Should We Use Linearized Models to Calculate Fiscal Multipliers?

We calculate the magnitude of the government consumption multiplier in linearized and nonlinear solutions of a New Keynesian model at the zero lower bound. Importantly, the model is amended with real rigidities to simultaneously account for the macroeconomic evidence of a low Phillips curve slope and the microeconomic evidence of frequent price changes. We show that the nonlinear solution is associated with a much smaller multiplier than the linearized solution in long-lived liquidity traps, and pin down the key features in the model which account for the difference. Our results caution against the common practice of using linearized models to calculate fiscal multipliers in long-lived liquidity traps.


Introduction
The magnitude of the …scal spending multiplier is a classic subject in macroeconomics. To calculate the magnitude of the multiplier, economists typically employ a linearized version of their actual nonlinear model. Does linearizing the nonlinear model matter for the conclusions about the multiplier? We document this may be the case, especially in long-lived liquidity traps. When interest rates are expected to be constrained by the zero (or e¤ective) lower bound for a protracted time period, the nonlinear solution suggests a much smaller multiplier than the linearized solution of the same model.
The …nancial crisis and "Great Recession" have revived interest in the magnitude of the …scal spending multiplier. A quickly growing literature suggests that the …scal spending multiplier can be very large when nominal interest rates are expected to be constrained by the zero (or e¤ective) lower bound (ZLB henceforth) for a prolonged period, see e.g. Eggertsson (2010), Davig and Leeper (2011), Christiano, Eichenbaum and Rebelo (2011), Woodford (2011), Coenen et al. (2012) and Leeper, Traum and Walker (2015). Erceg and Lindé (2014) show that in a long-lived liquidity trap …scal stimulus can be self-…nancing. Conversely, the results of the above literature suggest that it is hard to reduce government debt in the short-run through aggressive government spending cuts in long-lived liquidity traps: …scal consolidation can in fact be self-defeating in such a situation.
Importantly, the bulk of the existing literature analyzes …scal multipliers in models where all equilibrium equation have been linearized around the steady state, except for the ZLB constraint on the monetary policy rule. Implicit in the linearization procedure is the assumption that the linearized solution is accurate even far away from the steady state. However, recent work by Boneva, Braun, and Waki (2016) suggests that linearization produces severely misleading results at the zero lower bound. Essentially, Boneva et al. argue that extrapolating decision rules far away from the steady state is invalid.
Our paper provides a positive analysis of the e¤ect of spending-based …scal stimulus on output and government debt using a fully nonlinear model. We compare the …scal spending multipliers for output and government debt of the nonlinear and linearized solution as function of the liquidity trap duration. Moreover, our framework allows us to pin down the key features which account for the di¤erence between the multiplier schedule for the nonlinear and linearized solutions of the model. The New Keynesian model employed in our analysis features monopolistic competition and Calvo sticky prices. The central bank follows a Taylor rule subject to the ZLB constraint on the nominal interest rate. The key di¤erence to existing work is that we introduce real rigidities into the model using the Kimball (1995) aggregator. The Kimball aggregator aggregates intermediate goods into a …nal good. The Kimball aggregator is commonly used in New Keynesian models, see e.g. Smets and Wouters (2007), as it allows to simultaneously account for the macroeconomic evidence of a low Phillips curve slope and the microeconomic evidence of frequent price changes.
The key …nding of our paper is that in a long-lived liquidity trap the fully nonlinear model implies a much smaller …scal spending multiplier than the linearized version of the same model.
More precisely, when the ZLB binds for 12 quarters, the nonlinear model implies a multiplier of about 0.7 while the linearized version of the same model implies a multiplier in excess of 2. 1 Importantly, our analysis suggest that the nonlinear model is incapable of producing a multiplier that is close to or exceeds unity when government spending follows a standard AR(1) process.
What accounts for the large di¤erence between the nonlinear and linearized solutions in a prolonged liquidity trap? We document that the di¤erence can almost entirely be accounted for by the nonlinearities in the price setting block of the model -the Phillips curve. Key here is the nonlinearity implied by the Kimball aggregator. The Kimball aggregator implies that the demand elasticity for intermediate goods is state-dependent, i.e. the …rms'demand elasticity is an increasing function of its relative price. In short, the demand curve is quasi-kinked. While the fully nonlinear model takes this state-dependency explicitly into account, a linear approximation replaces that nonlinearity by a linear function. Put di¤erently, linearization replaces the quasi-kinked demand curve with a linear function. 2 Intuitively, in a deep recession that triggers the ZLB to bind for a long time, the Kimball aggregator carries the implication that …rms do not …nd it attractive to cut their prices much since that reduces the demand elasticity and thereby does not crowd in more demand. With more …scal spending in such a situation, …rms also …nd it less attractive to increase their prices. Thus -with policy rates stuck at zero -aggregate in ‡ation increases only little and therefore the real interest rate falls by little: the multiplier does not increase to the same extent with the duration of the ZLB. When the model is linearized, the response of aggregate in ‡ation is notably stronger due to the nature of a linear approximation of a quasi-kinked demand curve at the steady state with no dispersion. Hence, the drop in the real interest rate is elevated following a spending hike and the multiplier is magni…ed. The bottom line: the linearized version of the model exaggerates the rise in expected and actual in ‡ation due to a sizable approximation error and thereby elevates the magnitude of the …scal multiplier in long-lived liquidity traps.
We perform several robustness checks. In particular, we compare our benchmark results based on the Kimball (1995) aggregator to those when a Dixit-Stiglitz (1977) aggregator is used instead.
We also examine the importance of how the model economy is taken to the zero lower bound. In addition we also study the e¤ects of the price indexation for the resulting multiplier. Moreover, we investigate the sensitivity of our results with respect to the government spending process. Finally, and perhaps most importantly, we compare the sensitivity of our results with respect to the solution method of the nonlinear model. Our benchmark solution method is based on Fair and Taylor (1983).
That solution method solves the model by imposing certainty equivalence. As a robustness check, we also solve the model using global methods, i.e. solving the model without certainty equivalence.
In other words, we compare the deterministic solution of the linear and nonlinear model with the fully stochastic linear and nonlinear model solution in which agent's decision rules are a¤ected by the variance of shocks hitting the economy. 3 Importantly, we document that the …scal multiplier in the nonlinear model is little a¤ected by shock uncertainty. The nonlinearity of the Kimball aggregator and the low slope of the Phillips curve based on macro-and micoeconomic evidence are responsible for that result. By contrast, the linear model is a¤ected in a dramatic way by shock uncertainty: the …scal multiplier is even more elevated due to the approximation error. Hence, our basic …nding of an signi…cant di¤erence between the linearized and nonlinear solutions in long-lived liquidity traps is even further strengthened when allowing for future shock uncertainty.
We argue that the results based on the Kimball speci…cation appears to dominate those based on the Dixit-Stiglitz speci…cation for at least two reasons. First, in contrast to Dixit-Stiglitz, the Kimball speci…cation does not produce a 'missing de ‡ation' puzzle at the onset of the Great Recession. In other words, in ‡ation does not fall much in response to an adverse Great Recession type shock. Second, the small rise in in ‡ation expectations in response to …scal stimulus with the Kimball speci…cation is consistent with evidence provided by Dupor and Li (2015). These authors argue that expected in ‡ation reacted little to spending shocks in the United States during the Great Recession. By contrast, in ‡ation expectations react much more under the Dixit-Stiglitz speci…cation.
Our results have potentially important implications for the scope of …scal stimulus to be self-…nancing, and the extent to which …scal consolidations can be self-defeating. In the nonlinear 3 In the stochastic economy, the probability of hitting the ZLB is 10 percent in each period. model, …scal stimulus is never a "free lunch" and conversely, …scal consolidations are never selfdefeating. The linearized model arrives at the opposite conclusions: …scal stimulus can be self-…nancing in a su¢ ciently long-lived liquidity trap and …scal consolidations can be self-defeating.
These …ndings cast doubt on the existing literature on the …scal implications of …scal stimulus. It should be noted, however, that we study a model environment in which the …scal output multiplier is small in normal times (1/3 as mentioned earlier). Had we considered a medium-sized model with Keynesian accelerator e¤ects in which the multiplier is in the mid-range of the empirical evidence when monetary policy is unconstrained, the multiplier could be magni…ed su¢ ciently in a long-lived liquidity trap to obtain a "…scal free lunch" for a transient spending hike. 4 We elaborate more on this in the conclusions.
Our paper is related to Boneva, Braun and Waki (2016), Christiano and Eichenbaum (2012), Johannsen (2016), Fernandez-Villaverde et al. (2015), Eggertsson andSingh (2016) andNakata (2015). Importantly, none of the above papers considers the case of a Kimball (1995) aggregator. Boneva, Braun and Waki (2016) report that the multiplier is smaller in a fully nonlinear model. Their model features a Dixit-Stiglitz aggregator. Eggertsson and Singh (2016) report that the multipliers of the nonlinear and linearized model di¤er only very little. Their model features a Dixit-Stiglitz aggregator and assumes …rms-speci…c labor markets, implying that price dispersion is irrelevant for the nonlinear model dynamics. By contrast, our analysis shows how important these assumptions are: moving to the frequently used Kimball aggregator and allowing for price dispersion alters the conclusions about the multiplier substantially. Nakata (2015) and Fernández-Villaverde et al. (2015) show that shock uncertainty may have potentially important implications for the equilibrium dynamics of the model. As mentioned above, our robustness analysis shows that allowing for shock uncertainty has a quantitatively small impact on our results for the nonlinear model. Christiano and Eichenbaum (2012) and Christiano, Eichenbaum and Johannsen (2016) analyze multiplicity of equilibria in a nonlinear New Keynesian model. They document that there is a unique stable-under-learning rational expectations equilibrium in their model and that all other equilibriums are not stable under learning.
The remainder of the paper is organized as follows. Section 2 presents the New Keynesian model and Section 3 the results. Section 4 provides an in-depth robustness analysis. Section 5 discusses potential implications of our work for future empirical work. Finally, section 6 concludes. 4 A large empirical literature has examined the e¤ects of government spending shocks, mainly focusing on the post-WWII pre-…nancial crisis period when monetary policy had latitude to adjust interest rates. The bulk of this research suggests a government spending multiplier in the range of 0.5 to somewhat above unity (1.5). See e.g. Hall (2009), Ramey (2011), Blanchard, Erceg and Lindé (2016 and the references therein.

New Keynesian Model
The model that we study is very similar to the one developed Erceg and Lindé (2014). We deviate from Erceg and Lindé (2014) in so far as we allow for a Kimball (1995) aggregator which aggregates intermediate goods into a …nal good. The speci…cation of the Kimball aggregator nests the standard Dixit and Stiglitz (1977) speci…cation as a special case. Below, we outline the model environment.
All linearized and nonlinear equilibrium equations are available in the Appendix. 5

Households
The utility functional for the representative household is where the discount factor satis…es 0 < < 1: As in Erceg and Lindé (2014), the utility function depends on the household's current consumption C t in deviation from a "reference level"C t where C denotes steady state consumption and t is an exogenous consumption preference shock. 6 The utility function also depends negatively on hours worked N t : The household's budget constraint in period t states that its expenditure on goods and net purchases of (zero-coupon) government bonds B t must equal its disposable income: Thus, the household purchases the …nal consumption good at price P t . The household is subject to a constant distortionary labor income tax N and earns after-tax labor income (1 N ) W t N t .
The household pays lump-sum taxes net of transfers T t and receives a proportional share of the pro…ts t of all intermediate …rms.
We also have the following labor supply schedule: 5 A technical appendix with all derivations is available here: https://sites.google.com/site/mathiastrabandt/home/downloads/LindeTrabandt_Multiplier_TechApp.pdf 6 In the robustness section below we also examine the implications of a discount factor shock instead of the consumption preference shock. The impulse responses of both shocks are virtually identical. We use the consumption preference shock in our baseline speci…cation to remain as close as possible to Erceg and Lindé (2014). Equations (3) and (4) are the key equations for the household side of the model.

Firms and Price Setting
Final Goods Production The single …nal output good Y t is produced using a continuum of di¤erentiated intermediate goods Y t (f ). Following Kimball (1995), the technology for transforming these intermediate goods into the …nal output good is As in Dotsey and King (2005) and Levin, Lopez-Salido and Yun (2007), we assume that G ( ) is given by the following strictly concave and increasing function: . Firms that produce the …nal output good are perfectly competitive in product and factor markets. Thus, …nal goods producers minimize the cost of producing a given quantity of the output index Y t , taking as given the price P t (f ) of each intermediate good Y t (f ). Moreover, …nal goods producers sell units of the …nal output good at a price P t , and hence solve the following pro…t maximization problem: 7 The parameter used in Smets and Wouters (2007) to characterize the curvature of the Kimball aggregator can be mapped to our model using the following formula: = 1 : subject to the constraint (5). The …rst order conditions can be written as where t denotes the Lagrange multiplier on the aggregator constraint (6). Note that for = 0 it follows that t = 1 8t and the …rst-order conditions in (8) simplify to the usual Dixit and Stiglitz (1977) expressions by monopolistically competitive …rms, each of which produces a single di¤erentiated good. Each intermediate goods producer faces a demand schedule from the …nal goods …rms through the solution to the problem in eq. (7) that varies inversely with its output price P t (f ) and directly with aggregate demand Y t : Aggregate capital K is assumed to be …xed, so that aggregate production of the intermediate good …rm is given by Despite the …xed aggregate stock K R K (f ) df , fractions of the aggregate capital shock can be freely allocated across the f …rms, implying that real marginal cost, M C t (f )=P t is identical across …rms and equal to The prices of the intermediate goods are determined by Calvo (1983) style staggered nominal contracts. In each period, each …rm f faces a constant probability, 1 , of being able to re-optimize its price P t (f ). The probability that a …rm receives a signal to reset its price is assumed to be independent of the time that it last reset its price. If a …rm is not allowed to optimize its price in a given period, it adjusts its price as follows where is the steady state (net) in ‡ation rate andP t is the updated price. In the robustness section we examine the implications of not allowing for price indexation.
Given Calvo-style pricing frictions, …rm f that is allowed to re-optimize its price, P opt t (f ), solves the following problem where t;t+j is the stochastic discount factor (the conditional value of future pro…ts in utility units, recalling that the household is the owner of the …rms), and demand Y t+j (f ) from the …nal goods …rms is given by the equations in (8).

Monetary and Fiscal Policies
The evolution of nominal government debt is determined by the government budget constraint where G t denotes real government expenditures on the …nal good Y t . Following Erceg and Lindé (2014) we assume that net lump-sum taxes as share of nominal steady state GDP, t Tt PtY , stabilize government debt as share of nominal steady state GDP, b t Bt PtY : Here and b denote the steady states of t and b t : Finally, real government consumption, G t , is assumed to be exogenous.
Turning to the central bank, we assume that it sets the nominal interest rate by using the following Taylor rule that is subject to the zero lower bound: where Y pot t denotes the level of output that would prevail if prices were ‡exible, and i the steadystate net nominal interest rate, which is given by r + where r 1= 1.

Aggregate Resources
It is straightfoward to show that aggregate output Y t is given by where The variable p t 1 denotes the Yun (1996) aggregate price distortion term.
Aggregate output can be used for private consumption and government consumption so that: The price distortion term introduces a wedge between the use of production inputs and the output that is available for private and government consumption. Note, however, that p t vanishes when the model is linearized.

Parameterization
Our benchmark calibration essentially adopted from Erceg and Lindé (2014) is fairly standard at a quarterly frequency. We set the discount factor = 0:995; and the steady state net in ‡ation rate = 0:005 which implies a steady state nominal interest rate i = 0:01 (i.e., four percent at an annualized rate). 8 We set the capital share parameter = 0:3 and the Frisch elasticity of labor supply 1 = 0:4: We set the steady state value for the consumption preference shock = 0:01: 9 Three parameters determine the direct sensitivity of prices to marginal costs: the gross markup , the stickiness parameter and the Kimball parameter . We have direct evidence on two of these -and . A large body of microeconomic evidence, see e.g. Nakamura and Steinsson (2008), Klenow and Malin (2010) and the references therein, suggest that …rms change their prices rather frequently, on average somewhat more often than once a year. Based on this micro evidence, we set = 0:667, implying an average price contract duration of 3 quarters. We set the gross markup = 1:1 as a compromise between the low estimate of in Altig et al. (2011) and the higher estimated value by Smets and Wouters (2007). To pin down the Kimball parameter consider the log-linearized New Keynesian Phillips Curve in our model: where d mc t denotes the log-deviation of marginal cost from its steady state.^ t denotes the logdeviation of gross in ‡ation from its steady state. The parameter denotes the slope of the Phillips curve and is given by: (1 )(1 ) 1 1 : 8 We rule out steady state multiplicity by restricting our attention to the steady state with a positive in ‡ation rate. 9 By setting the steady value of the consumption taste shock to a small value, we ensure that the dynamics for the other shocks are roughly invariant to the presence of C t in the period consumption utility function.
The macroeconomic evidence suggest that the sensitivity of aggregate in ‡ation to variations in marginal cost is very low, see e.g. Altig et al. (2011). To capture this, we set the Kimball parameter = 12:2 so that the slope of the Phillips curve is = 0:012 given the values for , and discussed above. 10 This calibration allows us to match micro-and macroevidence about …rms'price setting behavior and is aimed to capture the resilience of core in ‡ation, and measures of expected in ‡ation, to a deep downturns such as the Great Recession.
Consistent with the pre-crisis U.S. federal debt level, we assume a government debt to annualized output ratio of 0:6. We assume that government consumption accounts for 20 percent of GDP: Further, we set net lump-sum taxes = 0 in steady state. The above assumptions imply a steady state labor income tax N = 0:33. The parameter ' in the tax rule (13) is set equal to 0:0101, which implies that the contribution of lump-sum taxes to the response of government debt is negligible in the …rst couple of years following a shock. For monetary policy, we use the standard Taylor (1993) rule parameters = 1:5 and x = 0:125.
In order to facilitate comparison between the nonlinear and linearized model, we specify processes for the exogenous shocks such that there is no loss in precision due to an approximation. In particular, the consumption preference and government spending shocks are assumed to follow AR (1) processes: where " G;t and " ;t are normally distributed iid shocks. In our baseline parameterization we assume = G = 0:95. We also investigate the sensitivity of our results when we assume moving average processes instead of autoregressive processes. Those results are reported in Appendix A.

Model Solution
Our benchmark results are based on the solution of the linearized and nonlinear model using the solution method developed in Fair and Taylor (1983). For robustness, we also compute the solution of the linearized and nonlinear model using the global solution method developed by Judd (1988) and (Coleman, 1990(Coleman, , 1991 which allows shock uncertainty to a¤ect the decision rules of households and …rms. 2.6.1. Benchmark Solution Method: Fair and Taylor (1983) The Fair and Taylor (1983) method solves the linearized and nonlinear equilibrium equationsincluding kinks such as the ZLB -by solving a two-point boundary value problem. The Fair and Taylor (1983) method is often referred to as 'extended path', 'deterministic simulation'or 'perfect foresight solution'. To solve a model, the method assumes that after a shock the model economy converges back to its steady state in a …nite number of periods. In addition, the solution method assumes certainty equivalence. That is, the variance of shocks does not a¤ect the decision rules of households and …rms. By imposing certainty equivalence on both the linearized and nonlinear model, the Fair and Taylor (1993) solution method allows us to trace out implications of using nonlinear equilibrium equations instead of linearized equilibrium equations for the resulting multiplier.
We check the robustness of our results by also using a global solution method which allows shock uncertainty to explicitly a¤ect the model solution. However, our benchmark results are based on Fair and Taylor (1983) for the following three reasons.
First, because much of the existing literature has often used a perfect foresight approach that imposes certainty equivalence to solve a model, retaining this approach allows us to parse out the e¤ects of going from a linearized to a nonlinear model framework. Second, the high degree of real rigidities we introduce in order to …t the micro-and macroeconomic evidence implies that expected in ‡ation adjusts slowly, which in turn means that the impact of future shock uncertainty is modest.
As shown below, allowing for shock uncertainty does not a¤ect the solution of the nonlinear model noticeably. By contrast, allowing for shock uncertainty in the linearized model a¤ects the model solution a lot implying even bigger di¤erences between the linearized and nonlinear model for the resulting multiplier. Third, the Fair and Taylor (1983) method allows us to solve the nonlinear model in fractions of a second while the nonlinear model solution with shock uncertainty takes several hours to calculate. Moreover, the Fair and Taylor (1983) method also allows to calculate the solution of larger scale models with many state variables very fast. So far, the solution algorithms used to solve models with shock uncertainty have typically not been applied to models with more than 4-5 state variables. 12 We use Dynare to solve the nonlinear and linearized model equations that are provided in the Appendix A. Dynare is a pre-processor and a collection of MATLAB routines which can solve linear and nonlinear models with occasionally binding constraints. The details about the implementation https://sites.google.com/site/mathiastrabandt/home/downloads/LindeTrabandt_Multiplier_Codes.zip of the algorithm used can be found in Juillard (1996). The perfect foresight simulation algorithm implemented in Dynare is Fair and Taylor (1983). To solve a model using it, one just has to use the 'simul' command. The algorithm can easily handle the ZLB constraint: one just writes the Taylor rule including the max operator in the model equations, and the solution algorithm reliably calculates the model solution in fractions of a second. Thus, except for perhaps obtaining intuition about the economics embedded into models, there is no need anymore to linearize models to solve and simulate them.
For the linearized model, we used the algorithm outlined in Hebden, Lindé and Svensson (2011) to check for uniqueness of the local equilibrium associated with a positive steady state in ‡ation rate and to impose the ZLB. 13 We did not …nd evidence of multiplicity of the local equilibrium in the nonlinear model.
As noted earlier, we rule the well-known problems associated with steady state multiplicity emphasized by Benhabib, Schmitt-Grohe and Uribe (2001)  these solutions are not stable under learning.

Alternative Solution Method: Global Solution
For robustness, we also solve the linearized and nonlinear model using the global solution method developed by Coleman (1990Coleman ( , 1991. This solution method is based on a time iteration method on the decision rules of households and …rms. With this method, the variance of shocks a¤ects the decision rules of households and …rms. The time iteration method has been used recently by e.g.

Nakata (2016) and Richter and Throckmorton (2017).
A growing body of work such as e.g. Billi (2006, 2007)  are not well anchored due to non-optimal monetary policy or when aggregate prices adjust slowly.
As we will show below, however, the solution of our nonlinear model is nearly una¤ected by the presence of substantial shock uncertainty due to the Kimball (1995) aggregator and an empirically realistic low slope of the Phillips curve.

Results
In this section, we report our benchmark results. Our aim is to compare spending multipliers in linearized and nonlinear versions of the model economy. Speci…cally, we seek to characterize how the di¤erence between the multiplier in the linearized and nonlinear solutions varies with the expected duration of the liquidity trap. We start by reporting how we construct a baseline in which the model economy is driven to the zero lower bound and then report the …scal multipliers.

Construction of Baseline
To construct a baseline where the nominal interest rate is bounded at zero for a number of periodssay ZLB DU R = 1; 2; 3; :::; M -we follow the …scal multiplier literature (e.g. Christiano, Eichenbaum and Rebelo, 2011) and assume that the economy is hit by a large adverse shock that triggers a deep recession and drives the nominal interest rate to zero. The longer the expected liquidity trap duration, i.e. the larger value of ZLB DU R , we want to consider, the larger the adverse shock has to be. The particular shock we consider is a negative realization of the consumption preference shock t discussed above. 14 As an example, Figure 2 shows the baseline generated by the adverse consumption preference shock in the linearized and nonlinear model when the ZLB is binding for eight quarters, i.e. In the robustness section below we show that the type of shock that we use to generate the baseline is immaterial for our results. For example, we show that if the baseline is generated by a discount factor shock instead of a consumption demand shock, the resulting …scal multipliers are nearly una¤ected.
1 5 Figure 2 also depicts a third line ("Nonlinear model with linearized NKPC and Resource Constraint"), which we will discuss further in Section 3.2.
(panel 5) has to drop much more than in the linearized model. Accordingly, the output gap (panel 1) is much more negative in the nonlinar model. Even so, and perhaps most important, we see that the drop in in ‡ation (panel 2) is substantially smaller in the nonlinear model. This suggests that the di¤erence between the linearized and nonlinear model is driven to a large extent by the nonlinearities embedded in the pricing setting equations.
Interestingly, the linearized model predicts a protracted period of de ‡ation in response to the Great Recession type shock. In the data, however, a long period of deep de ‡ation after the onset of the Great Recession was not observed. This observation is commonly referred to as the 'missing de ‡ation'puzzle, i.e. actual in ‡ation did not fall nearly as much as predicted by the linearized New Keynesian model. By contrast, the nonlinear model based on the Kimball speci…cation does not appear to produce the 'missing de ‡ation'puzzle. In ‡ation in the nonlinear model falls by much less and turns negative for a very brief period only before recovering relative to the linearized model.
Based on these results we argue that the 'missing de ‡ation' puzzle is not a puzzle: it arises due to an approximation error when one extrapolates the predictions of a linearized model to very large shocks. The underlying true nonlinear model predicts that macroeconomists should not have expected a long period of deep de ‡ation to occur in the aftermath of the Great Recession.
Given the above discussion, we seek to compare …scal multipliers in liquidity traps of same expected duration in both the linearized and nonlinear model. Accordingly, we allow for di¤erently sized shocks so that each model variant generates a liquidity trap with the same expected duration ZLB DU R = 1; 2; 3; :::; M .
denote matrices of time series with simulated variables of the linearized and nonlinear models where i indexes the set of time series for a given ZLB duration:The notation re ‡ects that the innovations, " ;i , to the consumption preference shock shock t , in eq. (19) are set so that

Marginal Fiscal Multipliers
Conditional on the set of baseline scenarios that we have constructed above, we add an increase of government spending g t in the period when the ZLB starts to bind.
denote matrices of time series with simulated variables of the linearized and nonlinear models where i indexes the set of time series for a given ZLB duration and " G denotes the positive government spending shock that hits the economy when the ZLB starts to bind.
We then compute the marginal impact of the …scal spending shock as where we write I linear (ZLB DU R ) and I nonlinear i (ZLB DU R ) to highlight their dependence on the liquidity trap duration. Notice that the …scal spending shock is the same for all i and is scaled such that ZLB DU R is the same as in the baseline. By setting the …scal impulse so that the liquidity trap duration remains una¤ected we calculate "marginal" spending multipliers in the sense that they show the impact of a small change in the …scal instrument. 16 Figure 3 contains the main results of the paper. The upper panels report results for the benchmark calibration with the Kimball aggregator. The lower panels report results under the Dixit-Stiglitz aggregator, in which case = 0. This parametrization implies a substantially higher slope of the linearized Phillips curve (see eq. 18) and thus a much stronger sensitivity of expected in- ‡ation to current and expected future marginal costs (and output gaps). We will …rst discuss the results under the Kimball parameterization, and then turn to the Dixit-Stiglitz results.
The left panels of Figure 3 report the impact GDP multiplier of government spending, i.e.
where the -operator represents the di¤erence between the scenario with the government spending change and the baseline without the spending change. We compute m i for ZLB DU R = 1; :::; 12. We also compute results for the case when the economy is at the steady state, so that ZLB DU R = 0.
The top left panel in Figure 3 reports that if the economy is close to or at the steady state (e.g. the ZLB is not binding, ZLB DU R = 0), the linearized and nonlinear multipliers coincide.
In other words, the linear approximation is accurate if the economy is close to or at the steady  Figure 3 show that the di¤erences between the linearized and nonlinear model are even more pronounced in this case. 19 The larger di¤erences in the Dixit-Stiglitz case are driven by a substantially higher slope of the New Keynesian Phillips curve (equation (17)) when setting = 0 and keeping all other parameters unchanged. In other words, expected in ‡ation reacts even more in response to …scal stimulus which implies an even larger …scal multiplier in longlived liquidity traps. Taken together, the results in Figure 3 suggest that the …ndings reported in the previous literature -which mostly relied on using linearized models -might be biased upward 1 7 For ease of interpretability, we have normalized the response of debt and in ‡ation so that they correspond to a initial change in government spending as share of steady state output by one percent.
1 8 The small rise in in ‡ation expectations in response to …scal stimulus with the nonlinear model speci…cation is consistent with the evidence provided by Dupor and Li (2015). These authors argue that expected in ‡ation reacted little to spending shocks in the United States during the Great Recession.
1 9 We only show results up to 8 quarters with the Dixit-Stiglitz aggregator to be able to show the di¤erences more clearly in the graph. from the perspective of the underlying nonlinear model.

Accounting for the Di¤erences
Given the results described above the following key questions arise: why are the GDP multipliers so di¤erent and why does expected in ‡ation respond so much more in the linearized solution, and particularly so in the Dixit-Stiglitz case? To shed light on these questions we simulate and report  Interestingly, as shown by the green dashed-dotted line in the top panels of Figure 4, it is almost su¢ cient to just linearize the NKPC to account for most of the di¤erences in terms of the …scal multipliers between the linearized and nonlinear solution with the Kimball aggregator. Therefore, the nonlinearities implied by the price dispersion term do not matter much quantitatively for the Kimball aggregator speci…cation of the nonlinear model. Even so, the fact that the price dispersion is elevated following an adverse shock implies that many …rms percieve that their demand elasticity is high, and they are therefore reluctant to change prices much (if at all) in response to impulses in marginal costs. In terms of Figure 1, the …rms move to the upper left quadrant. 20 On the other hand, the bottom panels in Figure 4 show that linearization of the New Keynesian Phillips curve alone is not su¢ cient to explain the large discrepancies between the linearized and nonlinear model when the Dixit-Stiglitz aggregator is used. Put di¤erently, with the Dixit-Stiglitz aggregator the tables are turned: the nonlinearities in the price dispersion term account for most of the di¤erences between the linearized and nonlinear models while the nonlinearities of the price setting block are of second order importance. The driving force behind the di¤erences between the Kimball and the Dixit-Stiglitz aggregators is that the price distortion variable moves much more for the latter speci…cation. Re ‡ecting the insights from Figure 1, re-optimizing …rms will adjust their prices much more under Dixit-Stiglitz compared to Kimball for a given value of . So in a model with Dixit-Stiglitz aggregation …rms adjust prices a lot when they re-optimize so that the bulk of the di¤erence between the linearized and nonlinear model is driven by movements in the price distortion term. By contrast, …rms adjust prices only little in response to shocks with the Kimball aggregator speci…cation so that the price distortion term is much less important.

Relation to Existing Work
Our results are very helpful to understand the di¤erences between the results reported in Boneva, Braun and Waki (2016) and Eggertsson and Singh (2016). The former authors argue that it is key to account for the price distortion term as the main di¤erence between the linearized and nonlinar solutions. Our results are in line with their …nding given that Boneva, Braun and Waki (2016) consider a model framework that incorporates the Dixit-Stiglitz aggregator. In terms of the magnitude of the multiplier it is important to note that we report lower multipliers in our nonlinear solution (the red dotted line in Figure 4) than Boneva, Braun and Waki (2016) for the same degree of price adjustment. The reason for our lower multipliers is due to our government spending process which is assumed to be a fairly persistent AR(1) process. As an alternative to our benchmark speci…cation we follow Boneva, Braun and Waki (2016) and assume that government spending follows a moving average (MA) process and is increased only when the nominal policy 2 0 Notice that this means that in ‡ation and output behaves asymmetrically in expansions and recessions under the Kimball aggregator. Recessions are associated with relatively modest declines in in ‡ation but booms can be associated with large upward swings in in ‡ation. rate is constrained by the ZLB. With this speci…cation we obtain a marginal multiplier of unity in both the linearized and nonlinear model already in a one-quarter liquidity trap. 21 More details about the results based on the MA process are provided in Section 4.2.5. There we show that the important di¤erences between the linearized and nonlinear model hold up for longer-lived liquidity traps.
Our results can also be used to understand the results reported by Eggertsson and Singh (2016).
These authors consider a model with a Dixit-Stiglitz aggregator and assume …rm-speci…c labor which implies that the price distortion term does not a¤ect equilibrium allocations.

Robustness
In this section, we examine the robustness of the results. We focus on the sensitivity of our results when solving the model with global methods to allow future shock uncertainty to a¤ect the decision rules of households and …rms. We also summarize the results of further robustness checks including the e¤ects of other shocks, the sensitivity of our results with respect to the baseline shock, the aggregator speci…cation (Kimball vs. Dixit-Stiglitz), price indexation and the exogenous process for government spending.

Global Solution Allowing for Shock Uncertainty
Jung, Teranishi and Watanbee (2005), Billi (2006, 2007), Fernández-Villaverde et al. focusing on the e¤ects of shock uncertainty on the decision rules of households and …rms. In this subsection we show that our key …ndings hold up -and are even strengthened -when we allow for substantial future shock uncertainty.
We solve the stochastic linearized and nonlinear models using the global solution method developed by Judd (1988) and (Coleman, 1990(Coleman, , 1991. This solution method is based on a time iteration method on the decision rules of households and …rms. The variance of shocks can a¤ect the policy functions. The time iteration method has been used recently by e.g. Nakata (2016)   where we subject the model to a negative consumption preference shock which generates an expected 8-quarter liquidity trap under the assumption that no further consumption preference shocks are realized during the transition back to the steady state. In the quarter in which the liquidity trap is expected to last for 8 quarters, we add a small positive government spending shock and compute the impulse responses in Figure 5 as the di¤erence between the simulation with government spending and the simulation with consumption preference shocks only. 24 The solid-black lines in panel A of Figure 5 correspond to the linearized model solved with the Fair and Taylor (1983) method, i.e. the deterministic solution of the linearized model. The impact of government consumption on GDP in period 1 is the same as the one depicted in the top panel of Figure 3, i.e. the impact multiplier is 1.2 in an 8-quarter ZLB episode. The red-dashed 2 3 We are grateful to Richter and Throckmorton (2017) for making their codes publicly available. Their codes provided us with a useful starting point for solving our model. 2 4 Bodenstein, Hebden and Nunes (2012) use the same approach when computing impulse responses in their paper. Although the assumption that no further shocks are realized on the transition path back to steady state is improbable, this way of computing the impulses makes them directly comparable with how they are computed in the deterministic solution. Importantly, the impulse responses still re ‡ect the impact of future shock uncertainty via the e¤ect that shock uncertainty has on the decision rules of households and …rms. lines in panel A correspond to the case when the linearized model is solved subject to future shock uncertainty. In this case, the impact multiplier increases to about 2.1. Evidently, shock uncertainty elevates the multiplier substantially in the linearized model. Panel B shows the comparison of the impulse responses in the deterministic vs. stochastic solution in the nonlinear model. The solid-black lines correspond to the deterministic solution of the nonlinear model. As in Figure 3, the multiplier is about 0.6 for an 8-quarter liquidity trap. Interestingly, the nonlinear model is not much a¤ected by shock uncertainty. The multiplier increases from 0.61 in the deterministic solution to 0.64 in the fully stochastic nonlinear model solution.
The solution of the nonlinear model is not much a¤ected due to the nonlinearities embedded in the Kimball aggregator together with the low slope of the Phillips curve. Both features reduce the incentive of …rms to change their prices in response to expectations of adverse shocks in the future even when the economy is stuck in a long-lived liquidity trap. By contrast, the linearized model incorrectly extrapolates the behavior of households and …rms such that in ‡ation reacts much stronger to shocks leading to an even higher multiplier than in the deterministic solution.
To sum up, our results indicate that the implications of uncertainty in the nonlinear model are quantitatively negligible. By contrast, the the multiplier in the linearized model is greatly elevated when shock uncertainty is allowed for in the solution of the model. Consequently, our conclusion of an important di¤erence between the linearized and nonlinear solution in long-lived liquidity traps holds up under shock uncertainty.

Additional Robustness Analysis
We perform a variety of additional robustness checks. Given space constraints, we summarize the key takeaways from the additional robustness analysis here. Appendix A.4 -A.8 contains further details.

E¤ects of Other Shocks
We examine the implications of the following four additional shocks for the linearized and nonlinar model: discount factor shock, technology shock, markup shock and monetary policy shock. Two key takeaways emerge from this analysis. First, for all shocks considered, there are substantial di¤erences between the linearized and nonlinear model. Second, in the linearized model, the responses of variables to the government consumption shock, the consumption demand shock, the discount factor shock and the technology shock are observationally equivalent. In the nonlinear model the same observation is arises, i.e. the responses of model variables are (nearly) observationally equivalent.
Appendix A.4 contains the details.

Choice of Baseline Shock
We study how our results are a¤ected when a discount factor shock or a technology shock is used to generate the baseline in which the ZLB is binding for a desired number of quarters. For the linearized model, the multiplier results are invariant with respect to the the choice of the baseline shock (see Erceg and Lindé, 2014, for analytical proofs). That is, the multiplier is identical when the baseline is generated either by a consumption preference shock or by a discount factor shock or by a technology shock. For the nonlinear model we show that the multiplier schedules are nearly invariant with respect to alternative baseline shocks. Appendix A.5 contains the details.

Kimball vs. Dixit-Stiglitz
In the linearized model, we show that the Kimball and Dixit-Stiglitz aggregators yield identical multiplier schedules when the degree of price stickiness and the Kimball elasticity parameter are parameterized such that the slope of the linearized New Keynesian Phillips curve ( in eq. 17) is kept constant. So going from Kimball to Dixit-Stiglitz by making prices more sticky yields identical multipliers in the linearized model. In the nonlinear model, the same conclusion is not true. There, a reparameterization of the Dixit-Stiglitz version of the model with higher price stickiness does not produce the same multipliers as under Kimball. This demonstrates that the modeling of price frictions matters importantly within a nonlinear framework. Appendix A.6 contains the details.

Price Indexation
We examine the consequences of not allowing prices of non-optimizing …rms to be indexed to the steady state rate of in ‡ation. We show that our benchmark results are little a¤ected by the indexation assumption. Appendix A.7 contains the details.

Government Spending Process
Finally, we examine the implications of adopting a moving average (MA) process for government spending at the ZLB instead of a general AR(1) process. We show that our benchmark results hold up well for a MA process for government spending. If anything, an MA process magni…es the di¤erences between the linearized and nonlinear model in terms of the multiplier. Appendix A.8 contains the details.

Empirical Implications
A key feature of the Great Recession in the United States and other advanced economies was a large, sharp and persistent fall in GDP of nearly 10 percent relative to the pre-crisis trend. As oil prices fell sharply in response to the recession, headline in ‡ation slowed down substantially.
However, measures of core in ‡ation -the relevant benchmark for standard macroeconomic models without commodities -slowed down only by a modest amount of about 1 percentage point (see e.g. Figure 8 in Christiano, Eichenbaum and Trabandt, 2015).
Estimated standard New Keynesian models which target to explain all variation in the data using full information Bayesian likelihood methods have di¢ culties to account for the low elasticity between output and in ‡ation observed during the Great Recession.
One way to account for the moderate drop in in ‡ation in the face of the large contraction in GDP is to resort to large o¤setting e¤ects on in ‡ation stemming from positive price markup shocks (see e.g. Lindé, Smets and Wouters, 2016). Some researchers have emphasized that …nancial frictions may be responsible for the small elasticity between output and in ‡ation witnessed during the crisis. Christiano, Eichenbaum and Trabandt (2015) use a model to show that the observed fall in total factor productivity and the rise in …rms'cost to borrow funds for working capital played critical roles in accounting for the small drop in in ‡ation that occurred during the Great Recession. Gilchrist, Schoenle, Sim and Zakrajsek (2016) develop a model in which …rms face …nancial frictions when setting prices in an environment with customer markets. Financial distortions create an incentive for …nancially constrained …rms to raise prices in response to adverse …nancial or demand shocks in order to preserve internal liquidity and avoid accessing external …nance. While …nancially unconstrained …rms cut prices in response to these adverse shocks, the share of …nancially constrained …rms is su¢ ciently large in their model to attenuate the fall in in ‡ation in response to ‡uctuations in GDP. Gilchrist, Schoenle, Sim and Zakrajsek (2016) examine a micro data set which supports the implications of their model.
The mechanism based on the nonlinear Kimball (1995) aggregator that we have identi…ed in our paper o¤ers an alternative explanation for understanding the small elasticity of in ‡ation and output observed during the Great Recession. To examine the empirical potency of the mechanism in a rigorous way, one would have to follow the work of Gust, Herbst, Lopez-Salido and Smith (2016), Richter and Throckmorton (2016) and Kulish and Pagan (2017) and estimate the nonlinear model with likelihood methods. Given the strong nonlinearities associated with the Kimball (1995) aggregator and the fact that embedding the nonlinear Kimball (1995) aggregator into a standard New Keynesian model requires the introduction of several endogenous state variables, this is will be a tough but potentially very rewarding challenge, as suggested by the recent work of Arouba, Boccola and Schorfheide (2017). To begin with, one could use the perfect foresight approach to likelihood evaluation developed by Iacoviello and Guerrieri (2016). An interesting extension in this context would also be to examine the possibility of the existence of a de ‡ationary regime, as in Arouba, Cuba-Borda and Schorfheide (2017), as this could have important implications for the size of the …scal multiplier.
Ideally, one should also complement the empirical approach based on macroeconomic data with …rm-level data on prices and quantities. Using micro data would allow to examine empirically the properties of the nonlinear Kimball (1995) aggregator shown in Figure 1. In addition, one could possibly also draw and extend empirical …ndings in the industrial organization literature to shed further light on the properties of the Kimball (1995) aggregator. While we are excited about these empirical applications we leave them to future research.

Conclusions
All told, our paper provides an example of a potential …rst-order policy mistake that is based on using a linear approximation to solve a model to calculate a …scal spending multiplier. The mistake involves a nearly three times as large multiplier (2 instead of 0.7) as well as an implication of a self-…nancing …scal stimulus in a long-lived liquidity trap. Our analysis of the true underlying nonlinear model arrives at very di¤erent conclusions: a small multiplier and no self-…nancing. Therefore, our results caution against the common practice of using linearized models to calculate …scal multipliers in long-lived liquidity traps. Using our benchmark model with real rigidities following Kimball (1995), we have documented that it is the linearization of the Phillips curve which accounts for the bulk of the di¤erence between the linearized and nonlinear model. The results in our model imply large di¤erences between the linearized and nonlinear model, supporting the …ndings in Boneva, Braun and Waki (2016). In contrast to Boneva et al. (2016), however, it is important to point out that we consider a model which matches macroeconomic evidence of a low Phillips curve slope and microeconomic evidence of frequent price changes by …rms.
Even so, the way nonlinearities are introduced into a model can matter. Speci…cally, our analysis has shown that it is possible to construct New Keynesian models in which the di¤erence between the linearized and nonlinear model is relatively small even in long-lived liquidity traps. More precisely, con…rming the results in Eggertsson and Singh (2016), our analysis documents that this is the case in the Eggertsson and Woodford (2003) "Yeoman farmer" New Keynesian sticky price model with …rm-speci…c labor. In that model the price dispersion term is irrelevant for equilibrium dynamics.
As this model …ts the macro-and microevidence on price setting equally well as our benchmark model using the Kimball (1995) aggregator, an important issue that remains to be studied is which of the competing frameworks best …ts the data.
It would also be interesting to study the robustness of our results in an empirically realistic framework such as Christiano, Eichenbaum and Evans (2005) where one would allow for a nonlinear Kimball (1995) aggregator in both price-and wage setting and nonlinearities originating from …nancial frictions following for example the Bernanke, Gertler and Gilchrist (1999) …nancial accelerator mechanism. Such a framework would allow to study the robustness of our …ndings in a framework which has a spending multiplier in the mid-range of the VAR evidence when monetary policy is unconstrained. Doing so is important for the substantive issue whether the spending multiplier can be su¢ ciently elevated in a long-lived liquidity trap so that a transient hike in spending is associated with a …scal free lunch (and conversely whether a spending cut could be self-defeating in a long-lived trap). We leave these extensions to future research. Below we state the nonlinear and linearized equilibrium conditions of the model. A.1 We also provide a detailed description of the additional robustness analyses which we summaried in the main text.

A.3. Details about the Global Solution Method
In order to solve the fully stochastic nonlinear and linearized model, we discretize the state space.
Solving the stochastic nonlinear model is computationally challenging due to the nonlinearities embedded in the Kimball aggregator, the size of the state space and the non-availability of closed gridpoints. It takes about one hour on above workstation to solve the linearized model.
We calibrate the two exogenous processes as follows: The autocorrelation of 0:95 and the standard deviation of 0:01 for the government consumption process are estimated using the cyclical component of hp-…ltered U.S. data from 1955Q1 to 2017Q2.
The autocorrelation of the consumption demand shock is set to 0:80 following Nakata (2016). To introduce the discount factor shock t we specify the household utility function as follows: The technology shock z t is introduced into the production function: The markup shock % t is introduced into the equation for marginal cost: The monetary policy shock t is introduced into the Taylor rule: All shocks are assumed to follow AR(1) processes with autocorrelation 0.95 except the monetary policy shock which we assume to have an autocorrelation of 0.7. We subject the linearized and the nonlinar models to the same shock. We size the shock such that the in ‡ation rate in the nonlinear model falls from its steady state of 2 percent to 0 percent in response to each shock. observationally equivalent in the nonlinear model.

A.5. Robustness: Choice of Baseline Shock
In line with Erceg and Lindé (2014) we use the consumption preference shock v t to generate our baselines. A negative shock to v t implies that both potential output and the real interest rate fall (see Figure 2). In contrast, most papers in the literature on …scal multipliers have assumed that an increased desire to save, represented by a higher discount factor, causes the natural real rate to fall below zero and thereby triggers the economy to enter into a liquidity trap. A higher discount factor leaves potential output unchanged, and consequently has the ‡avor of a negative demand shock when output (and the output gap) contracts because monetary policy cannot cut the policy rate below zero to mimic the fall in the natural real rate.
To ensure that our results hold up when we follow the bulk of the literature, we present results when the recession is assumed to be triggered by a discount factor shock as used in the seminal papers by Eggertsson and Woodford (2003) and Christiano, Eichenbaum and Rebelo (2011). For the linearized model, we establish that the results are invariant with respect to the the choice of the baseline shock (see Erceg and Lindé, 2014, for analytical proofs). For the nonlinear model, Figure Erceg and Lindé (2014) by showing that the …scal spending multiplier is independent of the shock driving the baseline when the model is linearized as long as the di¤erent baseline shocks generate an equally long-lived ZLB episode. So our choice to work with the consumption preference shock t instead of the discount factor shock t has no consequences for our results in the linearized model.
As for the nonlinear model, the lower panels in Figure A.3 show that the results are very similar even in the nonlinear solution, so our choice of the baseline shock appears to be unproblematic.

A.6. Robustness: Kimball vs. Dixit-Stiglitz Aggregator
To further tease out the di¤erence between the Kimball vs. Dixit-Stiglitz aggregator, Panel A in Even so, the nonlinear solutions shown in Panel A in Figure A.4 di¤er. In particular, we see that the Dixit-Stiglitz aggregator implies that expected in ‡ation and the output multiplier respond more when the duration of the liquidity trap increases. Thus, when the Kimball parameter goes toward zero, the more will expected in ‡ation and the output multiplier respond when ZLB DU R increases; conversely, increasing more negative and lowering ‡attens the output multiplier schedule even more. The explanation behind this …nding is that a more negative value of induces the elasticity of demand to vary more with the relative price di¤erential among the intermediate good …rms as shown in Figure 1, and this price di¤erential increases when the economy is far from the steady state. Thus, intermediate goods …rms which only infrequently are able to re-optimize their price will optimally choose to respond less to a given …scal impulse far from the steady state when price di¤erentials are larger as they perceive that they may have a much larger impact on their demand for a given change in their relative price. As a result, aggregate in ‡ation and expected in ‡ation are less a¤ected far from the steady state in the Kimball case relative to the Dixit-Stiglitz case for which the elasticity of demand is independent of the relative price di¤erential. This demonstrates that the modeling of price frictions matters importantly within a nonlinear framework, especially so when nominal wages are ‡exible.

A.7. Robustness: Price Indexation
So far, we have followed the convention in the literature and assumed that non-optimizing …rms index their prices to the steady state rate of in ‡ation, see eq. (11). This is a convenient benchmark modelling assumption as it simpli…es the analysis by removing steady state price distortions. However, the indexation assumption has been criticized for being inconsistent with the microeconomic evidence on price setting behavior of …rms. According to micro evidence on price setting, prices set by …rms remain unchanged for several quarters. By contrast, the indexation scheme in our model (as well as in most of the literature) implies that prices changes in each quarter -either because …rms can choose an optimal price or because of mechanical indexation of the price set in the previous period.
To examine the importance of the indexation assumption for the resulting …scal multiplier we re-formulate the model: In particular, following e.g. Ascari and Ropele (2007) and Trabandt (2015, 2016) we do not allow non-optimizing …rms to index their prices:These …rms must keep their price unchanged, i.e. From the panels, we see that abandoning the conventional assumption of full indexation results in a somewhat steeper …scal multiplier schedule:The steeper …scal multiplier schedule is due to the higher sensitivity of expected in ‡ation in the "no-indexation" model since …rms take into account in their price setting decisions that their prices will not automatically adjust in response to shocks.
We veri…ed that the …scal marginal multipliers in the linearized model are also roughly unchanged (not shown in the …gure). All told, our benchmark results are robust with respect to the price indexation assumption.

A.8. Robustness: Government Spending Process
Another aspect we want to understand is how our results di¤er from Boneva, Braun, and Waki (2016) due to our AR(1) assumption for government spending instead of the MA-process they work with. Figure A.5 assess this issue by comparing the results of our benchmark AR(1) process for G t against a moving average (MA) in which G t is elevated to a higher level as long as the policy rate is bounded at zero and set to its steady state value otherwise. Apart from the fact that our benchmark solution procedure does not account for shock uncertainty, this approach of modeling government spending is identical to Boneva, Braun, and Waki (2016) who in turn follow Eggertsson (2010).
As can be seen from the upper panels of Figure A.5, the MA-process increases the marginal spending multiplier at the ZLB substantially relative to the AR(1) process. The multiplier is higher because increases in government spending have very benign e¤ects on the potential real interest rate when the duration of the spending hike equals the expected duration of the liquidity trap (see e.g. Erceg and Lindé, 2014). For a one quarter liquidity trap the multiplier equals unity, as shown analytically by Woodford (2011). Our fairly persistent AR(1) process tends to dampen the multiplier schedule since a relatively large fraction of spending occurs when the ZLB is no longer binding. This feature explains why the AR(1) multiplier is substantially lower in a short -lived liquidity trap. However, the AR(1) process is also associated with a substantially lower multiplier even in a fairly long-lived trap compared to the MA process because its has less benign e¤ects on the potential real rate.
All this is well-known in the body of work focusing on linearized models. However, the results for the non-linear model, shown in the lower panels of Figure A.5, are much less explored. We have already discussed the AR(1) case at length in the text. What we see is that the results for the MA process are quite di¤erent for longer ZLB durations, because the MA schedule for the nonlinear model stays essentially ‡at at unity, in line with the …ndings of Boneva, Braun, and Waki (2016); for a 12-quarter trap the multiplier only increases to 1.03 from a multiplier of unity in a one-period liquidity trap. This is in sharp contrast to the multiplier schedule for the linearized model where the multiplier is as high as 5 in a liquidity trap lasting 3 years. All told, the results show that our benchmark results hold up well for a MA-process for government spending. If anything, an MA process magni…es the di¤erences between the linearized and nonlinear solution in terms of the multiplier. Moreover, the linear and nonlinear model results in Figure