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When is the Government Spending Multiplier
Large?
Lawrence Christiano, Martin Eichenbaum, and Sergio Rebelo
Northwestern University
December 2010
Abstract We argue that the government-spending multiplier can be much larger than one when the zero lower bound on the nominal interest rate binds. The larger is the fraction of government spending that occurs while the nominal interest rate is zero, the larger is the value of the multiplier. After providing intuition for these results, we investigate the size of the multiplier in a dynamic, stochastic, general equilibrium model. In this model the multiplier eect is substantially larger than one when the zero bound binds. Our model is consistent with the behavior of key macro aggregates during the recent financial crisis.
(^) We thank the editor, Monika Piazzesi, Rob Shimer, and two anonymous referees for their comments.
1. Introduction
A classic question in macroeconomics is: what is the size of the government- spending multiplier? There is a large empirical literature that grapples with this question. Authors such as Barro (1981) argue that the multiplier is around 0. 8 while authors such as Ramey (2008) estimate the multiplier to be closer to 1. 2.^1 There is also a large literature that uses general-equilibrium models to study the size of the government-spending multiplier. In standard new-Keynesian models the government-spending multiplier can be somewhat above or below one depend- ing on the exact specification of agent’s preferences (see Gali, López-Salido, and Vallés (2007) and Monacelli and Perotti (2008)). In frictionless real-business-cycle models this multiplier is typically less than one (see e.g. Aiyagari, Christiano, and Eichenbaum (1992), Baxter and King (1993), Burnside, Eichenbaum and Fisher (2004), Ramey and Shapiro (1998), and Ramey (2011)). Viewed overall it is hard to argue, based on the literature, that the government-spending multiplier is substantially larger than one. In this paper we argue that the government-spending multiplier can be much larger than one when the nominal interest rate does not respond to an increase in government spending. We develop this argument in a model where the multiplier is quite modest if the nominal interest rate is governed by a Taylor rule. When such a rule is operative the nominal interest rate rises in response to an expansionary fiscal policy shock that puts upward pressure on output and inflation. There is a natural scenario in which the nominal interest rate does not respond to an increase in government spending: when the zero lower bound on the nominal interest rate binds. We find that the multiplier is very large in economies where (^1) For recent contributions to the VAR-based empirical literature on the size of the government- spending multiplier see Fisher and Peters (2010) and Ilzetzki, Mendoza, and Vegh (2009). Hall (2009) provides an analysis and review of the empirical literature.
large fall in output. Consider now the eect of an increase in government spending when the zero bound is strictly binding. This increase leads to a rise in output, marginal cost and expected inflation. With the nominal interest rate stuck at zero, the rise in expected inflation drives down the real interest rate, which drives up private spending. This rise in spending leads to a further rise in output, marginal cost, and expected inflation and a further decline in the real interest rate. The net result is a large rise in output and a large fall in the rate of deflation. In eect, the increase in government consumption counteracts the deflationary spiral associated with the zero-bound state. The exact value of the government-spending multiplier depends on a variety of factors. However, we show that this multiplier is large in economies in which the output cost associated with the zero-bound problem is more severe. We argue this point in two ways. First, we show that the value of the government-spending multiplier can depend sensitively on the model’s parameter values. But, parameter values which are associated with large declines in output when the zero bound binds are also associated with large values of the government-spending multiplier. Second, we show that the value of the government-spending multiplier is positively related to how long the zero bound is expected to bind. An important practical objection to using fiscal policy to counteract a contrac- tion associated with the zero-bound state is that there are long lags in implement- ing increases in government spending. Motivated by this consideration, we study the size of the government-spending multiplier in the presence of implementation lags. We find that a key determinant of the size of the multiplier is the state of the world in which new government spending comes on line. If it comes on line in future periods when the nominal interest rate is zero then there is a large eect on current output. If it comes on line in future periods where the nominal interest
rate is positive, then the current eect on government spending is smaller. So our analysis supports the view that, for fiscal policy to be eective, government spending must come online in a timely manner. In the second step of our analysis we incorporate capital accumulation into the model. For computational reasons we consider temporary shocks that make the zero bound binding for a deterministic number of periods. Again, we find that the government-spending multiplier is larger when the zero bound binds. Allowing for capital accumulation has two eects. First, for a given size shock it reduces the likelihood that the zero bound becomes binding. Second, when the zero bound binds, the presence of capital accumulation tends to increase the size of the government-spending multiplier. The intuition for this result is that, in our model, investment is a decreasing function of the real interest rate. When the zero bound binds, the real interest rate generally rises. So, other things equal, saving and investment diverge as the real interest rate rises, thus exacerbating the meltdown associated with the zero bound. As a result, the fall in output necessary to bring saving and investment into alignment is larger than in the model without capital. The simple models discussed above suggest that the multiplier can be large in the zero-bound state. The obvious next step would be to use reduced form meth- ods, such as identified VARs, to estimate the government-spending multiplier when the zero bound binds. Unfortunately, this task is fraught with diculties. First, we cannot mix evidence from states where the zero bound binds with evi- dence from other states because the multipliers are very dierent 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. Almost surely gov- (^2) To see how critical this step is, suppose that the government chooses spending to keep output exactly constant in the face of shocks that make the zero bound bind. A naive econo- metrician who simply regressed output on government spending would falsely conclude that the
smaller is the value of the multiplier. Consistent with the theoretical analysis above, this result implies that for government spending to be a powerful weapon in combating output losses associated with the zero-bound state, it is critical that the bulk of the spending come on line when the lower bound is actually binding. Fourth, we find that the model generates sensible predictions for the current crisis under the assumption that the zero bound binds. In particular the model does well at accounting for the behavior of output, consumption, investment, inflation, and short-term nominal interest rates. As emphasized by Eggertsson and Woodford (2003), an alternative way to escape the negative consequences of a shock that makes the zero bound binding is for the central bank to commit to future inflation. We abstract from this possibility in this paper. We do so for a number of reasons. First, this theoretical possibility is well understood. Second, we do not think that it is easy in practice for the central bank to credibly commit to future high inflation. Third, the optimal trade-o between higher government purchases and anticipated inflation depends sensitively on how agents value government purchases and the costs of anticipated inflation. Studying this issue is an important topic for future research. Our analysis builds on Christiano (2004) and Eggertsson (2004) who argue that increasing government spending is very eective when the zero bound binds. Eggertsson (2011) analyzes both the eects of increases in government spending and transitory tax cuts when the zero bound binds. The key contributions of this paper are to analyze the size of the multiplier in a medium-size DSGE model, study the model’s performance in the financial crisis that began in 2008, and quantify the importance of the timing of government spending relative to the timing of the zero bound. Our analysis is related to several recent papers on the zero bound. Bodenstein, Erceg, and Guerrieri (2009) analyze the eects of shocks to open economies when
the zero bound binds. Braun and Waki (2006) use a model in which the zero bound binds to account for Japan’s experience in the 1990s. Their results for fiscal policy are broadly consistent with our results. Braun and Waki (2006) and Coenen and Wieland (2003) investigate whether alternative monetary policy rules could have avoided the zero bound state in Japan. Our paper is organized as follows. In section 2 we analyze the size of the government-spending multiplier when the interest follows a Taylor rule in a stan- dard new-Keynesian model without capital. In section 3 we modify the analysis to assume that the nominal interest rate does not respond to an increase in govern- ment spending, say because the lower bound on the nominal interest rate binds. In section 4 we extend the model to incorporate capital. In section 5 we discuss the properties of the government-spending multiplier in the medium size DSGE model proposed by ACEL and investigate the performance of the model during the recent financial crisis. Section 6 investigates the sensitivity of our conclusions to the presence of distortionary taxes. Section 7 concludes.
2. The standard multiplier in a model without capital
In this section we present a simple new-Keynesian model and analyze its implica- tions for the size of the “standard multiplier,” by which we mean the size of the government-spending multiplier when the nominal interest rate is governed by a Taylor rule.
Households The economy is populated by a representative household, whose life-time utility, U , is given by:
U = E 0
^
t=
t
[C (^) t (1 N (^) t ) 1 ^ ]^1 ^ 1 1 +^ v^ (G^ t^ )
where N (^) t (i) denotes employment by the ith^ monopolist. We assume there is no entry or exit into the production of the ith^ intermediate good. The monopolist is subject to Calvo-style price-setting frictions and can optimize its price, P (^) t (i), with probability 1 . With probability the firm sets:
P (^) t (i) = P (^) t 1 (i).
The discounted profits of the i th^ intermediate good firm are:
Et
^
j=
j^ (^) t+j [P (^) t+j (i) Y (^) t+j (i) (1 ) W (^) t+j N (^) t+j (i)] , (2.5)
where = 1/ denotes an employment subsidy which corrects, in steady state, the ineciency created by the presence of monopoly power. The variable (^) t+j is the multiplier on the household budget constraint in the Lagrangian representation of the household problem. The variable W (^) t+j denotes the nominal wage rate. Firm i maximizes its discounted profits, given by equation (2.5), subject to the Calvo price-setting friction, the production function, and the demand function for Y (^) t (i), given by equation (2.4).
Monetary policy We assume that monetary policy follows the rule:
R (^) t+1 = max(Z (^) t+1 , 0), (2.6)
where
Z (^) t+1 = (1/)(1 + (^) t ) ^1 (1^ R^ )^ (Y (^) t /Y ) ^2 (1^ R^ )^ [ (1 + R (^) t )]^ R^ 1.
Throughout the paper a variable without a time subscript denotes its steady state value, e.g. the variable Y denotes the steady-state level of output. The variable (^) t denotes the time-t rate of inflation. We assume that 1 > 1 and 2 (0, 1).
According to equation (2.6) the monetary authority follows a Taylor rule as long as the implied nominal interest rate is non-negative. Whenever the Taylor rule implies a negative nominal interest rate, the monetary authority simply sets the nominal interest rate to zero. For convenience we assume that steady-state in- flation is zero. This assumption implies that the steady-state net nominal interest rate is 1 / 1.
Fiscal policy As long as the zero bound on the nominal interest rate is not binding, government spending evolves according to:
G (^) t+1 = G t exp( (^) t+1 ). (2.7)
Here G is the level of government spending in the non-stochastic steady state and (^) t+1 is an i.i.d. shock with zero mean. To simplify our analysis, we assume that government spending and the employment subsidy are financed with lump-sum taxes. The exact timing of these taxes is irrelevant because Ricardian equivalence holds under our assumptions. We discuss the details of fiscal policy when the zero bound binds in Section 3.
Equilibrium The economy’s resource constraint is:
Ct + G (^) t = Y (^) t. (2.8)
A ‘monetary equilibrium’ is a collection of stochastic processes,
{Ct , Nt , W (^) t , Pt , Yt , Rt , Pt (i) , Yt (i) , Nt (i) , (^) t , Bt+1 , (^) t },
such that for given {G (^) t } the household and firm problems are satisfied, the mone- tary and fiscal policy rules are satisfied, markets clear, and the aggregate resource constraint is satisfied.
Similarly, combining equations (2.11) and (2.12) and using the fact that Nˆ (^) t = Yˆ (^) t we obtain:
Y^ ˆ (^) t g [ ( 1) + 1] Gˆ (^) t = (2.14) Et
(1 g) [ (R (^) t+1 R) (^) t+1 ] + Yˆ (^) t+1 g [ ( 1) + 1] Gˆ (^) t+
As long as the zero bound on the nominal interest rate does not bind, the linearized monetary policy rule is given by:
R (^) t+1 R = (^) R (R (^) t R) +^1 ^ ^ R
1 (^) t + 2 Yˆ (^) t
Whenever the zero bound binds, R (^) t+1 = 0. We solve for the equilibrium using the method of undetermined coecients. For simplicity, we begin by considering the case in which (^) R = 0. Under the assumption that 1 > 1 , there is a unique linear equilibrium in which (^) t and Yˆ (^) t are given by:
(^) t = A Gˆ (^) t , (2.15) Y^ ˆ (^) t = AY Gˆ (^) t. (2.16)
The coecients A and AY are given by:
A = (^1)
1 g +^
N
1 N
A (^) Y (^1) g g
A (^) Y = g (^ ^ ^1 )^ ^ ^ [^ (^ ^ 1) + 1] (1^ ^ ) (1^ ^ ) (1 ) [ 1 (1 g) 2 ] + (1 g) ( 1 )
1 g +^1 NN
The eect of an increase in government spending Using equation (2.12) we can write the government-spending multiplier as:
dY (^) t dG (^) t^ =
g
Y^ ˆ (^) t G^ ˆ (^) t^ = 1 +
1 g g
C^ ˆt G^ ˆ (^) t^.^ (2.19)
This equation implies that the multiplier is less than one whenever consumption falls in response to an increase in government spending. Equation (2.16) implies that the government-spending multiplier is given by: dY (^) t dG (^) t^ =^
AY
g.^ (2.20) To analyze the magnitude of the multiplier outside of the zero bound we con- sider the following baseline parameter values:
= 0. 85 , = 0. 99 , 1 = 1. 5 , 2 = 0, = 0. 29 , g = 0. 2 , = 2, (^) R = 0, = 0. 8. (2.21) These parameter values imply that = 0. 03 and N = 1/ 3. Our baseline parameter values imply that the government-spending multiplier is 1. 05. In our model Ricardian equivalence holds. From the perspective of the repre- sentative household, the increase in the present value of taxes equals the increase in the present value of government purchases. In a typical version of the standard neoclassical model we would expect some rise in output driven by the negative wealth eect on leisure of the tax increase. But in that model the multiplier is generally less than one because the wealth eect reduces private consumption. From this perspective it is perhaps surprising that the multiplier in our base- line model is greater than one. This perspective neglects two key features of our model: the frictions in price setting and the complementarity between con- sumption and leisure in preferences. When government purchases increase, total demand, Ct + G (^) t , increases. Since prices are sticky, price over marginal cost falls after a rise in demand. As emphasized in the literature on the role of monopoly power in business cycles, the fall in the markup induces an outward shift in the labor demand curve. This shift amplifies the rise in employment following the rise in demand. Given our specification of preferences, > 1 implies that the marginal utility of consumption rises with the increase in employment. As long as
by: dY (^) t dG (^) t^ =
[ ( 1) + 1] (1 )
1 + (1 g) 2 >^0. Note that when 2 = 0 the multiplier is greater than one as long as is greater than one. When prices are perfectly flexible ( = ) the markup is constant. In this case the multiplier is less than one:
dY (^) t dG (^) t^ =^
1 + (1 g) (^1) NN^ <^1.
This result reflects the fact that with flexible prices an increase in government spending has no impact on the markup. As a result, the demand for labor does not rise as much as in the case in which prices are sticky. Third, the multiplier is a decreasing function of 1. The intuition for this eect is that the expansion in output increases marginal cost which in turn induces a rise in inflation. According to equation (2.6) the monetary authority increases the interest rate in response to a rise in inflation. The rise in the interest rate is an increasing function of 1. Higher values of 1 lead to higher values of the real interest rate which are associated with lower levels of consumption. So, higher values of 1 lead to lower values of the multiplier. Fourth, the multiplier is a decreasing function of 2. The intuition underlying this eect is similar to that associated with 1. When 2 is large there is a substantial increase in the real interest rate in response to a rise in output. The contractionary eects of the rise in the real interest rate on consumption reduce the size of the multiplier. Fifth, the multiplier is an increasing function of (^) R. The intuition for this result is as follows. The higher is (^) R the less rapidly the monetary authority increases the interest rate in response to the rise in marginal cost and inflation that occur in the wake of an increase in government purchases. This result is
consistent with the traditional view that the government-spending multiplier is greater in the presence of accommodative monetary policy. By accommodative we mean that the monetary authority raises interest rates slowly in the presence of a fiscal expansion. Sixth, the multiplier is a decreasing function of the parameter governing the persistence of government purchases, . The intuition for this result is that the present value of taxes associated with a given innovation in government purchases is an increasing function of . So the negative wealth eect on consumption is an increasing function of .^4 Our numerical results suggest that the multiplier in a simple new-Keynesian model can be above one for reasonable parameter values. However, it is dicult to obtain multipliers above 1. 2 for plausible parameter values.
3. The constant-interest-rate multiplier in a model without
capital
In this section we analyze the government-spending multiplier in our simple new- Keynesian model when the nominal interest rate is constant. We focus on the case in which the nominal interest rate is constant because the zero bound binds. Our basic analysis of the multiplier builds on the work of Christiano (2004) and Eggertsson (2004) and Eggertsson and Woodford (2003). 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. (^4) We redid our calculations using a forward-looking Taylor rule in which the interest rate re- sponds to the one-period-ahead expected inflation and output gap. The results that we obtained are very similar to the ones discussed in the main text.
We can solve for Yˆ l^ using equation (2.13) and the following version of equation (2.14), which takes into account the discount factor shock:
Y^ ˆ (^) t g [( 1) + 1] Gˆ (^) t (3.4) = Et
Yˆ (^) t+1 g [( 1) + 1] Gˆ (^) t+1 (1 g) (R (^) t+1 r (^) t+1 ) + (1 g) (^) t+
We focus on the case in which the zero bound binds at time t, so R (^) t+1 = 0. Equations (2.13) and (3.4) can be re-written as:
Y^ ˆ l^ = g [ ( 1) + 1] Gˆ l^ +^1 ^ g 1 p
r l (^) + p l (^) , (3.5)
l^ = p l^ +
1 g +^
N
1 N
Y^ ˆ l^ g 1 g ^ Gˆ l^. (3.6)
Equations (3.5) and (3.6) imply that l^ and Yˆ l^ are given by:
l^ =
(1 g)
1 g +^1 NN
r l +g(1p)
1 g +^1 NN
( 1) + 1 NN
Gˆ l^. (3.7)
Y^ ˆ l^ = (1^ ^ p) (1^ ^ g)r^
l +
(1 p) (1 p) [ ( 1) + 1] p g^ Gˆ l^ , (3.8)
where: = (1 p) (1 p) p
1 + (^1) N N (1 g)
Since r l^ is negative, a necessary condition for the zero bound to bind is that > 0. If this condition did not hold inflation would be positive and output would be above its steady state value. Consequently, the Taylor rule would call for an increase in the nominal interest rate so that the zero bound would not bind. Equation (3.8) implies that the drop in output induced by a change in the discount rate, which we denote by , is given by:
= (1^ ^ p) (1^ ^ g)r^
l .^ (3.9)
By assumption > 0 , so < 0. The value of can be a large negative number for plausible parameter values. The intuition for this result is as follows. The basic shock to the economy is an increase in agent’s desire to save. We develop the intuition for this result in two steps. First, we provide intuition for why the zero bound binds. We then provide the intuition for why the drop in output can to be very large when the zero bound binds. To understand why the zero bound binds, recall that in this economy saving must be zero in equilibrium. With completely flexible prices the real interest rate would simply fall to discourage agents from saving. There are two ways in which such a fall can occur: a large fall in the nominal interest rate and/or a substantial rise in the expected inflation rate. The extent to which the nominal interest rate can fall is limited by the zero bound. In our sticky-price economy a rise in the rate of inflation is associated with a rise in output and marginal cost. But a transitory increase in output is associated with a further increase in the desire to save, so that the real interest rate must rise by even more. Given the size of the shock to the discount factor, there may be no equilibrium in which the nominal interest rate is zero and inflation is positive. So the real interest rate cannot fall by enough to reduce desired saving to zero. In this scenario the zero bound binds. Figure 1 illustrates this point using a stylized version of our model. Saving (S) is an increasing function of the real interest rate. Since there is no investment in this economy saving must be zero in equilibrium. The initial equilibrium is represented by point A. But the increase in the discount factor can be thought of as inducing a rightward shift in the saving curve from S to S ^. When this shift is large, the real interest rate cannot fall enough to re-establish equilibrium because the lower bound on the nominal interest rate becomes binding prior to reaching that point. This situation is represented by point B. To understand why the fall in output can be very large when the zero bound
