Fiscal Multipliers in Recessions

The Great Recession, and the fiscal response to it, has revived interest in the size of fiscal multipliers. Standard business cycle models have difficulties generating multipliers greater than one. And they also cannot produce any significant state-dependence in the size of the multipliers over the business cycle. In this paper we employ a variant of the Curdia-Woodford model of costly financial intermediation and show that fiscal multipliers can be strongly state dependent in a countercyclical manner. In particular, a fiscal expansion during a recession may lead to multiplier values exceeding two, while a similar expansion during an economic boom would produce multipliers falling short of unity. This pattern obtains if the spread (the financial friction) is more sensitive to fiscal policy during recessions than during expansions, a feature that is present in the data. Our results are consistent with recent empirical work documenting the state contingency of multipliers.


Introduction
Keynes advocated a fiscal stimulus during the Great Depression, and since then governments have routinely implemented fiscal expansions during recessions as a means of stimulating economic activity. Standard business cycle models offer scant support for this practice. Much of the criticism levelled at the Obama administration's stimulus plan was based on the implication of these models that government spending is ineffective. In a nutshell, this implication rests on the argument that an increase in government spending raises consumers' expected tax burden, and this negative wealth effect largely curtails the expansion of aggregate demand. The associated multipliers are small, hovering around one.
Moreover, as we elaborate below, these models also imply that fiscal policy is ineffective even during very severe downturns.
The objective of this paper is to propose a model that can generate large and asymmetric multipliers, thus potentially justifying countercyclical fiscal policy. The model represents an extension of the Curdia andWoodford [2009, 2010] model of costly financial intermediation with borrowers and lenders. Our main extension is to allow financial frictions to vary over the business cycle, and in particular, to be countercyclical. This possibility has been long recognized in financial economics 1 and is also confirmed by our empirical analysis.
More recently, Cromb and Vayanos [2011] develop a model that has this implication. In particular, they show that when the wealth of financial intermediaries decreases, intermediation becomes less effective (more costly) because of margin constraints and spreads increase.
We find that countercyclical financial frictions can make multipliers large during recessions and small during expansions. 2 Multipliers are large during recessions due to the presence of a strong financial accelerator mechanism. An increase in output mitigates the financial friction, increasing the consumption of the borrowers as well as total consumption. Higher consumption leads to further output gains which further alleviates the financial friction and increases consumption and so forth.
As is well known, the empirical estimation of fiscal multipliers is a hazardous affair due to identification and data problems. No consensus view exists in the profession regarding their size. Nonetheless our calibrated model gives rise to multipliers that, for increases 1 See, for instance, the detailed discussion in Mishkin [2001], chapters 8 and 25, about how the cyclicality of firm net worth, of household liquidity etc. induces countercyclical variation in moral hazard and adverse selection problems.
2 Collard and Dellas [2008] calculate fiscal multipliers in the model of Bernanke et al. [1999]. They find that they are small and exhibit limited cyclical variability. They attribute this to the limited cyclical variability in the degree of the financial friction present in this model even when one solves the non linear model. Similarly, in a model with financial frictions, Fernández-Villaverde [2010] finds output multipliers of about 1.
in government spending of the order of 1%, are very close to those recently estimated by Auerbach and Gorodnichenko [2010] and Bachmann and Sims [2011]. Auerbach and Gorodnichenko [2010] use regime switching SVAR's to show that output multipliers are countercyclical. They find that the point estimates of the maximum output multiplier (over the first 20 quarters) are 0.57 during expansions and 2.48 during recessions. When they ignore the distinction between recessions and expansions they obtain an estimate of 1.00, which is typical of estimates in the more recent empirical literature. 3 Countercyclical fiscal policy is problematic for the currently popular New-Keynesian models. Cogan et al. [2010] (CCTW hereafter) used the Smets and Wouter's [2007] model to compute consumption and output multipliers. They consider several alternative experiments (such as permanent vs temporary government spending increases, the particular case of the Obama administration American Recovery and Reinvestment Act, different lengths of time for the zero bound constraint, etc.). They report that the maximum output multiplier is about unity (and typically much smaller) and consumption and investment multipliers are negative. More importantly from the point of view of this paper and in line with the findings of Collard and Dellas [2008], CCTW do not find any significant variation in the multiplier over the business cycle when solving the non-linear version of the model. In particular, using an output gap of 6.5%, and letting the zero bound become endogenous hardly affects the output multipliers; if anything, it made them slightly smaller.
The existing literature is not unanimous regarding the role of the zero nominal interest rate bound in making fiscal multipliers larger. While CCTW find no role, Eggertsson [2001] and Christiano et al. [2009] find that it can make a big difference for the multipliers. 4 Erceg and Lindé [2010] fall in between CCTW and Christiano et al. [2009]. But independently of the effects of the zero bound on the fiscal multiplier, there seems to be a need for a supplementary or perhaps more general explanation of the large multipliers during recessions because nominal interest rates have not been at the zero bound for most of the recessions in the post world war II period. 5 In addition to generating large and cyclically variable fiscal multipliers, our analysis has other useful implications. For instance, it implies that the size of the fiscal intervention matters for the magnitude of the multiplier. In particular, while a 1% increase in govern-3 It should be noted that the standard errors of the estimates are rather large 4 The mechanism is as follows. Normally, nominal and real interest rates would rise following an increase in government spending, chocking off the expansion. But if the nominal interest rate is stuck at zero, this channel does not operate.
5 Although the mechanism presented in this paper relies on the existence of financial frictions, it does not only apply to recessions induced by financial shocks. Most business cycle shocks will give rise to non-linear, countercyclical multipliers in our model. ment purchases during a recession produces a multiplier between 2 and 3, an increase of 5% or 10% gives rise to multipliers between 1.5 and 2. The reason large fiscal interventions are less effective than smaller ones is that the -negative-marginal wealth effect due to the higher tax liabilities is increasing in the size of the fiscal intervention while the -positivemarginal effect on the borrower from the reduction in the finance premium is decreasing in the size of the fiscal expansion.
Another implication is that multipliers during recessions remain greater than one even when the government finances higher spending through taxes. But as in the IS-LM analysis, the multipliers are even bigger for debt financed spending. The reason is that while higher government spending sets in motion the financial accelerator, higher taxes partly counter this by reducing the quantity of funds available to financially constrained individuals.
The rest of the paper proceeds as follows: In Section ??, we outline the model, and describe it's calibration; at the end of the section, we discuss a financial accelerator that is created by our countercyclical intermediation friction. In Section ??, we present our results for consumption and output multipliers. We show that they involve multipliers falling short of unity during expansion and exceeding unity during recessions. In Section ??, we conclude.

The Model
Our model adopts the Curdia andWoodford [2009, 2010] framework of financial intermediation. Our main point of departure from that model is that we assume credit market frictions are countercyclical. This is the mainstream view in the profession, but we will present empirical evidence that offers direct support to this hypothesis, as it is crucial to our results. And we will use our empirical findings to calibrate the parameters of the function describing the financial friction.
It is not easy to formulate a dynamic general equilibrium model with private borrowers and lenders. Keeping track of the wealth of heterogeneous agents can be a daunting task.
So, Curdia and Woodford devise a rather ingenious insurance scheme to make the solution tractable. In the Curdia-Woodford framework, households have access to complete financial markets, but only on a random and infrequent basis. During the intervening periods, the household only has access to limited -and costly -financial intermediation: savers can deposit funds at a bank (or hold government bonds) and borrowers can obtain loans from the bank. Banks are competitive, and they maximize profits period by period. But more importantly, banks incur a cost when making a loan. It turns out that households' infrequent access to complete financial markets makes wealth dynamics tractable, while 4 costly financial intermediation adds the financial friction that is at the heart of our results.

Households
In each period t an individual agent, i, has a type µ(i) ∈ {b, s}. The household's type may vary over time in a manner that is described below. Household i's preferences in period t are represented by (1) where u µ (·, ·) is the utility of consumption of type µ household. The consumption good is a CES aggregate of the outputs of a continuum of firms, indexed by f . Members of household i work at all of these firms, and v µ (·, ·) is the type µ household's disutility for the hours worked at each firm. ξ t is the vector of preference shocks: specific preference shocks for borrowers and savers, and aggregate shocks to the disutility of hours worked.
The difference between type b and type s agents lies in the fact that type b agents have a higher marginal utility of current consumption, that is, for all c and all ξ. In equilibrium, the type b agents will borrow while the type s will save.
We will also refer to type b (resp. s) agents as impatient (resp. patient).

Evolution of Household Types
As explained previously, the type of an agent can change from one period to the next. 6 The type change is governed by a simple stochastic process. In each and every period an agent either keeps his type with probability δ ∈ [0, 1) or redraws a type with probability 1 − δ. In the latter event, the agent draws type b -becomes a borrower-with probability π b or type s with probability π s = 1 − π b . The law of large numbers implies that π b and π s will be the -unchanging-fractions of borrowers and savers in the economy. The type drawing process for somebody who is at present a saver is described in Figure 1 (a similar process applies to a current borrower). Given that agents can switch type with probability 1 − δ, the number of household histories goes to infinity, potentially generating a formidable heterogeneity of wealth. Curdia and Woodford [2009] develop an insurance scheme that makes it possible to aggregate across agents in a tractable way. Agents can sign state contingent contracts that allow them to transfer to or receive resources from an insurance agency when and only when they have been selected to draw a new type.
These contracts -which are optimal in the context of the Curdia-Woodford model-have the property that they eliminate all history dependence for those drawing a new type.
In particular, wealth is redistributed in such a way that all agents who end up drawing the same type after visiting the insurance agency are identical. Curdia and Woodford then show that the consumption and employment decisions within each type is the same, independent of individual history. The distribution of wealth across agents at any point in time becomes irrelevant.

The Household's Budget Constraint
The net wealth of household i at the end of period t is where Π f t (i) and Π b t (i) are the profits received by the household as the owner of firms and banks, T t (i) is a transfer from the insurance fund (0 unless the household had access to the agency at the beginning of the period), τ g t (i) is a real lump sum tax, and A t (i) denotes agent i's nominal assets at the beginning of period t; that is, is the nominal interest rate on bank loans and i d t−1 is the interest rate on bank deposits created in period t − 1. Note that government bonds compete with bank deposits, and the government bond rate, i g t−1 , is equal to the deposit rate in equilibrium.

Bank Intermediation
Banks issue one period deposits to households that save and make one period loans to households that borrow. Unlike the operation of the insurance agency, bank intermediation is costly: a bank expends real resources to make loans. Its costs are given by where y t = yt−y y denotes the relative deviation of output from its steady state level. ξ Ψ,t is a cost shock. Like Curdia and Woodford, we assume that the cost is convex in the (real) amount of loans made, b t . 7 But, in addition, we assume that banking costs vary as a function of the business cycle (the output gap). We use this as a proxy for agency problems (default risk) in credit markets that become more severe during recessions 8 . This assumption implies, that loan rates will have a countercyclical spread over deposits rates.
There are compelling theoretical reasons for this countercyclicality 9 which also received strong empirical support. 10 Banks are competitive -they take the deposit and lending rates, i d t and i b t , as given -and they maximize profits period by period. Real bank profits in period t are The bank chooses d t and b t to maximize profits subject to 11 Let us define the spread between the lending and the deposit rate, ω t , by 1 , then the bank's first order condition is The cost of making an additional dollar loan (the RHS) is equal to the benefit (the LHS).
Using (5), the bank's first order condition can be written as 7 Using the loan to GDP ratio in place of bt does not affect the implications of the model. See the technical appendix. 8 The worsening of agency problems may require greater screening/monitoring efforts on the part of the intermediaries. 9 See Mishkin [2001], for a detailed discussion of how reductions in net worth and cash flows exacerbate adverse selection and moral hazard problems in lending to firms. Unfortunately, the existing ways of modelling these agency problems in macroeconomics do not easily apply to models with heterogenous agents. Curdia and Woodford use the shock to the cost of banking to represent exogenous variation in the probability of default. We adopt their approach and use the endogenous output gap in place of their exogenous shock to capture the same variation in default (agency problems).
10 See, for instance, Gilchrist and Zakrajsek [2011], Figure 1. 11 Following Curdia and Woodford we let banks select deposits and loans subject to this equation. As i d t is smaller than i s t , dt > bt. The difference between the volume of deposits and loans is used to pay for the intermediation costs and is the source also of bank profits.

7
Our assumption that spreads are countercyclical (α > 0) will play a crucial role in what follows. As noted above, this represents the conventional view in the profession. Nonetheless, given the role it plays in our analysis, we also present empirical evidence that supports it. Table 1 reports estimates of the long-run elasticities of the spread with respect to total loans, η, and the output gap, α, obtained from the regression Output is measured by real GDP and loans correspond to total loans at commercial banks. 12 For both variables we use deviations of the log levels from a linear trend. The long-run elasticities are obtained as The spread is measured as the difference between corporate bond rates, either AAA or BAA, and either the federal funds rate, FFR, or the Treasury bill rate, TBILL. 13 We 12 Using instead either consumer loans or business loans produces very similar results, but leads to higher estimates of the degree of countercyclicality of spreads, α.
13 Data sources are reported in Appendix A.
present two sets of regressions, since monetary policy is generally thought to have changed fundamentally around 1980. The estimates of the marginal cost parameters, α and η , are very stable over time. All of the regressions produce a positive and significant estimate of α. We utilize these regression results to calibrate the parameters in the cost function (5).

Firms
A continuum of monopolistically competitive firms, indexed by f, produce intermediate goods using the technology where h t (f ) is a CES aggregate of the households' labor and ξ y,t is an auto-regressive aggregate productivity shock. Competitive retailers buy the intermediate goods at price P t (f ) and bundle them into the final good, y t , using a CES aggregator with elasticity θ.
The final good is then sold, at price P t = 1 0 P t (f ) 1−θ df 1 1−θ , to households and the government.
Wages are flexible, but prices are not. In particular we employ the popular Calvo price setting scheme. In each period, an intermediate good firm gets the opportunity to re-set price optimally with probability 1− γ. As is well known, a dispersion of intermediate good prices distorts household consumption patterns and the efficient use of labor. So, aggregate output is df > 1 when γ > 0. When γ = 0, prices are flexible and there is no price dispersion; that is, ∆ t = 1.
In equilibrium

Government
The consolidated government flow budget constraint is where i g t is the interest rate on government bonds; it will be recalled that i g t = i d t (since savers are indifferent between holding bank deposits and public debt). Government spending follows an auto-regressive process where ξ g,t is an innovation. Increases in government spending are initially bond financed, but lump sum taxes increase over time to stabilize the debt Note that Ricardian equivalence does not hold in our model. In particular, borrowers discount future liabilities at a rate that exceeds the interest rate on public debt. 14 A tax cut financed by an increase in government debt generates a positive wealth effect for them.
Monetary policy follows a standard interest rate rule where π t is the rate of inflation and ξ i,t is a policy shock.

Model Calibration
The baseline calibration of our model's parameters closely follows Curdia andWoodford [2009, 2010] 15 and is reported in Table 2. In what follows, we will let The curvature parameters of the utility functions, σ b and σ s , are set so that the average curvature parameter is 6.25 and the ratio of the curvature parameters is σ b /σ s = 5.
The levels of ξ b c and ξ s c are set in a way that guarantees that borrowers always have a higher marginal utility than the savers (see equation 2). The value of the labor elasticity parameter is as in Curdia and Woodford. The discount factor, β, is set so that the nominal deposit rate is 1% per quarter. Households' access to the insurance agency is infrequent: δ = 0.975. But once there, the household has a 50-50 chance of changing type: π b = π s = 1 2 . On the firm side, the inverse labor elasticity is set to ψ = 0.75, and the elasticity of substitution between intermediate goods is set so that the markup rate is 15%. The Calvo parameter and the production parameters are standard in the literature. Setting γ = 2/3 means that price settings last 3 quarters on average. The parameters of the interest rate rule and the process for government spending are also representative of those used in the literature.
14 Savers also discount the future at a rate exceeding that on public debt because of the possibility of switching type.
15 More precisely, the parameters are set using the same methodology as in Curdia and Woodford. However, since our model departs slightly from theirs in several minor ways (for example, we do not have sales taxes) some of our parameters differ from theirs.  Gilchrist and Zakrajsek, 2011). This also implies that steady state financial costs represent 0.24% of output. Note that we later conduct a thorough sensitivity analysis on these parameters.
The model is solved under perfect foresight using the non-linear method proposed by Laffargue [1990] and Boucekkine [1995] as implemented in DYNARE.

Cyclical Fiscal Multipliers
We can compute multipliers over the business cycle for cycles generated by any of the shocks in the model. Let ξ x denote a shock to the exogenous variable x. Let us denote by ξ R x , respectively ξ E x , the value of the shock to the exogenous variable x that triggers a recession, respectively expansion. In our benchmark experiment, we choose a ξ R x that is large enough to make output fall by 2.5%; then, we choose a ξ E x that will make output rise by 2.5%. 17 In order to evaluate the effectiveness of fiscal policy over the business cycle we generate an expansion or recession and then immediately induce a fiscal response to it by having the government spending shock, ξ g,t , respond by one percent. Let z t+i (ξ x , g) denote the path of either consumption or aggregate output following a shock to the exogenous variable x which is accompanied by a fiscal response, and let z t+i (ξ x ) denote the path in the absence of a fiscal response. Then the cumulative multiplier h quarters after the shock is computed 16 Note that the value chosen is rather conservative. As noted previously, when consumer loans or business loans are used in place of total loans, higher values of α are obtained. Using those values would result in even higher multipliers during recessions (in the range of 3 to 4).
17 Note that the two shocks need not be of the same size (in absolute value) since the model is not linear.

Financial Market Shocks and Multipliers
In our benchmark simulations, we study business cycles caused by the shock to the spread, ξ Ψ . Figure 2 shows impulse response functions (IRF) for output in the absence of a fiscal response. The dark (or black) IRF is generated by a positive shock that is large enough to cause a output to fall by 2.5%. The light (or red) IRF is generated by a negative shock that would cause a 2.5% expansion; the graph for an expansion has been inverted for easier comparison with the recession case.

Expansion Recession
The IRF's are not symmetric. In particular, output reverts to its steady state value more quickly in the case of a recession; a fact which is consistent with the empirical evidence (see Hamilton [1989], Beaudry and Koop [1993], Acemoglu and Scott [1997]). This is due to the fact that any given change in output has a more powerful effect on the premium during recessions, which serves to accelerate the process of recovery during recessions.
The reason for this lies in the existence of a powerful financial accelerator during recessions.
An increase in output of a given size translates into a larger reduction in the financial friction and thus a greater output gain during recessions than during expansions.This is due to the fact that at the depth of a recession, the spread, i b  An increase in government spending that is partly financed by higher taxes raises the present and future tax burden on all agents. This by itself has a negative wealth effect on households. In the standard model, this is the only effect, and the Ricardian households are induced to work harder and/or consume less in order to meet their higher tax obligations.
In our model, however, there is an additional effect that operates through the credit friction. The reduction of the spread caused by higher government spending has a positive effect on the consumption of the credit constrained agent. If this effect is large enough relative to the negative wealth effect associated with the higher expected taxes, then the borrowers end up increasing their consumption (while the savers' consumption drops).

Expansion Recession
An expansion in government spending during a severe recession (a period of high spreads) has thus the potential to lead to an increase in the consumption of borrowers that exceeds the reduction in the consumption of the savers. Aggregate consumption rises. This is sufficient to produce output multipliers that are greater than one. But in expansions, the increase in the consumption of the borrowers is smaller, and in our calibration, aggregate consumption falls; output multipliers are less than one. The reason for this result lies in the asymmetric cyclical variation of the spread. As can be seen in Figure 5, the spread, , widens disproportionately during a recession while it contracts in an expansion. That is, any amelioration in the financial friction is much more stimulating for the borrowers -who play the crucial role for the multiplier-in bad than in good times. Figure 6 provides some direct suggestive evidence concerning the relationship between spreads and government spending (the government spending to GDP share) that supports our theoretical analysis. We classify periods as either a "recession" or as an "expansion" depending on whether output in that period is above or below the H-P trend. The red dots in the graph correspond to recessions and the black ones to expansions. We then fit a regression line for each category of the state of the business cycle using instrumental variables (past values of the regressors). The figure exhibits three features. First and consistent with the empirical findings reported in section 1.2, spreads are on average higher during recessions than during expansions. Second, spreads are negatively related to government spending. And third and more important from the point of view of the properties of the model discussed above, there is a cyclical asymmetry. In particular, the effect of any change in government spending (as a share of GDP) on spreads is considerably more pronounced during recessions relative to booms (the slope of the red line is steeper than that of the black line). Related information on correlations between g/y and spreads is reported in Table 3. It is worth noting that the resources used in the intermediation process are a small share of output (0.24% of steady state output) and, moreover, they do not directly matter   Note: E denotes an expansion and R denotes a recession.

Debt vs Tax Finance of Government Spending and Multipliers
In our benchmark simulations, the lump sum tax rule (15) stabilizes debt dynamics.
This means that the increase in government spending is partially bond financed. Figure 7 shows how Figure 3 would change if the debt-tax rule were replaced by a balanced budget rule. The cumulative multipliers in Figure 7 are now smaller than those shown in Figure 3.
The reason is that the increase in the consumption of the borrowers is lower (See Figure  8). As in the case with partly debt financed spending, government spending expands output and closes the output gap, which makes the credit spread decrease and generates a positive wealth effect for the borrowers. But unlike the case of debt financed spending, the borrower is taxed in the current period and so has fewer funds to spend on consumption.
This implies a weaker consumption response and a smaller multiplier.

Expansion Average Recession
In contrast, the savers' consumption drops by less under a balanced government budget.
This is due to the difference in interest rates across the two schemes of financing government spending. When no debt is issued the deposit rate is lower than when debt is issued (due to the violation of Ricardian equivalence). With a lower interest rate there is less of an incentive to reduce current consumption. Nonetheless, the differential effect on the consumption of the savers is much smaller than that on the borrowers, so total consumption increases by less, leading to lower multipliers.
While the mechanisms are different, this result is reminiscent of a similar result in the traditional IS-LM, Keynesian model, namely, that the size of the multiplier varies with the method used to finance government spending. And that the greater the reliance on debt, the greater the multipliers.

The Size of the Fiscal Shock and Multipliers
Does the size of the multiplier vary with the size of the fiscal expansion? Figure 9 shows that the multiplier is decreasing in the size of the fiscal intervention. For instance, the impact multiplier for a 5% or 10% intervention is lower than that for 1% (1.85, 1.65 and 2.25 respectively). The reason that large amounts of government spending may prove less effective than smaller amounts is that the -negative-marginal wealth effect due to the higher tax liabilities is increasing in the size of the fiscal intervention while the -positivemarginal effect on the borrower from the reduction in the premium is decreasing in the size of the fiscal expansion. 19

Multipliers and the Source of the Business Cycle
Expansions and recessions can have a variety of origins and the size of multipliers may well depend upon the source of the business cycle. 20 Table 5 reports cumulative output multipliers for various types of shocks: the first three are preference shocks (to the marginal utility of the impatient and patient households and the disutility of labor), the fourth is the financial shock used in the benchmark scenario above, the fifth is a productivity shock (ξ y,t ), and the sixth is a monetary policy shock (ξ i,t ). In all cases the size of the shock is such that it generates a recession (resp. expansion) of 2.5%.
There is some variation in the impact multipliers; our benchmark shock gives the largest impact multiplier. Importantly, no matter the source of the business cycle, multipliers are larger in recessions (about 2) and smaller (around one or less) in expansions. After the first year, the cause of the business cycle does not seem to matter any more.
19 Note that our model is silent on normative issues such as the optimal size of the fiscal intervention. 20 Hereafter and unless clearly specified, we will refer to output multipliers as multipliers

Sensitivity to Parameters
In this section we examine whether the size of the multipliers implied by our model is sensitive to the calibration used. We consider variation in: i) the degree of price rigidity, ii) the amplitude of the business cycle, iii), the parameters of the monetary policy rule and, iv) the parameters in the bank lending cost function. 21 . In no case do minor perturbations make a big difference for the size of the multipliers. The sensitivity analysis will be conducted only under the benchmark bank lending cost shock as the last section showed that the source of the cycle did not make much of a difference. Figure 10 shows that cumulative output multipliers rise as the degree of price rigidity, γ (the Calvo parameter), increases and that they reach their maximum at about γ = 0.8.

The Degree of Price Rigidity
Our benchmark setting is γ = 0.67, that is, prices are reset on average every 3 quarters.
In the New Keynesian literature, common values for γ are 0.67 and 0.75. In this range the multiplier are large in recessions and small in expansions, and of a magnitude consistent with the findings of Auerbach and Gorodnichenko [2010]. The reason that the multiplier is increasing in the degree of price rigidity is that, the more rigid the prices, the bigger the effect of government spending on closing the output gap and hence the larger the decline in the spread. Under our calibration, this effect peaks at about γ = 0.8 and then it declines somewhat (but remains large). The reason for this non-monotonicity seems to be that under extreme degrees of price rigidity, monetary policy is more potent and it closes more of the output gap by itself, leaving less room for the fiscal stimulus to manifest its

Amplitude of the Business Cycle
The model being non-linear, the size of the multiplier ought to depend on the amplitude of the business cycle. Figure 11 shows that this is indeed the case: the size of multipliers in a recession grows with the amplitude of the cycle, while the size of multipliers in an expansion falls with an increase in the amplitude. In our benchmark case, we chose shocks that made output rise or fall by 2.5%, which may be deemed a normal amplitude for business cycles. The impact multiplier during a recession was 2.17. But for a deeper The reason for this can be found in, yet again, the cyclical variation of the spreads.
The deeper the recession the larger the interest rate spread, i b t − i g t , and also and more importantly the larger the elasticity of the spread to a variation in y. 22 Hence after an increase in fiscal expenditures, the amelioration of the financial friction will be larger in deeper recessions. The potential output gains from a fiscal stimulus are therefore magnified. On the contrary, the greater the expansion, the smaller the elasticity and hence the smaller the gains from the mitigation of the friction.

Multipliers and the Conduct of Monetary Policy
As the literature on the zero bound has shown, multipliers are not independent of the conduct of monetary policy. Figure 12 shows how monetary policy, through its concern for inflation and output fluctuations, can affect cumulative multipliers.

Expansion Recession
Panel (a) suggests that an increase in the reaction of monetary authorities to the output gap lowers the size of the multiplier. This is because a stronger reaction means that monetary policy closes more of the output gap and hence lowers the spread by more. As we have shown before, fiscal policy is less effective when applied to a smaller spread, so the multipliers are decreasing in the level of κ y .
22 This elasticity is given by −α yt.
Panel (b) depicts the multiplier as a function of the reaction to inflation, κ π . In order to facilitate the exposition we employed a policy rule with κ y = 0. 23 An increase in the weight placed on price stability means a smaller multiplier. The reason is as follows.
Consider a negative financial shock. Both output and inflation decrease. The central bank cuts interest rates as inflation is below target, and the cut is larger the larger κ π .
A more expansionary monetary policy means a smaller -negative-output gap and thus a smaller spread. But with a smaller spread, the effects of fiscal policy on output are smaller. That is, a more aggressive countercyclical monetary policy limits the contribution of countercyclical fiscal policy.

The Role of Banking Parameters
As argued before, the existence of a sizable multiplier lies in the presence of the "financial accelerator mechanism" described at the end of section 3.2. One measure of the degree of financial friction is ω = i b − i d , the steady state level of the spread between borrowing and deposit rates. Figure 13 shows that the cumulative output multipliers in a recession vary significantly with perturbations to the steady state spread. For instance, while in our benchmark calibration -an annual spread of 2%-the recession multiplier is about 2.2, raising the spread by just 20 basis points increases the recession multiplier by about 50% (around 3.25).

Expansion Recession
The reason is that a larger steady state spread corresponds to a larger gap between the rates used to discount future consumption streams and tax liabilities. Hence the potential positive wealth effects for borrowers -and hence the multipliers-are larger the greater is the spread. 23 As expected in light of the previous discussion, this leads to much larger multipliers.
The elasticity of bank lending costs to the output gap, α, is a parameter that is fundamental to our quantitative results. A larger α means that the spread is more sensitive to the state of the business cycle, and thus that fiscal policy is more effective: An increase in aggregate demand during a recession has a large impact on the spread, generating large positive wealth effects on borrowers and driving the size of the multiplier up. Figure 14 shows that even small perturbations in α can have a big effect on the cumulative output multipliers. We chose α = 23 in our benchmark setting as suggested by our empirical estimates (reported in Table 1). This value produced multipliers consistent with the multipliers found by Auerbach and Gorodnichenko [2010] in the data. However, our regressions can not rule out even larger values of α that would give rise to even larger multipliers.
For instance, a value of α of 29 gives a multiplier in a recession of about 4.

Expansion Recession
One can similarly analyze the role of η for the multiplier. In general, a higher η means a larger spread for any given level of debt. Hence its implications for the multiplier are quite similar to those discussed above for the steady state spread.

Conclusion
Countercyclical fiscal policy represents a puzzle. Policymakers routinely fight economic downturns by using budget deficits, presumably because they think that fiscal multipliers are large. While this is in line with Keynes's original recommendation and is consistent with traditional IS-LM type of thinking, there exists preciously little in terms of modern economic thinking that supports multiplier values exceeding unity during recessions.
Some recent work has suggested that the zero nominal interest bound may make multipliers large during recessions (exceeding one) even when they are quite small during expansion.
But, except for the recent recession, countercyclical policy does not seem to coincide with the presence of such constraint on monetary policy, so the zero bound cannot be the full story.
In this paper we have proposed an alternative, more general theory of large and cyclically variable multipliers. Our theory is based on the postulate that financial frictions matter for the business cycle. And how much they matter varies over the business cycle. There exists a presumption in the literature that the variation in the financial frictions is countercyclical, and we have provided empirical evidence supporting this presumption. We show that countercylical financial frictions can make government spending quite effective during recessions, in particular when financed by debt. The main mechanism is similar to the old-Keynesian idea that providing financially strapped agents (households and firms) with funds creates a positive wealth effect for them even when they take into account any increase in their future tax liabilities. The more severe and widespread the financial constraints, the larger this wealth effect and thus the higher the likelihood of a positive aggregate consumption response to a fiscal stimulus. While our analysis relies on spread movements rather than on the relaxation of quantitative borrowing constraints the logic is the same.