Download The Government Transfer Multiplier

When is the Government Transfer Multiplier Large?∗
Eric Giambattista and Steven Pennings†
This draft: 7 November 2012
Abstract
Government transfers to individuals were a larger part of the 2009 US fiscal stimulus
package than government purchases. In this paper, we use a two-agent model with nominal
rigidities to investigate the impact of transfers to financially constrained households on output.
We find that the government transfer multiplier can be large—defined as larger than the
purchases multiplier, or larger than one—especially if fiscal policy is short-lived, the share
of credit-constrained households is not too low, or the Zero Lower Bound (ZLB) on nominal
interest rates binds. The mechanism involves the tendency for a positive output gap to
increase labour demand and the wages of credit-constrained households, which then boosts
aggregate demand as the extra income is spent hand-to-mouth (the “disposable income effect”).
A government transfer, unlike a government purchase, leads to a reduction in the labour
supply of constrained households due to wealth effects (a force which weakens when wages
are sticky). When the disposable income effect outweighs the central bank’s efforts to lean
against a positive output gap by raising interest rates, the economy’s aggregate demand curve
inverts, such that a reduction in supply from the fiscal transfer provides a second-round boost
to output. In our model, the multiplier is about one for both transfers and purchases if the
fiscal stimulus only lasts a quarter, or around 0.4 (0.6) for transfers (purchases) if the change
in fiscal policy is persistent. At the ZLB (which lasts for five years) and with persistent fiscal
policy, the transfers (purchases) multiplier rises to about 1.7 (1.2).
1
Introduction
In the years preceding the Global Financial Crisis, the role of macroeconomic management had
largely fallen to central banks, with fiscal policy playing a secondary role. But with the sheer
∗
JEL: E63 E62 E52. Keywords: Fiscal Transfers, Fiscal policy, Fiscal stimulus, Government spending, multipliers, New-Keynesian models, Zero Lower Bound, monetary policy. Helpful comments have been received from
Mariano Kulish, Jonathan Kearns, Tommaso Monacelli, Mark Gertler, Taisuke Nakata, Alex Heath and seminar
participants at the Reserve Bank of Australia and New York University.
†
Department of Economics, New York University (NYU) 19 W. 4th St, 6th Floor, New York, NY, 10009, USA.
Email: [email protected] and [email protected]
1
magnitude of the global recession, and the Zero Lower Bound (ZLB) on nominal interest rates
binding in the United States and other countries, fiscal policy has now taken a more prominent
role in policymakers’ attempts to stimulate the economy. This has lead to a renewed interest in the
response of output to an increase in government purchases: the government purchases multiplier.
Despite the focus on the government purchase multiplier in the literature (Woodford 2010,
Christiano et al 2011, Cogan et al 2010), the majority of the increase in government spending during
the global financial crisis was government transfers to households, not government purchases.
According to Oh and Reis (2011), 75 per cent of the increase in US government spending between
2007 and 2009 was transfers, slightly above the OECD median of 64 per cent. Using a more
conservative classification of transfers (described in the Appendix A), we find that during the ten
quarters from 2009:1, transfers accounted for around US$250 billion or more than 50 per cent
of the spending component of the the American Recovery and Reinvestment Act (ARRA). In
representative household models, government transfers have no effect. This has led Cogan and
Taylor (2010) to conclude: “Basic economic theory implies that temporary increases in transfer
payments have a much smaller impact than government purchases” (p22).
This paper examines the determinants of the government transfer multiplier in a two-agent
model with nominal rigidities where around a third of the population are financially constrained.
The model examines the effect of a package similar to the ARRA, in a closed economy setup that is
commonly used to analyse the US economy. In the model, the fiscal package consists of a transfer
to financially constrained households, funded by lump-sum taxes on the unconstrained households.
Because the unconstrained households are Ricardian, the timing of tax payments and the size of
the government deficit do not affect the economy.
We find that the government transfer multiplier can be large, which we define as (i) larger than
the purchases multiplier, or (ii) larger than one. The transfer multiplier is likely to be large (by
either definition) when fiscal policy is short-lived, the share of credit-constrained households is not
too low (when monetary policy follows a Taylor Rule). For unanticipated transfers lasting only
a quarter, the transfer multiplier is usually larger than the purchases multiplier, though for more
persistent fiscal stimulus, the purchases multiplier is usually larger. When the ZLB binds, as has
been the case the in the United States since the fourth quarter of 2008, the transfer multiplier is
usually larger than the purchase multiplier — even when the fiscal stimulus is persistent.
We show analytically in a simplified model (Section 3) that the size of the transfer and purchases
multiplier depends crucially on the relative strength of “disposable income effect” and the “Taylor
Principle effect”. The disposable income effect is the tenancy for a positive output gap to boost the
wages of credit-constrained households, which then boosts aggregate demand as the extra income
is spent hand-to-mouth. The “Taylor principle effect” works in the opposite direction: a positive
output gap causes the central bank to raise interest rates, reducing the demand of the Ricardian
household. Whenever the disposable income effect is stronger, the transfer multiplier is larger than
the purchases multiplier.
2
While both transfers and purchases boost aggregate demand directly, transfers also reduce aggregate supply (through a wealth effect on the labour supply of constrained households). Normally,
one would think the reduction in aggregate supply would lower the transfer multiplier below the
purchases multiplier. However, we show analytically in Section 3 that when the “disposable income
effect” dominates the “Taylor principle effect”, the economy’s aggregate demand curve becomes inverted, causing a reduction in supply from the fiscal transfer to give a second-round boost to output.
In Section 4, we show this reduction in supply is weakened by sticky wages—as household’s labour
supply decisions are determined by past wage rates rather than wealth effects—leading the transfer
multiplier to approach the purchase multiplier in the limit.
In Section 5, we calculate the transfer multiplier in a calibrated medium-scale DSGE model
with capital and sticky wages. We find that a once-off 1 per cent of GDP fiscal stimulus from
either transfers or purchases raises the present value of output by about 1 per cent. Policies with a
persistence similar to that of the ARRA (auto-correlation of 0.9), have a present value multiplier
of around 0.4 for transfers or 0.6 for purchases. If monetary policy is constrained by the ZLB for
five years, the transfer multiplier is around 1.3 for a once-off stimulus, and 1.7 for a persistent
stimulus (with purchase multipliers being around 1.2-1-3 in either case).
For US policymakers, our results suggest that transfers can be used to stimulate the economy,
but are most effective when (i) the ZLB binds during the time of the fiscal stimulus, (ii) they are
targeted those that are financially constrained, who are more likely to spend them, and (iii) they
are once-off (if monetary policy is not constrained by the ZLB).1
Our approach is guided by the empirical results of Johnson et al (2006), who examine the effect
the 2001 Bush tax rebates on consumption. The timing of these lump-sum rebates was randomised
by social security number and so rigorous identification of their impact is possible. Despite the
program being pre-announced, Johnson et al (2006) find that in total around 20-40 per cent of the
rebates were spent in the months that they arrived. This is inconsistent with a standard frictionless
model, where it is only the present value of the payments — and not their timing — that should
affect consumption. Moreover, the marginal propensity to consume (MPC) is statistically greater
than zero — and insignificantly different from unity — for the most financially constrained and
lowest third of households by income, but insignificantly different from zero for other households.
Our model is consistent with these empirical facts: around a third of households in the model
are financially constrained (in the default calibration) and have a MPC of one, and the other two
thirds of households are Ricardian.2
1
Three caveats (and areas for future research) are that in the model taxation is lump-sum (rather than distortionary), the economy is closed, and that we rely on linearisation and thus ignore the fact that the effectiveness
of stimulus may depend on how far the economy is from steady state. During the most recent recession many
policymakers justified stimulus by noting that the economy was far below potential, with a large output gap and
high unemployment — our linearised environment necessarily ignores differential effects of policy when the economy
is far away from steady state.
2
Based on the life-cycle and differences in returns, Kaplan and Violante (2011) show that wealthy households
can behave in a hand-to-mouth fashion if they hold low levels of liquid assets.
3
Although there are many recent papers examining government purchases in DSGE models (for
example, Christiano et al (2011), Cogan et al (2010), Woodford (2010) and Uhlig (2010)), there are
few papers that consider transfers in a setting similar to ours. Closest to our paper is contemporaneous work by Monacelli and Perotti (2011) and Mehrota (2011). Monacelli and Perotti (2011)
present a New Keynesian saver-borrower DSGE model, and find that the government purchase
multiplier is larger when taxes are levied on the savers (rather than the borrowers). They briefly
discuss the effects of fiscal transfers, finding a positive impact multiplier. Their model differs from
our analytical model (Section 3) primarily as they assume the labour of constrained and unconstrained households is perfect substitutes in production, whereas we use Cobb-Douglas production
function — though these differences become less important with sticky wages. Mehrota (2011) also
uses a two-agent model with nominal rigidities to examine the effect of transfers and purchases,
but he assumes a debt-elastic interest rate spread for the constrained borrower as in Curdia and
Woodford (2010), and also assumes labour of the different agents is perfect substitutes in production. He examines a largely debt-funded lump-sum rebate to all households (an untargeted
transfer) that has a small multiplier — rather than the targeted transfer we consider here.
More broadly, our paper is related to Oh and Reis (2011) who document the size of transfers in
the ARRA, and find the transfer multiplier to be stimulatory using a heterogeneous agent model
where a continuum of agents suffer health and employment shocks. Their model is different to
ours in that they do not model monetary policy, which is a focus of our paper. Eggertsson and
Krugman (2012) model the effects of debt reduction in a model similar to our simple model in
Section 3. They also find an upward sloping aggregate demand curve, but only when the ZLB
is binding, rather than with a Taylor rule as we find in Section 3. Bilbiie (2008) shows that the
presence of non-Ricardian households in a model with nominal rigidities can lead to “inverted
aggregate demand logic”, though this is conceptually different from the upward sloping aggregate
demand curve presented here (his paper also does not consider fiscal policy).
2
Model
We examine the effect of government transfers and purchases in a New Keynesian DSGE model
with two agents that differ in their access to financial markets. The Ricardian consumer (agent 1)
has full access to financial markets and the constrained consumer (agent 2) is completely cut off
from financial markets and so consumes his entire income each period in a hand-to-mouth fashion
as in Gali et al (2007). The government levies lump-sum taxes on the Ricardian consumer to pay
for government purchases, as well as transfers to the constrained household.
The Ricardian household owns capital (which they rent to the intermediate goods firm) and
owns retailers who transform intermediate goods into final goods. Retailer’s prices are sticky in the
Calvo sense and so aggregate demand and monetary policy will matter for real outcomes. Wages
are sticky as in Erceg, Henderson, and Levin (2000).
4
In the model, the Ricardian consumer represents higher-income households who pay net taxes,
and the constrained households represent lower income households who are the main recipients of
government transfers. We present evidence in the Appendix which shows that the typical recipient
of transfers from the ARRA was lower income.
Unlike Gali et al (2007), we assume that labour of the two households is differentiated by skill
level — agent 1 and 2 are skilled and semi-skilled respectively (with a Cobb-Douglas production
function). This is similar to Iacoviello (2005) who also has constrained and unconstrained households with differentiated labour.3 Data presented in Appendix B supports this differentiation:
using Hall’s (2011) definition of financial constraints, people with lower levels of education tend
to be more financially constrained. Although the differentiated and perfect substitutes production
functions differ a little in the quantitative predictions of the stripped-down model in Section 3,
once sticky wages are included in Section 4 and Section 5 the differences are small.4 Qualitatively,
the results of the two models are similar.
2.1
Ricardian household’s problem
The Ricardian agent chooses real consumption (c1,t ), desired labour hours (L1t ), real debt (bt ) and
investment (It ) to maximize his utility, taking real interest rates (RRt ), lump-sum taxes (T1,t ),
real wages (w1t ), the real gross rate of return on capital (M P Kt ) and profits from retailers (Πt ) as
given. In the model with flexible wages, desired labour hours equal actual hours. However, if wages
are sticky, then actual hours are determined by the firm at the given (sticky) real wage. LSSS is
a steady state transfer from Ricardian households to the constrained households to stop profits
from changing the steady state income distribution in the simple model (equals zero in the full
2
t−1
t
− δ K2δ
.
model).5 Changing the level of capital is subject to adjustment costs of ACKt = ψ KIt−1
Therefore the Ricardian household’s problem is:
max{c1,t ,b1,t ,It ,L1,t } E0
∞
X
t=0
Lη1,t
β [ln(c1,t ) −
]
η
t
(1)
such that:
c1t + It + ACKt + bt = RRt b,t−1 + M P Kt Kt−1 + w1t L1t + Πt − T1,t − LSSS
(2)
Kt = (1 − δ)Kt−1 + It
(3)
3
We thank Matteo Iacoviello for making his Dynare code publicly available.
The main differences are driven by the labour supply decisions of the two households, which are muted when
wages are sticky.
5
Specifically, LSSS = (1 − α)ΠSS
4
5
2.2
Constrained household’s problem
The constrained consumer’s problem is much simpler than that of the Ricardian consumer: he
only has to choose how much to work each period (L2,t ) as he can not smooth consumption over
time. Real consumption (c2,t ) is equal to labour income plus lump-sum transfers ( T2t ) from the
government.
max{c2,t ,L2,t } E0
∞
X
t=0
Lη2,t
]
β2 [ln(c2,t ) −
η
t
(4)
such that:
w2,t L2t + T2,t + LSSS = c2,t
2.3
(5)
Sticky wages
The government transfer multiplier depends crucially on the labour supply response of different
types of agents. Christiano et al (2005) argue that sticky wages are important in fitting the
response of a monetary policy shock to the data, and Gali et al (2007) argue that some form of
wage rigidity (a union in their case) is needed to fit the response of consumption to a government
purchases shock. Because wage stickiness necessitates adding additional state variables (lagged
real wages), we assume flexible wages in the analytical model (but include sticky wages in the full
model in Section 5).
We model wage stickiness as in Erceg, Henderson, and Levin (2000). The Ricardian and
constrained households are each composed of a continuum of identical agents, indexed by j. The
production function is Cobb-Douglas as before, except now L1 and L2 are now CES composites of
the different types of differentiated labour inputs (with a share α of total labour supplied by the
Ricardian agent):
´ 1−α
´α
w
w
1
1
L1t = [ 0 L1t (j)1− w dj] w −1 ] and L2t = [ 0 L2t (j)1− w dj] w −1 ]
The wage indices are defined as follows:
´ 1−α
´α
1
1
W 1t = [ 0 W 1t (j)1−w dj] 1−w and W 2t = [ 0 W 2t (j)1−w dj] 1−w .
As with Calvo pricing, the household is allowed to reset its nominal wage with constant probability 1 − θw in each period. Since each household possesses market power in their labour supply
decision, they are able to set their wage at a markup above their marginal rate of substitution.
Given the wage-setting decisions by households which re-optimize, and the fact that households
which do not re-optimize must keep their nominal wages at the last period’s value, there is an
analogue of a Phillips curve for each of the types of household, expressed in terms of nominal
wage inflation for each household π̂ w
t,i = logWi,t − logWi,t−1 , i = 1, 2 as a function of expected
wage inflation tomorrow and the deviations of the households marginal rate of substitution from
its steady state level (hereafter, all variables with hats denote deviations from steady-state). We
assume that the steady state distortion from wage differentiation is small.
6
with λi =
2.4
(1−θw )(1−θw β)
,
θw (1+(η−1))
w
w
π̂ w
i,t = βEt π̂ i,t+1 − λi µ̂i,t
(6)
µ̂w
i,t = ŵi,t − (η − 1)L̂i,t − ĉi,t
(7)
and where i = 1, 2 and ŵi,t = ŵi,t−1 + π̂ w
i,t − π̂ t is the real wage.
Sticky prices, Retailers, Intermediate and Final Output
Intermediate output is Cobb-Douglas in capital, and the labour from each of the agents, with A
being a productivity parameter6 .
(1−µ)α
Yt = AKtµ L1t
(1−µ)(1−α)
L2t
(8)
As in Bernanke et al (1999) and Iacoviello (2005), final output is produced by a unit continuum of
retailers, indexed by i, who buy intermediate output in competitive market, costlessly differentiate
it, and sell a variety of final output Yi,t . Aggregate final output is given by the index Ytf =
σ
´
σ−1
σ−1
1
σ
Y
. Each retailer faces a downward sloping demand curve for his variety, and he
di
0 i,t
must chose the optimal nominal price for his variety subject to a Calvo probability θ that he
may not be able to change his price. The pricing problem of retailers leads to a standard New
Keynesian Phillips curve (Equation 9), which is shown in log deviation from steady state, where
π̂t = lnPt −lnPt−1 is the inflation rate (steady state inflation is zero), X̂t = lnXt −lnX is deviation
σ
and κ = (1 − θ)(1 − βθ)/θ).
in the retailer’s average markup from steady state (where X = σ−1
The variable κ can be thought of at the slope of the Phillips curve — the higher κ, the more
responsive inflation (and less responsive output) is to a given shift in demand. With flexible prices
κ → ∞, and so shifts in demand affect prices and not output. With more sticky prices (higher θ)
most firms are unable to change their prices to move markups towards their desired level, resulting
in a muted response of inflation and a boost in output to increases in government purchases or
transfers.
π̂t = βEt π̂t+1 − κX̂t
(9)
The price of intermediate output in terms of final output is the inverse of the retailer’s average
1
P int
markup t =
. As such, the marginal product of labour or capital in terms of intermediate
Pt
Xt
goods must be divided by the markup to generate the real marginal product. Following Iacoviello
(2005), intermediate and final output move closely in the neighbourhood of the steady state, and
6
A is normalized so that steady-state output is equal to 1. This is chosen solely for computational convenience.
7
so we assume that Ytf ≈ Yt .
wj =
2.5
1 Yt
, j = 1, 2
Xt Ljt
(10)
Monetary and Fiscal Policy
The central bank follows a Taylor Rule (in linearised form) with interest rate smoothing, where
R̂t = lnRt − lnR is the log deviation of the nominal interest rate from its steady state level. In
the full model, we allow for the possibility that the central bank is constrained by the ZLB and
keeps the nominal rate fixed at zero for a certain number of periods before resuming the Taylor
rule (Equation 11). The degree of interest rate smoothing is governed by the parameter rR.
R̂t = rR.R̂t−1 + (1 − rR)(φπ π̂t + φY Ŷt )
2.5.1
(11)
Fiscal policy
Government expenditures consist of unproductive government purchases gt , and targeted transfers
to the constrained households T2t . Government expenditure is financed by lump sum tax on the
Ricardian households T1t . The government runs a balanced budget each period. Note here that
throughout the paper, T̂1t and ĝt are government purchases and transfers as a share of GDP,
1t
and ĝt ≡ YGSSt . This simplifies the expressions for multipliers and allows for the
i.e. T̂1t ≡ YTSS
possibility that purchases are zero in steady state. This only applies to transfers and purchases:
other variables with “hats” are log deviations from their respective steady-states.
Whether the government runs a balanced budget does not matter for the path of the economy
as taxes are only levied on the unconstrained households, who are Ricardian — it is only the timing
of the transfers and purchases that affect allocations.7
T1t = T2t + Gt
(12)
The path of T̂2t and ĝt are exogenous and we assume here that they take an AR(1) process. As
we show in Appendix A.1, the path of transfers in the ARRA closely resembles an AR(1) process.
T2t+1 = ρT2t + eT 2,t+1
(13)
gt+1 = ρgt + eG,t+1
(14)
The model is closed by adding a standard aggregate resource constraint:
Yt = c1t + c2,t + It + Gt
7
This is not the case if taxes are levied on the constrained households, or if taxes are distortionary.
8
(15)
2.6
Solution method and steady state
To solve the model, we log-linearise around the non-stochastic steady state. We solve the full
model numerically using Dynare. Where the ZLB binds for a predetermined number of periods,
we solve the model using the deterministic response to an initial fiscal policy shock using the same
method as Cogan et al (2010).
In order to solve the model analytically in Section 3 (and keep expression simple), we assume
(i) wages are flexible (λj → ∞, j = 1, 2), (ii) the capital share goes to zero in the Cobb-Douglas
production function µ → 0, (iii) the central bank only respond to inflation (φY ), and (iv) that the
steady state consumption share of each agent is equal to their share of wage income. To achieve
the last condition, we set steady state government purchases to zero (Gss = 0), and assume a
small lump-sum transfer LSSS = (1 − α)ΠSS to constrained consumers to offset the effect of
c1,ss
= α and
retailer’s profits on the income distribution.8 In the simple model (without capital),
Yss
c2,ss
= 1 − α , steady state inflation is zero, bond holdings are zero, and the real interest rate is
Yss
β −1 . In this case, the only state variables are government purchases and transfers, and these follow
AR(1) processes. As such, all variables depend on these AR(1) processes, depend linearly on Gt
and T2,t , and the multiplier is constant.
We can make the steady state more realistic for the full model, as we no longer need an
analytical solution. Specifically, we keep government transfers T2,t as zero in steady state, but set
LSSS = 0. Steady state government purchases are 20 per cent of GDP (GSS = 0.2) and we assume
that these are funded by lump sum taxes, which happen to be proportional to the labour and
capital income of each agent, net of depreciation.9
2.7
Parameters
Table 1 lists the parameters in the full model, and which ones are set to zero in the simple model.
Most are taken from Iacoviello (2005) and are fairly standard in the literature. Parameters taken
from elsewhere include wage stickiness parameters (taken from Christiano et al 2005), and the
Frisch elasticity of labour supply of 2, which implies η = 1.5, is the average of the value taken in
Bernanke, Gertler and Gilchrist (1999) and Christiano, Eichenbaum and Evans (2005) (3 and 1
respectively). α = 0.64 is the labour share of Ricardian agents in the economy, and is taken from
the estimated value in Iacoviello (2005). This is between the value of 0.5 from Hall (2011) and
Campbell and Mankiw (1989), and Cogan et al (2010) estimated value of 0.74. See Appendix B
for a discussion of the evidence on financial constraints.
8
An isomorphic approach used by Bilbiie (2008) is to assume fixed costs of operating that exactly offset profits
in steady state.
9
Thus, the Ricardian Agent’s share is given by (α(1 − µ) + µ − ISS ) / (1 − ISS ). This reflects that all income
earners pay payroll taxes etc. that fund government purchases in steady state.
9
Table 1: Parametrization and Steady State
Parameter
Symbol
Simple Model
Full Model
Panel A: Parameters in Analytical Model and Full Model
Discount Rate
β
0.99
0.99
Utility Parameter, leisure
η
1.5
1.5
Labour share Ricardian household
α
0.64
0.64
Calvo Prob. constant price
θ
0.75
0.75
Inflation Coefficient (Taylor rule)
φπ
1.27
1.27
σ
)
σ−1
X
1.05
1.05
Steady State Markup (X =
Panel B: Parameters only in Full model
Capital share
µ
0
0.3
Capital adjustment cost
ψ
-
2
Capital depreciation rate
δk
-
0.03
Calvo Prob. constant wage
θw
-
0.75
Sticky Wage CES elasticity
ε
-
21
Output coefficient (Taylor rule)
φY
0
0.13
Interest rate smoothing (Taylor rule)
rR
0
0.73
Steady State Government Purchases
GSS
0
0.2
Sources: Most of the model’s parameters come from Iacoviello (2005). Exception are η
(leisure utility parameter) which leads to a Frisch Elasticity of labour supply of 2 that is
the average of the values in Bernanke et al (1999) and Christiano et al (2005). Wage
stickiness parameters (Calvo probability and CES elasticity) are taken from Christiano
et al (2005). The steady state government share is the same as Christiano et al (2011).
Notes: Steady state GSS funded by SS labour and capital income share in the full
model. Ricardian consumer share=(α(1 − µ) + µ − ISS )/(1 − ISS ) = 0.68.
10
3
When is the transfer multiplier larger than the purchases
multiplier? Analytical results from a simple model
So long as fiscal policy is not too persistent and the share of hand-to-mouth consumers is not
too small, the transfer multiplier is larger than the purchases multiplier over a wide region of the
parameter space. In this section, we use a stripped down model (described in Section 3.1) to
show analytically the conditions under which the transfer multiplier is larger than the purchases
multiplier (see Section 3.3). For the most part, we focus on the transfers vs purchases multiplier in
the same model economy. However, the purchases multiplier is sensitive to many model parameters,
so we also compare the transfer multipliers to the flex-price purchases multiplier, which only
depends on the Frisch elasticity of labour supply (see Section 3.2 for the flex-price benchmark).
The analytical model also allows us to to discuss why the transfer multiplier is larger than the
purchases multiplier, which we do in Section 3.4.
3.1
The simple model
To derive intuition, we assume, for the moment, that (i) wages are flexible (λj → ∞, j = 1, 2),
(ii) the capital share is zero in the Cobb-Douglas production function µ → 0, and (iii) that the
steady state consumption share of each agent is equal to their share of wage income (requiring
LSSS = (1 − α)ΠSS and Gss = 0). For simplicity, we also assume the central bank only responds
to inflation contemporaneously. The log-linearised production function (Equation 16), the agents’
Labour-Leisure FOC (Equation 17) and the FOC of firms (Equation 18), combine to form an
supply relation, which is the first equation of the system of 8 equations (^ denotes percentage
deviation from steady state).
Ŷt = αL̂1t + (1 − α)L̂2t
(16)
ŵjt = ĉjt + (η − 1)L̂jt , ∀j = 1, 2
(17)
Ŷt − X̂t = ŵj,t + L̂j,t , ∀j = 1, 2
(18)
11
Equations in the Analytical Model (A1-A8)
1
[A1 Supply] Ŷt = − η−1
(X̂t + αĉ1,t + (1 − α)ĉ2,t )
[A2 Resource constraint] Ŷt = αĉ1,t + (1 − α)ĉ2,t + ĝt
[A3 Cons. Constrained] (1 − α)ĉ2,t = T̂t + (1 − α)(Ŷt − X̂t )
P
[A4 Cons Ricardian] ĉ1,t = −rr
ˆ t + Et ĉ1,t+1 = −Et ∞
ˆ t+i
i=0 rr
[A5 Taylor Rule] R̂t = φπ π̂t
[A6 Phillips Curve] π̂t = βEt π̂t+1 − κX̂t
[A7 Fisher relation] rr
ˆ t = R̂t − Et π̂t+1
[A8 Fiscal policy] T̂2t+1 = ρT̂2t + êT 2,t+1 or ĝt+1 = ρĝ2t + êG,t+1
Equations A1-A4 can be combined to form an IS curve (Equation 19), which when combined
with the Phillips curve (Equation A6) and the Taylor Rule (Equation A5) define the dynamic
system. Due to limited asset market participation of the constrained consumers, the coefficient
multiplying interest rates is no longer 1, as is the case in the standard New Keynesian IS curve
(with log preferences).10 As in a standard representative agent NK model, an increase in real
interest rates leads to a reduction in aggregate demand.
Ŷt = Et Ŷt+1 −
3.2
T̂t + αĝt
T̂t+1 + αĝt+1
α
rr
ˆt +
− Et
1 − (1 − α)(1 + η)
1 − (1 − α)(1 + η)
1 − (1 − α)(1 + η)
(19)
The Flexible price benchmark
When prices are flexible, the transfer multiplier is 0 and the government purchase multiplier is 1/η
(in the simple model). Equation (A1) and Equation (A2) combine to form Equation (20). When
price are flexible (κ → ∞), retailers keep their markups at the profit-maximizing optimum, so
X̂t = 0 and Ŷt = η1 ĝt , with no role for transfers in determining output. As the persistence of the
government policy (ρ), and share of credit constrained households (1 − α) and other parameter
do not affect the purchases multiplier, we use it as a robust lower bound for considering when a
multiplier is “large” (Section 3.3).
1
Ŷt = [ĝt − X̂t ]
η
(20)
To get some intuition for why transfers have no impact on output with flexible prices, note that
transfers increase demand (through ĉ2 ), increases in real interest rates lower ĉ1 , and that ĉ1 and ĉ2
enter both the supply and demand equations co-linearly. Hence, if prices are flexible, the increased
demand by the hand-to-mouth agent due to the rise in transfers (through ĉ2 ) is completely offset
by the fall in consumption of the Ricardian agent due to the rise in the path of real interest rates:
P
T̂2t = α ∞
ˆ t+i (with ĝt = 0). The positive labour supply effect of the extra taxes on the
i=0 rr
Ricardian consumer (which makes him work harder and boosts output), is perfectly offset by the
10
As α → 1, (the labour share of the Ricardian households approaches 1), this coefficient approaches 1.
12
reduced labour supply of the constrained household who is made wealthier and thus works less. In
contrast, the government spending multiplier is positive because the wealth effect on the hand-tomouth agent is absent, while the positive wealth effect on the Ricardian household remains (who
must pay the taxes to fund either the increased government expenditure or the transfers to the
hand-to-mouth agent).11
3.3
Multiplier under a Taylor Rule with Sticky Prices
When prices are sticky, the government transfer multiplier is often larger the government purchases multiplier, both in the simple model discussed here, and also in the full model discussed
in Section 5. The conditions under which is the case are summarized in Result 1. Using a guess
and verify approach that all variables follow an AR(1) process with persistence ρ, Equation (21)
show that output is a linear function of contemporaneous
government purchases
and transfers.12
i
h
The government transfer multiplier is γ = 1 − (1 − α)(1 + η) +
o
n
(φπ −ρ)κ
.
multiplier is γα 1 + (1−βρ)(1−ρ)
Ŷt = γ T̂t + γα 1 +
(φπ − ρ)κ
(1 − βρ)(1 − ρ)
α(φπ − ρ)κ
η
where γ = 1 − (1 − α)(1 + η) +
(1 − βρ)(1 − ρ)
3.3.1
α(φπ −ρ)κ
η
(1−βρ)(1−ρ)
−1
and the purchases
ĝt
(21)
−1
The importance of sticky prices
As seen in Equation (21) and Figure (1), the transfer multiplier is much more sensitive to price
stickiness than the purchases multiplier and is almost always larger as prices become very sticky
(also true in the full model).13 Price stickiness (θ) enters the expression for the multipliers via
the slope of the Phillips curve κ = (1 − θ)(1 − βθ)/θ: with increasing price stickiness θ → 1 and
κ → 0 (with increasing flexibilityθ → 0 and κ → ∞). Crucially, κ enters only the denominator
of the transfers multiplier, but both the numerator and denominator of the purchases multiplier.
11
Note that the results above are a special case when transfers in steady state are zero, and steady state consumption shares equal labour income share. If steady state transfers are positive, then the negative wealth effect on the
Ricardian households is larger than the positive wealth effect on the constrained households and so the multiplier
rises. This effect is stronger the larger the share of steady state transfers, and the higher the Frisch elasticity of
labour supply, and decreases with the elasticity of substitution between the two types of labour in production.
Likewise negative steady state transfers (for example, if the constrained household has debt in steady state), will
reduce the multiplier. These results are similar in the simple model with sticky prices or the full model in Section 5.
SS
For example, a steady state transfer of α(1−µ)+µ−I
GSS = 0.064 of GDP to the constrained households to offset
1−ISS
his share of steady state taxes (increasing their consumption by 20 per cent) raises all multipliers by about 0.14.
12
To do this, first solve for the present
value of real interest rates
and then substitute into Equation (19).
P∞
(φπ − ρ)κ
(φπ − ρ)κ
φY
ˆ t+i = −
ĝt +
η+
Ŷt
i=0 rr
(1 − βρ)(1 − ρ)
(1 − βρ)(1 − ρ)
1−ρ
13
Parameters used to generate Figure 1: ρ = 0.9 (from the ARRA), θ = 0.75, η = 1.5 and φπ = 1.27, though the
results hold analytically for any parameters in the simple model so long as α < 1.
13
Figure 1: How transfer (solid) and purchase multipliers (dashed) vary with price stickiness
Hence, when prices are very sticky (flexible), the transfer multiplier becomes larger (smaller) than
the purchases multiplier. As prices become perfectly sticky, the terms with κ go to zero and
so the transfer multiplier becomes [1 − (1 − α)(1 + η)]−1 and the purchases multiplier approaches
α [1 − (1 − α)(1 + η)]−1 , which will be smaller than the transfers multiplier so long as the the share
of credit constrained households is non-zero.
Sticky prices boost multipliers through changes in firm’s mark-ups and the wages of creditconstrained households—with transfers being more sensitive than purchases (see Section 3.4 for a
discussion of the difference). With sticky prices, firms can no longer set markups at their optimal
level. When aggregate demand increases—whether through transfers or purchases—firms must
increase their output rather than simply raising prices and in order to meet the higher demand,
so firms have higher demand for labour so wages rise. In the presence of hand-to-mouth agents,
higher wage income is consumed, providing a second-round boost to aggregate demand missing
in standard representative-agent models. Since wages (marginal cost) rises, firms that can raise
prices will do so (leading to higher inflation), whereas firms that can’t raise prices will see their
markups (and average markups in the economy) fall. Thus high aggregate demand, high inflation,
low markups and high wages all occur simultaneously in the model.
3.3.2
The importance of the persistence of fiscal policy and the share of credit constrained households
Other things equal, the transfer multiplier is larger than the purchases multiplier when (i) fiscal
policy is not too persistent and (ii) when the share of credit constrained households is not too
small. (The Zero Lower Bound also has has a large effect but is discussed in Section 5.2). From
rearranging Equation (21), the condition where the government purchase multiplier is larger is
given by Equation (22), which is more easily met for larger 1 − α (share of credit constrained
households) and a lower ρ. Figure (2A) shows the area (in white) where Equation (22) holds.
For a wide range of values of 1 − α commonly estimated in the literature (between 0.25-0.5), the
transfer multiplier is larger than the purchases multiplier when fiscal policy has low persistence.
14
Figure 2: Regions of parameter space where the transfer multiplier (white), where there purchases
multiplier is larger (grey/blue), or where the simple model is indeterminate (black) .
Figure 2B shows the region where the transfer multiplier is larger than the flexible price purchases
multiplier (dŶt /dT̂ > 1/η). With this slightly weaker condition, the transfers multiplier is larger for
the whole region where ρ < 0.5 (approx). The results are qualitatively similar with the full model
(see Section 5.1). The black (darkest) region is where the simple model becomes indeterminate
(Blanchard-Khan conditions fail) however the the full model is determinate for all values (off
corners) of α and ρ. As Bilbiie (2008) shows, for simple New-Keynesian models with sufficiently
high levels of rule-of-thumb consumers, determinacy is guaranteed by setting the Taylor coefficient
on inflation φπ sufficiently low — the opposite of the standard “Taylor Principle” logic.
(1 − α)
(1 − βρ)(1 − ρ)
κ
+ α [−(φπ − ρ)] > 0
(22)
Result 1 : The transfer multiplier is large when:
1. Prices are sticky.
2. The fiscal policy is not very persistent.
3. The share of credit constrained households is not extremely small.
These results apply to the simple model and the full model (under a Taylor Rule), and to
whether large is defined as greater than purchases multiplier (simple or full model) or greater
than equal to one (full model).
Persistence of fiscal policy
For both transfers and purchases, the multiplier is greater for less persistent fiscal policy
(ρ), though transfers are more sensitive to ρ. With ρ = 0, the transfer multiplier approaches
15
[1 − (1 − α)(1 + η) + αφπ κη]−1 and the purchases multiplier approaches α {1 + φπ κ}. Clearly the
transfers multiplier will be larger than the purchases multiplier if α < [1 + φπ κ]−1 , which holds for
most value of α given that κ is typically fairly small in sticky price models. In contrast, as fiscal
policy becomes permanent (in the limit) the transfer multiplier and purchases multiplier approach
the flex-price multipliers from Section 3.2 (zero and 1/η respectively).
In response to stimulatory fiscal policy, firms must meet increased demand by some combination
of higher prices or higher output. If the increase in demand is not very persistent, firms will be
reluctant to raise their prices significantly, and will largely increase their labour demand. For the
hand-to-mouth consumers who spend all their labour income, this will lead to a second round
boost in aggregate demand. In contrast, with a persistent increase in fiscal policy, firms will be
much more willing to increase their prices, implying a smaller increase in labour income for the
hand-to-mouth households, and thus a smaller increase in output. The monetary policy response
(which reduces demand) is also much greater when fiscal policy is more inflationary.
3.4
Why is the transfer multiplier larger than the purchases multiplier:
the inverted aggregate demand curve
Generating an upward sloping demand curve
Result (2) states that the transfers multiplier will be greater than the purchases multiplier
whenever the economy’s “old Keynesian” aggregate demand curve is inverted. The slope of this
curve depends on the relative strength of the two opposing Disposable Income and Taylor Principle
effects as seen by the terms in curly and square brackets (respectively) of Equation (22).14
The Taylor Principle effect is the standard negative relationship between the consumption
demand of the Ricardian household and the inflation rate. A rise in inflation from stimulatory
fiscal policy leads to an increase in real interest rates, which “chokes off” demand. One can see
that the negative relation between demand and inflation is ensured by the Taylor principle φπ > 1,
such that −(φπ − ρ) < 0 (as 0 ≤ ρ < 1) in Equation (22).
Since times of high inflation (driven by a demand shock) also means low markups, high labour
demand and high wages, credit constrained households will have higher incomes, which will be
spent, leading to higher aggregate demand [the disposable income effect]. Constrained households
do not respond to an increase in real interest rates as their consumption is income-determined.
14
Thisncan be see by
o rewriting the demand curve with T̂t = ĝt = 0 for simplicity, as α(1−ρ)Ŷt = α [−(φπ − ρ)] π̂t +
(1−βρ)(1−ρ)
(1 − α)
π̂t
κ
16
Result 2 [Inverted aggregate demand curve]: In a simple New Keynesian model :
1. The transfer multiplier is greater than the purchase multiplier whenever Equation (22)
holds, or equivalently, whenever the disposable income effect dominates the Taylor Principle effect, where:
• Taylor Principle effect: the tendency for higher inflation associated with a
positive output gap to “choke off” the demand of Ricardian households when the
central bank raises nominal and real interest rates.
• Disposable income effect: the tendency for higher inflation associated with
a positive output gap to boost, labour demand and real wages, which are then
spent by hand-to-mouth households to generate a second-round boost to aggregate
demand.a
2. The transfer multiplier is greater than the purchase multiplier whenever the economy’s
“Old Keynesian” aggregate demand curve is inverted (a positive relation between inflation/prices and aggregate demand).
3. Transfers and purchases have the same first-round boost to demand, but transfers also
lead to fall in supply. When the aggregate demand curve is inverted, the fall in supply
will provide a second-round boost to output (and transfers multiplier>purchases multiplier), whereas when it is downward sloping (as is standard) the fall in supply will
reduce output.
The logic applies to the full model of Section 5, in particular Zero Lower Bound, which
weakens the Taylor principle effect.
a
The output gap here equals Ŷt − ŶtF LEX = Ŷt − Ĝt /η = −X̂t /η
The strength of these two forces changes with some of the key parameters outlined above.
First, an increase in the share of credit constrained households (1 − α) clearly boosts the relative
strength of the disposable income effect simply because there are more consumers who spend their
new higher disposable income. When α = 0, as in standard representative agent model, the
Disposable Income effect doesn’t exist. Second, stickier prices reduce movements in inflation for a
given movement in mark-ups/the output gap, which strengthens the disposable income effect (as
extra output translates to higher wages and disposable income) and weakens the Taylor Principle
effect. Second, when fiscal policy is not very persistent κ/(1 − ρβ) in π̂ = −[κ/(1 − ρβ)]X̂t will be
small, and so movements in markups will be much larger than movements in inflation (ten times
larger with ρ = 0 and our default parametrisation).15 Given the relatively small movement in
inflation for a given movement in markups, the central bank doesn’t need to raise interest rates
15
As ρ → 1 with our default parameters, κ/(1 − ρβ) becomes large.
17
much,which weakens the Taylor Principle effect relative to the disposable income effect. Finally, a
more aggressive central bank (as characterized by a higher φπ ) directly strengthens Taylor Principle
effect. In contrast, when the zero lower bound on nominal interest rates is binding, the Taylor
Principle effect moves in the opposite direction (a rise in inflation reduces real interest rates,
which boosts demand from the Ricardian HHs), which means that the transfer multiplier is almost
always larger than the purchases multiplier (see Section 5.2).16
The aggregate demand and supply curves are shown in Equation (23) and Equation (24). These
equations are relations between output and inflation. Given last period’s log price, inflation moves
one-for-one with today’s aggregate price level as in “Old Keynesian” aggregate supply and demand
curves. Equation (22) determines the sign of the denominator of the aggregate demand curve.17
When the condition in Equation (22) is met, the denominator of Equation (23) will be positive,
and so an increase in inflation is associated with an increase in output — the aggregate demand
curve will be inverted. Clearly, when Equation (22) is not satisfied, the denominator aggregate
demand curve is negative and so it will slope down as usual. Note that this is guaranteed in a
model with no hand-to-mouth households: α = 1 will make the second term in the denominator
equal zero.18
π̂t =
(1 − ρ)
+α{(1 − ρ)}
n
o Ŷt −
n
o [T̂t + ĝt ] (23)
(1−βρ)(1−ρ)
(1−βρ)(1−ρ)
α [−(φπ − ρ)] + (1 − α)
α [−(φπ − ρ)] + (1 − α)
κ
κ
π̂ =
[(1 − ρ)(η − α)]
(1 − ρ)
n
o Ŷt +
n
o T̂t
(1−βρ)(1−ρ)
α(φπ − ρ) + α (1−βρ)(1−ρ)
α(φ
−
ρ)
+
α
π
κ
κ
(24)
Effects of an inverted demand curve
With an aggregate demand curve that is upward sloping (but still steeper than the supply
curve), a decrease in supply increases output. This is the difference between transfers and purchases: they both boost demand by the same amount but transfers also decrease supply, which
boosts output when the demand curve is upward sloping.
This is shown in Figures 3A and 3B. In these figures, inflation is on the vertical axis, as a
16
Note that an upward sloping aggregate demand curve is very different from the “inverted aggregate demand
logic” (IADL) in Bilbiie (2008) as discussed above. Bilbiie’s (2008) condition relates real interest rates to output.
Our result relates inflation to output, and inflation and real interest rates rarely move one-for-one (The exception
(with φY = 0 ) is the knife edge case φπ = 1 + ρ). Moreover, Equation (24) depends on price stickiness (through
κ), monetary policy (through φπ ) and the persistence of fiscal shocks (through ρ), none of which determine IADL.
Likewise, IADL depends on preferences over leisure (through η), which do not enter Equation (24).
17
To solve for the Aggregate Demand Curve, substitute equations A3-A7 into equation A2. Substitute out for
consumption and the path of real interest rates using X̂t = −(1 − βρ)π̂t /κ. Note that that supply curve will always
be upward sloping by natural restrictions on the parameters: η > 1 ≥ α, φπ > 1 > ρ, κ > 0
18
Eggertsson and Krugman (2012) also include an upward sloping demand curve in a related model with handto-mouth consumers. They focus on the case where the ZLB binds (“Topsy-turvy economics”), while in our model
the demand curve can be upward sloping even when the central bank follows a Taylor rule.
18
Panel A: Purchase Multiplier
Agg supply & Demand; High Persistence
(Baseline; Supply=red, Demand=blue )
Agg supply & Demand; Low Persistence
(Baseline; Supply=red, Demand=blue )
0.2
0.15
0
dY/dG
-0.05
-0.1
Aggregate
Demand
0.1
0.05
Inflation
Inflation
0.1
0.2
0.15
Aggregate
Demand
Aggregate
Supply
0.05
dY/dG
0
-0.05
Aggregate
Supply
-0.1
-0.15
-0.15
-0.2
-1
-0.5
0
0.5
% Change Output
-0.2
-1
1
-0.5
0
0.5
% Change Output
1
Panel B: Transfer Multiplier
Agg supply & Demand; Low Persistence
(Baseline; Supply=red, Demand=blue )
0.2
0.2
0.15
0.15
0.1 dY/dTr
0.05
0
dY/dG
-0.05
0.05
-0.1
-0.1
-0.15
-0.5
0
0.5
% Change Output
-0.2
-1
1
dY/dG
0
-0.05
-0.15
-0.2
-1
dY/dTr
0.1
Inflation
Inflation
Agg supply & Demand; High Persistence
(Baseline; Supply=red, Demand=blue )
-0.5
0
0.5
% Change Output
1
Figure 3: Aggregate demand and supply when the transfer multiplier is larger (low persistence of
fiscal policy) and when the purchase multiplier is larger (high persistence of fiscal policy). The
purchase multiplier is illustrated in Panel A, and the transfers multiplier is in Panel B.
function of the deviation of output from its steady state level. We consider two levels of persistence
of the fiscal shock: high persistence (Figure 3A, ρ = 0.9) where the government spending multiplier
is slightly larger and low persistence (Figure 3B, ρ = 0.5) where the transfer multiplier is larger.
Equation (23) and Equation (24) are the demand curve and supply curves which are plotted in
both figures. Comparing the aggregate demand curves, the effect of reducing ρ is clear: with a
high ρ the aggregate demand curve is downward sloping (left graphs), as standard in Keynesian
models. However, with a lower persistence the aggregate demand curve becomes upward sloping
(right graphs).
Panel A shows the effect of a government purchases shock, whereas panel B shows a transfers
shock of equal size (0.2 per cent of GDP) and compares it with the purchases shock. A x per cent
of GDP increase in transfers has the same effect on aggregate demand as a x per cent increase in
government purchases. As such, they both shift the demand curves right by the same amount.
The difference between transfers and purchases shocks is in supply: the government purchase
shock does not shift aggregate supply (Panel A), but the transfers shock shifts the supply curve
to the left (Panel B). This is because transfers increase the income and hence consumption of
the constrained households, which leads to reduced labour supply through a wealth effect. When
the demand curve is downward sloping, as is standard (and with high persistence), a decrease in
supply increases inflation and reduces output, as one would expect. However, with lower persistence
19
the decrease in supply (with transfers) combines with the upward sloping demand curve to push
marginal cost and inflation higher, further boosting output (as the higher wages of the hand-tomouth household are consumed).
4
Sticky Wages
The differences between the analytical model and the full model are mostly accounted for by the
addition of sticky wages (which are excluded from the analytical model). Figure 4 adds sticky
nominal wages to the analytical model of Section 3. With θw = 0 multipliers are the same as the
analytical model, but the addition of sticky wages changes the multiplier substantially.19 Note,
however, that sticky wages do not alter the order of transfer and purchase multipliers — but merely
changes their magnitudes. This suggests that the upward sloping aggregate demand model still
explains relative magnitudes.
Result 3 [Sticky Wages]
• Sticky wages weakens the reduction in labour supply caused by a government fiscal
transfer.
• This makes the transfer and purchases multiplier closer, with most of the adjustment
in the transfer multiplier and for less persistence fiscal shocks.
To gain some intuition for Result 3, note that in the model with sticky wages, the real wage this
period is last period’s wage plus the difference between nominal wage inflation and price inflation:
w
ŵjt = π̂j,t
− π̂t + ŵjt−1 . The firm chooses labour demand consistent with this real wage, and the
household supplies the amount of labour demanded. As such, the transfer to the constrained agent
that reduces his desired hours only creates upward pressure on nominal wages via Equation (7),
rather than increasing them directly as when wages are flexible. This weakens the negative supply
effect of transfers that drives the difference between the transfer and purchase multipliers. Since
changes in government purchases do not induce the labour supply response for the constrained
household that transfers do, the government spending multiplier is less sensitive to the amount of
wage stickiness, and the transfer multiplier converges to the government purchase multiplier.
Result (3) highlights the larger fall in the transfer multiplier when wages are sticky (e.g. from
3.4 to 1.4 for transfers with ρ = 0).20 The multiplier when ρ = 0 falls the most because households
are reluctant to raise nominal wages today, as demand will fall again tomorrow. As such, real
wages become much smoother, and so labour supply (and output) moves by a smaller amount in
the period the shock hits, leading to a smaller multiplier.
19
With sticky wages, the multiplier is no longer constant over time, and so we use the present value multiplier
defined in Equation (25) in the next section. In the case that wages are perfectly flexible, the present value multiplier
is the same as the analytical multiplier in section 3.
20
Note that the simple model is more qualitative than quantitative, so these numbers for the multiplier are purely
illustrative.
20
5
Rho=.9, Transfer Mult.
Rho=.9, Purchases Mult.
Rho=0, Transfer Mult.
Rho=0, Purchases Mult.
Multiplier
4
3
2
1
0
0
0.2
0.4
0.6
θw
0.8
1
Figure 4: Wage stickiness and the multiplier (simple model + sticky wages)
5
The transfer multiplier in a medium-scale DSGE model
By adding realistic features such as sticky wages, capital, and the ZLB (relative to the simple
model), the full model allows us to show (i) that the transfer multiplier is often large in quantitative
sense (greater than or equal to one), (ii) that the ZLB increases the transfer multiplier more than
the purchases multiplier, and (iii) that the results of Section 3 hold qualitatively in a richer model.
See Table 1 for parameters, and Section 2 for a statement of the full model.21 ,22
In the full model, output will not be a constant multiple of the fiscal stimulus (as there are
additional state variables), and so instead we report the present value multiplier. This is the
discounted sum of increases to output relative to the discounted sum of fiscal expenditure, where
the discounting is at the agents’ discount rate β
P∞
P V M ultipler =
i
i=0 β Ŷt+i
P∞ i
i=0 β T̂2,t+i
P∞ i
β Ŷt+i
or Pi=0
∞
i
i=0 β ĝt+i
(25)
We assume the central bank follows a Taylor rule with a degree of interest rate smoothing
that has been estimated in the literature,23 except when the ZLB binds and nominal rates are
21
Relative to the simple model, we also add nominal interest rate smoothing to the Taylor rule, and remove the
small steady state transfer of profits.
22
In earlier versions of the paper, we assumed that the constrained household was able to borrow against consumer
durables or housing. With adjustment costs around the average of those those used in the literature, the inclusion
of consumer durables or housing did not substantially effect the results. In that model, the Ricardian households
are more patient than constrained households which ensured that constraints bind in the neighbourhood of the
steady state.
23
For our baseline model, we assume that the central bank sets R̂t = rY R̂t−1 + (1 − rY )(φπ π̂t + φY Ŷt ), where
rY = 0.73 (taken from Iacoviello 2005). The smoothing of interest rates has the effect of reducing the present value
multiplier on transfers and purchases by about 0.08 (if ρ = 0.9) to 0.17 (if ρ = 0). We did not use interest rate
smoothing in the simple model of Section 3 because it adds an extra state variable to the model (lagged interest
rates).
21
Figure 5: Regions of parameter space where the transfer multiplier is large (white) by various
definitions (Panel A: greater than 1, Panel B: greater than the purchases multiplier, Panel C:
greater than the flex price purchases multiplier).
constant.24 We find that the multipliers in this section are much more robust than those in the
simple models in previous sections.
The numbers produced by this model can be thought of as our best estimate of what an increase
in output might be in response to a transfer policy. Although we motivate our approach by the
ARRA, it is important to note that some of the transfers in that package were not targeted to the
credit constrained or to those in labour force. As such, we see the multipliers in this section as
indicative of a more general transfers policy, rather than the ARRA in particular.
Back-of-the-envelope calculations with imperfect targeting are provided in Appendix C and
suggest the multiplier would be around 1.3 to 1.5 for the transfers component of the ARRA. The
high estimate is mostly because the ARRA was well targeted at households likely to be credit
constrained and the ZLB was binding at the time the package was implemented. In contrast the
Bush tax rebates in 2001 have multipliers of around 0.3-0.6 because they were not targeted at the
financially constrained, and the ZLB was not binding in 2001.
5.1
Results from the full model
The transfer multiplier in the full model (when policy follows a Taylor rule) is large when the fiscal
policy is not very persistent or the share of credit constrained households is not extremely small
—as in Result 1. This is easiest seen as the lower left hand region in Figure 6 (Panel A), where
“large” is defined quantitatively as a multiplier greater than or equal to one. This region expands or
contracts when using a stricter definition of large (greater than the purchases multiplier, panel B)
or weaker definition (greater than the Flex-price purchases multiplier, panel C), but is qualitatively
similar.25
The first row of Table 2 presents the transfer and purchase multipliers when the central bank
24
We also set the share of steady state profit share of the patient households to one (rather than α as in Section
3), which lowers multipliers by about 0.04.
25
With the full model there are no regions (away from corners) of the α, ρ space that are indeterminate.
22
Table 2: Full Model - Present Value Multipliers
A. Transfers
B. Purchases
Fiscal persistence:
ρ=0
ρ = 0.9
ρ=0
ρ = 0.9
Taylor Rule
1.0
0.4
1.0
0.6
2 years ZLB (2011Q2)
1.3
1.0
1.3
0.9
5 years ZLB (2014Q2)
1.3
1.7
1.3
1.2
Shock: 1 per cent GDP fiscal stimulus. 2009Q2 starting date for ARRA.
follows a Taylor Rule. When fiscal policy is a once-off — like the Bush 2001 tax rebates — both
transfer and purchase multipliers are around 1. With a persistence ρ = 0.9 similar to the transfers
component of the ARRA, the transfer multiplier is around 0.4 and the purchase multiplier is larger
at around 0.6. Sticky wages have a large effect on the multiplier (Result 3), but adding capital
and steady state government spending does not change the multiplier substantially.26 Christiano
et al (2011) also find a similar result for capital.
5.2
Zero Lower Bound (ZLB)
Since December 2008, the Federal Reserve has maintained nominal interest rates at 0-0.25 per
cent, and recent announcements suggest that the policy rate will be maintained at zero until
mid-2015.27 It is well documented in the literature that government purchases are much more
potent when monetary policy is at the ZLB; for example, Christiano et al (2011) find a purchase
multiplier well above two in the case that the ZLB binds. Despite the focus on the ZLB itself, in a
linear model it is the path of nominal rates that determines the multiplier, rather than the reason
nominal rates take that path (Christiano et al 2011). Given that the conditions under which the
ZLB binds have been modelled in other papers, we simply assume the central bank commits to
keeping the nominal interest rate constant for a certain number of periods (and then returns to a
Taylor rule).28
26
This is conditional on steady state purchases not affecting the income distribution, as both agents pay steady
state taxes in proportion to their income share.
27
“In particular, the Committee also decided today to keep the target range for the federal funds rate at 0 to 1/4
percent and currently anticipates that exceptionally low levels for the federal funds rate are likely to be warranted
at least through mid-2015.” FOMC Statement September 13, 2012
28
We implement this using the same methodology as Cogan et al (2010) — and we thank them for making their
Dynare code publicly available. The number of periods of constant interest rates is known by agents and the central
bank is believed to be credible.
23
Result 4 [Zero Lower bound]:
1. The Zero Lower Bound (ZLB) increases the transfer multiplier by much more the purchases multiplier, with the effects being even stronger for more persistent fiscal policy.
2. When the ZLB is binding for more than two years, the transfer multiplier is larger than
the purchases multiplier in most situations—even when the share of credit constrained
households is very small, or fiscal policy is very persistent.
3. Intuitively, this is because the ZLB weakens the Taylor Principle Effect (from Result
2) because the real interest rates falls (rather than rises) in response to an inflationary
fiscal stimulus.
While the ZLB increases the purchases multiplier, it increases the transfers multiplier by much
more (see Result 4). Figure 6 shows in white the regions of the (ρ, α)-space that where the transfer
multiplier is greater than the purchases multiplier as the ZLB binds for one, two or three years.
One can see that ZLB dramatically increases the area where the transfer multiplier is larger than
purchases multiplier, and for three years of ZLB, the transfer multiplier is always larger (Result
4.2).
Intuitively, the Zero Lower Bound weakens the Taylor Principle Effect because the real interest
rates falls (rather than rises) in response to an inflationary fiscal stimulus (Result 4.3). Hence the
economy’s aggregate demand curve is more likely to be upward sloping, and so the transfer multiplier greater than the purchases multiplier (although we can’t derive these relations analytically
for the full model).
Table 2 shows that with constant interest rates for two years (as considered by Cogan et al 2010),
all multipliers are close to or greater than one — with a particularly large increase for persistent
transfers and purchases. With five years of constant rates — until 2014Q2 — the multipliers are
now large for ρ = 0.9, specifically 1.7 for transfers and 1.2 for purchases.29 Particularly striking
are the increases for more persistent transfer-based stimulus: the transfer multiplier for ρ = 0.9
increases by 1.3 going from a Taylor rule to five years ZLB, whereas the purchases multiplier only
increases by 0.6, and less persistent fiscal policy increases by 0.3 for both purchases and transfers
(Result 4.1). As shown by Woodford (2011), multiplier is very sensitive to fiscal policy that occurs
after the ZLB stops binding. Hence longer horizons of the ZLB binding have a larger effect on the
multipliers for very persistent fiscal policy.30
29
The purchase multiplier is lower than that in Christiano et al (2011), perhaps because the stimulus policy here
lasts longer than the ZLB.
30
This suggests that one of the reasons more persistent fiscal policy was less stimulatory was that it invoked a
stronger monetary policy response, partly because retailers were more likely to increase prices in response to a more
permanent shock.
24
Full model (1yrZLB)−White Region is where the transfer multiplier is larger
Full model (2yrZLB)−White Region is where the transfer multiplier is larger
Full model (3yrZLB)−White Region is where the transfer multiplier is larger
1
1
1
0.9
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
α0.5
α0.5
α0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0
0
0.2
0.4
ρ
0.6
0.8
1
0
0.1
0
0.2
0.4
ρ
0.6
0.8
1
0
0
0.2
0.4
ρ
0.6
0.8
1
Figure 6: Effect of the Zero Lower Bound on the region where the transfer multiplier is larger than
the purchases multiplier (White): 1 yr ZLB (Panel A), 2 years ZLB (Panel B) or 3 years ZLB
(Panel C).
5.3
Impulse Responses
Figure 87 plots the impulse responses of the key variables for the economy as a whole (top 6
graphs) and for each type of household (bottom 3 graphs) for a 1 per cent of GDP transfers shock
with persistence ρ = 0.9. As expected, the variables move further from steady state when the
ZLB binds (dashed lines) as compared to when the central bank follows a Taylor rule (solid line).
A transfer increases output by raising marginal cost (reducing markups) which increases firms’
demand for labour (relative to steady state). In the case of the ZLB, a decrease in the real interest
rate encourages investment, whereas in the baseline model, investment falls slightly as real interest
rates rise.
Looking across households in the bottom three panels, the hand-to-mouth household increases
his consumption substantially, and the Ricardian household cuts back on consumption largely in
order to pay his tax bill. Both households work harder (at least initially), though the increase is
larger for the Ricardian household as it becomes poorer. Note the relatively muted movement in
real wages — due to wage stickiness — which limits wealth effects.
6
Conclusion
Government transfers were a larger share of the 2009 ARRA than government purchases. At the
same time, with depressed growth prospects in the United States and other economies, there has
been a debate about the efficacy of fiscal stimulus. We have demonstrated that, in a New-Keynesian
model modified to have two types of agents that differ in their access to financial markets, the
transfer multiplier is often larger and one, and larger than the government purchase multiplier.
Using a simplified model that we can solve analytically, we show that the transfer multiplier will
be larger than the purchase multiplier when the “disposable income effect” dominates the standard
“Taylor principle effect”, leading the economy’s aggregate demand curve to become inverted. In
this case, a reduction in aggregate supply from the wealth effect of the fiscal transfer further boosts
output. We show that sticky wages weaken wealth effects, leading the transfer multiplier to be
25
Path of Trasfers (% of GDP)
3
Baseline
2
2yrs ZLB
Output (% from SS)
1
0
2
4
6
8
0
2
-0.5
1
-1
0
10
Markup (% from SS)
3
Investment (% GDP from SS)
0.1
2
4
6
8
10
Inflation (% from SS)
-1.5
2
4
6
8
10
Nominal interest rate (% from SS)
0.2
0.2
0.1
0.1
0
-0.1
2
4
6
8
0
10
Consumption (% GDP from SS)
2
4
6
8
10
0
Labor (% from SS)
2
4
6
8
10
Real wage (% from SS)
4
1
2
0.5
0
0
2
1
0
2
4
6
8
Ricardian (C, W or L)
10
-2
0
5
Ricardian +ZLB
10
-0.5
0
Constrained
5
10
Constrained+ ZLB
Figure 7: Impulse response functions: Full model, 1 per cent of GDP transfers shock with ρ = 0.9
26
similar to the purchases multiplier. In normal times (when the central bank follows a Taylor rule),
fiscal policy must not be too persistent and the share of credit constrained households must not
be too small for the transfers multiplier to be large. When the ZLB binds, the transfer multiplier
is usually larger than the purchase multiplier because constant nominal interest rates weaken the
“Taylor principle effect”.
We conclude with two areas for future research. First, we have ignored open-economy considerations. One concern here is that a large portion of the transfer payments to constrained
households could be spent on imports, which could lead to a smaller impact on output. Second,
we have considered the case in which government spending is financed by lump-sum taxation on
unconstrained, Ricardian households. To the extent that the government finances transfers with
distortionary taxation, the multiplier may be smaller. However, as Eggertsson (2010) has emphasised, distortionary taxation can actually increase the size of the multiplier when the ZLB binds,
as the increase in inflation expectations lowers the real interest rate and stimulates spending.
Appendix
A
Transfers in the American Recovery and Reinvestment
Act
Throughout the paper, we approximate the transfers component of the ARRA as a 1 per cent of
GDP shock, with persistence ρ = 0.9. In this section, we analyse data on the ARRA from the
Bureau of Economic Analysis (BEA) which show that this approximation is quite good (Figure 8).
We document that transfers were a large and quantitatively important component of the ARRA,
and Appendix B argues that most of these transfers went to people who might have been financially
constrained.
Our calculations indicate that transfers from the Federal government to individuals were about
about 40 per cent of the total ARRA (which include expenditures and tax cuts) and over 50 per
cent of the expenditures component of the package. We classify all of the BEA category “social
transfers” as transfers, as well as the lump-sum tax cuts component of reduced tax receipts. Capital
transfers and various other tax cuts (such as those to corporations) are clearly not transfers to
individuals. More controversially, we do not classify all payments to state and local governments
— most of which are for Medicaid — as transfers. This a more conservative approach than Oh
and Reis (2011) who do classify these as transfers to individuals. We take this line because intergovernmental transfers are somewhat fungible — if a state government didn’t receive a Medicaid
transfer, it may have still made the same Medicaid payments but raised taxes or reduced other
expenditures. To the extent that state and local governments would not have cut back elsewhere,
our estimates can be seen as a lower bound on the size of transfers. The largest categories of
27
ARRA Transfers and AR(1) approx.
Percent of US GDP
%
1.2
%
1.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
Transfers component of the ARRA
AR(1), rho=0.9
0.2
0.2
0
0
2009Q1
2009Q3
2010Q1
2010Q3
2011Q1
Source: BEA
Figure 8: AR(1) approximation to the Transfers component of the ARRA
transfers are the $400 Making Work Pay tax credit, and $400 refundable tax credits, which are
essentially lump sum payments. Also large is payment of benefits to the unemployed, and in the
second quarter of 2009 a once-off $250 payment to social security recipients. Food stamps and
other programs were small. Over 2009:1-2011:2, around US$244 billion of transfer payments had
been made, 38 per cent of the cumulative total of the ARRA package.
B
Financial constraints and skill
In this paper, we model transfers to financially constrained households who have a different type of
labour from non-financially constrained households. In this section we examine empirical evidence
on financial constraints, which suggests that the lower income households are (i) more likely to
receive ARRA transfers, (ii) more likely to be financially constrained, (iii) more likely to be in
lower skilled occupations, and this justifies the assumptions of the model.
Hall (2011) defines a family as “liquidity constrained” if it has less than 2 months income of
liquid assets. Using the 2007 Survey of Consumer Finances, he finds that illiquid households earned
58 per cent of all income, and that 74 per cent of households where illiquid because “lower-income
households are more likely to be constrained” (p436). The key point is that recipients of these
programs are likely to be of lower income, and so fit into Hall’s category of the constrained. The
$400 payments were per-capita, and so benefit the working poor on average. The $250 payments
were to those who receive social security benefits. And many of the recently unemployed would
have been in difficult financial circumstances. In line with Hall’s analysis, we model this group of
recipient as being so financially constrained that they cannot borrow at all, and so they consume
hand-to-mouth. For a group that includes food-stamp recipients, this doesn’t seem implausible.
Empirically, professionals and those with a college degree are less likely to be financially constrained. Data from the Survey of Consumer finances suggest that those in professional occupations
28
have around three times the liquid assets than non-professionals ($8500 for professional compared
to $2000-$3000 for non-professionals in technical, sales, services or other occupations).31 Likewise, those with a college degree have more than three times the liquid assets on hand than those
without a college degree (college graduates have $9500 compared to $1200-$2700 for non-college
graduates). This is in line with Johnson et al (2006), who find that low-income consumers (who
are more likely to work in semi-skilled occupations) spend around 63 percentage points more of
their 2001 tax rebate.
As such, we divide the agents in our model into two groups: the first represents skilled workers,
who are not financially constrained and don’t receive many ARRA transfers (and who are more
likely to pay the taxes funding the package). The second group represents lower income households,
who are lower skilled, financially constrained and tend to receive (and spend) lump sum rebates. As
skilled and semi-skilled labour is not perfectly substitutable, we assume a Cobb-Douglas production
function, rather than perfect substitutes.
C
Imperfect targeting and financially constrained households
In the sections above, we modelled transfers as being perfectly targeted at the financially constrained. While this is naturally a simplification, rough calculations here suggest for ARRA, the
majority of recipients are likely to be financially constrained, and so the transfers multiplier on
the ARRA was still large, even through it was not perfectly targeted.
The level of targeting of fiscal policy is difficult to calculate, but sits between two extremes.
On one end, the transfer policy can be completely untargeted, and so each household receives the
same payment. This approximately corresponds to the Bush tax rebates in 2001 examined by
Johnson et al (2006). On the other extreme, payments can be “targeted” at those that are likely
to be financially constrained. The ARRA contained some of these targeted payments — to food
stamp recipients for example — but also included untargeted lump-sum rebates and tax credits.
For these untargeted rebates/credits, we need to estimate the proportion of credit constrained
household in the population. A detailed discussion of these issues is beyond the scope of this
paper, but we provide some rough back-of-the-envelope estimates in order to get a ball-park figure
for the multiplier when transfers are not perfectly targeted at the financially constrained.
For the ARRA, we assume all social transfers to individuals, except the $400 rebates, are
transfers to the financially constrained. These consist of payments to the unemployed (48 per
cent), food stamp recipients (20 per cent), students (14 per cent), social security recipients (11 per
cent) and other people (7 per cent). These “targeted” programs make up around half of the transfers
components of ARRA (US$120 billion). For the other half of the transfers component comprising
lump sum US$400 rebates and tax credits, we assume that the proportion of financially constrained
31
Source: “Amount in Transaction Account” form Survey of Consumer Finances (2009) Appendix Table 2.B.
29
Table 3: Imperfect Targeting of Transfers - Present Value Multipliers
Transfers
A. Perfect Targeting
B. Untargeted
C. Partially Targeted
Benchmark
e.g. Bush 2001
e.g. ARRA 2009
Fiscal persistence:
ρ=0
ρ = 0.9
ρ=0
ρ = 0.9
ρ=0
ρ = 0.9
Taylor Rule
1.0
0.4
[0.3,0.6]
[0.1,0.3]
[0.7, 0.9]
[0.29,0.34]
2 years ZLB (2011Q2)
1.3
1.0
[0.4,0.8]
[0.3,0.62]
[1.0, 1.1]
[0.7, 0.9]
5 years ZLB (2014Q2)
1.3
1.7
[0.4,0.8]
[0.5,1.01]
[1.0,1.1]
[1.3, 1.5]
Shock: 1 per cent GDP fiscal stimulus. 2009Q2 starting date for ARRA.
households is the same as the general population, somewhere between 27 per cent (Cogan et al
2010), and 74 per cent (Hall 2011). Cogan et al (2010) estimate a model similar to Smets-Wouters
(2007) to get their estimate, whereas Hall’s estimate is the proportion of households with less than
2 months income of liquid assets. As credit constrained households are more likely to be poor,
Hall estimates these households earn 57 per cent of all income. The estimate of Iacoviello (2005)
of 36 per cent of labour income, is more towards Cogan et al’s estimate, whereas Campbell and
Mankiw’s (1987) estimate of 50 per cent is in the middle of the range.
Combining these numbers, we calculate the proportion of stimulus going to the financially
constrained is 27 to 74 per cent for the 2001 Bush rebates, and 64 to 87 per cent for the ARRA
transfers. The remaining proportion (e.g. 13 to 36 per cent for the ARRA) is transferred from the
Ricardian household to itself, and so has no effect on economic outcomes. As the model is linear,
reducing the size of the transfers to the financially constrained by a factor of x scales the output
profile by x, and hence the multiplier by x.
The results are presented in Table 3 for untargeted transfers (column B) and partially targeted
transfers (column C). The 2001 Bush rebates were once-off, and at a time when the Federal Reserve
was not constrained by the ZLB. As such, we estimate multiplier between 0.3 and 0.6 for these
transfers, with the wide range due to the untargeted nature of the transfers and the wide range
of estimates in the literature of the proportion of credit constrained households. In contrast, the
ARRA (ρ = 0.9) was enacted when the ZLB was binding (and bound for several years afterwards),
and was more targeted than the Bush tax rebates. As such the multiplier is much higher, around
1.3 to 1.5.
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