = 1 1 τ w ) 1 ε 1) 16) and Φ is called overall economic disorion. I will discuss his meric in a secion below Aggregae condiions Aggregae oupu saisfies: Y = N 17) where is called he relaive price dispersion and is defined as: ) ε P i) = di 18) 10 P

11 or in a recursive form: = θπ ε θ) p ) ε 19) I wrie he price level 7) in a recursive form and divide boh sides by P o obain he opimal relaive price: 1 p θπ = 1 θ ) 1 1 ε 20) Plugging his opimal relaive price in he relaive price dispersion equaion 19) we obain: = 1 θ) 2.4. Overall economic disorion 1 θπ ) ε + θπ ε 1 θ 1 21) In his secion, I discuss he overall economic disorion, which is defined as in equaion 16). To undersand more abou he meaning of his noaion, le us consider an economy wih flexible price. In his economy, he marginal cos, ϕ, equals he inverse of markup or 1 ε 1 )). From equaion 4), 9), and 17) we ) compue he equilibrium flexible-price oupu and he equilibrium efficien oupu Y ) as follows: Y f Y f = N 1 Φ) 1 η+γ 22) Y = N 23) where N is he long-run efficien oupu/labor. The percenage deviaion of he flexible-price oupu from he efficien oupu equals: Y f Y Y ) η + γ Φ

12 The larger he overall economic disorion, he smaller he flexible-price oupu relaive o he efficien oupu. I is imporan o noe ha while he overall economic disorion is zero, i does no mean here is no any ype of economic disorions. Insead, i means ha we can aain he efficien oupu level by designaing a labor income subsidy o fully offse he monopoly power, given no price sickiness. I is also imporan o emphasize ha when he overall economic disorion is large or he inverse labor elasiciy and risk aversion are small, he flexible price oupu is far below he efficien oupu level. Under discreion, he cenral bank ends o creae posiive inflaion o ry o aain he efficien oupu. In equilibrium, he greaer he overall economic disorion, he smaller he flexible-price oupu relaive o he efficien oupu, and he greaer he inflaion he cenral bank ends o creae. 3. Opimal discreionary policy problem under he ZLB In he discreionary framework, he cenral bank is able o re-opimize is problem every period, and he economic agens ake his informaion ino accoun when hey form heir expecaions. Given his common knowledge, he cenral bank s problem is o maximize he sociey s or he represenaive household s) expeced presen-discouned lifeime uiliy subjec o he opimaliy condiions of he economic agens, he aggregae condiions, he law of moion for he sae variables, and he explici ZLB on he nominal ineres rae. The FNL model feaures an endogenous sae variable - he relaive price dispersion. As a resul, he cenral bank akes ino accoun how oday s price dispersion may affec he agens expecaions of fuure inflaion, consumpion, ec. 4 4 The cenral bank can manipulae privae expecaions by commiing o a pah of curren and fuure inflaion, ineres raes, ec. However, he ime-inconsisency issue arises, and he commimen may no be credible. 12

13 The problem can be saed in he form of a Bellman equaion: { C 1 γ χn1+η V 1, β ) = max {i,c,n,s,f,π, } 1 γ 1 + η + β E V } ), β +1 24) subjec o i) Households and firms opimaliy condiions, and aggregae condiions. ii) Law of moion for sae variables. iii) ZLB on he nominal ineres rae i 0). iv) No commimen o fuure policy ha is made in he pas. 5 The soluion of he above nonlinear sysem is called he Markovian invarian policy funcion of he sae, s = 1, β ), where 1 is an endogenous sae and β is an exogenous one. In he paper, I solve he above FNL model using a global mehod called he collocaion mehod. Firs, I use equidisan collocaion nodes o solve he model and find ou policy funcion. Based on his informaion, I invesigae poenial kinks. I hen redisribue he nodes by clusering hem around hese poenial kinks and resolve he model. I employ he ime-ieraion mehod. A each collocaion node, I solve a complemenariy problem using he Newon mehod and he semi-smooh roo-finding algorihm as described in Miranda and Fackler 2002). I also provide an analyical Jacobian marix compued from he approximaing funcions. 6 Moreover, I wrie my code using a parallel compuing mehod ha allows me o spli up a large number of collocaion nodes ino smaller groups ha hen are assigned o differen processors o solve simulaneously. These compuaional characerisics help o significanly increase he rae of convergence and make he 5 See Appendix A for how o wrie down he problem in deail. 6 See Appendix C for he analyical Jacobian marix. 13

14 soluion mehod very reliable. 7 I also use he LQ approach, as described in Woodford 2003), o simplify he FNL model which I hen solve using he same mehod. 8 Specifically, according o he LQ approach, he cenral bank s objecive funcion is quadraically approximaed and all he consrains and law of moion are log)linearly approximaed around he seady sae values associaed wih zero inflaion. The endogenous variables in he LQ framework are defined as below: π = log 1 + π ) log1) 25) î = log 1 + i ) log1/β) 26) x = logy ) logy ) ) ) logy f ) logy f ) 27) where β is he seady sae discoun facor; Y and Y f are he seady sae sickyprice and flexible-price oupus when he overall economic disorion presens; x is he oupu gap. Firs, I solve for he policy funcion in he LQ framework, including î, π,and x. Then, I back ou he policy funcion for i, π, Y using equaions 25) 27), which I call LQ resuls. 7 See Appendix B for how o solve he model in deail, including he error repored from checking he soluion. 8 See Appendix D for he simplified LQ model. Noe ha i) he LQ model wih he ZLB is also a nonlinear model and I have o use he global mehod o solve i; ii) he LQ approach is acually no applicable when he overall economic disorion, Φ, is large. 14

15 4. Resuls 4.1. Parameer calibraion I calibrae he seady sae quarerly ime discoun facor, β, o be 0.993, corresponding o a real ineres rae of 2.8% per year. The relaive risk aversion γ) is 4, as in Nakov 2008), and he inverse elasiciy of labor wih respec o wages η) is 1. The monopoly power parameer ε) is calibraed o be 10, corresponding o a 11% markup ha is in he range found by Diewer and Fox 2008). The probabiliy ha a firm keeps is price unchanged each quarer, θ, is chosen o be 0.75 so ha firms keep heir prices for 4 quarers on average. This value is commonly used in he lieraure, such as Anderson e al. 2010). Table 1. Parameerizaion Symbol Descripion Values β Quarerly discoun facor γ Consan relaive risk aversion 4 η Inverse elasiciy of labor wih respec o real wage 1 ε Monopoly power 10 θ Probabiliy ha a firm keeps is price unchanged each quarer 0.75 σ β Sandard deviaion of preference preference shocks percen) 0.4 ρ β AR-coefficien of preference shocks 0.8 Φ Overall economic disorion 0; 0.20 χ Parameer associaed wih he disuiliy of labor 1 I calibrae he persisence of he preference shock o be 0.8, which is consisen wih he persisence of he naural rae of ineres rae as in Adam and Billi 2007). The difficuly is deermining how o calibrae he variance of he preference shock. In 15

16 his paper, I calibrae his parameer o be 0.42% per quarer ha enables he model o generae he uncondiional probabiliy of hiing he ZLB of around 6%. This value is a lile small compared wih he fac ha he nominal ineres in he U.S. has been a he ZLB since December 2008 and ha i is projeced o be a he bound unil mid However, his value is sill he upper value of he range 5% 6% found in he empirical sudies before he las financial crisis, as in Fernandez-Villaverde e al. 2012). The overall economic disorion, Φ, is calibraed o be eiher 0 or The firs value corresponds o he well-known non-disored seady sae. The second value corresponds o he case where labor income ax is se o be 11%. Alhough, he ax rae is conservaive, i is sill in he range found by Diewer and Fox 2008). As I show below, a higher value of Φ only makes he LQ model more inaccurae Seady sae The seady sae values depend on he overall economic disorion Φ). Wih a labor income subsidy designed o fully offse he monopolisic disorion, he overall economic disorion is zero. In his case, he seady sae inflaion and gross ineres rae are 0 and 1/β respecively. However, in he case of posiive overall economic disorion, i is difficul o compue he seady sae values. Figure 1 shows opimal gross inflaion and relaive price dispersion as a funcion of iniial relaive price dispersion, given he seady sae preference of β. The seady sae relaive price dispersion is he value ha equals he iniial relaive price dispersion. In his example, hey are or abou 1.2% annually). Using his value, we can compue he seady sae gross inflaion o be or 2.4% per year). See he Appendix D for he formula ha can be used o compue he seady sae inflaion using he LQ mehod. 16

17 o Gross inflaion Π ) Gross relaive price disperion Δ ) ,1.006) Δ =Δ 1 = Previous relaive price dispersion Δ 1 ) Figure 1: Inflaion and curren relaive price dispersion. Noe ha preference is a he seady sae β SS =β = 0.993) and overall economic disorion Φ) is To illusrae he impac of nonlineariy and overall economic disorion, I compue he deerminisic seady sae inflaion in boh LQ and FNL models wih respec o differen values of he overall economic disorion. 9 The resuls are presened in Figure 2 and are similar o hose of Anderson e al. 2010). Two ineresing feaures in Figure 2 are worh being addressed. Firs, here is a posiive relaionship beween he seady sae inflaion and he size of overall 9 Noe ha he monopolisic disorion is always 0.1. For each value of overall economic disorion Φ), we can compue a corresponding value of income ax using equaion 16). For example, if Φ = 0, τ w = 11%; if Φ = 0.2, τ ω = 11% 17

18 4.5 4 FNL LQ % per year Size of overall economic disorion Φ) Figure 2: Seady sae inflaion. economic disorion in boh FNL and LQ models. Inuiively, he larger he size of he overall disorion, he higher he marginal benefi of inflaion: a higher inflaion rae can help o lower he real markup, simulaing oupu and employmen. However, his comes a a cos. In fac, a higher inflaion rae induces a firm o se a higher price when i has a chance o do so. This is because he firm knows ha i may no be able o adjus is price in he fuure and ha a higher inflaion rae will erode is relaive price and profi. This fron-loading behavior in price seing causes he dispersion in relaive prices o increase and lower he aggregae oupu, as in equaion 17). 18

19 Second and more imporanly, Figure 2 shows ha he seady sae ineres rae in he rue FNL) model is a convex funcion wih respec o he size of overall economic disorion. However, he seady sae ineres raes in he LQ model are only he firs order approximaion of he rue value around Φ = 0. Due o he convexiy of he rue funcion, he LQ model always underesimaes he rue seady sae value. 10 When he size of he overall economic disorion increases, he underesimaion increases a an increasing rae. Surprisingly, i is no difficul o prove ha, under commimen, he seady sae inflaion rae in boh LQ and FNL models is zero regardless of he size of overall economic disorion. 11 Therefore, he seady sae ineres rae is always equal o he seady sae real ineres rae, which is 2.8% annually. Wih inflaion and ineres raes being smaller han he rue ones, he ZLB is more likely o be reached in he LQ model. Therefore, given preference shocks ha cause he ZLB o bind in boh models, he LQ mehod generaes more sizeable oupu losses han he FNL mehod. We will see his more clearly in he nex secion Opimal oupu, inflaion, and ineres rae policy When here is a posiive preference shock, households value heir fuure consumpion more. In oher words, hey are more paien so hey end o save more and consume less oday, puing downward pressure on oupu and he price level. To resore consumpion and oupu, we need a lower real rae. If he cenral bank was no resrained by he ZLB, i could adjus he nominal ineres rae so ha he acual real ineres rae is he same as he naural real rae. However, because he 10 While he degree of convexiy depends negaively on he curvaure of he labor supply η) and he risk aversion parameer γ), i is posiively relaed o he price sickiness. 11 See Schmi-Grohe and Uribe 2010) for he proof. 19

20 ZLB is allowed, a big posiive preference shock causes he ZLB o bind. As a resul, he acual real rae will be larger han he naural real rae because he nominal ineres rae canno be negaive, resuling in a sizable oupu loss. For comparison, I experimen wih differen levels of overall economic disorion and relaive price dispersion The case of a non-disored seady sae In his efficien economy, here exiss a labor income subsidy designed o fully offse he monopoly power so he overall economic disorion Φ) is zero. Figure 3 shows he policy funcion a each sae of he preference ha is presened as a percenage deviaion from he seady sae β, given he iniial relaive price dispersion a he seady sae. The resuls are annualized. The solid blue lines represen he resuls from he FNL model wih he ZLB, while he dashed green lines represen hose from he FNL model wihou he ZLB. The do-dashed red lines show he resuls from he LQ model wih he ZLB. The resuls from he FNL model have he same characerisics as hose in Adam and Billi 2007) and Nakov 2008). Firs, in he absence of he ZLB, he cenral bank can achieve he arge efficien oupu and price sabilizaion by adjusing he nominal ineres rae as much as possible, even o is being negaive. Second, when he ZLB presens, he cenral bank canno sabilize oupu and inflaion under shocks ha cause he ZLB o bind. Third, he cenral bank cus he nominal ineres rae more aggressively, especially when he economy is near he ZLB, in he model wih he sochasic ZLB han in he model wihou he ZLB or wih perfec foresigh binding ZLB. The aggressiveness occurs due o he risk of falling ino he liquidiy rap associaed wih deflaion ha forces he cenral bank o cu he ineres rae 20

21 % per year A. Ineres rae No ZLB LQ FLN B. Oupu deviaion C. Inflaion 0.2 D. Relaive price dispersion 0.15 % per year Preference shock % per year) Preference shock % per year) Figure 3: Opimal policy in he economy wih a non-disored seady sae Φ = 0). Gross iniial relaive price dispersion is a he seady sae 1 = SS = 1). The preference shock is annualized percenage deviaion from he seady sae β SS =β = 0.993). more han i would be wihou he risk. 12 Adam and Billi 2007) ask wheher a fully-nonlinear model migh generae a policy funcion differen from heirs. By solving boh he FNL and LQ models, I am able o answer ha quesion. 13 The do-dashed red lines in Figure 3 presen he 12 See Adam and Billi 2007) and Nakov 2008) for more deailed explanaion. 13 Fernandez-Villaverde e al. 2012) claim ha resuls from a fully nonlinear model are very differen from hose in he LQ model. However, hey model moneary policy using a Taylor rule wih an inflaion arge of 2% insead of zero inflaion arge as in his par. Judd e al. 2011) compare he resuls from heir nonlinear mehod wih hose from he perurbaion mehod, no wih he LQ mehod. 21

22 policy funcion using he LQ framework. Wihou a paricularly posiive shock, he opimal policy is very similar in he wo models. The finding is robus o he parameers and he naure of shocks. The reason is ha, when Φ = 0, he seady sae inflaion and ineres raes are he same in boh models regardless of he parameers and he naure of shocks. Also, he relaive price dispersion is zero in he wo models. However, when a paricularly posiive shock occurs and he ZLB binds, he cenral bank canno sabilize he price level, so he curren relaive price dispersion increases from he seady sae and sars playing is role, as a negaive echnology shock, in he FNL model. In his case, he FNL model generaes more oupu loss and more decline in he price level han does he LQ model, which keeps he relaive price dispersion consan regardless of he sae of he economy, as in Panel D of Figure 3. However, he differences beween he FNL and LQ models are no significan. To invesigae he role of iniial relaive price dispersion under he ZLB, I plo he opimal policy using differen values of iniial relaive price dispersion, as in Figure 4. The solid blue lines show he policy funcion when he iniial relaive price dispersion is 0% annually, while he dash-doed red lines and dashed green lines show he policy funcion when he iniial relaive price dispersion is a 3.5% and 10% respecively. Noe ha he iniial relaive price dispersion 1 ) can be very high due o a change in he ax regime, alhough he responses of curren relaive price dispersion ) are relaively small under preference shocks. For example, when he labor income ax changes from τ w = 16.67% iniially o τ w = 11.0% as in he case of a non-disored seady sae, he iniial relaive price dispersion is 5.2% per year. The larger he ax change, he greaer he inial relaive price dispersion. As shown in Figure 4, when he iniial relaive price dispersion is 3.5%, he nominal ineres rae is abou 0.7% higher han i would oherwise be if he iniial 22

23 % per year % per year A. Ineres rae Δ 1 =0 %) Δ 1 =3.5 Δ 1 = C. Inflaion Preference shock % per year) B. Oupu deviaion D. Relaive price dispersion Δ) Preference shock % per year) 4 5 Figure 4: Opimal policy in he economy wih a non-disored seady sae Φ = 0), wih differen values of iniial relaive price dispersion, in he FNL model. The preference shock is annualized percenage deviaion from he seady sae β SS =β = 0.993). relaive price dispersion is zero. The curren relaive price dispersion is abou 2.5%, which is 1.0% lower han he iniial value. The oupu loss is abou 1.0%, due o high dispersion of relaive prices. The economy experiences deflaion. Inuiively, when he iniial relaive price dispersion is large, he inefficiency wedge is high, and he cenral bank would implemen highly conracionary moneary policy by pursuing higher nominal ineres raes on average han i would oherwise. By doing so, he cenral bank can lower he fron-loading price seing behavior of firms and, as a resul, lower he curren relaive price dispersion. In his case, he moneary 23

24 policy is so conracionary ha i creaes oupu losses and disinflaion or deflaion in his case). Ineresingly, due o high nominal ineres raes on average, he ZLB is less likely o bind. For example, in he case of 3.5% iniial relaive price dispersion, he ZLB binds only when a shock wih a magniude of a leas 2.5% occurs, compared wih 2% in he zero iniial relaive price dispersion The case of a disored seady sae As explained in he calibraion secion, in his case, he overall economic disorion Φ) in his economy is This means ha a he seady sae, he economy produces much less han he efficien oupu level. Wih his overall economic disorion, he cenral bank no longer arges zero inflaion. The deerminisic seady sae inflaion is abou 2.4% ha is associaed wih he seady sae ineres rae and price dispersion of 5.2% and 1.2%, respecively. Figure 5 shows he opimal policy where he relaive price dispersion is se a he seady sae of or 1.2% annually). The solid blue lines show he policy funcion in he FNL model. We can easily see ha on average he cenral bank pursues higher inflaion and nominal ineres raes in he FNL model han in he LQ model. Wihou a paricularly posiive preference shock, he cenral bank implemens he inflaion rae and ineres rae of around 2.4% and 5.2% respecively. The higher he average inflaion and ineres raes, he less likely he ZLB will bind. In he firs case of zero overall economic disorion, he seady sae inflaion and ineres raes are 0 and 2.8% respecively. A posiive preference shock wih a magniude of 1.2 sandard deviaions or 2% per year), which reduces he naural real rae by 2% annually, can cause he ZLB o bind in he firs case. However, in his case of a disored seady sae, wih 2.4% inflaion arge and 5.2% seady sae ineres rae, a much more severe shock is required o drive he economy o he ZLB 24

25 4 A. Ineres rae 0.5 C. Oupu deviaion % per year No bound LQ No bound FLN LQ FLN % per year B. Inflaion Preference shock % per year) D. Relaive price dispersion Preference shock % per year) Figure 5: Opimal policy in he economy wih a disored seady sae Φ = 0.20). Noe ha he iniial relaive price dispersion a he deerminisic seady sae 1 = SS = , or 1.2% per year). The preference shock is annualized percenage deviaion from he seady sae β SS =β = 0.993). - abou 3 sandard deviaions or reducing he naural real rae by 5% annually). More imporanly, when he economy is near he ZLB, he cenral bank in he economy wih zero overall economic disorion cus ineres rae more aggressively han in he economy wih posiive disorion. The inuiion is ha, a large overall economic disorion in his paper incenivizes he cenral bank o inflae he economy and, as a resul, generae higher inflaion and ineres raes on average. When he economy is near he ZLB, a paricularly adverse preference shock migh have occurred. Because he preference process is mean-revering, i is raher unlikely 25

26 ha anoher adverse shock will happen and push he economy ino he liquidiy rap wih large oupu losses and low inflaion. As a resul, he downward pressure on he condiional expeced inflaion is very small, generaing small pressure on furher lowering he nominal ineres rae. Therefore, an ineres rae cu is no as big as i would be in an economy wih a smaller overall economic disorion. The logic also helps explain why inflaionary biases are reduced subsanially in he case of a sufficienly-disored seady sae. Nakov 2008) sudies he case of a non-disored seady sae, and argues ha, under discreionary opimizaion, he presence of he ZLB biases privae secor expecaions of inflaion and he oupu gap downwards, resuling in average inflaion below he arge. Depending on he specific parameerizaion his negaive bias can be quie large. However, as explained above, he downward pressure on boh condiional and uncondiional expeced inflaion is significanly smaller in his case of a disored seady sae, reducing he inflaionary bias subsanially. The lieraure concerning he case of a non-disored seady sae, including Adam and Billi 2007), indicaes ha even when he economy escapes from a liquidiy rap or when he oupu gap is posiive), he cenral bank sill keeps he nominal ineres rae a zero for some ime unil he risk of falling back o he rap is no considerable. Specifically, from Panels A and B of Figure 3, when he preference is beween 2% and 2.8% higher han he seady sae, he nominal ineres rae is kep a zero even hough he economy is no a he liquidiy rap. However, wih he posiive overall economic disorion, he cenral bank is less likely o keep he nominal ineres rae a zero when he economy escapes from he liquidiy rap, as seen in Panels A and C of Figure 5. Unlike he case of a non-disored seady sae, in his case he ZLB and liquidiy rap are no necessarily associaed wih deflaion because of higher expeced inflaion. 26

27 Even if a paricularly adverse demand shock occurs and pushes he economy ino he liquidiy rap wih oupu loss and binding ZLB, downward pressure on he price level may no be big enough o offse he high expeced inflaion. Hence, he acual inflaion is posiive. The resuls also shed ligh on he recen discussion abou he missing disinflaion, e.g. Coibion e al. 2012). For example, a he preference of 6% higher han he seady sae value or he naural real rae is 6% lower han is seady sae rae), oupu loss is 0.3% ha is associaed wih 2.3% inflaion rae. The resuls wih posiive overall economic disorion are consisen wih wha we have observed since he las recession where he federal funds rae FFR) is echnically zero and inflaion is moderaely posiive. The policy funcion from he LQ framework is presened by he dash-doed red lines in Figure 5, alhough hey are less accurae compared wih he rue policy in he FNL model. Specifically, in he LQ model, a shock wih a magniude of 2.3 sandard deviaions or 4% per year) is required o drive he economy o he ZLB, while i requires a shock wih a magniude of a leas 3 sandard deviaions or 5.2% per year) o make he ZLB binding in he FNL model. In addiion, given ha he ZLB is binding in boh models, he oupu loss and inflaion decline are much larger in he LQ model. Specifically, when he ime discoun facor is 8% higher han is seady sae value or he naural real rae is 8% smaller han is seady sae), he bounds are binding in boh models. 14 The oupu falls by 2.2%, associaed wih an inflaion rae decline of 1.7% in he LQ model, compared o 1.25% and 0.4% in he FNL model. The cenral bank in he LQ approach pursues lower inflaion and nominal ineres raes on average because of he inaccuracy of he LQ approach. Inuiively, he 14 The shock is slighly above hree sandard deviaions. 27

28 inaccuracy of he LQ model comes from he fac ha he LQ mehod eliminaes an endogenous sae variable called relaive price dispersion. As shown in Panel D of Figure 5, in he LQ framework, he relaive price dispersion is always 0% annually, while i is 1.2% in he FNL model. We know ha higher relaive price dispersion is associaed wih higher inflaion, as in equaion 21). Thus, an adverse shock ha causes he ZLB o bind in he FNL model mus generae more slackness in he LQ model. As a resul, he oupu loss and decline in inflaion are greaer in he LQ model han in he FNL model. Figure 5 also shows ha when he economy is near he ZLB, he ineres rae is cu more aggressively in he LQ model han in he FNL model. The inuiion is he same as above. Wih a larger overall economic disorion, inflaion and nominal ineres raes in he LQ model are smaller han hose in he rue model, which is he FNL model. Therefore, he ZLB in he LQ model is more likely o bind and he economy is more likely o be pushed in he liquidiy rap wih lower inflaion. When he economy is near he ZLB, downward pressure on he condiional expeced inflaion in he LQ model is larger han in he FNL model, resuling in a more aggressive cu in ineres raes. To provide a more deailed comparison of he FNL and he LQ model, i is useful o answer he quesion: o which exen are he equilibrium responses in he case of a disored seady sae really driven by he presence of he ZLB? From Figure 5, he cenral bank can obain he inflaion and oupu arges using boh FNL and LQ models in he absence of he ZLB. There are only demand shocks or preference shocks) in he model, so here is no rade-off beween oupu and inflaion sabilizaion. However, he arges based on he LQ mehod are inaccurae because his mehod does no capure he full nonlineariy of he model. Specifically from Panel B of Figure 5, he equilibrium inflaion rae is always 2.4% per year in he FNL 28

29 model and 1.6% in he LQ model. The difference of 0.8% beween he wo models is solely due o he nonlineariy. In he presence of he ZLB, he equilibrium responses in he FNL and LQ model are differen due o wo facors. The firs facor is he nonlineariy effec as explained above. The second facor is he presence of he ZLB. To see he second effec, le us examine Panel B of Figure 5 again. Under a shock ha increases he preference by 6% per year and he ZLB binds, he equilibrium inflaion is abou 2.3% in he FNL model and 1% in he LQ model. The oal difference is abou 1.3%. Because he nonlineariy accouns for approximaely 0.8%, he ZLB presence accouns for approximaely 0.5%. To exend he resuls from Yun 2005), I invesigae he role of iniial relaive price dispersion and repor he resuls in Figure 6. When he iniial dispersion is greaer han he seady sae value of 1.2%, boh oupu gap and inflaion fall furher han hey would if he iniial dispersion were kep a he seady sae value. The opposie resuls occur if he iniial dispersion is smaller han he seady sae value. Again, his happens because he relaive price dispersion is posiively correlaed wih he iniial dispersion as we see in Figure 1. In addiion, he relaive price dispersion plays he role of endogenous echnology in he aggregae producion funcion. The higher he relaive price dispersion, he lower he echnology and he lower he oupu. Therefore, he addiional oupu loss gain) depends on wheher he iniial relaive price dispersion is greaer smaller) han is seady sae value. From Figure 5 we also see ha he greaer he iniial relaive price dispersion, he higher he nominal ineres rae. As a resul, he ZLB is less likely o bind. 29

30 4 A. Ineres rae 0.5 B. Oupu deviaion % per year % per year Δ 1 =0 %) Δ 1 =1.2 Δ 1 = C. Inflaion Preference shock % per year) D. Relaive price dispersion Preference shock % per year) Figure 6: Opimal policy in he economy wih a disored seady sae Φ = 0.2), wih differen values of iniial relaive price dispersion, in he FNL model. The preference shock is annualized percenage deviaion from he seady sae β SS =β = 0.993) Wha is he opimal inflaion rae? Since he lae 1990s when Japan fell ino he liquidiy rap wih binding ZLB, economiss, such as Krugman 1998), have debaed wheher cenral banks should arge a significanly posiive inflaion arge and wha he opimal inflaion rae is. These opics are even more imporan oday as he U.S. federal funds rae has been a zero since December 2008 and he U.S. economy is experiencing is greaes slump since he Grea Depression. Blanchard e al. 2010) sugges ha policymakers migh consider an opimal inflaion arge of around 4%. The suggesion lies under 30

31 he argumen ha, in he presence of he ZLB, significanly posiive inflaion creaes leeway for he cenral bank o deal wih a paricularly adverse demand shock ha would drive he economy ino he liquidiy rap wih a binding ZLB. However, posiive inflaion comes a a price. Higher inflaion is always associaed wih more fron-loading behavior of firms when hey have a chance o rese heir prices. As a resul, i is associaed wih higher relaive price dispersion and lower oupu. This occurs because if he firms know ha inflaion is high and hey canno adjus heir prices flexibly in he fuure, hey will se higher prices oday, causing higher relaive price dispersion. In his secion, I am going o use he FNL model o answer a very imporan policy quesion - wha is he opimal inflaion arge he social planner should pursue by seing he size of overall economic disorion or ax rae accordingly, as in equaion 16)? For example, he social planner can choose a 2% inflaion arge by seing overall economic disorion of 0.18, or 8.89% income ax. I is reasonable o hink of he social planner having wo separae decision-making bodies. One is he Treasury Deparmen ha conducs ax policy, and he oher is he cenral bank ha conducs moneary policy. The social planner is assumed o be able o choose and commi o a labor income ax policy knowing ha he will implemen opional discreionary moneary policy laer. The social planner s problem boils down o comparing he social welfare for each ax/subsidy policy under discreionary moneary policy, choosing he bes ax policy once, hen commiing o he policy. The seup is very similar o he case of discreionary policy wihou model misspecificaion as in Billi 2011), where he social planner chooses he inflaion goal once and for all a ime =0. Absen he ZLB, he social planner should choose a zero inflaion goal and, as a resul, zero opimal long-run inflaion. However, in he presence of he ZLB, he social planner should 31

32 choose a posiive inflaion arge as a guard agains he incidence of a binding ZLB. 8 Probabiliy of hiing he ZLB Inflaion Uncondiional relaive welfare 0.5 Inflaion, Prob. of hiing he ZLB %) Welfare gain %) Size of overall economic disorion Φ) Figure 7: Uncondiional welfare gain relaive o he case of non-disored seady sae. The probabiliy of hiing he ZLB is around 6% in he non-disored case. To find ou he opimal inflaion rae, I firs solve he FNL model wih respec o differen values of he overall economic disorion. Then, based on he opimal policy, I compue he corresponding long-run inflaion, which is he average inflaion from a simulaion of 300, 000 periods. I also compue he simulaed probabiliy of hiing he ZLB using hese 300, 000 periods. As shown in Figure 7 wih he y-axis on he lef, he higher he overall economic disorion, he higher he long-run inflaion, and 32

33 he lower he probabiliy of hiing he ZLB. For example, if he overall economic disorion is zero, or he income ax rae is 11%, he average long-run inflaion is around 0.02% and he probabiliy of hiing he ZLB is around 6%. The solid red line in Figure 7, wih he y-axis on he righ, presens he uncondiional welfare relaive o he one associaed wih he non-disored seady sae, as he funcion of overall economic disorion. To compue his uncondiional relaive welfare, for each value of overall economic disorion, I firs solve for he value funcion as he funcion of he iniial relaive price disorion and he preference shock. Then, I ake a random sample of 300, 000 preference shocks, and I compue he average welfare, given he iniial relaive price disorion a he seady sae. Evenually, I compue he uncondiional welfare gain as a percenage change from he one associaed wih he non-disored seady sae. I is surprising ha he uncondiional welfare is decreasing in he size of he overall disorion. In oher words, he uncondiional expeced welfare is maximal when he overall economic disorion is around zero. This happens even when he average long-run inflaion is approximaely zero and he probabiliy of hiing he ZLB is he greaes. Therefore, he opimal inflaion rae is approximaely zero. The inuiion is ha : alhough he benefi of a significanly posiive inflaion arge relaive o he non-disored case is large when a paricularly adverse demand shock occurs, as shown by he red lines in Figure 8, such a shock is raher unlikely o occur. Wihou such a shock, he economy has o incur welfare loss associaed wih posiive inflaion every period. In sum, he uncondiional welfare declines in inflaion and overall economic disorion. For example, wih he overall economic disorion of 0.02 or he average long-run inflaion of 0.25% per year, he welfare is smaller han he non-disored welfare by abou in almos all saes of he preference shock, as presened by he hick solid red line in Figure 8. Only wih a shock of a 33

34 Φ=0 Φ=0.01 Φ=0.02 Condiional relaive welfare Preference shock % per year) Figure 8: Condiional welfare relaive o he case of non-disored seady sae. The probabiliy of hiing he ZLB is around 6% in he non-disored case. leas 6.7% or 4 sandard deviaions), he welfare gain relaive o he non-disored case is posiive. Therefore, on average, he uncondiional welfare corresponding o he long-run inflaion of 0.25% per year is smaller han he one wih he inflaion rae of 0.02% Sensiiviy analysis For a robusness check, in his secion, I raise he variance of he preference shock such ha he uncondiional probabiliy of hiing he ZLB increases from 6% o around 9.5% in he economy wih a non-disored seady sae. In his case, 34

35 10 Probabiliy of hiing he ZLB Inflaion Uncondiional relaive welfare 1 Inflaion, Prob. of hiing he ZLB %) 5 0 Welfare gain %) Size of overall economic disorion Φ) Figure 9: Uncondiional welfare gain relaive o he case of non-disored seady sae. The probabiliy of hiing he ZLB is 9.5% in he non-disored case. alhough he opimal inflaion arge is no longer zero, i is no significanly greaer han zero. Figure 9 shows ha he uncondiional welfare increases in inflaion when inflaion is small, and, afer a poin, i decreases in inflaion. The welfare is maximal when he overall economic disorion is se around 0.01, corresponding o he average long-run inflaion of 0.02% per year. Figure 10 shows ha, when he size of he overall economic disorion is zero or he long-run inflaion is around 0.07% per year, a small increase in he size of 35