Optimal Government Spending at the Zero Bound: Nonlinear and Non-Ricardian Analysis

Transcription

1 Opimal Governmen Spending a he Zero Bound: Nonlinear and Non-Ricardian Analysis Taisuke Nakaa New York Universiy Firs Draf: May 9 This Draf : April Absrac This paper characerizes opimal governmen spending when moneary policy is consrained by he zero lower bound under a variey of assumpions abou a se of fiscal insrumens available o finance governmen spending. The privae secor of he model is given by a sandard New Keynesian model. In response o a large and persisen ime-preference shock, governmen chooses a sequence of nominal ineres rae and governmen spending, which can be financed by eiher lump-sum ax, a mix of labor income ax and deb, or a mix of consumpion ax and deb. There are four main findings. Firs, opimal governmen spending policy is characerized by an iniial expansion followed by a sharp reducion during he period of zero nominal ineres raes. Second, opimal dynamics of deb and primary balance depend on he available disorionary ax and he iniial level of deb. Third, welfare gain of having governmen spending as an addiional policy insrumen depends imporanly on he available disorionary ax, bu is generally much smaller han welfare gain of having deb insrumen or disorionary ax. Finally, welfare gains of various fiscal insrumens are larger in he economy wih larger iniial deb. aisuke.nakaa@nyu.edu The firs draf was circulaed under he ile Opimal Moneary Policy and Governmen Expendiure under a Liquidiy Trap. I hank Tim Cogley, Mark Gerler, and Thomas Sargen for heir advice, and Gaui Eggersson for kindly sharing his code a he early sage of his research projec.

2 Inroducion The recen recession led governmens across he world o provide large fiscal simulus. This has generaed a renewed ineres in he effec of fiscal policy among economiss and policymakers. In paricular, some auhors have examined he effec of governmen spending, and have noed ha governmen spending is effecive in raising oupu and consumpion when nominal ineres rae is consrained a he zero lower bound. While hese sudies consider he effec of exogenous governmen spending shock, oher sudies have invesigaed opimal governmen spending policy a he zero nominal ineres rae and found ha, no only can governmen spending increase oupu and consumpion, i can also increase welfare a he zero bound. However, normaive analyses on governmen spending have hus far focused on he economy in which lump-sum ax is available o finance governmen spending 3. I would be useful o consider oher financing schemes involving disorionary axes and deb. Accordingly, his paper characerizes opimal governmen spending a he zero nominal ineres rae under a variey of assumpions abou a se of fiscal insrumens available o finance governmen spending. Privae secor of he model is given by a sandard New Keynesian model. In response o a discoun facor shock, governmen chooses and commis iself o a sequence of nominal ineres rae and governmen spending, which is financed by eiher lump-sum ax, labor income ax/deb, and consumpion ax/deb. While his paper sudies he model in which governmen can commi, my ongoing research sudies he model wihou commimen. Following Levin e al. (), I consider a large and persisen discoun facor shock inended o capure he severiy of he Grea Recession. In he same New Keynesian model in which governmen can commi, bu nominal ineres rae is he only available policy insrumen, hey find a large decline in oupu a he zero bound when he discoun facor shock is large and persisen, and sugges poenial roles for addiional policy insrumens. This paper sudies he poenial role of governmen spending under he same environmen. I documen how governmen can improve allocaions if i can opimally choose governmen spending in addiion o nominal ineres rae, and evaluae welfare gain from having governmen spending as an addiional policy insrumen using he represenaive household s uiliy as he welfare measure. Since he focus of he paper is on fiscal policy, I do no assume ha governmen provides lumpsum-ax-financed labor-income subsidy o he represenaive household o offse he disorion arising from monopolisic compeiion, which is ofen assumed in he analysis of opimal fiscal and moneary policy using he New Keynesian model. Thus, he seady-sae of he economy is no efficien. For his reason, and also because policy funcions are no differeniable due o he zero-lower bound consrain, accurae evaluaion of opimal policy and welfare requires nonlinear soluion mehod. I apply a modified Newon mehod described in Julliard e al. (998) o solve he model in is original nonlinear form. This urns ou o maer quaniaively in he welfare Chrisiano, Eichenbaum, and Rebelo (), Erceg and Linde (), Eggersson (a), and Uhlig () o name a few. 3 Examples are Eggersson (), Chrisiano e al. (), and Woodford (). Eggersson (b) inroduces a quadraic adjusmen cos of collecing lump-sum axes o break he Ricardian-Equivalence.

3 evaluaion. There are four main findings. Firs, opimal governmen spending policy is characerized by an iniial expansion followed by a sharp reducion o he level below he seady-sae. This reducion occurs while he nominal ineres rae is sill bounded a zero. Governmen spending evenually rises o a new seady-sae level as he zero nominal ineres rae ceases o bind. An increase in governmen spending is desirable as i leads o he increased demand for labor by he firms, which resuls in an increase in he real wage and inflaion. While his is rue in all models, he variaion in governmen spending is much smaller in he model wih consumpion ax and deb han in he oher wo models, and reflecs a concern o align marginal uiliy of governmen spending wih hose of consumpion and leisure. Second, opimal dynamics of deb and primary balance 4 depends on he available disorionary ax and he iniial level of deb. In he model wih labor income ax and deb, primary balance decreases by a small amoun a ime one, and spends a few quarers a a level below is erminal seady sae level. Neverheless, due o lower ineres expenses, governmen can reduce he real value of deb. In he model wih consumpion ax and deb, primary balance drops by a large amoun a ime one, and gradually increases hereafer. Accordingly, he real value of deb increases a ime one and says above he iniial level for several quarers. In he long-run, deb converges o a level smaller han he iniial level if he iniial deb is posiive, regardless of which disorionary ax is available. If governmen iniially has asse (i.e., negaive deb), hen he asse converges o a smaller level. This long-run dynamics can be explained by he fac ha he nominal ineres rae says below is seady-sae level for a very long period afer he discoun facor shock his. Third, welfare gain of having governmen spending as an addiional policy insrumen is small in absolue erms, and is also small in relaive erms compared o welfare gains of disorionary ax and deb insrumens. Welfare gain from any of hese fiscal insrumens is small relaive o welfare gain of moving from an economy in which all policy variables are se according o simple rules o an economy in which governmen can opimally choose nominal ineres rae, bu canno opimally choose fiscal insrumens. Tha is, given ha nominal ineres is chosen opimally, here are no much more governmen spending can do o improve allocaions. However, I should noe ha his resul is specific o he commimen case, and a preliminary analysis in my ongoing research suggess ha his resul is overurned in he model wihou commimen. Finally, welfare gains of various fiscal insrumens are larger in he economy wih higher iniial deb or asse. For example, in he model wih labor income ax and deb, welfare gains of governmen spending, labor income ax, and deb insrumens are.,.5, and.7 percen in he economy wih iniial deb-o-annualized oupu raio of 5 percen, bu hey are.5,.7, and.9 percen in he economy wih iniial deb-o-annualized oupu raio of percen, and.,.8, and.5 percen in he economy wih iniial asse-o-annualized oupu raio of percen. Similar resuls obain in he model wih consumpion ax and deb. The reason is as follows. Regardless of he iniial deb level, opimal moneary policy is o se nominal ineres rae below is seadysae level for a long period. Wih posiive iniial deb or asse, such changes in nominal ineres 4 Primary balance is defined as ax-revenue minus governmen spending. 3

4 raes requires adjusmen in fiscal insrumens in order o keep governmen budge consrain in equaliy. Thus, fiscal insrumens have a dual role of miigaing he effec of discoun facor shock and mainaining governmen budge consrain in response o sudden changes in nominal ineres raes. This second ask is large in he economy wih larger iniial deb and asse because he same change in nominal ineres raes leads o larger changes in he ineres expenses or receips. Under such environmen, resricing one of fiscal insrumens increases he burden for he res of fiscal insrumens, and resuls in a large welfare loss.. Relaion o he Lieraure There are several papers closely relaed. Le me summarize hem, and explain how his paper differs from hem. Relaion o Eggersson and Woodford (4) and Eggersson (, 6, a, b): Eggersson and Woodford (4) sudy opimal fiscal and moneary policy wih commimen a he zero nominal ineres rae. They sudy opimal pah of labor income ax and deb, and also consumpion ax and deb, bu hey do so while holding governmen spending consan. Eggersson () sudies opimal governmen spending financed by lump-sum ax, and my resuls on he model wih lump-sum ax essenially reproduce his findings. Eggersson (6 and b) solves for opimal deb, governmen spending, and nominal ineres rae. He breaks Ricardian equivalence by inroducing quadraic ax-collecion coss o he lump-sum ax, and does no consider disorionary axes. Eggersson (a) sudies he effec of governmen spending, labor income ax, and consumpion ax a he zero bound, one a a ime, assuming ha here is lump-sum ax o mainain budge balance and ha moneary policy is no opimal. Overall, wo papers by Eggersson (6 and b) are he closes o his paper. My paper differs from hem in wo ways. Firs, i breaks Ricardian Equivalence by inroducing disorionary axes, insead of ax-collecion cos on lump-sum ax. Second, i focuses on he model wih commimen. The focus of Eggersson (6 and b) is on he governmen s abiliy o use deb o overcome he lack of commimen. Relaion o Woodford () and Chrisiano e al. (): While heir analyses are mainly on he governmen spending muliplier, hey do consider wheher increasing governmen spending during he zero-bound period enhances welfare. They boh analyically derive opimal increase in governmen spending during he zero bound period. However, hey do so by assuming ha (i) cenral bank follows a runcaed Taylor rule, (ii) governmen spending is se o he seadysae level soon afer he zero-bound period, and (iii) governmen spending is financed hrough lump-sum ax. I will no assume any of hem. In paricular, nominal ineres rae is chosen opimally joinly wih fiscal insrumens in his paper. Relaion o Correia e al. (): Correia e al. () show ha, once we allow for he simulaneous use of labor income ax, consumpion ax, and lump-sum ax, he efficien allocaion can be aained even a he zero bound, and sugges ha here is no role for governmen spending. In arriving his resul, i is crucial ha he governmen have access o all of he hree axes. This paper finds ha, even if he governmen does no have access o all of he hree axes, welfare gain 4

5 of using he governmen spending ends o be small.. Organizaion of he Paper Secion describes he model and defines he equilibrium. Secion 3 formulaes governmen s problem, discusses he seady-sae of he economy, and defines he welfare measures. Secion 4 discusses calibraion and soluion mehod. Secion 5 presens main resuls. Secion 6 discusses addiional resuls. Secion 7 concludes. There are hree appendices. Appendix A describes nonlinear soluion mehod. Appendix B describes piecewise linear soluion mehod. Appendix C liss firs-order necessary condiions of he Lagrangian problem associaed wih governmen s problem. Tables and figures follow. Model The firs few subsecion describes he privae secor of he model, which is given by he New Keynesian model. The economy sars a =. The model has one exogenous variable, discoun facor shock, bu is deerminisic. The price-seing environmen is given by he Calvo model. Since he model is widely known, he presenaion is kep o minimum. The model differs from he sandard model because governmen spending eners ino uiliy funcion of he household. I do his in order o have he seady-sae level of governmen spending share o be posiive 5. I will consider hree versions of he model presened below. The firs version allows only for lump-sum ax, he second version allows for labor income ax and deb, and he hird version allows for consumpion ax and deb. I will describe he model wih all of hese fiscal insrumens, bu i should be undersood ha a subse of hem is se o zero in any version of he model.. Household There is a represenaive household who maximizes he uiliy funcion. = β s= [ C χ c N +χn, G χg, ] δ s χ n, + χ g, χ c + χ n, χ g, subjec o ( + τ c, )P C + R B ( τ n, )W N + B P T + Γ and B given. C is consumpion of final goods, N is labor supply, P is price of consumpion goods, and W is nominal wage. T is lump-sum ax, B is he quaniy of risk-free one-period bonds carried over from period, and paying one uni of money in period +. R denoe he gross nominal reurn on bonds purchased in period, and Γ is profis from he inermediae-good producing firms. τ n, is labor income ax, and τ c, is consumpion ax. 5 Oherwise, I would need o include an inequaliy consrain on governmen consumpion. 5

6 {δ } = is he discoun facor shock ha affecs how he represenaive household values he period uiliy a ime + relaive o he period uiliy a ime. {δ } = is exogenously given, and saisfies δ = δ = + ǫ δ, δ = + ρ δ (δ ) for ǫ δ, is revealed a he beginning of = before agens in he model make decisions. δ muliplies all period uiliies, and is normalized o one. The Lagrangian problem is given by L(B ) = = β = s= β [C χc N +χn, G χg, ] δ s χ n, + χ g, χ c + χ n, χ g, s= δ s λ [ ( + τc, )P C + R B (( τ n, )W N + B P T + Γ ] FONCs wih respec o C, N, and B are given by ( + τ c, )P λ = C χc ( τ n, )W λ = χ n, N χ n, λ = βr E δ λ + labor, Now, by aking he raio of he FONCs wih respec o consumpion and FOC wih respec o Le w W P. Then we can wrie ( τ n, )W = χ n, N χ n, ( + τ c, )P τ n, w = χ n, N χ n, + τ c, Using he FONC wih respec o consumpion o eliminae λ from he FONC wih respec o C χc C χc 6

7 bond holding, we obain ( + τ c, )P C χc + τ c,+ C χc + τ c, = βr E δ ( + τ c,+ )P + C χc + = βr E δ C χc + Π +. Final-Goods Producing Firms There are a coninuum of monopolisically compeiive firms producing differeniaed inermediae goods indexed by i [, ]. These inermediae goods, Y (i), are used as inpus by a represenaive perfecly compeiive firm o produce a final good, Y. Producion echnology of he final-goods producing firm is given by [ Y = ] Y (i) θ θ θ θ di where Y (i) is he quaniy of inermediae good j used as an inpu. Profi maximizaion of he final goods firm, aking he final good price P and he prices for inermediae goods P (i) as given, yields he se of demand funcions and he zero profi condiion Y (i) = [ P = [ P (i) P ] θ Y ] P (i) θ θ di.3 Inermediae-Goods Producing Firms An inermediae-goods producing firm ses is price and produces he oupu demanded by he final-goods producing firms a ha price. Labor is chosen o o mee he oupu demand in a cos minimizing way. The producion funcion is given by Y (i) = N (i).3. Cos minimizaion problem The inermediae-goods producing firms choose labor o produce Y (i) in a cos minimizing way. min s.. W N (i) Y (i) = N (i) 7

8 The soluion is Thus, N (i) = Y (i) Cos(Y (i)) = W Y (i) and.3. Opimal Price MC = Cos(Y (i)) Y (i) = W Inermediae goods firms are assumed o se nominal prices in a saggered fashion, as in Calvo (983). Each firm opimizes is price wih probabiliy ζ p each period, independenly of he ime elapsed since he las adjusmen. Opimizing-firms maximize he discouned sum of he fuure profis where he discoun facor comes from he sochasic discoun facor in he household s Euler equaion. max P (i)e s= subjec o he sequence of demand funcion D,+s is given by.4 Governmen Y +s (i) = D,+s = β+s λ +s +s k= δ k β λ k= δ k ζ s pd,+s [ P (i) W +s ]Y +s (i) [ P (i) P +s ] θ Y+s = β s δ δ + δ +s λ +s λ The se of poenial policy insrumens is [R,G,T,τ n,,τ c,,b ]. I will someimes refer o R as moneary insrumen, and he res as fiscal insrumens. I ake he zero-lower bound on he nominal ineres rae explicily ino accoun. R As menioned previously, I consider hree versions of he model, each associaed wih a unique se of fiscal insrumens. Firs version is when governmen has access o lump-sum ax. Governmen budge consrain is given by P G = P T Dividing boh sides by P, we have G = T 8

9 Second version is when here are labor income ax and deb. In his case, governmen budge consrain is given by Dividing boh sides by P, we obain B + P G = τ n, W N + R B b Π + G = τ n, w N + R b where Π = P P and b B P. Finally, he hird version is when here are consumpion ax and deb. In his case, governmen budge consrain is given by Dividing boh sides by P, we obain B + P G = τ c, P C + R B.5 Marke Clearing b Π The labor marke clearing condiion is given by The resource consrain is given by + G = τ c, C + R b N = N (i)di C + G = Y The bond marke clearing condiion is embedded in he noaion already as I use he same noaion, B, in he represenaive household s budge consrain and governmen budge consrain..6 An Equilibrium Given an iniial level of deb B, he disribuion of iniial prices P i, for all i [, ], and a sequence of discoun facor shocks {δ } =, an equilibrium of his economy consiss of {C, N, Y, P i,, G, R, T, τ n,, τ c,, B } = such ha. {C, N, B } = solves he household problem.. {P i, } = solves he firms problem. 3. Governmen budge consrain is saisfied. 4. Markes clear. 9

10 I is sraighforward o show ha an equilibrium is characerized by {C, N, Y, Π, p,s, C d,, C n,, R, G, T, τ n,, τ c,, b } = saisfying he following se of equaions: + τ c,+ C χc + τ c, = βr δ C χc + Π + τ n, w = χ n, N χ n, + τ c, = ( ζ p ) [ p p θ C n, = θ C d, C χc s = ( ζ p ) [ p θ ] + ζp Π θ s ] θ + ζp Π θ C n, = C d, = Y s Y = N GBC = Y w C χc + ζ p βδ Π θ + τ +C n,+ c, Y C χc + ζ p βδ Π θ + + τ C d,+ c, = C + G R where GBC = G T when lump-sum ax is used o finance governmen spending, GBC = b Π + G τ n, w N R b when labor income ax and deb are used o finance governmen spending, GBC = b Π + G τ c, C R b when consumpion ax and deb are used o finance governmen spending. C n, and C n, are auxiliary variables inroduced o describe he equilibrium condiions recursively, p is he price se by he opimizing firms normalized by he aggregae price, and s is cross-secional price dispersion. Noice ha an equilibrium depends only on he real deb b and he price dispersion s. My main resuls will be based on nonlinear soluion mehod. However, log-linearized equaions are useful in describing he main mechanism behind he resuls. I will pu hem here for laer reference.

11 Ŷ = Ŷ+ + Γ R (ˆΠ + ˆR ˆδ ) + Γ G (Ĝ Ĝ+) Γ τ,c (ˆτ c, ˆτ c,+ ) ˆΠ = Ω Y Ŷ Ω G Ĝ + Ω τ,cˆτ c, + Ω τ,nˆτ n, + β ˆΠ + where he coefficiens are shown in he foonoe 6. 3 Governmen s Problem The governmen s problem is o selec an equilibrium ha generaes he highes uiliy for he household. Tha is, where max {u} T β = = s= [ C χ c δ s N +χn, G χg, ] χ n, + χ g, χ c + χ n, χ g, u = [C, N, Y, w, p,π, C d,, C n,, R, G, eiher of (T, {τ c,, b }, {τ n,, b })] subjec o he se of equaions described above, and b and s are given. 6 The coefficiens of he log-linearized equaions as funcions of srucural parameers are: Γ R = χ c C ss Y ss, Γ G = Gss Y ss, Γ τ,c = χ c C ss Y ss τ c,ss + τ c,ss Y ss G ss Ω Y = κ(χ N, + χ c ), Ω G = κχ c, Ω τ,c = κ C ss C ss τc,ss, Ω τ,n = κ τn,ss + τ c,ss τ n,ss and κ = ( ζp)( βζp) ζ p.

12 The associaed Lagrangian problem is L(b, s ) = min {ω} = max {d,u } = + β = s= β = s= [ C χ c δ s χ n, N +χ n, χ c + χ n, [ δ s + ω, [ τ n, w χ n, N χ n, C χc ] + τ c, G χ g, ] + χ g, χ g, ω, [ + τ c,+ C χc βδ R C χc + + τ Π + ] c, + ω 3, [s ( ζ p ) [ p θ ] ζp Π θ s ] + ω 4, [ ( ζ p ) [ p ] θ ζp Π θ ] + ω 5, [p θ C n, ] θ C d, [ + ω 6, Cn, Y w C χc ζ p βδ Π θ ] + τ +C n,+ c, [ + ω 7, Cd, Y C χc ζ p βδ Π θ + + τ C ] d,+ c, + ω 8, [Y s N ] + ω 9, [Y C G ] + ω, GBC + ω, [R ] ] and ω, = if R > and ω, > if R =. Firs order necessary condiions for his Lagrangian problem a are given in Appendix C. 3. The modified governmen s problem As in many Ramsey problems, he firs order necessary condiions (FONCs) a = differ from he FONCs a. Therefore, even wihou any discoun facor shock, he soluion exhibis a peculiar iniial dynamics before i converges o a seady sae 7 (I will define he Ramsey seadysae and discuss is properies in he following subsecion). In order o focus our aenion on he economy s response o he discoun facor shock, I will modify he governmen s problem so as o eliminae he iniial dynamics ha would exis in he absence of any shock. Specifically, following Kahn, King and Wolman (3), he governmen s objecive funcion is modified o include penaly erms involving a hypoheical ime-zero Lagrange mulipliers. The hough experimen behind his modified governmen s problem is as follows. Suppose ha governmen solved he Ramsey problem a long ime ago, and ha he economy is a is Ramsey seady-sae a =. Now, if governmen 7 This is wha makes he Ramsey allocaion ime-inconsisen. In his model, governmen will reduce he level of deb a ime one by levying high labor income or consumpion ax.

13 were o re-opimize again a =, governmen would use his opporuniy o improve he welfare from ha poin on by deviaing from wha was promised a =. The erms involving he ime- Lagrangian muliplier ha appear in he modified objecive funcion penalizes such deviaion so ha, in he absence of any unanicipaed shocks, governmen coninues o choose he same allocaion and policy as hose chosen a = 8. The modified objecive funcion of he governmen is given by = β s= [ C χ c N +χn, G χg, ] δ s χ n, +χ g, ω, R C χc χ c + χ n, χ g, and he modified Lagrangian problem is given by Π ω 6,ζ p Π θ C n, ω 7, ζ p Π θ C d, L(b, s, R, ω,, ω 6,, ω 7, ) = min {ω} = max {u } = + = β s= [ C χ c δ s χ n, N +χ n, G χ g, ] + χ g, χ c + χ n, χ g, ω, R C χc Π ω 6, ζ p Π θ C n, ω 7, ζ p Π θ C d, [ β δ s ω, [ + τ c,+ C χc βδ R C χc + + τ Π + ] c, = s= + ω, [ τ n, w χ n, N χ n, C χc ] + τ c, + ω 3, [s ( ζ p ) [ p θ ] ζp Π θ s ] + ω 4, [ ( ζ p ) [ p ] θ ζp Π θ ] + ω 5, [p θ C n, ] θ C d, [ + ω 6, Cn, ν Y w C χc ζ p βδ Π θ ] + τ +C n,+ c, + ω 7, [ Cd, + ω 8, [Y s N ] + ω 9, [Y C G ] + ω, GBC + ω, [R ] Y C χc ζ p βδ Π θ + + τ C ] d,+ c, ] where he problem is now indexed by ω,, ω 6,, ω 7,, and R in addiion o b and s. FONCs of his modified Lagrangian problem are he same as he FONCs of he original Lagrangian problem a, and are given in Appendix C. 8 Modifying he governmen s objecive funcion in his way is ofen said o be aking imeless perspecive. 3

14 3. The consrained governmen s problem In order o documen he consequence of various fiscal insrumens, I consider various consrained Ramsey problems in which governmen is consrained o keep one of is fiscal insrumen consan. Such consrained problems have addiional erms each period in he Lagrangian problem above. For example, he Lagrangian problem in which governmen is consrained o keep is governmen spending consan has ω, [G Ḡ] added. 3.3 The Ramsey equilibrium and he Ramsey seady-sae Given he iniial values (b, s, R, ω,, ω 6,, and ω 7, ), he Ramsey equilibrium consiss of {u } = and {ω j,} = saisfying he FONCs of he Lagrangian problem above. I define he Ramsey seady-sae as a se of values, {C ss, N ss, Y ss, w ss, p ss, Π ss, C d,ss, C n,ss, R ss, G ss, b ss, τ n,ss, τ c,ss, ω,ss, ω,ss,, ω,ss } saisfying he FONCs of he modified Lagrangian problem. The Ramsey equilibrium and he Ramsey seady-sae in he consrained economies are defined in he same way. Le me make several remarks abou he Ramsey seady-sae. Firs, while here is a unique Ramsey-seady sae in he model wih lump-sum ax, here are infiniely many Ramsey seadysaes in he model wih deb, each indexed by he rae of disorionary ax 9. Figure plos he period uiliy and deb level associaed wih he Ramsey seady-saes wih differen ax raes in he model wih labor income ax and deb, and Figure does he same for he model wih consumpion ax and deb. Firs, noice from he righ panels ha, for he range of ax raes shown in he figures, here is one-o-one mapping beween he ax rae and he deb level. Second, he lef panels show ha ha differen Ramsey seady-saes yield differen period uiliies for he household. The Ramsey seady-sae wih highes period uiliy is he one in which governmen has a large asse, large enough o pay for labor income subsidy (or consumpion subsidy) ha eliminaes he disorion from monopolisic compeiion among inermediae-goods firms. In his economy, governmen s incenive o inflae away nominal deb o reduce he real value of deb, or alernaively he incenive o deflae so ha real value of he asse reaches o his bes level, is counered by he producion inefficiency associaed wih price dispersion caused by inflaion or deflaion. Consrained by he given iniial deb level, governmen canno move o his bes Ramsey seady-sae. Finally, in any Ramsey seady-sae, Π ss = s ss = p ss = and R ss = β. This is because price dispersion leads o producion inefficiency (Y s = N ). 9 This can be proved by showing ha he seady-sae value of he Lagrangian muliplier on governmen budge consrain is indeerminae. Adam () makes he same observaion. The deb level associaed wih a very high ax rae beyond ones shown in he figure are also consisen wih he deb level associaed wih a lower ax rae. This is due o he Laffer-curve effec. 4

15 3.4 Welfare Measures I define he welfare gain of governmen spending insrumen, denoed by WG G, as he one ime consumpion ransfer I need o make o he household a ime one in he economy in which governmen is consrained o keep governmen spending a is iniial level in order o equae he welfare of he consrained economy o he welfare of he unconsrained economy. Specifically, WG G is defined hrough = u(c C, + WG G C ss, N C,, Ḡ) + = β = β s= δ s u(c UC,, N UC,, G UC, ) + penaly erm s= δ s u(c C,, N C,, Ḡ) + penaly erm where he lef hand side is welfare in he consrained economy afer he consumpion ransfer and he righ hand side is welfare in he unconsrained economy. X C, refers o he variable X a ime in he consrained economy where he governmen is consrained o se G a Ḡ, and X UC, refers o he variable X a ime in he unconsrained economy. Noice ha welfare of an economy is defined o be he objecive funcion of he governmen, which is he sum of he household s uiliy and he penaly erm punishing governmen from deviaing from he iniial Ramsey seady-sae. C ss is he iniial Ramsey seady-sae consumpion (which are he same in boh economies). Since we are comparing alernaive policy responses o one-ime shock, his definiion is more naural han ha based on he consumpion variaion applied o all ime periods. Similarly, I define he welfare gain of deb insrumen, denoed by WG b, as he consumpion ransfer required o equae he welfare of he economy where governmen is consrained o keep is iniial deb level wih he welfare of he unconsrained economy. Finally, I define he welfare gain of disorionary ax insrumen, denoed by eiher WG τ,n or WG τ,c, as he consumpion ransfer required o equae he welfare of he economy where governmen is consrained o keep boh deb and governmen spending a heir iniial levels wih he welfare in he benchmark economy. Since hese welfare gains can be inerpreed as measuring he welfare losses from losing he abiliy o vary fiscal insrumens, I will someimes refer o hem as he welfare loss when i is more naural o do so. In order o pu hese welfare measures ino perspecive, I compue welfare in he model in which governmen ses is policy insrumens according o some simple rules. In his non-opimizing governmen regime, nominal ineres rae is se according o a runcaed Taylor rule augmened wih price level arge: R = max[, [ Π Π ] ρπ [P ] ρp ] P I include he price level gap in he policy rule because sandard Taylor rules leads o large deflaion and oupu decline wih small χ c considered in his paper, and even leads o non-exisence of he To compare he number here wih welfare gain number based on perpeual ransfer (as in Lucas welfare calculaion), you need o divide he number here by (= ), or muliply he Lucas number by. β 5

16 equilibrium under cerain parameer values. Leing nominal ineres rae responds o he price level gap improves he oucome considerably. In he model wih lump-sum ax, non-opimizing governmen ses G = G ss and ses T o saisfy governmen budge consrain. In he model wih labor income ax and deb, non-opimizing governmen ses G = G ss N N ss τ n, = τ n,ss + α n (b b ) and ses b o saisfy governmen budge consrain. Gss, Nss, and τ n,ss are he Ramsey-seadysae values of G, N, and τ n, associaed wih b. Finally, in he model wih consumpion ax and deb, non-opimizing governmen ses G = G ss N N ss τ c, = τ c,ss + α c (b b ) and ses b o saisfy governmen budge consrain. τ c,ss is he Ramsey-seady-sae value of τ c, associaed wih b. I impose ha, in he model wih disorionary axes, non-opimizing governmen chooses governmen spending so as o keep is raio o oupu consan, whereas non-opimizing governmen ses governmen spending consan in he model wih lump-sum ax. This imposiion is necessary because here urns ou o be no equilibrium consisen wih consan level of governmen spending in his simple-rule regime model under cerain parameer values. Parameer values for he policy rules are lised in Table. 4 Calibraion and Soluion Mehod 4. Calibraion Table liss parameer values considered. When muliple values are lised, he bold number represens he benchmark calibraion. Since his paper aims o conribue o he lieraure on he zero nominal ineres rae, I calibrae he parameer so ha hey are in he range of he values considered in his lieraure. Here, I discuss some key parameers. Governmen spending eners ino he household s uiliy funcion. I choose χ G,, he inverse of he ineremporal elasiciy of subsiuion for governmen spending, o be uniy, and χ G,, he weigh on he uiliy from governmen spending relaive o he uiliy from privae consumpion, In he model wihou fiscal policy insrumens, Eggersson and Woodford (3) show ha a version of price level argeing can replicae he Ramsey oucome. 6

17 o be. so ha he seady-sae level of governmen spending o oupu raio is roughly abou /6. Alernaive values of χ G, =.5 and χ G, =. are also considered o sudy he effec of a large governmen secor. I consider hree alernaive values for χ c, he inverse of he ineremporal elasiciy of subsiuion for privae consumpion. The benchmark is χ c = 6 from Jung e al. (3) and Levin e al. (). I also consider χ c = which corresponds o log-uiliy funcion, and χ c = which is he value used in Eggersson s work. For he inverse labor supply elasiciy, I use χ n, = as he benchmark, and consider alernaive values of χ n, =.5 and χ n, =.. For he shock process, I use ǫ δ, =., which makes he annualized nominal ineres ha would neuralize his shock -4 percen a ime one. This is roughly he same magniude of shock considered by Levin e al. () in heir experimen inended o capure he severiy of he Grea Recession, and is slighly larger han he value considered in Eggersson and Woodford (4). For he persisence of he shock, I se ρ δ =.9, which is slighly larger han he value considered in Levin e al. (). Eggersson and Woodford (4) consider a sochasic wo-sae Markov process for he discoun facor shock, and hus he persisence of heir shock canno direcly ino he deerminisic geomeric pah of δ considered in his paper. However, heir process implies he average duraion of he naural rae being negaive o be abou quarers, which ranslaes ino ρ δ =.93 in my seing. I also consider larger shocks (ǫδ, =.5 and ǫ δ, =.5) and more persisen shocks (ρ δ =.95 and ρ δ =.95). As discussed above, he model wih nominal deb possesses an infinie number of Ramsey seady saes, and he iniial level of deb deermines he he erminal Ramsey seady-sae o which he economy converges. Thus, I need o calibrae he iniial level of deb. In he benchmark calibraion, I choose b so ha he raio of deb o annualized oupu a ime zero is.5, which is slighly above he raio of publicly-held deb o GDP in he U.S. during he fiscal year 7, a period jus before he Grea Recession. The raio was.36 in he U.S., and some oher developed counries who are now consrained by he zero-lower bound had higher deb-o-gdp raios. I consider alernaive values of he iniial deb b = [,,,, ], and sudy how he iniial level of deb affecs opimal policy and welfare. 4. Soluion Mehod Excep in he Ramsey seady-sae associaed wih a large governmen asse, he allocaions are inefficien. For his reason, and also because policy funcions are no differeniable due o he zero lower bound consrain on nominal ineres rae, we need o solve he model in is original nonlinear form in order o accuraely evaluae opimal policy and welfare. Appendix A describes nonlinear soluion mehod. The mehod is a furher modificaion of a modified Newon algorihm by Julliard e al. (998). They modify a sandard Newon algorihm so as o ake advanage of he recursive srucure of he problem, and I embed heir algorihm in a shooing algorihm where he erminal Ramsey seady-sae is searched. 7

18 5 Resuls 5. Opimal Policy wih Lump-Sum Tax Figure 3 shows wo impulse response funcions for he model wih lump-sum ax. Solid black lines are he impulse response funcions wih unconsrained governmen and dashed red lines are he impulse response funcions wih governmen consrained o keep is spending a he iniial Ramsey seady-sae level. Dashed red lines reproduce he resul from Jung e al. (5) and Levin e al. (). Opimal moneary policy is characerized by he exended period of holding he nominal ineres rae a zero. For > 6, he nominal ineres rae ha would neuralize he discoun facor shock is above, and he governmen could achieve he seady-sae allocaions from ha period on. However, keeping nominal ineres rae a he zero for a longer period increases expeced inflaion, which prevens a large drop in inflaion and oupu a he beginning of he recession. Neverheless, as emphasized in Levin e al. (), a combinaion of large and persisen shock wih high ineremporal elasiciy of subsiuion resuls in a sharp decline in oupu and consumpion. Noice ha inroducing governmen spending policy does no affec his feaure of opimal ineres rae. In he unconsrained Ramsey allocaion, governmen spending iniially jumps and declines gradually during mos of he binding zero-bound period. Governmen spending declines below is seady-sae level around =5, and reaches is boom a =7. Afer ha, i sars rising and comes back o he seady-sae level as he zero bound ceases o bind. The inuiion for his dynamics can be illusraed wih he following log-linearized version of he privae secion equilibrium. Ŷ = Ŷ+ + Γ R (ˆΠ + ˆR ˆδ ) ˆΠ = Ω Y Ŷ Ω G Ĝ + β ˆΠ + where coefficiens are funcions of srucural parameers (see Secion.5). Firs equaion is he Euler equaion, and second equaion is he forward-looking Phillips curve. An expeced decline in governmen spending (an increase in (Ĝ Ĝ+)) raises oupu oday according o he Euler equaion. An increased oupu demand will increase he demand for labor by he inermediae good firms, which increases he real wage and inflaion oday, which is capured by he firs erm in he Phillips Curve. Today s increase in inflaion reduces he real ineres rae facing he household in he previous period, which works o offse he effec of discoun facor shock. Therefore, expeced reducion in governmen spending has he effec of miigaing deflaion and oupu collapse. Alhough increasing oday s governmen spending lowers oday s inflaion hrough Ω G erms, his effec is quaniaively small. Since governmen wans o keep is spending closer o he seadysae level, i is opimal o moderaely increase governmen spending iniially, and le i decline below is seady-sae, raher han increasing i by a large amoun and reducing i sraigh o he seady-sae level. 8

19 Variaions in governmen spending have limied effecs on allocaions. The iniial increase in governmen spending is abou 5 percen of is iniial seady sae level, which is less han percen of iniial seady sae oupu. Oupu muliplier is roughly abou one, and governmen spending does no have any significan effec on consumpion. One imporan difference beween he consrained and unconsrained economies is in he response of price dispersion. Price dispersion is always smaller in he unconsrained economy han in he consrained economy. This is a reflecion of slighly more subdued inflaion under he unconsrained economy during he period of zero nominal ineres raes. In order o furher highligh he limied effecs of governmen spending, Figure 4 shows he impulse response funcions associaed wih governmen following a runcaed Taylor rule and seing governmen spending consan described in Secion 3.4. Dashed blue lines are he allocaions associaed wih such non-opimizing governmen, and solid black line and dashed red lines are he same as in Figure 3. Under non-opimizing governmen, nominal ineres rae sars rising a =6. Wihou governmen commiing iself o an exend period of zero nominal ineres rae, here is large oupu collapse and deflaion. I is clear from his figure ha, compared o he effecs of opimally choosing nominal ineres raes, he addiional effec of opimally choosing governmen spending are small. We will see his poin again laer in he welfare calculaion. 5. Opimal Policy wih Labor Income Tax and Deb Figure 5 shows wo impulse response funcions for he model wih labor income ax and deb. Solid black lines are he impulse response funcions wih unconsrained governmen, and dashed red lines are he impulse response funcions wih governmen consrained o keep iniial deb level. As in he model wih lump-sum ax, opimal moneary policy involves an exended period of he zero nominal ineres rae, and opimal governmen spending is characerized by an iniial expansion followed by a sharp reversal. Labor income ax follows dynamics similar o governmen spending. I increases a ime one, declines below he iniial seady-sae rae, and rises back o is new seady-sae rae as he zero bound ceases o bind. We can again obain an inuiion for such pah of labor income ax hrough he log-linearized privae secor equilibrium condiions. Ŷ = Ŷ+ + Γ R (ˆΠ + ˆR ˆδ ) + Γ G (Ĝ Ĝ+) ˆΠ = Ω Y Ŷ Ω G Ĝ + Ω τ,nˆτ n, + β ˆΠ + According o he Phillips curve, an increase in labor income ax leads o an increase in oday s inflaion. This is because higher labor income ax reduces he household s willingness o work, which increases he real wage and hus inflaion. As discussed earlier, such increase in oday s inflaion reduces he real ineres rae facing he household in he previous period, which works o miigae he effec of he discoun facor shock 3. 3 Eggersson has discussed his expansionary effec of he labor income ax increase in various papers. See 9

20 The iniial jump in labor income ax does no generae revenues large enough o cover he iniial increase in governmen spending. As a resul, primary balance (= τ n, w N G ) decreases. Neverheless, deb declines slighly a ime one because he decline in nominal ineres rae reduces he ineres expense of he governmen. Thereafer, deb coninues o decline unil i converges o a new seady-sae level, which is lower han he iniial deb level. The deb is smaller in he erminal Ramsey seady-sae because he ineres expense on deb paid by governmen is lower han he level associaed wih he iniial Ramsey seady-sae for all periods afer he shock due o below-rend nominal ineres raes. Again, variaions on governmen spending have limied effecs on allocaion. As before, he iniial increase in governmen spending is around 5 percen of is iniial seady-sae level, which is less han percen of iniial seady-sae oupu. The iniial drop in he primary balance is smaller and hus he iniial drop in deb is larger in he consrained economy han in he unconsrained economy. As in he model wih lump-sum ax, price dispersion is smaller and inflaion is slighly more conained a he peak in he unconsrained economy. Figure 6 shows he impulse response funcions for hree differen levels of iniial deb. Dashed red lines are for he economy wih high iniial deb (b /(4Y ) = ), solid black lines are for he economy wih no iniial deb, and he dashed red lines are for he economy wih negaive iniial deb(b /(4Y ) = ). For mos variables, he impulse response funcions exhibi similar dynamics. However, here are several quaniaively imporan differences. Firs, he iniial increase in governmen spending is much larger, and he iniial increase in labor income ax is much lower, in he economy wih large iniial deb. In he large-deb economy, governmen spending jumps by percen while i jumps only by 5 percen in he large-asse economy or no-deb economy. Labor income ax does no flucuae much in he large-deb economy, bu i increases by abou 7 percen in he large-asse economy. The differences in he dynamics of governmen spending and labor income ax lead o a sark difference in he dynamics of primary balance. Primary balance decreases in he large-deb economy, whereas i increases in he large-asse economy. In he long-run, he high-deb economy converges o a erminal Ramsey seady-sae wih lower deb, and he high-asse economy converges o a erminal Ramsey seady-sae wih lower asse. As explained earlier, he reason for he economy wih posiive iniial deb o converge o a erminal seady-sae wih lower deb is he below-rend nominal ineres rae. This below-rend nominal ineres rae also leads he high-asse economy o sele in a erminal Ramsey seady-sae wih lower asse. Afer he shock, governmen receives he ineres paymens from he household ha are smaller han he level associaed wih he iniial Ramsey seady-sae because of he lower nominal ineres raes, and governmen s asse decreases as a resul. 5.3 Opimal Policy wih Consumpion Tax and Deb Figure 7 shows he impulse response funcions for he model wih consumpion ax and deb. Solid black lines are he impulse response funcions wih unconsrained governmen, and dashed red lines are he impulse response funcions wih consrained governmen. As in he previous wo Eggersson and Woodford (4), Eggersson (a and c)

21 models, i is opimal for governmen o se nominal ineres raes o zero for an exended period of ime. Dynamics of consumpion ax is characerized by an iniial drop followed by he gradual increase. Afer reaching is peak, i gradually declines o is erminal seady sae level. The iniial drop in consumpion ax is abou 4 percen. An inuiion for such consumpion ax dynamics can be again illusraed by he log-linearized version of he privae secor equilibrium condiions. Ŷ = Ŷ+ + Γ R (ˆΠ + ˆR ˆδ ) + Γ G (Ĝ Ĝ+) Γ τ,c (ˆτ c, ˆτ c,+ ) ˆΠ = Ω Y Ŷ Ω G Ĝ + Ω τ,cˆτ c, + β ˆΠ + According o he Euler equaion, an increase in omorrow s consumpion ax rae relaive o oday s consumpion ax rae (an increase in (ˆτ c,+ ˆτ c, )) increases oupu oday. This is because an expeced increase in consumpion ax induces he household o save less and consume more oday, which neuralizes he effec of discoun facor shock. Such increase in oupu demand by he household leads he firms o demand more labor, which pushes up he real wage and hus inflaion, and his effec is capured Ω Y erm in he Phillips Curve. A reducion in consumpion ax oday can lead o lower inflaion oday hrough he Ω τ,c erm, bu his erm is small, and he incenive o reduce consumpion ax oday dominaes. In he Euler equaion, he expeced increase in consumpion ax and he expeced reducion in governmen spending acs in a similar way o miigae he discoun facor shock, holding everyhing else equal. This explains he reduced role of governmen spending when consumpion ax is available. Governmen spending pah in his model is qualiaively similar in he previous wo models, bu he variaion is much smaller in size. The iniial increase is less han.5 percen while i was around 5 percen in he previous wo models. Governmen chooses o use variaion in consumpion ax o miigae he discoun facor shock and vary governmen spending in a way o mainain he equaliy of is marginal uiliy wih he marginal uiliy of privae consumpion. Consumpion a ime one acually increases by a small amoun due o a combinaion of a large reducion in oday s consumpion ax and an expeced increase in consumpion ax in he fuure. Since he variaion in governmen spending is very small, he differences beween consrained and unconsrained economies are very small. An expeced increase in consumpion ax and a reducion in nominal ineres rae are close subsiues o governmen because hey boh affec he household s ineremporal decision in he same way wihou perurbing he resource consrain. Thus, in he model wih consumpion ax, he zero lower bound consrain is less of a problem for governmen han in he models wihou i. This is refleced in much smaller values of he Lagrangian muliplier (ω, ) on he zero lower bound consrain. This Lagrangian muliplier capures he shadow value of furher decreasing he nominal ineres rae from zero. In he previous wo models, his Lagrangian muliplier jumps a ime one o 3, and reaches o 5 a he peak (see Figure 3 and 5). In he model wih consumpion

22 ax and deb, i jumps by less han, and reaches o around 3 a he peak. This is because he value of relaxing his inequaliy is smaller as he variaion in consumpion ax can accomplish wha he furher reducion in nominal ineres rae could accomplish. A combinaion of a very small increase in governmen spending and a large drop in consumpion ax means ha primary balance drops a ime one. As a resul, deb iniially increases and says above he iniial level during he firs several quarers. As consumpion ax rises above is iniial level, he deb sars o decline and slowly converges o a new seady-sae level. As we saw in he model wih labor income ax, he deb in he erminal Ramsey seady-sae is below is iniial level as governmen s reduced ineres rae expenses allows governmen o reduce deb level due o he below-rend nominal ineres raes. Figure 8 shows he impulse response funcions for hree differen levels of iniial deb. Dashed red lines are for he economy wih high iniial deb (b /(4Y ) = ), solid black lines are for he economy wih no iniial deb, and he dashed red lines are for he economy wih negaive iniial deb, or equivalenly posiive iniial asse (b /(4Y ) = ). Boh in he large-deb and large asse economies, i is opimal o increase he nominal ineres rae a ime one. From ime wo on, he dynamics of nominal ineres rae is he same as in he benchmark model wih moderae amoun of iniial deb or in he model wih no iniial deb. As discussed earlier, a reducion in nominal ineres rae and an expeced increase in consumpion ax are close subsiues o governmen. In he model wih large iniial deb or asse, governmen chooses o combine a large increase in nominal ineres rae and a large expeced increase in consumpion ax a ime one. Governmen achieves his large expeced increase in consumpion ax a ime one by reducing he consumpion ax rae a ime one wihou changing ime-wo consumpion ax rae much. In boh high-deb and high asse economies, he iniial decrease in consumpion ax is more han percen. Finally, as in he model wih labor income ax and deb, a long period of below-rend nominal ineres raes leads he economy wih posiive iniial deb and he economy wih posiive iniial asse o converge o erminal Ramsey seady-saes wih lower deb and asse, respecively. 5.4 Welfare Table, 3 and 4 repor he welfare gains for, respecively, he model wih lump-sum ax, he model wih wih labor income ax and deb, and he model wih consumpion ax and deb. For he models wih deb, I compue welfare gains for alernaive iniial deb levels. In he model wih lump-sum ax, welfare gain of governmen spending insrumen is.. This is subsanially smaller han he welfare gain of moving from an economy wih non-opimizing governmen o an economy in which governmen chooses nominal ineres rae opimally, which is consisen wih wha we saw earlier in Figure 4. Thus, in his economy, here is no much addiional improvemen governmen spending can make if nominal ineres rae is chosen opimally. In he model wih labor income ax and deb, welfare gain of governmen spending insrumen is.7, slighly larger han in he model wih lump-sum ax. Welfare gain of governmen spending is smaller han hose of labor income ax and deb. As before, condiional on opimally chosen nominal ineres raes, addiional welfare gain from any of he fiscal insrumens is small. Finally, in

23 he model wih consumpion ax and deb, welfare gain from governmen spending is even smaller. This is wha we would expec from he impulse response funcions above; Opimal variaions in governmen spending in his model are much smaller han in he previous wo models. Finally, welfare gain of consumpion ax is larger han welfare gain of deb in his model. Welfare gains depend on he iniial level of deb or asse. According o he second panels of Table 3 and 4, he larger he iniial deb or asse is, he larger he welfare gains of fiscal insrumens are, regardless of he available disorionary ax. Why are he welfare gains of various fiscal insrumens larger in he economy wih large iniial deb or asse? The inuiion is as follows. Regardless of he level of iniial deb or asse, we have seen ha opimal moneary policy involves an exended period of zero nominal ineres rae followed by he gradual reversal owards is seady-sae level. In he model wih deb, such change in he nominal ineres rae perurbs governmen budge consrain as i affecs he ineres expenses or receips. If he economy sars wih a posiive iniial deb, eiher deb has o decrease, governmen spending has o increase, or labor income ax has o decrease, in order o mainain governmen budge consrain. Therefore, fiscal insrumens have a dual role of miigaing he effec of he discoun facor shock and mainaining governmen budge consrain. When he iniial deb or asse level is small, changes in he nominal ineres rae does no affec he governmen budge consrain much. However, when he iniial level of deb or asse is high, he variaion in fiscal insrumens needed o balance governmen budge consrain is large. Under such circumsance, consraining a fiscal insrumen forces oher fiscal insrumens o adjus more o mainain governmen budge consrain, which can conflic wih heir role of miigaing he discoun facor shock and lead o larger welfare losses. 6 Discussion This secion summarizes addiional resuls. 6. Sensiiviy Analysis This subsecion describes how opimal allocaion and welfare change wih alernaive parameer values. The qualiaive resuls are unchanged, bu alernaive parameer values lead o quaniaively differen impulse response funcions. Insead of presening a large number of impulse response funcions, I will presen hree summary saisics (he iniial increase in governmen spending as a percenage of is iniial seady-sae level, he iniial increase in disorionary axes, and he las period a which nominal ineres rae is zero) and welfare gains, for each parameer configuraions and for each model. I focus on he sensiiviy of he resuls on he following six parameers; Inverse IES for consumpion good (χ n, ), Calvo parameer (ζ p ), inverse labor supply elasiciy (χ n, ), he uiliy weigh on governmen spending (χ g, ), he magniude of he discoun facor shock (ǫ δ, ), and persisence of he shock (ρ δ ). 3