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1 When Is the Government Spending Mutipier Large? Author(s): Lawrence Christiano, Martin Eichenbaum, Sergio Rebeo Source: Journa of Poitica Economy, Vo. 119, No. 1 (February 2011), pp Pubished by: The University of Chicago Press Stabe URL: Accessed: 08/09/ :22 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, avaiabe at. JSTOR is a not-for-profit service that heps schoars, researchers, and students discover, use, and buid upon a wide range of content in a trusted digita archive. We use information technoogy and toos to increase productivity and faciitate new forms of schoarship. For more information about JSTOR, pease contact support@jstor.org. The University of Chicago Press is coaborating with JSTOR to digitize, preserve and extend access to Journa of Poitica Economy.

2 When Is the Government Spending Mutipier Large? Lawrence Christiano Northwestern University and Nationa Bureau of Economic Research Martin Eichenbaum Northwestern University and Nationa Bureau of Economic Research Sergio Rebeo Northwestern University and Nationa Bureau of Economic Research We argue that the government-spending mutipier can be much arger than one when the zero ower bound on the nomina interest rate binds. The arger the fraction of government spending that occurs whie the nomina interest rate is zero, the arger the vaue of the mutipier. After providing intuition for these resuts, we investigate the size of the mutipier in a dynamic, stochastic, genera equiibrium mode. In this mode the mutipier effect is substantiay arger than one when the zero bound binds. Our mode is consistent with the behavior of key macro aggregates during the recent financia crisis. I. Introduction A cassic question in macroeconomics is, what is the size of the government-spending mutipier? There is a arge empirica iterature that grappes with this question. Authors such as Barro (1981) argue that the mutipier is around 0.8 whereas authors such as Ramey (2011) estimate We thank the editor, Monika Piazzesi, Rob Shimer, and two anonymous referees for their comments. [Journa of Poitica Economy, 2011, vo. 119, no. 1] 2011 by The University of Chicago. A rights reserved /2011/ $

3 government spending mutipier 79 the mutipier to be coser to There is aso a arge iterature that uses genera equiibrium modes to study the size of the governmentspending mutipier. In standard new-keynesian modes the governmentspending mutipier can be somewhat above or beow one depending on the exact specification of agents preferences (see Gai, López-Saido, and Vaés 2007; Monacei and Perotti 2008). In frictioness rea business cyce modes this mutipier is typicay ess than one (see, e.g., Aiyagari, Christiano, and Eichenbaum 1992; Baxter and King 1993; Ramey and Shapiro 1998; Burnside, Eichenbaum, and Fisher 2004; Ramey 2011). Viewed overa, it is hard to argue, on the basis of the iterature, that the government-spending mutipier is substantiay arger than one. In this paper we argue that the government-spending mutipier can be much arger than one when the nomina interest rate does not respond to an increase in government spending. We deveop this argument in a mode in which the mutipier is quite modest if the nomina interest rate is governed by a Tayor rue. When such a rue is operative, the nomina interest rate rises in response to an expansionary fisca poicy shock that puts upward pressure on output and infation. There is a natura scenario in which the nomina interest rate does not respond to an increase in government spending: when the zero ower bound on the nomina interest rate binds. We find that the mutipier is very arge in economies in which the output cost of being in the zero-bound state is aso arge. In such economies it can be sociay optima to substantiay raise government spending in response to shocks that make the zero ower bound on the nomina interest rate binding. We begin by considering an economy with Cavo-stye price frictions, no capita, and a monetary authority that foows a standard Tayor rue. Buiding on Eggertsson and Woodford (2003), we study the effect of a temporary, unanticipated rise in agents discount factor. Other things equa, the shock to the discount factor increases desired saving. Since investment is zero in this economy, aggregate saving must be zero in equiibrium. When the shock is sma enough, the rea interest rate fas and there is a modest decine in output. However, when the shock is arge enough, the zero bound becomes binding before the rea interest rate fas by enough to make aggregate saving zero. In this mode, the ony force that can induce the fa in saving required to reestabish equiibrium is a arge, transitory fa in output. Why is the fa in output so arge when the economy hits the zero bound? For a given fa in output, margina cost fas and prices decine. With staggered pricing, the drop in prices eads agents to expect future 1 For recent contributions to the vector autoregression (VAR) based empirica iterature on the size of the government-spending mutipier, see Izetzki, Mendoza, and Vegh (2009) and Fisher and Peters (2010). Ha (2009) provides an anaysis and review of the empirica iterature.

4 80 journa of poitica economy defation. With the nomina interest rate stuck at zero, the rea interest rate rises. This perverse rise in the rea interest rate eads to an increase in desired saving, which partiay undoes the effect of a given fa in output. So, the tota fa in output required to reduce desired saving to zero is very arge. This scenario resembes the paradox of thrift originay emphasized by Keynes (1936) and recenty anayzed by Krugman (1998), Eggertsson and Woodford (2003), and Christiano (2004). In the textbook version of this paradox, prices are constant and an increase in desired saving owers equiibrium output. But, in contrast to the textbook scenario, the zero-bound scenario studied in the modern iterature invoves a defationary spira that contributes to and accompanies the arge fa in output. Consider now the effect of an increase in government spending when the zero bound is stricty binding. This increase eads to a rise in output, margina cost, and expected infation. With the nomina interest rate stuck at zero, the rise in expected infation drives down the rea interest rate, which drives up private spending. This rise in spending eads to a further rise in output, margina cost, and expected infation and a further decine in the rea interest rate. The net resut is a arge rise in output and a arge fa in the rate of defation. In effect, the increase in government consumption counteracts the defationary spira associated with the zero-bound state. The exact vaue of the government-spending mutipier depends on a variety of factors. However, we show that this mutipier is arge in economies in which the output cost associated with the zero-bound probem is more severe. We argue this point in two ways. First, we show that the vaue of the government-spending mutipier can depend sensitivey on the mode s parameter vaues. But parameter vaues that are associated with arge decines in output when the zero bound binds are aso associated with arge vaues of the government-spending mutipier. Second, we show that the vaue of the government-spending mutipier is positivey reated to how ong the zero bound is expected to bind. An important practica objection to using fisca poicy to counteract a contraction associated with the zero-bound state is that there are ong ags in impementing increases in government spending. Motivated by this consideration, we study the size of the government-spending mutipier in the presence of impementation ags. We find that a key determinant of the size of the mutipier is the state of the word in which new government spending comes on ine. If it comes on ine in future periods when the nomina interest rate is zero, then there is a arge effect on current output. If it comes on ine in future periods in which the nomina interest rate is positive, then the current effect on government spending is smaer. So our anaysis supports the view that, for

5 government spending mutipier 81 fisca poicy to be effective, government spending must come on ine in a timey manner. In the second step of our anaysis we incorporate capita accumuation into the mode. For computationa reasons we consider temporary shocks that make the zero bound binding for a deterministic number of periods. Again, we find that the government-spending mutipier is arger when the zero bound binds. Aowing for capita accumuation has two effects. First, for a given size shock it reduces the ikeihood that the zero bound becomes binding. Second, when the zero bound binds, the presence of capita accumuation tends to increase the size of the government-spending mutipier. The intuition for this resut is that, in our mode, investment is a decreasing function of the rea interest rate. When the zero bound binds, the rea interest rate generay rises. So, other things equa, saving and investment diverge as the rea interest rate rises, thus exacerbating the metdown associated with the zero bound. As a resut, the fa in output necessary to bring saving and investment into aignment is arger than in the mode without capita. The simpe modes discussed above suggest that the mutipier can be arge in the zero-bound state. The obvious next step woud be to use reduced-form methods, such as identified VARs, to estimate the government-spending mutipier when the zero bound binds. Unfortunatey, this task is fraught with difficuties. First, we cannot mix evidence from states in which the zero bound binds with evidence from other states because the mutipiers are very different in the two states. Second, we have to identify exogenous movements in government spending when the zero bound binds. 2 This task seems daunting at best. Amost surey government spending woud rise in response to arge output osses in the zero-bound state. To know the government-spending mutipier we need to know what output woud have been had government spending not risen. For exampe, the simpe observation that output did not grow quicky in Japan in the zero-bound state, even though there were arge increases in government spending, tes us nothing about the question of interest. Given these difficuties, we investigate the size of the mutipier in the zero-bound state using the empiricay pausibe dynamic stochastic genera equiibrium (DSGE) mode proposed by Atig et a. (2011). This mode incorporates price- and wage-setting frictions, habit formation in consumption, variabe capita utiization, and investment adjustment costs of the sort proposed by Christiano, Eichenbaum, and Evans (2005). 2 To see how critica this step is, suppose that the government chooses spending to keep output exacty constant in the face of shocks that make the zero bound bind. A naive econometrician who simpy regressed output on government spending woud fasey concude that the government-spending mutipier is zero. This exampe is, of course, just an appication of Tobin s (1970) post hoc, ergo propter hoc argument.

6 82 journa of poitica economy Atig et a. estimate the parameters of their mode to match the impuse response function of 10 macro variabes to a monetary shock, a neutra technoogy shock, and a capita-embodied technoogy shock. Our key findings based on the Atig et a. mode can be summarized as foows. First, when the centra bank foows a Tayor rue, the vaue of the government-spending mutipier is generay ess than one. Second, the mutipier is much arger if the nomina interest rate does not respond to the rise in government spending. For exampe, suppose that government spending goes up for 12 quarters and the nomina interest rate remains constant. In this case the impact mutipier is roughy 1.6 and has a peak vaue of about 2.3. Third, the vaue of the mutipier depends criticay on how much government spending occurs in the period during which the nomina interest rate is constant. The arger the fraction of government spending that occurs whie the nomina interest rate is constant, the smaer the vaue of the mutipier. Consistent with the theoretica anaysis above, this resut impies that for government spending to be a powerfu weapon in combating output osses associated with the zero-bound state, it is critica that the buk of the spending come on ine when the ower bound is actuay binding. Fourth, we find that the mode generates sensibe predictions for the current crisis under the assumption that the zero bound binds. In particuar, the mode does we at accounting for the behavior of output, consumption, investment, infation, and short-term nomina interest rates. As emphasized by Eggertsson and Woodford (2003), an aternative way to escape the negative consequences of a shock that makes the zero bound binding is for the centra bank to commit to future infation. We abstract from this possibiity in this paper. We do so for a number of reasons. First, this theoretica possibiity is we understood. Second, we do not think that it is easy in practice for the centra bank to crediby commit to future high infation. Third, the optima trade-off between higher government purchases and anticipated infation depends sensitivey on how agents vaue government purchases and the costs of anticipated infation. Studying this issue is an important topic for future research. Our anaysis buids on the work by Christiano (2004) and Eggertsson (2004), who argue that increasing government spending is very effective when the zero bound binds. Eggertsson (2011) anayzes both the effects of increases in government spending and transitory tax cuts when the zero bound binds. The key contributions of this paper are to anayze the size of the mutipier in a medium-size DSGE mode, study the mode s performance in the financia crisis that began in 2008, and quantify the importance of the timing of government spending reative to the timing of the zero bound.

7 government spending mutipier 83 Our anaysis is reated to severa recent papers on the zero bound. Bodenstein, Erceg, and Guerrieri (2009) anayze the effects of shocks to open economies when the zero bound binds. Braun and Waki (2006) use a mode in which the zero bound binds to account for Japan s experience in the 1990s. Their resuts for fisca poicy are broady consistent with our resuts. Braun and Waki (2006) and Coenen and Wieand (2003) investigate whether aternative monetary poicy rues coud have avoided the zero-bound state in Japan. In onine Appendix B, we anayze the sensitivity of our concusions to the presence of distortionary taxes on abor and capita. Eggertsson (2010, 2011) shows that the effects of distortionary taxes can be very different depending on whether the zero ower bound binds or not. Indeed, some distortionary taxes that ower output when the zero ower bound does not bind actuay raise output when the zero bound does bind. Of course, if the tax that finances government spending actuay increases output, then the government-spending mutipier is actuay increased. We quantify the effects of distortionary abor taxes in Atig et a. s study when the zero ower bound binds. In addition, we discuss the effects of different types of capita income taxes. We argue that our concusions are robust to aowing for different types of distortionary taxes. Our paper is organized as foows. In Section II, we anayze the size of the government-spending mutipier when the interest foows a Tayor rue in a standard new-keynesian mode without capita. In Section III, we modify the anaysis to assume that the nomina interest rate does not respond to an increase in government spending, say because the ower bound on the nomina interest rate binds. In Section IV, we extend the mode to incorporate capita. In Section V, we discuss the properties of the government-spending mutipier in the medium-size DSGE mode proposed by Atig et a. (2011) and investigate the performance of the mode during the recent financia crisis. Section VI presents concusions. II. The Standard Mutipier in a Mode without Capita In this section we present a simpe new-keynesian mode and anayze its impications for the size of the standard mutipier, by which we mean the size of the government-spending mutipier when the nomina interest rate is governed by a Tayor rue. Househods. The economy is popuated by a representative househod, whose ifetime utiity, U, is given by { } g 1 g 1 j [C t(1 N t) ] 1 t 0 t tp0 U p E b v(g ). (1) 1 j

8 84 journa of poitica economy Here E0 is the conditiona expectation operator, and Ct, Gt, and Nt denote time t consumption, government consumption, and hours worked, respectivey. We assume that j 1 0, g (0, 1), and v(7) is a con- cave function. The househod budget constraint is given by PC t t Bt 1 p B(1 t R t) WN t t T t, (2) where T t denotes firms profits net of ump-sum taxes paid to the government. The variabe B t 1 denotes the quantity of one-period bonds purchased by the househod at time t. Aso, P t denotes the price eve and Wt denotes the nomina wage rate. Finay, Rtdenotes the one-period nomina rate of interest that pays off in period t. The househod s probem is to maximize utiity given by equation (1) subject to the budget constraint given by equation (2) and the condition E im B /[(1 R )(1 R ) (1 R )] 0. 0 t t t r Firms. The fina good is produced by competitive firms using the technoogy 1 /( 1) ( 1)/ Yt p [ Y(i) t di ], 1 1, (3) 0 where Y(i) t, i [0, 1], denotes intermediate good i. Profit maximization impies the foowing first-order condition for Y(i): t 1/ Y t Y(i) ] t P(i) p P, (4) t t[ where P(i) t denotes the price of intermediate good i and Pt is the price of the homogeneous fina good. The intermediate good, Y(i) t, is produced by a monopoist using the foowing technoogy: Y(i) t p N(i), t where N(i) t denotes empoyment by the ith monopoist. We assume that there is no entry or exit into the production of the ith intermediate good. The monopoist is subject to Cavo-stye price-setting frictions and can optimize its price, P(i) t, with probabiity 1 v. With probabiity v the firm sets P(i) p P (i). t The discounted profits of the ith intermediate-good firm are t 1 j t t j t j t j t j t j jp0 E bu [P (i)y (i) (1 n)w N (i)], (5)

9 government spending mutipier 85 where n p 1/ denotes an empoyment subsidy that corrects, in steady state, the inefficiency created by the presence of monopoy power. The variabe u t j is the mutipier on the househod budget constraint in the Lagrangian representation of the househod probem. The variabe W t j denotes the nomina wage rate. Firm i maximizes its discounted profits, given by equation (5), subject to the Cavo price-setting friction, the production function, and the demand function for Y(i) t, given by equation (4). Monetary poicy. We assume that monetary poicy foows the rue R t 1 p max (Z t 1, 0), (6) where f 1(1 r R) f 2(1 r R) rr Z p (1/b)(1 p) (Y /Y ) [b(1 R )] 1. t 1 t t t Throughout the paper a variabe without a time subscript denotes its steady-state vaue; for exampe, the variabe Y denotes the steady-state eve of output. The variabe p t denotes the time t rate of infation. We assume that f1 1 1 and f2 (0, 1). According to equation (6), the monetary authority foows a Tayor rue as ong as the impied nomina interest rate is nonnegative. Whenever the Tayor rue impies a negative nomina interest rate, the monetary authority simpy sets the nomina interest rate to zero. For convenience we assume that steady-state infation is zero. This assumption impies that the steady-state net nomina interest rate is 1/b 1. Fisca poicy. As ong as the zero bound on the nomina interest rate is not binding, government spending evoves according to r Gt 1 p Gt exp (h t 1). (7) Here G is the eve of government spending in the nonstochastic steady state and h t 1 is an independent and identicay distributed shock with zero mean. To simpify our anaysis, we assume that government spending and the empoyment subsidy are financed with ump-sum taxes. The exact timing of these taxes is irreevant because Ricardian equivaence hods under our assumptions. We discuss the detais of fisca poicy when the zero bound binds in Section III. Equiibrium. The economy s resource constraint is Ct Gt p Y t. (8) A monetary equiibrium is a coection of stochastic processes {C t, N t, W t, P, t Y t, R t, P(i), t Y(i), t N(i), t u t, B t 1, p t} such that for given {G t } the househod and firm probems are satisfied, the monetary and fisca poicy rues are satisfied, markets cear, and the aggregate resource constraint is satisfied. To sove for the equiibrium we use a inear approximation around the nonstochastic steady state of the economy. Throughout, denotes Ẑ t

10 86 journa of poitica economy the percentage deviation of Z t from its nonstochastic steady-state vaue, Z. The equiibrium is characterized by the foowing set of equations. The Phiips curve for this economy is given by p p E (bp kmc t t t 1 t), (9) where k p (1 v)(1 bv)/v. In addition, MC t denotes the rea margina cost, which, under our assumptions, is equa to the rea wage rate. Without abor market frictions, the percentage deviation of rea margina cost from its steady-state vaue is given by ˆ N MC p C N ˆ t t t. (10) 1 N The inearized intertempora Euer equation for consumption is ˆ N [g(1 j) 1]C (1 g)(1 j) Nˆ t t p 1 N (11) { } ˆ N E b(r R) p [g(1 j) 1]C (1 g)(1 j) N ˆ t t 1 t 1 t 1 t 1. 1 N The inearized aggregate resource constraint is Ŷ p (1 g)cˆ gg ˆ, (12) t t t where g p G/Y. Combining equations (9) and (10) and using the fact that N ˆ ˆ t p Yt, we obtain [( ) ] 1 N g p p be (p ) k Yˆ G ˆ t t t 1 t t. (13) 1 g 1 N 1 g Simiary, combining equations (11) and (12) and using the fact that Nˆ ˆ t p Yt, we obtain Yˆ g[g ( j 1) 1]Gˆ t t p (14) E { (1 g)[b(r R) p ] Yˆ g[g(j 1) 1]G ˆ t t 1 t 1 t 1 t 1}. As ong as the zero bound on the nomina interest rate does not bind, the inearized monetary poicy rue is given by 1 r R R p r (R R) R (f p f Y ˆ t 1 R t 1 t 2 t). b Whenever the zero bound binds, R t 1 p 0. We sove for the equiibrium using the method of undetermined coefficients. For simpicity, we begin by considering the case in which rr p 0. Under the assumption that f1 1 1, there is a unique inear equi- ibrium in which p and Y ˆ are given by t t

11 government spending mutipier 87 p p AG ˆ t p t (15) and Ŷ p AG. ˆ t Y t (16) The coefficients A and A are given by p Y [( ) ] k 1 N g A p p AY (17) 1 br 1 g 1 N 1 g and A Y p (18) (r f 1)k [g(j 1) 1](1 r)(1 br) g. (1 br)[r 1 (1 g)f 2] (1 g)(r f 1)k[1/(1 g) N/(1 N )] The effect of an increase in government spending. Using equation (12), we can write the government-spending mutipier as dy 1 Yˆ 1 gcˆ t t t p ˆ p 1 ˆ. (19) dgt g Gt g Gt This equation impies that the mutipier is ess than one whenever consumption fas in response to an increase in government spending. Equation (16) impies that the government-spending mutipier is given by dyt AY p. (20) dgt g To anayze the magnitude of the mutipier outside of the zero bound, we consider the foowing baseine parameter vaues: v p 0.85, b p 0.99, f1 p 1.5, f2 p 0, g p 0.29, (21) g p 0.2, j p 2, rr p 0, r p 0.8. These parameter vaues impy that k p 0.03 and N p 1/3. Our baseine parameter vaues impy that the government-spending mutipier is In our mode Ricardian equivaence hods. From the perspective of the representative househod, the increase in the present vaue of taxes equas the increase in the present vaue of government purchases. In a typica version of the standard neocassica mode we woud expect some rise in output driven by the negative weath effect on eisure of the tax increase. But in that mode the mutipier is generay ess than one because the weath effect reduces private consumption. From this perspective it is perhaps surprising that the mutipier in our baseine mode is greater than one. This perspective negects two key features of our mode: the frictions in price setting and the compementarity between consumption and eisure in preferences. When government purchases

12 88 journa of poitica economy increase, tota demand, Ct Gt, increases. Since prices are sticky, price over margina cost fas after a rise in demand. As emphasized in the iterature on the roe of monopoy power in business cyces, the fa in the markup induces an outward shift in the abor demand curve. This shift ampifies the rise in empoyment foowing the rise in demand. Given our specification of preferences, j 1 1 impies that the margina utiity of consumption rises with the increase in empoyment. As ong as this increase in margina utiity is arge enough, it is possibe for private consumption to actuay rise in response to an increase in government purchases. Indeed, consumption does rise in our benchmark scenario, which is why the mutipier is arger than one. To assess the importance of our preference specification, we redid our cacuations using the basic specification for the momentary utiity function commony used in the new-keynesian DSGE iterature: 1 1 c u p (C t 1)/(1 ) hn t /(1 c), (22) where, h, and c are positive. The key feature of this specification is that the margina utiity of consumption is independent of hours worked. Consistent with the intuition discussed above, we found that, across a wide set of parameter vaues, dy/dg is aways ess than one with this preference specification. 3 To provide additiona intuition for the determinants of the mutipier, we cacuate dy/dg for various parameter configurations. In each case we perturb one parameter at a time reative to the benchmark parameter vaues. Our resuts can be summarized as foows. First, we find that the mutipier is an increasing function of j. This resut is consistent with the intuition above, which buids on the observation that the margina utiity of consumption is increasing in hours worked. This dependence is stronger the higher j is. Second, the mutipier is a decreasing function of k. In other words, the mutipier is arger the higher the degree of price stickiness. This resut refects the fa in the markup when aggregate demand and margina cost rise. This effect is stronger the stickier prices are. The mutipier exceeds one for a k! In the imiting case in which prices are perfecty sticky ( k p 0), the mutipier is given by dy t [g(j 1) 1](1 r) p 1 0. dg 1 r (1 g)f t 2 Note that when f2 p 0, the mutipier is greater than one as ong as j is greater than one. When prices are perfecty fexibe ( k p ), the markup is constant. 3 See Monacei and Perotti (2008) for a discussion of the impact of preferences on the size of the government-spending mutipier in modes with Cavo-stye frictions when the zero bound is not binding.

13 government spending mutipier 89 In this case the mutipier is ess than one: dyt 1 p! 1. dgt 1 (1 g)[n/(1 N )] This resut refects the fact that with fexibe prices an increase in government spending has no impact on the markup. As a resut, the demand for abor does not rise as much as in the case in which prices are sticky. Third, the mutipier is a decreasing function of f 1. The intuition for this effect is that the expansion in output increases margina cost, which in turn induces a rise in infation. According to equation (6), the monetary authority increases the interest rate in response to a rise in infation. The rise in the interest rate is an increasing function of f 1. Higher vaues of f 1 ead to higher vaues of the rea interest rate, which are associated with ower eves of consumption. So higher vaues of f 1 ead to ower vaues of the mutipier. Fourth, the mutipier is a decreasing function of f 2. The intuition underying this effect is simiar to that associated with f1. When f2 is arge, there is a substantia increase in the rea interest rate in response to a rise in output. The contractionary effects of the rise in the rea interest rate on consumption reduce the size of the mutipier. Fifth, the mutipier is an increasing function of r R. The intuition for this resut is as foows. The higher r R, the ess rapidy the monetary authority increases the interest rate in response to the rise in margina cost and infation that occurs in the wake of an increase in government purchases. This resut is consistent with the traditiona view that the government-spending mutipier is greater in the presence of accommodative monetary poicy. By accommodative we mean that the monetary authority raises interest rates sowy in the presence of a fisca expansion. Sixth, the mutipier is a decreasing function of the parameter governing the persistence of government purchases, r. The intuition for this resut is that the present vaue of taxes associated with a given innovation in government purchases is an increasing function of r. So the negative weath effect on consumption is an increasing function of r. 4 Our numerica resuts suggest that the mutipier in a simpe new- Keynesian mode can be above one for reasonabe parameter vaues. However, it is difficut to obtain mutipiers above 1.2 for pausibe parameter vaues. 4 We redid our cacuations using a forward-ooking Tayor rue in which the interest rate responds to the one-period-ahead expected infation and output gap. The resuts that we obtained are very simiar to the ones discussed in the text.

14 90 journa of poitica economy III. The Constant Interest Rate Mutipier in a Mode without Capita In this section we anayze the government-spending mutipier in our simpe new-keynesian mode when the nomina interest rate is constant. We focus on the case in which the nomina interest rate is constant because the zero bound binds. Our basic anaysis of the mutipier buids on the work of Eggertsson and Woodford (2003), Christiano (2004), and Eggertsson (2004). As in these papers, the shock that makes the zero bound binding is an increase in the discount factor. We think of this shock as representing a temporary rise in agents propensity to save. A discount factor shock. We modify agents preferences, given by (1), to aow for a stochastic discount factor { } g 1 g 1 j [C t(1 N t) ] 1 0 t t tp0 U p E d v(g ). (23) 1 j The cumuative discount factor,, is given by d t t 1 d p 1 r1 1 r2 1 rt t (24) {1 t p 0. The time t discount factor, r, can take on two vaues, r and r t, where r! 0. The stochastic process for r t is given by Pr [r p r Fr p r ] p p, t 1 t Pr [r t 1 p rfr t p r ] p 1 p, (25) Pr [r t 1 p r Fr t p r] p 0. The vaue of rt 1 is reaized at time t. We define b p 1/(1 r), where r is the steady-state vaue of r t 1. We consider the foowing experiment. The economy is initiay in the steady state, so r p r. At time 0, r takes on the vaue r t 1. Thereafter, r t foows the process described by equation (25). The discount factor remains high with probabiity p and returns permanenty to its norma vaue, r, with probabiity 1 p. In what foows we assume that r is sufficienty high that the zero-bound constraint on nomina interest rates binds. We assume that Gˆ p Gˆ 0 in the ower bound and Gˆ t t p 0 otherwise. To sove the mode we suppose (and then verify) that the equiibrium is characterized by two vaues for each variabe: one vaue for when the zero bound binds and one vaue for when it does not. We denote the vaues of infation and output in the zero bound by p and Yˆ, respec- tivey. For simpicity we assume that rr p 0, so there is no interest rate smoothing in the Tayor rue, (6). Since there are no state variabes and

15 government spending mutipier 91 Ĝt p 0 outside of the zero-bound state, as soon as the zero bound is not binding, the economy jumps to the steady state. We can sove for Ŷ using equation (13) and the foowing version of equation (14), which takes into account the discount factor shock: Ŷ g[g(j 1) 1]Gˆ t t p (26) E {Yˆ g[g(j 1) 1]Gˆ t t 1 t 1 b(1 g)(r t 1 r t 1) (1 g)p t 1}. We focus on the case in which the zero bound binds at time t, so R t 1 p 0. Equations (13) and (26) can be rewritten as 1 g ˆ Ŷ p g[g(j 1) 1]G (br pp ) (27) 1 p and 1 N g ˆ ˆ p p bpp k( ) Y kg. (28) 1 g 1 N 1 g Equations (27) and (28) impy that p and Yˆ are given by and (1 g)k[1/(1 g) N/(1 N )]br p p D (29) [1/(1 g) N/(1 N )]g(j 1) N/(1 N ) gk(1 p) Gˆ D (1 bp)(1 g)br (1 bp)(1 p)[g(j 1) 1] pk ˆ Ŷ p gg, (30) D D where N D p (1 bp)(1 p) pk[ 1 (1 g). 1 N ] Since r is negative, a necessary condition for the zero bound to bind is that D 1 0. If this condition did not hod, infation woud be positive and output woud be above its steady-state vaue. Consequenty, the Tayor rue woud ca for an increase in the nomina interest rate so that the zero bound woud not bind. Equation (30) impies that the drop in output induced by a change in the discount rate, which we denote by V, is given by (1 bp)(1 g)br V p. (31) D By assumption D 1 0, so V! 0. The vaue of V can be a arge negative number for pausibe parameter vaues. The intuition for this resut is as foows. The basic shock to the economy is an increase in agents desire to save. We deveop the intuition for this resut in two steps. First,

16 92 journa of poitica economy Fig. 1. Simpe diagram we provide intuition for why the zero bound binds. We then provide the intuition for why the drop in output can be very arge when the zero bound binds. To understand why the zero bound binds, reca that in this economy saving must be zero in equiibrium. With competey fexibe prices the rea interest rate woud simpy fa to discourage agents from saving. There are two ways in which such a fa can occur: a arge fa in the nomina interest rate and/or a substantia rise in the expected infation rate. The extent to which the nomina interest rate can fa is imited by the zero bound. In our sticky-price economy a rise in the rate of infation is associated with a rise in output and margina cost. But a transitory increase in output is associated with a further increase in the desire to save, so that the rea interest rate must rise by even more. Given the size of the shock to the discount factor, there may be no equiibrium in which the nomina interest rate is zero and infation is positive. So the rea interest rate cannot fa by enough to reduce desired saving to zero. In this scenario the zero bound binds. Figure 1 iustrates this point using a styized version of our mode. Saving (S) is an increasing function of the rea interest rate. Since there is no investment in this economy, saving must be zero in equiibrium. The initia equiibrium is represented by point A. But the increase in the discount factor can be thought of as inducing a rightward shift in

17 government spending mutipier 93 the saving curve from S to S. When this shift is arge, the rea interest rate cannot fa enough to reestabish equiibrium because the ower bound on the nomina interest rate becomes binding prior to reaching that point. This situation is represented by point B. To understand why the fa in output can be very arge when the zero bound binds, reca that equation (29) shows how the rate of infation, p, depends on the discount rate and on government spending in the zero-bound state. In this state D is positive. Since r is negative, it foows that p is negative, and so too is expected infation, pp. Since the nom- ina interest rate is zero and expected infation is negative, the rea interest rate (nomina interest rate minus expected infation rate) is positive. Both the increase in the discount factor and the rise in the rea interest rate increase agents desire to save. There is ony one force remaining to generate zero saving in equiibrium: a arge, transitory fa in income. Other things equa, this fa in income reduces desired saving as agents attempt to smooth the margina utiity of consumption over states of the word. Because the zero bound is a transitory state of the word, this force eads to a decrease in agents desire to save. This effect has to exacty counterbaance the other two forces, which are eading agents to save more. This reasoning suggests that there is a very arge decine in income when the zero bound binds. In terms of figure 1, we can think of the temporary fa in output as inducing a shift in the saving curve to the eft. We now turn to a numerica anaysis of the government-spending mutipier, which is given by dy (1 bp)(1 p)[g(j 1) 1] pk p. (32) dg D In what foows we assume that the discount factor shock is sufficienty arge to make the zero bound binding. Conditiona on this bound being binding, the size of the mutipier does not depend on the size of the shock. In our discussion of the standard mutipier, we assume that the first-order seria correation of government spending shocks is 0.8. To make the experiment in this section comparabe, we choose p p 0.8. This choice impies that the first-order seria correation of government spending in the zero bound is aso 0.8. A other parameter vaues are given by the baseine specification in (21). For our benchmark specification the government-spending mutipier is 3.7, which is roughy three times arger than the standard mutipier. The intuition for why the mutipier can be arge when the nomina interest rate is constant, say because the zero bound binds, is as foows. A rise in government spending eads to a rise in output, margina cost, and expected infation. With the nomina interest rate equa to zero, the rise in expected infation drives down the rea interest rate, eading

18 94 journa of poitica economy to a rise is private spending. This rise in spending generates a further rise in output, margina cost, and expected infation and a further decine in the rea interest rate. The net resut is a arge rise in infation and output. The increase in income in states in which the zero bound binds raises permanent income, which raises desired expenditures in zero-bound states. This additiona channe reinforces the intertempora channe stressed above. Since the zero-bound probem is temporary, we expect that the importance of this channe is reativey sma. We now consider the sensitivity of the mutipier to parameter vaues. The first row of figure 2 dispays the government-spending mutipier and the response of output to the discount rate shock in the absence of a change in government spending as a function of the parameter k. The circe indicates resuts for our benchmark vaue of k. This row is generated assuming a discount factor shock such that r is equa to 2 percent on an annuaized basis. We graph ony vaues of k for which the zero bound binds, so we dispay resuts for 0.02 k Three key features of this figure are worth noting. First, the mutipier can be very arge. Second, without a change in government spending, the decine in output is increasing in the degree of price fexibiity; that is, it is increasing in k as ong as the zero bound binds. This resut refects that, conditiona on the zero bound binding, the more fexibe prices are, the higher the expected defation and the higher the rea interest rate. So, other things equa, higher vaues of k require a arge transitory fa in output to equate saving and investment when the zero bound binds. 5 Third, the government-spending mutipier is aso an increasing function of k. The second row of figure 2 dispays the government-spending mutipier and the response of output to the discount rate shock in the absence of a change in government spending as a function of the parameter p. The asterisk indicates resuts for our benchmark vaue of p. We graph ony vaues of p for which the zero bound binds, so we dispay resuts for 0.75 p Two key resuts are worth noting. First, without a change in government spending, the decine in output is increasing in p. So the onger the expected duration of the shock, the worse the output consequences of the zero bound being binding. Second, the vaue of the government-spending mutipier is an increasing function of p. Figure 2 shows that the precise vaue of the mutipier is sensitive to the choice of parameter vaues. But ooking across parameter vaues, we see that the government-spending mutipier is arge in economies 5 The basic ogic here is consistent with the intuition in De Long and Summers (1986) about the potentiay destabiizing effects of margina increases in price fexibiity.

19 Fig. 2. Government-spending mutipier when the zero bound is binding (mode with no capita)

20 96 journa of poitica economy in which the drop in output associated with the zero bound is aso arge. Put differenty, fisca poicy is particuary powerfu in economies in which the zero-bound state entais arge output osses. One more way to see this resut is to anayze the impact of changes in N, which governs the easticity of abor suppy, on dy/dg and V. Equations (31) and (32) impy that dy (1 bp)(1 p)[g(j 1) 1] pk p V. (33) dg (1 bp)(1 g)br From equation (31) we see that changes in N that make D converge to zero impy that V, the impact of the discount factor shock on output, converges to minus infinity. It foows directy from equation (33) that the same changes in N cause dy/dg to go to infinity. So, again we concude that the government-spending mutipier is particuary arge in economies in which the output costs of being in the zero-bound state are very arge. 6 Sensitivity to the timing of government spending. In practice, there is ikey to be a ag between the time at which the zero bound becomes binding and the time at which additiona government purchases begin. A natura question is, how does the economy respond at time t to the knowedge that the government wi increase spending in the future? Consider the foowing scenario. At time t the zero bound binds. Government spending does not change at time t, but it takes on the vaue G 1 G from time t 1 on, as ong as the economy is in the zero bound. Under these assumptions, equations (13) and (26) can be written as 1 N p p bpp k Y ˆ t ( ) t (34) 1 g 1 N and ˆ ˆ ˆ Yt p (1 g)br py g[g(j 1) 1]pG (1 g)pp. (35) Here we use the fact that Gˆ,, ˆ ˆ t p 0 E t(p t 1) p pp E t(g t 1) p pg, and E ˆ ˆ. The vaues of and ˆ t(y t 1) p py p Y are given by equations (29) and (30), respectivey. Using equation (30) to repace Ŷ in equation (35), we obtain dy 1 g p dp t,1 p. (36) ˆ dg g 1 p dg Here the subscript 1 denotes the presence of a one-period deay in impementing an increase in government spending. So dy t,1/dg rep- resents the impact on output at time t of an increase in government spending at time t 1. One can show that the mutipier is increasing 6 An exception pertains to the parameter j. The vaue of dy /dg is monotonicay increasing in j, but dy ˆ /dr is independent of j.

21 government spending mutipier 97 in the probabiity, p, that the economy remains in the zero bound. The mutipier operates through the effect of a future increase in government spending on expected infation. If the economy is in the zero bound in the future, an increase in government purchases increases future output and therefore future infation. From the perspective of time t, this effect eads to higher expected infation and a ower rea interest rate. This ower rea interest rate reduces desired saving and increases consumption and output at time t. Evauating equation (36) at the benchmark vaues, we obtain a mutipier equa to 1.5. Whie this mutipier is much ower than the benchmark mutipier of 3.7, it is sti arge. Moreover, this mutipier pertains to an increase in today s output in response to an increase in future government spending that occurs ony if the economy is in the zerobound state in the future. Suppose that it takes two periods for government purchases to increase in the event that the zero bound binds. It is straightforward to show that the impact on current output of a potentia increase in government spending that takes two periods to impement is given by dy 1 g dp 1 dp t,2 t,1 p p. ( dg g dgˆ 1 p dgˆ ) Here the subscript 2 denotes the presence of a two-period deay. With our benchmark parameters, the vaue of this mutipier is 1.44, so the rate at which the mutipier decines as we increase the impementation ag is reativey ow. Consider now the case in which the increase in government spending occurs ony after the zero bound ends. Suppose, for exampe, that at time t the government promises to impement a persistent increase in government spending at time t 1 if the economy emerges from the zero bound at time t 1. This increase in government purchases is j 1 governed by Gˆ ˆ t j p 0.8 Gt 1 for j 2. In this case the vaue of the mutipier, dy t/dgt 1, is ony 0.46 for our benchmark vaues. The usua objection to using fisca poicy as a too for fighting recessions is that there are ong ags in gearing up increases in spending. Our anaysis indicates that the key question is, in which state of the word does additiona government spending come on ine? If it comes on ine in future periods when the zero bound binds, there is a arge effect on current output. If it comes on ine in future periods when the zero bound is not binding, the current effect on government spending is smaer. Optima government spending. The fact that the government-spending mutipier is so arge in the zero bound raises the foowing question: taking as given the monetary poicy rue described by equation (6), what is the optima eve of government spending when the representative

22 98 journa of poitica economy agent s discount rate is higher than its steady-state eve? In what foows we use the superscript L to denote the vaue of variabes in states of the word in which the discount rate is r. In these states of the word the zero bound may or may not be binding, depending on the eve of government spending. From equation (29) we anticipate that the higher government spending is, the higher expected infation is and the ess ikey the zero bound is to bind. L We choose G to maximize the expected utiity of the consumer in states of the word in which the discount factor is high and the zero bound binds. For now we assume that in other states of the word Ĝ is L zero. So we choose G to maximize ( ) } } t L g L 1 g 1 j p [(C )(1 N ) ] 1 L L U p v(g ) 1 r { tp0 1 j (37) r L g L 1 g 1 j 1 r [(C )(1 N ) ] 1 L p v(g ). 1 r p { 1 j L To ensure that U is finite, we assume that p! 1 r. Note that L L ˆ L Y p N p Y(Y 1), ˆ L L L C p Y(Y 1) G(G 1). Substituting these expressions into equation (37), we obtain r ˆ L ˆ L g ˆ L 1 g 1 j 1 r {[N(Y 1) Ng(G 1)] [1 N(Y 1)] } 1 L U p 1 r p ( 1 j ) r 1 r L v[ng(gˆ 1)]. 1 r p L L We choose the vaue of Ĝ that maximizes U subject to the intertempora Euer equation (eq. [14]), the Phiips curve (eq. [13]), and L L L L L Yˆ ˆ, ˆ, ˆ t p Y Gt p G E(G t t 1) p pg, pt 1 p p, E t(p t 1) p pp, and L R t 1 p R, where L L R p max (Z,0) and 1 1 L L ˆ L Z p 1 (f1p f2y ). b b The ast constraint takes into account that the zero bound on interest rates may not be binding even though the discount rate is high. Finay, for simpicity we assume that v(g) is given by ˆ

23 government spending mutipier 99 G 1 j v(g) p w. g 1 j We choose wg so that g p G/Y p 0.2. Since government purchases are financed with ump-sum taxes, the optima eve of G has the property that the margina utiity of G is equa to the margina utiity of consumption: j g(1 j) 1 (1 g)(1 j) wg g p gc N. This reation impies g(1 j) 1 (1 g)(1 j) j wg p g{[n(1 g)]} N (Ng). Using our benchmark parameter vaues, we obtain a vaue of w g equa to L L L L L L Figure 3 dispays the vaues of U, Yˆ, Z, Cˆ, R, and p as a function of Ĝ L. The asterisk indicates the eve of a variabe corresponding to the optima vaue of Ĝ L. The circe indicates the eve of a variabe corre- L sponding to the highest vaue of Ĝ that satisfies Z 0. A number of L features of figure 3 are worth noting. First, the optima vaue of Ĝ is very arge: roughy 30 percent (reca that in the steady state government purchases are 20 percent of output). Second, for this particuar parameterization the increase in government spending more than undoes the effect of the shock that made the zero-bound constraint bind. Here, government purchases rise to the point where the zero bound is marginay nonbinding and output is actuay above its steady-state eve. These ast two resuts depend on the parameter vaues that we chose and on our assumed functiona form for v(g t ). What is robust across different assumptions is that it is optima to substantiay increase government purchases and that the government-spending mutipier is arge when the zero-bound constraint binds. 7 The zero bound and interest rate targeting. Up to now we have emphasized the economy being in the zero-bound state as the reason why the nomina interest rate might not change after an increase in government spending. Here we discuss an aternative interpretation of the constant interest rate assumption. Suppose that there are no shocks to the economy but that, starting from the nonstochastic steady state, government spending increases by a constant amount and the monetary authority deviates from the Tayor rue, keeping the nomina interest rate equa to its steady-state vaue. This poicy shock persists with probabiity p. It is easy to show that the government-spending mutipier is given by 7 We derive the optima fisca poicy taking monetary poicy as given. Nakata (2009) argues that it is aso optima to raise government purchases when monetary poicy is chosen optimay. He does so using a second-order Tayor approximation to the utiity function in a mode with separabe preferences in which the natura rate of interest foows an exogenous stochastic process.

24 Fig. 3. Optima eve of government spending in the zero bound

25 government spending mutipier 101 equation (32). So the mutipier is exacty the same as in the case in which the nomina interest rate is constant because the zero bound binds. Of course there is no reason to think that it is sensibe for the centra bank to pursue a poicy that sets the nomina interest rate equa to a positive constant. For this reason, a binding zero bound is the most natura interpretation for why the nomina interest rate might not change after an increase in government spending. IV. A Mode with Capita In the previous section we use a simpe mode without capita to argue that the government-spending mutipier is arge whenever the output costs of being in the zero-bound state are aso arge. Here we show that this basic resut extends to a generaized version of the previous mode in which we aow for capita accumuation. As above we focus on the effect of a discount rate shock. 8 The mode. The preferences of the representative househod are given by equations (23) and (24). The househod s budget constraint is given by k P(C t t I t) Bt 1 p B(1 t R t) WN t t Pr t tkt T t, (38) k where It denotes investment, Kt is the stock of capita, and rt is the rea renta rate of capita. The capita accumuation equation is given by K t 1 p It (1 d)k t D(I t, I t 1, K t), (39) where the function D(I t, I t 1, K t) represents investment adjustment costs. To assess robustness we consider two specifications for these adjustment costs. The first specification is the one considered in Lucas and Prescott (1971): ( ) 2 ji It D(I t, I t 1, K t) p d K t. (40) 2 K t The parameter ji 1 0 governs the magnitude of adjustment costs to capita accumuation. As ji r, investment and the stock of capita become constant. The resuting mode behaves in a manner very simiar to the one described in the previous section. The second specification is the one considered in Christiano et a. (2005) and in Section V: [ ( )] I t t t 1 t I t 1 t D(I, I, K ) p 1 S I. (41) 8 In a previous version of this paper, avaiabe on request, we aso anayze the effect of a neutra and an investment-specific technoogy shock.