The Government Spending Multiplier in a Model with the Cost
Channel
Salem Abo-Zaid∗
Department of Economics, Texas Tech University
Abstract
This paper studies the government spending multiplier in the presence of the cost channel
of the nominal interest rate. I find that the spending multiplier of normal times declines
markedly when this channel is introduced. The drop in the multiplier is bigger for more
flexible prices and stronger responses to inflation in the interest-rate rule. The rise in government spending leads to a rise in the nominal interest rate and, with the cost channel, to
a rise in the marginal cost and inflation. In turn, this leads to a bigger rise in the nominal interest rate and the expected real interest rate, hence a lower multiplier, than in the
model that abstracts from the cost channel. On the other hand, to the extent that firms’
borrowing costs rise at the zero lower bound following government spending shocks, the cost
channel makes the spending multiplier larger than in a model that does not account for this
channel. When the nominal interest rate is fixed, the cost channel induces a bigger drop in
the expected real interest rate and thus a larger government spending multiplier. Therefore,
by ignoring the cost channel, the government spending multiplier is overestimated in normal
times and underestimated during liquidity-trap episodes. Since liquidity-trap periods are
rare, however, the spending multiplier is mostly lower than in previous estimates.
Keywords: Government Spending Multiplier; Cost Channel; Fiscal Policy; Monetary
Policy; Zero Lower Bound.
JEL Classification: E52; E58; E62; E63.
1. Introduction
The main goal of this paper is to study the government spending multiplier in a model with
the cost channel of the nominal interest rate. It is found that the cost channel matters a
great deal for the size of the spending multiplier but the exact effects depend on the episode
considered. In normal times, during which the nominal interest rate is not fixed, the cost
channel reduces the multiplier considerably. Depending on the strength of this channel, the
multiplier can drop from above one to zero or even a negative value just because of the
∗
248 Holden Hall, Department of Economics, Texas Tech University, Lubbock, TX, 79409.
salem.abozaid@ttu.edu.
Email:
March 6, 2016
mere introduction of the cost channel. This effect is stronger for less rigid prices, stronger
response of the nominal interest rate to inflation, more persistent government spending and
lower intertemporal elasticity of substitution in the utility function.
However, when the nominal interest rate is fixed, as can happen because of the zero lower
bound (ZLB) constraint, the cost channel makes the spending multiplier larger provided that
the borrowing costs of firms rise in response to government spending shocks. For plausible
calibration of the model, the liquidity-trap spending multiplier more than doubles in the
presence of the cost channel compared to its value in a model that does not account for this
channel. This effect is stronger when prices are less rigid and the responses of the firms’
borrowing interest rates to government spending shocks are stronger.
The intuition behind these results is as follows. When the nominal interest rate is free to
adjust, the rise in government spending raises this interest rate and, as a result, the marginal
costs of firms. In turn, this leads a larger increase in inflation and inflation expectations than
in a model that abstracts from this channel. And, since the nominal interest rate increases
by more than one-for-one with inflation (and the expected inflation rate), the rise in inflation
triggers even a larger increase in the nominal interest rate. The result is a larger increase
in the expected real interest rate, which induces a bigger decline (or at least a muted rise)
in consumer spending. In turn, the multiplier will be smaller than in an otherwise model
without the cost channel.
When the ZLB constraint is binding, however, the behavior of the expected (and actual)
real interest rate is reversed. Higher borrowing costs of firms raise the marginal costs,
and thus inflation and expected inflation by more than in the absence of the cost channel.
However, at the ZLB, the nominal interest rate that is relevant for consumer spending
remains unchanged (and it is unaffected by the cost channel). Therefore, the decline in the
expected real interest rate is larger than in the absence of this channel, which in turn implies
a larger spending multiplier during this type of episodes. Therefore, previous estimates of
the multiplier that did not account for the cost channel underestimated the multiplier of
stress times and overestimated the multiplier of normal times, implying a bigger asymmetry
when the cost channel is accounted for.
This study extends an otherwise standard New Keynesian (NK) model by allowing for
the cost channel to be operative. Following Christiano et al. (2005) and Ravenna and Walsh
(2006), among others, I assume that the wage bills of firms are paid before production takes
place, thus giving rise to borrowing by firms (which, in turn, leads to supply-side effects of
the nominal interest rate on inflation as in the standard cost channel of monetary policy).
In particular, the marginal cost of firms becomes a function of the nominal interest rate
on borrowing. The basic setup assumes one interest rate in the economy, but to address
the change in firms’ borrowing interest rates at the ZLB, I assume that these rates differ
from the policy interest rate. The deviation between these interest rates can possibly result
from market power in the intermdiation sector, adjustment costs of loans and other costs
associated with changes in the scale of operation of banks. See Elyasiani et al. (1995), Freixas
and Rochet (1997) and Kopecky and VanHoose (2012), among others, for a discussion.
2
The paper adds to the recent literature on the government spending multiplier. Christiano et al. (2011) first show that the multiplier is modest (slightly above 1) when the
nominal interest rate is governed by a standard Taylor rule. However, they show that the
government-spending multiplier can be much larger than one if the nominal interest rate
does not respond to an increase in government spending, as can happen when the nominal
interest rate hits the zero lower bound. The rise in government spending raises output and
inflation expectations. When the nominal interest rate is at the lower bound, that leads to
a decline in the real interest rate and to a further rise in consumption, output and expected
inflation, leading to a loop of rising inflation expectations and output. The multiplier, thus,
can be large. Quantitatively, the benchmark calibration of their model with the ZLB delivers a multiplier of more than three times the size of the multiplier of normal times. In
a related study, Carrillo and Poilly (2013) show that financial frictions increase the size of
the government spending multiplier, particularly at the ZLB. The liquidity-trap multiplier
increases by 2.8 and the nonliquidity-trap multiplier increases by roughly 0.8 compared to a
model without financial frictions.
Carlstrom et al. (2014) consider the effects of the duration of interest rate peg on the
size of the spending multiplier within a New-Keynesian framework. They compare the
results when the monetary-fiscal expansion lasts a certain number of periods (“deterministic
duration”) with the results when the expansion is stochastic (with a mean that is equal
to the fixed number of periods under the deterministic case). They find that the size of
the stochastic multiplier is large and clearly larger than the deterministic multiplier. For
example, when the interest rate is fixed for 4 periods (or the mean is 4), the stochastic
multiplier is 66.9 and the deterministic multiplier is 2.3.
Using a model similar to Carlstrom et al. (2014), Dupor and Li (2015) study the “expected
inflation channel” and its implications for the government spending multiplier. They show
that having a large multiplier requires a large response of expected inflation to government
spending. However, according to their empirical evidence, that has not been the case in the
U.S. over the periods 1959-1979 and 1981-2002 and, therefore, the expected inflation channel
has no support in U.S. data or it has been too small to generate large fiscal multipliers.
Furthermore, their analyses distinguish between “passive” monetary policy, which occurs
when the central bank increases the nominal interest rate by less than one-for-one with
the change in the expected inflation rate, and “active” monetary policy, when the opposite
occurs. It is found that under a passive monetary policy the multiplier exceeds one, and
under an active policy it is less than one.
The size of the fiscal multiplier at the ZLB has recently been questioned by the empirical findings of Ramey and Zubairy (2015). The authors construct quarterly U.S. data for
1889-2013 to test whether government spending multipliers differ according to the amount
of slack in the economy and being near the ZLB. They find that the amount of slack in the
economy does not affect the size of the multiplier. Further, the analysis does not point out
to any statistically different multipliers between normal times and ZLB episodes, particulary
when the full sample is used. Therefore, contrary to recent findings, government spending
3
multipliers during the Great Recession were not necessarily higher than the average. Auerbach and Gorodnichenko (2012), on the other hand, use U.S. data for 1947:Q1-2008:Q4 and
find large differences in the size of spending multipliers between recessions and expansions:
government spending is considerably more effective in recessions than in expansions, which
aligns with the predictions of quantitative models. My analyses indicate that the asymmetry
predicted by a quantitative model is bigger when the cost channel is introduced. Caggiano
et al. (2015) show that the size of the spending multiplier depends on the strength of the
business cycle: when the authors distinguish between deep and mild recessions, and weak
and strong expansions, they find that fiscal multipliers are larger in deep recessions than in
strong expansionary periods.
Existing literature also considered other aspects that can be important for the size of the
fiscal multiplier. Gali et al. (2007) extend a standard NK model to account for rule-of-thumb
consumers. They show that the exact agent’s preferences matter for the size of the multiplier,
including being larger than or less than unity. Monacelli and Perotti (2008) first document
that variations in government purchases generate a rise in consumption, the real wage and the
product wage, and a fall in the markup. They then build an otherwise standard business cycle
model with price rigidity, in which preferences can be consistent with an arbitrarily small
wealth effect on labor supply, and show that the model can well match the empirical evidence
on the effects of government spending shocks. This highlights again the role of preferences
for the size of the multiplier. Zubairy (2014) studies the government spending multiplier
in a stochastic general equilibrium model that features other distortionary taxes and finds
that the impact government spending multiplier is 1.07. In addition, private consumption
responds positively to government spending shocks and the multipliers for labor and capital
taxes are 0.13 and 0.34, respectively. Bouakez et al. (2015) find a multiplier of 2.3 in a
model with public investment and suggest that ignoring the investment component of the
2009 Stimulus Package leads to underestimation of the multiplier by roughly a half.
In light of the wide range of findings about the fiscal multiplier, Leeper et al. (2011)
study the fiscal multipliers in five nested models: a simple real business cycle (RBC) model,
the RBC model with real frictions, a standard new Keynesian model, the new Keynesian
model with hand-to-mouth agents and an open economy version of the new Keynesian model.
They use Bayesian prior predictive tools and conclude that model specifications and prior
distributions can tightly limit the range of possible spending multipliers. Moreover, a passive
monetary policy yields strong multipliers for output, consumption and investment.
Despite its empirical validity, the cost channel of the nominal interest rate has not been
addressed in the context of the government spending multiplier. This paper aims for filling
that gap in the literature. To this end, I first provide more updated evidence regarding the
existence of a cost channel and then examine the fiscal multiplier in a computable model
with this channel and compare it to a model that abstracts from it. I find that the cost
channel became more significant since 1984, that the firms’ credit spread surged during the
current ZLB episode and that firms’ borrowing costs are responsive to government spending
shocks, which supports the choice of a model that distinguishes between the policy interest
4
rate and the borrowing rates of firms as a platform to study the implications of the cost
channel for the size of the multiplier when the ZLB constraint is binding.
The effects of the cost channel on the fiscal multiplier are robust to multiple specifications about agent’s preferences, the degree of price rigidity, the responses to inflation in the
interest-rate rule, the strength of the cost channel (in the empirically relevant range) and
other parameters. In particular, the differences between the multipliers with and without
the cost channel are bigger for less rigid prices. This happens because the coefficient of the
nominal interest rate in the New-Keynesian Phillips Curve is bigger for lower price rigidity.
When prices approach full stickiness, the cost channel does not affect the size of the fiscal
multiplier as the marginal cost, through which the cost channel operates, becomes immaterial
for the behavior of inflation.
The results of the paper are compared to those of Christiano et al. (2011), Carlstrom
et al. (2014) and Dupor and Li (2015). I conduct the analyses for passive and active central
banks as defined in the latter study. Interestingly, I find that the cost channel weakens
the multiplier in the presence of “active” monetary policy but strengthens the multiplier
of a “passive” monetary policy. The reason for these results are as follows. When the
interest rate initially rises in response to government spending shocks, the marginal cost
rises, and so do inflation and the expected inflation rate. With a passive central bank, the
nominal interest rate rises by less than one-for-one with the rise in the expected inflation
rate, leading to a bigger drop in the expected real interest rate compared to the model with
no cost channel. The result will be a higher spending multiplier. With an active monetary
policy, the nominal interest rate rises by more than one-for-one with the rise in the expected
inflation rate, which makes the rise in the expected real interest rate with the cost channel
lower than in an otherwise model without it, leading a lower multiplier. When the central
bank is neutral, in the sense that the nominal interest rate rises by precisely one-for-one with
the expected inflation rate, the cost channel does not affect the size of the multiplier.
The remainder of the paper proceeds as follows. Section 2 outlines the model economy.
Section 3 provides empirical evidence on the existence of the cost channel. Section 4 presents
the government spending multiplier of normal times. Section 5 presents robustness analyses
using alternative preferences and monetary policy rule. The spending multiplier of a constant interest rate and evidence regarding the responsiveness of firms’ borrowing costs to
government spending shocks are presented in Section 6. Section 7 concludes.
2. The Model Economy
The economy is populated by a continuum of infinitely-lived households who derive utility
from consumption and leisure. Households also hold cash and they are engaged in the
intermediation sector by holding some of their assets in the form of interest-earning deposits.
Intermediate-good firms operate in an environment of monopolistic competition. They hire
labor as the only input to produce differentiated products, set prices in a staggered fashion
and borrow from intermediaries to finance their wage bill as in a standard model with the
5
“working capital” requirement. The products of these firms are sold to final-good firms
(retailers) who package them into final goods using a one-to-one technology. The basic
setup through which the cost channel is introduced follows Ravenna and Walsh (2006) and
Christiano et al. (2005), and the reader may refer to these studies for more details.
2.1. Households
The problem of the representative household is to choose consumption ct , labor supply nt ,
deposits Dt and cash Mt to maximize:
∞
[cγt (1 − nt )1−γ ]1−σ − 1
E0 ∑ β (
)
1−σ
t=0
t
(1)
where β is the subjective discount factor, E0 is the expectation operator, σ > 0 and γ ∈ (0, 1).
The assumption of non-separable utility function follows Christiano et al. (2011), but I will
also show the results with seperable preferences in what follows.
At the beginning of the period, households make deposits at the financial intermediary
and the remaining amount Mt + Wt nt − Dt , with Wt being the nominal wage, is used for
consumption. At the end of the period, households receive profits from the financial intermediary and intermediate-good firms as well as the principle and interest on their deposits at
the intermediary. The corresponding constraints that households face and a full description
of their problem is presented in Appendix A.1.
The optimization by households leads to the following standard consumption Euler equation and the labor supply condition, respectively, expressed in log deviations from the steady
state:
n
n
̂t −Et [̂
n̂t = R
πt+1 − [γ(1 − σ) − 1]̂
ct+1 + (1 − γ)(1 − σ)
n̂
t+1 ]
1−n
1−n
(2)
n
̂t
ĉt +
n̂t = w
(3)
1−n
where x denotes the non-stochastic steady state and x̂t is the log deviation from this steady
state for any variable xt , πt is the inflation rate, wt is the real wage and Rt is the nominal
policy (or market) interest rate. As expected, these conditions hold regardless of the cost
channel and they are similar to the corresponding conditions in Christiano et al. (2011).
With logarithmic preferences (σ = 1), condition (2) restores the more familiar form of the
log-linearized Euler condition as preferences become separable in labor and consumption.
[γ(1−σ)−1]̂
ct −(1−γ)(1−σ)
2.2. The Production Sector
As is standard in the literature, two types of firms operate in this sector: monopolistically
competitive intermediate-good firms who produce differentiated products and perfectly competitive firms who transform intermediate goods into final goods using a constant return to
scale technology.
6
2.2.1. Final-Good Firms
Firms in this sector operate in a perfectly competitive environment. They purchase a continuum of intermediate goods from intermediate-good producers, indexed by j ∈ (0, 1), and
assemble them into final goods using the following technology:
1
yt = (∫
0
ε−1
ε
ε
ε−1
yj,t dj)
(4)
with yj,t being the quantity of intermediate-good j that is purchased by a final-good firm
and ε > 1 is the elasticity of substitution between two differentiated types of intermediate
goods.
Profit maximization by intermediate-good producers gives the following downward-sloping
demand function for the product variety j:
yj,t = (
1
Pj,t −ε
) yt
Pt
(5)
1
1−ε
where Pt = (∫0 Pj,t
dj) 1−ε is the Dixit-Stiglitz aggregate price level that results from cost
minimization.
2.2.2. Intermediate-Good Firms
Firms in this sector are monopolistically competitive. Each firm j hires labor as the only
input to produce a differentiated product using the following technology:
yj,t = nj,t .
(6)
Firms in this sector borrow from the intermediary at the beginning of each period at an
interest rate of RL,t in order to pay their wage bills. This is in line with the “working capital”
requirement as in Christiano et al. (2005), Tillmann (2008) and Ravenna and Walsh (2006),
among others. Following these studies, I assume in this section that the intermediation
sector is perfectly competitive, complete pass through from the policy interest rate to the
loan rate and no adjustment costs for the intermediary. Therefore RL,t = Rt . However, in
what follows I also show the results when the loan rate is a markup over the policy rate.
Each period, a firm j has the probability 1 − θ of changing its price as in Calvo (1983).
With this characterization of the intermediate-good sector, the time t real profit function of
firm j is given by:
Πfj,t = yj,t − Rtδ wt nj,t
(7)
subject to (5)-(6). As in Tillmann (2008), the parameter δ measures the strength of the cost
channel and it will be key to the analyses in what follows.
As is standard in the NK literature, optimization leads to the following forward-looking
Phillips Curve:
π̂t = β Et π̂
mct
(8)
t+1 + κ̂
7
and labor demand condition:
̂t
̂
̂t + δ R
m
ct = w
(9)
with κ = (1−θ)(1−βθ)
. Letting st = wyt tnt be the labor share of output and using the log-linearized
θ
version of the production function, condition (9) can be rewritten as:
̂t
̂
m
ct = ŝt + δ R
(10)
which is the standard condition of the real marginal cost in models with the cost channel
(see, for example, Ravenna and Walsh (2006) and Tillmann (2008)). The Phillips Curve
then reads:
̂t )
π̂t = β Et π̂
st + δ R
(11)
t+1 + κ (̂
and, thus, the nominal interest rate directly affects the behavior of inflation. Introducing
the parameter that governs the strength of the cost channel (δ) appears here of particular
importance. First, setting this parameter to zero allows straightforward comparisons with
the model that abstracts from the cost channel. Second, if δ was imposed to be one, then
condition (11) implies that the response of inflation to changes in the nominal interest is
captured by κ, which may not be necessarily the case. Indeed, as will be illustrated in the
next section, this parameter mostly differs from one.
2.3. Market Clearing and Monetary Policy
Government expenditures are financed via lump-sum taxes (gt = Tt ). In equilibrium, the
resource constraint of the economy reads:
yt = ct + gt
(12)
Denoting the steady-state values of government spending and output by g and y, respectively,
log linearization of the resource constraint yields:
ŷt = (1 − g)̂
ct + g ĝt
(13)
where g = yg is the steady-state government spending-output ratio.
Government spending evolves according to the following AR(1) process:
ĝt = ρ̂
gt−1 + ut
(14)
with ρ being the persistence parameter and ut ∼ N (0, σu2 ).
In addition, monetary policy is governed by the following interest-rate rule:
̂t = φπ π̂t + φy ŷt
R
(15)
In what follows, I check the robustness of the results with an alternative interest-rate rule.
8
3. The Cost Channel: Empirical Evidence
This section provides some evidence about the existence of the cost channel using pre-crisis
U.S data (1960:Q1-2007:Q3). The sample ends in the third quarter of 2007 to avoid any
bias in the results during the recent financial crisis and the subsequent reach of the nominal
interest rate the zero lower bound. The estimation is then provided for the second half of the
sample (1984:Q1-2007:Q3) to asses whether the cost channel has become weaker or stronger
over time. I choose 1984 as the beginning of the second period because it is well known that
many macroeconomic aggregates have experienced changes since that year. It also happens
to be in the middle of the sample.
I estimate condition (11) and then calculate the implied value of δ. To do so, I use the
3-month treasury bill as a proxy for Rt , the non-farm business labor share of output for st
and the rate of change in the GDP deflator for πt . Following Ravenna and Walsh (2006), the
model is estimated using the Generalized Method of Moments with instrumental variables.
All data have been detrended using the Hodrick and Prescott (HP) filter using a standard
smoothing parameter of 1600. The results are reported in Table 1.
Description
1960:Q1-2007:Q3
1984:Q1-2007:Q3
κ
0.034
(0.015, 0.054)
0.023
(0.005, 0.040)
δ
1.087
(0.991, 1.113)
2.258
(1.818, 5.939)
Price Duration
6.065
(4.567, 7.848)
7.387
(5.148, 12.297)
J-Test (p-value)
0.724
0.927
κ
0.050
(0.029, 0.070)
0.030
(0.012, 0.049)
Price Duration
5.094
(4.089, 5.869)
6.409
(4.765, 8.431)
J-Test (p-value)
0.757
0.968
Obs.
191
95
Table 1: Empirical results- estimation of condition (11). The list of instruments follows Ravenna and Walsh
(2006) and includes four lags of non-farm business sector real unit labor cost, output gap, inflation rate
(GDP deflator), commodity price index inflation, 10-year minus 3-month U.S. government bond spread,
non-farm business sector hourly compensation inflation and 3-month T-bill rate. 95% confidence intervals
in parentheses. The J-Test tests the hypothesis that the over-identifying restrictions are satisfied.
The point estimates of κ for the full sample period are in line with previous findings and
with the standard values of this parameter that are used in the literature on the government
spending multiplier (e.g. Christiano et al., 2011, Carlstrom et al., 2014 and Dupor and Li,
9
2015). The cost channel is very strong and the value of δ is about unity, which is similar to
the findings of Ravenna and Walsh (2006) and Chowdhury et al. (2006). Furthermore, the
p-values of the J-Test indicate that the over-identifying restrictions are satisfied.
Over the period 1984:Q1-2007:Q3, the cost channel became significantly stronger: δ
exceeds 2 and the 95% confidence interval indicates that it can reach roughly 6. Given these
findings, my initial estimates of the spending multiplier will consider values of δ between 1
and 6, with the focus being on the range 1-3 where the point estimates of this parameter lie.
The bottom panel shows the estimation of the Phillips Curve using the (non-farm business) unit labor cost (ULC), which is defined as total labor compensation divided by real
output. This amounts to estimating condition (8). Given our linear technology, the ULC
is essentially the real marginal cost. The results indicate a slightly higher κ than in the
estimations with the labor share and the interest rate, and they also show a reduction in the
size of κ in the second sub period.
The table also reports the corresponding price durations, which can be found by using the
1
condition κ = (1−θ)(1−βθ)
and the fact that the price duration is given by 1−θ
. All cases indicate
θ
that the price duration is above 4 quarters, mostly 6 quarters or more, and can reach as high
as 12 quarters. These estimates are clearly higher than micro evidence about the duration of
price contracts: Bils and Klenow (2004) find that, even after excluding temporary price cuts
(sales), half of prices last up to 5.5 months. Nakamura and Steinsson (2008) report a higher
price duration: with the exclusion of product substitutions, the uncensored durations of
regular prices are between 8-11 months. Allowing for price changes associated with product
substitutions, the median duration until either the price changes or the product disappears
is 7-9 months. Christiano et al. (2005) find a price duration of 2.5 quarters and Smets and
Wouters (2007) suggest an average duration of prices of 3 quarters.
All of these estimates lie within the 1-4 quarters range. Therefore, to be consistent with
the literature on government spending multipliers, my analyses will first use the values of κ
as in Table 1. Then, I will use other values of κ that are consistent with the existing evidence
about the average duration of prices, particularly the micro evidence, which suggests average
price durations between 2 and 4 quarters.
For robustness, Table B.1 presents the results using the Baxter-King (BK) Band Pass
Filter as the filtering method and with Moody’s Seasoned Baa Corporate Bond Yield as the
alternative measure for firms’ costs. With the BK filter, κ is lower, but δ is higher (and for
the most part, δ tends to be higher with lower κ, thus keeping the coefficient of the interest
rate κδ more stable across specifications), but δ remains within the 1-6 interval. With the
Baa series, κ is similar to the previous estimates and δ is different from one. The average
price durations in either case are similar to their values in the benchmark analyses.
Table B.2 reports the results with Maximum Likelihood estimation. The values of κ and
δ are largely within the ranges that have been found in Table 1, and δ appears to be larger
than one. Furthermore, as the Likelihood Ratio (LR) tests shows, the model without the
cost channel is strongly rejected in favor of the model that accounts for it. The estimates
for average price durations are also in line with their benchmark values.
10
4. The Government Spending Multiplier
The main analyses regarding the government spending multiplier are presented in this section. By using conditions (3), (8) and (9), the Phillips Curve can be written as:
π̂t = β Et π̂
ct +
t+1 + κ (̂
n
̂t )
n̂t + δ R
1−n
(16)
Following the literature, the model is solved using the Method of Undetermined Coefficients. Under the assumption that φπ > 1, the solution yields:
ŷt = Â
gt
(17)
where:
A=g
κ(φπ − ρ) + (1 − ρ)(1 − βρ − δκφπ )[γ(σ − 1) + 1]
n
1
+ 1−n
+ δφy ]
(1 − βρ − δκφπ )[1 + (1 − g)φy − ρ] + (1 − g)(φπ − ρ)κ [ 1−g
(18)
The government spending multiplier is then given by:
dyt A
=
dgt g
(19)
1 − g ĉt
dyt
.
=1+
dgt
g ĝt
(20)
or, alternatively:
By setting δ = 0, we get the exact government spending multiplier as in Christiano et al.
(2011). Clearly, the multiplier is affected by the presence of the cost channel, but the exact
effect is not possible to tell without further information and it may depend on the size of δ
as well as other parameters. There is one case, however, when the role of the cost channel
is easy to see: if prices approach full stickiness (θ → 1), κ approaches zero, and the cost
channel will not affect the size of the multiplier. This result is as expected since, according
to condition (9), the interest rate does not affect the behavior of the marginal cost (hence,
inflation), and the cost channel is immaterial. As will be outlined below, the size of κ is very
important for the size of the multiplier and for the effects of the cost channel on it.
As expected, condition (20) makes it clear that the size of the multiplier depends on
the response of consumption to government spending disturbances. If consumption rises
(“crowding in”), then the multiplier will be larger than one. If it falls (“crowding out”),
the multiplier will be less than one. And, if consumption does not respond to shocks to
government spending, then the multiplier will be exactly one.
4.1. Results
Table 2 summarizes the parameter values that I use in the benchmark analyses. These values
mostly follow Christiano et al. (2011), with the only exception being κ, which is set based
11
on the empirical analysis that has been presented above. These parameter values are widely
accepted in the literature, but I will conduct comprehensive robustness analyses by changing
the values of these parameters and examine how the cost channel affects their effects on the
spending multiplier. I will also compare my results to Christiano et al. (2011) using their
value of κ = 0.03.
Parameter
Value
β
σ
γ
ρ
κ
g
φπ
φy
n
0.99
2.00
0.29
0.80
0.034
0.20
1.50
0.00
1/3
Table 2: Values of the parameters.
Figure 1 shows the government spending multiplier for values of δ between 0 and 6.
Without the cost channel, the multiplier is slightly above one (as in Christiano et al., 2011),
but it declines rapidly as the significance of the cost channel rises. At roughly δ = 6, the
multiplier becomes negative. This figure confirms the conjectures regarding the effects of
the cost channel on the size of the multiplier and it suggests that not only this effect can
be large, but it rises exponentially as this channel becomes stronger within the empirically
plausible range. Furthermore, the fact that the multiplier can turn negative suggests that
the cost channel may matter for more than just the numerical value of the multiplier.
1.2
1
Multiplier
0.8
0.6
0.4
0.2
0
-0.2
0
1
2
3

4
5
6
Figure 1: The government spending multiplier for various values of δ.
Within the range of δ = 1-3, the multiplier drops from 1.03 to 0.85, which is relatively
muted. Notice, however, that this calculation is based on a value of θ = 0.835, which implies
an average price duration of more than 6 quarters. As discussed above, this lies outside the
range of average price durations that have been found in empirical work, particularly at the
micro level. For this reason, in what follows I experiment on the price rigidity parameter
12
θ within the plausible range of average durations of prices and consider the implications of
changing this parameter for the size of the spending multiplier.
As in Christiano et al. (2011), the multiplier (with no cost channel) exceeds one due to
the complementarity between consumption and labor. A rise in government spending raises
output and labor and, due to this complimentarily, pushes for higher consumption. This
effect clashes with the substitution effect that results from a higher expected real interest
rate. With σ = 2, the first effect dominates, and consumption end up rising in the aftermath
of a surge in government spending (implying a multiplier above one). In what follows, I show
that this result depends on the size of σ. Interestingly, the cost channel itself can reverse
this result and that will be proven true for other values of σ.
The key intuition behind the effects of the cost channel on the spending multiplier can be
seen through the response of the expected real interest rate to government spending shocks.
As shown in Appendix A.2, the expected real interest rate is given by:
gt
Et r̂
t+1 = (φπ − ρ)Q̂
(21)
and, therefore, its response to a government spending shock depends on the response of
inflation to this shock (captured by Q). In particular, since φπ > ρ, the expected real interest
rate will rise by more than in the model without the cost channel as the response of inflation
to spending shocks is bigger. Indeed, Figure B.1 shows that the response of the expected
real interest rate to spending shocks is an increasing function of δ. With higher values of
δ, the rise in the expected real interest rate is bigger and, therefore, the multiplier is lower.
Furthermore, the shape of this graph is exactly the opposite of the multiplier (Figure 1),
which further supports the intiution behind the results of this subsection.
The rise in the actual real interest rate is also bigger when the cost channel is operative.
The actual real interest rate will rise provided that φπ > 1, which is the standard assumption
in this class of models. This results from the fact that the nominal interest rate rises by
more than one-for-one with the inflation rate. The cost channel makes Q larger and thus
leads to a bigger real interest rate. The derivations are shown in Appendix A.2.
When the nominal policy interest rate responds to output alongside inflation (φy > 0),
the same arguments hold. In this case, the expected real interest rate is given by:
̂t
Et r̂
t+1 = [(φπ − ρ)Q + φy A] g
(22)
which, other things equal, shows a larger increase in the expected real interest rate than in
the model with response to inflation only. Therefore, the government spending multiplier
in this case is lower than in the model with φy = 0 even in the absence of the cost channel,
which is in line with the findings of Christiano et al. (2011). Figure B.2 shows the numerical
results with the cost channel, and they are very similar to the results of Figure 1. Since the
effects of the cost channel are similar to the model with no response to output, I proceed
with the assumption that the nominal interest rate responds to inflation only.
13
4.2. Changing the Price Rigidity Parameter
Figure 2a reports the results of varying δ for three values of the price rigidity parameter.
1
Since the duration of prices is given by 1−θ
, these values correspond to price durations of
2, 3 and 4 quarters, which are the most consistent with the micro empirical evidence. The
impact of the cost channel is significantly stronger than with higher degrees of price rigidity.
With 4 quarters, the multiplier falls from 0.90 to 0.29 in this range of δ. With shorter price
durations, the multiplier drops faster and it even becomes negative at values of δ between 2
and 3. Furthermore, the impact of the cost channel is magnified as the price duration drops
and δ rises. This is well illustrated by the fact that the difference between the bottom and
the middle lines is significantly larger than the difference between the first and the second
lines (even though the difference between the values of θ is smaller).
(a)
(b)
1
1
0.5
0.5
0
Multiplier
Multiplier
0
-0.5
-1
-0.5
-1
-1.5
=0
=1
-1.5
-2
-2.5
0
=0.50
=0.67
=0.75
0.5
=2
=3
-2
1
1.5

2
2.5
0
3
0.1
0.2
0.3
0.4
0.5
1-
Figure 2: The government spending multiplier, the cost channel and price rigidity.
In Figure 2b, I show the results by varying the degree of price flexibility (1 − θ) for the
relevant values of δ. I use values of θ between 0.5 (a price duration of 2 quarters) and 1
(fully-rigid prices). The main insights can be summarized as follows. First, regardless of the
cost channel, a decrease in price rigidity leads to a decline in the multiplier. For example,
with no cost channel, the multiplier drops from 1.29 to 0.76. In fact, even when prices have
a high duration of 4 quarters (1 − θ = 0.25), the multiplier is 0.90, which is significantly lower
than the multiplier of fully-rigid prices.
Second, when prices approach full rigidity, the cost channel is immaterial, in line with
our earlier discussion. Intuitively, when κ approaches zero, the nominal interest rate does
not affect the pricing behavior of firms. Third, the cost channel magnifies the drop in the
multiplier as prices become more flexible. With a relatively small value of δ, the multiplier
drops to nearly 0.50 at a price duration of 2 quarters and to 0.80 at a price duration of 4
14
quarters. With higher values of δ, the multiplier drops to (roughly) zero or even a considerably negative value. With an intermediate value of δ, the multiplier is only 0.64 at a price
duration of 4 quarters, which is half of its value with fully-rigid prices.
4.3. Changing other Parameter Values
This subsection shows that the effects of the cost channel on the size of the multiplier depends
on other parameters as well and that this channel strengthens or weakens their effects on
the multiplier. The results are summarized in Figure 3. Consider first the response of the
nominal interest to inflation (φπ ). A bigger response of the nominal interest rate to inflation
leads to a higher rise in the nominal interest following a government spending shock, which
lowers the size of the multiplier, with and without the cost channel. The cost channel makes
this effect stronger, particularly as δ rises.
1.2
1.1
1.1
1
1
Multiplier
Multiplier
0.9
0.8
0.9
0.7
0.8
0.6
0.5
0.4
=0
=1
0.7
=2
=3
1.5
2
2.5
3

3.5
4
4.5
0.6
0.5
5
0.55
0.6
0.65

1.2
1.5
1.15
1.4
1.1
0.75
0.8
0.85
0.9
1.3
1.05
1.2
Multiplier
Multiplier
0.7

1
0.95
1.1
1
0.9
0.9
0.85
0.8
0.8
0.75
0.7
1
1.5
2

2.5
3
0
0.2
0.4
0.6
0.8

Figure 3: The government spending multiplier for various values of the model’s parameters.
15
1
The spending multiplier is lower for more persistent government spending, which results
from a bigger negative wealth effect on consumption when government spending is more
persistent as the present discounted value of taxes needed to finance these expenditures is
bigger. This effect is stronger in the presence of the cost channel: the bigger negative wealth
effect is coupled by a bigger substitution effect that results from a larger rise in the interest
rate, thus leading to a bigger drop in consumption and to a lower multiplier.
With σ = 1, the government spending multiplier is less than one even in the absence of
the cost channel. In this case, the lack of complementarity between labor and consumption
leads to a decline in consumption following a rise in the nominal interest rate (hence, the
expected real interest rate), which results in a multiplier of less than unity.
The multiplier is increasing in both σ and γ. A higher γ increases the marginal utility of
consumption and thus leads to a stronger response of consumption to government spending
shocks. Given the non-separable preferences in consumption and labor, the marginal utility
of consumption is increasing in the amount of labor supplied. This effect is stronger for
higher values of σ. The cost channel in this case weakens this effect: the desire to consume
more is partially offset by the bigger rise in the interest rate in the presence of the cost
channel. In fact, with δ = 4 the multiplier becomes virtually unaffected by the rise in γ, for
δ = 5 the multiplier is decreasing in γ (but remains slightly positive), and with δ = 6 the
multiplier drops markedly with the rise in γ and becomes negative even for γ = 0.29. Similar
results hold for σ. Therefore, not only that the cost channel renders the multiplier lower,
but it also flattens its rise in σ and γ, and even reverses the effects of these two parameters
on the size of the multiplier.
5. Robustness Analyses: Model Specification
In this section, I redo the analyses using a separable utility function as in Carlstrom et al.
(2014) with the rest of the model’s specifics remaining unchanged. I then use this model
with an alternative interest-rate rule; the second experiment considers an interest-rate rule
where the central bank responds to the expected inflation as opposed to the actual inflation
rate. This modeling follows Dupor and Li (2015). The analyses with the cost channel in this
section will allow for good comparisons with these two recent studies.
5.1. Separable Utility Function
The period utility function in this setup is given by:
u(ct , nt ) =
ct1−σ
n1+ν
−χ t
1−σ
1+ν
(23)
which yields the following Euler and labor supply conditions:
̂t − Et π̂
̂
̂t )
R
t+1 = σ(Et c
t+1 − c
(24)
̂t .
ν n̂t + σ̂
ct = w
(25)
16
By using conditions (8), (9) and (25), the Phillips Curve can then be written as:
̂t )
̂t + σ̂
π̂t = β Et π̂
ct + δ R
t+1 + κ (ν n
(26)
The solution for the spending multiplier, under the assumption φπ > 1, then yields:
κσ(φπ − ρ) + σ(1 − ρ)(1 − βρ − δκφπ )
dyt
=
dgt κ [ν(1 − g) + σ + δ(1 − g)φy ] (φπ − ρ) + [σ(1 − ρ) + (1 − g)φy ] (1 − βρ − δκφπ )
(27)
Once more, the multiplier is affected by the cost channel, κ matters markedly, and the
cost channel will not affect the size of the multiplier when κ = 0. Furthermore, the multiplier
is clearly less than one, and that holds even when the cost channel is absent (δ = 0). With
ρy = 0 and ν = 0, which implies an infinite labor supply elasticity, the multiplier is exactly
one, and the cost channel does not change this conclusion.
1
0.9
Multiplier
0.8
0.7
0.6
0.5
0.4
0
1
2
3

4
5
6
Figure 4: The government spending multiplier for various values
of δ with separable utility function.
I set β = 0.99, σ = 2, ν = 0.50, ρ = 0.80, κ = 0.034, φπ = 1.5 and g = 0.20. The first four
parameters are as in Carlstrom et al. (2014), κ follows the empirical findings and the rest
follow Christiano et al. (2011).1 As in the model with non-separable preferences, the spending
multiplier is decreasing in the size of δ, and effects of the cost channel on the multiplier are
exponentially rising in the size of this parameter. Overall, the shape of this graph is similar
to that of Figure 1 and, therefore, the particular assumption about preferences is not behind
the main findings of the paper.
1
Carlstrom et al. (2014) analyze the multiplier at the ZLB and their multiplier does not include the last
three parameters. For this reason, I use the values from the other study.
17
As before, the multiplier is very sensitive to changes in the average price duration as the
cost channel becomes more significant for less rigid prices. For example, at δ = 3 and a price
duration of 3 quarters, the multiplier is 0.27 as opposed to 0.87 in the model without the
cost channel. For δ = 3 and a price duration of 2 quarters, the multiplier is -5.26 compared
to 0.85 in the model that abstracts from this channel.
5.2. An Alternative Interest-Rate Rule: Active vs. Passive Monetary Policy
The model is similar to the one from subsection 5.1, but with one modification. Following
Dupor and Li (2015), the nominal interest rate is governed by the following rule:
̂t = (1 + ψ)Et π̂
R
t+1
(28)
Monetary policy is “active” if the nominal interest rate rises by more than one-for-one
with the expected inflation rate (which occurs when ψ > 0), “passive” if the interest rate
rises by less than one-for-one (ψ < 0), and “neutral” if the interest rate rises by exactly
one-for-one (ψ = 0). This specification directly addresses the expected inflation rate channel
that has been frequently mentioned in the fiscal multiplier literature.2
The government spending multiplier is then given by:
dyt
κρψσ + σ(1 − ρ)[1 − βρ − δκρ(1 + ψ)]
=
dgt κρψ[σ + ν(1 − g)] + σ(1 − ρ)[1 − βρ − δκρ(1 + ψ)]
(29)
Setting δ = 0 restores the exact government spending multiplier of Dupor and Li (2015).
The spending multiplier has a value of 1 when the central bank is neutral (ψ = 0) and less than
1 for an active monetary policy (ψ > 0). The multiplier is larger than 1 if monetary policy
is passive (ψ < 0) provided that the denominator is positive, which happens for most values
of ψ that are between (-1) and zero. Clearly, ψ = −1 is the lower bound on this parameter,
otherwise the nominal interest rate will decline following a rise in inflation expectations,
which is implausible. These observations hold whether or not the cost channel exists.
The intuition behind these findings is as follows: following a rise government spending,
an active central bank raises the nominal interest rate by more than one-for-one in response
to expected inflation, thus increasing the expected real interest rate and reducing consumer
̂t ) declines. A passive central bank increases the nominal
spending. In particular (Et ĉ
t+1 − c
interest rate by less than one-for-one, thus leading to a rise in consumer spending. And, a
neutral central bank does not influence the expected real interest rate, thus leading to no
2
As noted by Dupor and Li (2015), with ψ < 0, the equilibrium is not unique (as the coefficient of inflation
in the interest-rate rule is ρ(1 + ψ) < 1). In this case, the minimum-state-variable (MSV) equilibrium, which
has been frequently used in rational expectations models since McCallum (1983), is analyzed. The MSV
limits solutions to those that are linear functions of the minimum set of “state variables”, which are either
predetermined or exogenous determinants of current endogenous variables. McCallum (1999) argues that
the MSV approach generally identifies a single solution that can “reasonably be interpreted as the unique
solution that is free of bubble components”.
18
change in consumer spending: the left-hand side of equation (24) does not change, and the
right-hand side does not change either.
The implications of the cost channel for the size of the multiplier can be seen in condition
(29). With ψ < 0, both the numerator and denominator are larger than in the model without
the cost channel. But since κρψσ < κρψ[σ + ν(1 − g)], the numerator rises by more than the
denominator when the cost channel is introduced, leading to a rise in the multiplier. With
ψ > 0, the opposite happens, and the multiplier is lower. And, with ψ = 0, the cost channel
is immaterial for the size of the multiplier. In this respect, this setup allows for a clearer
illustration of the role of the cost channel in determining the size of the multiplier.
(a)
Passive
(b)
Active
0.75
4.2
4
3.8
0.7
3.4
Multiplier
Multiplier
3.6
3.2
0.65
0.6
3
2.8
0.55
2.6
2.4
2.2
0
1
2
3

4
5
0.5
6
0
1
2
3

4
5
6
Figure 5: The government spending multiplier and the cost channel: active vs. passive monetary policy.
The parameterization is based on Dupor and Li (2015) to allow for good comparisons
with the latter study. I set β = 0.995, σ = 1, ν = 4, ρ = 0.75, κ = 0.025 and g = 0.2, ψ = −0.5
for a passive central bank and ψ = 0.5 for an active central bank. The starting point of each
graph replicates exactly the multipliers of Dupor and Li (2015): a multiplier of 2.25 with a
passive central bank and a multiplier of 0.71 with an active central bank. As expected, for a
passive central bank, the cost channel strengthens the impact of government spending shocks
on consumption and increases the multiplier. The reason for this finding is the following:
the cost channel increases the expected inflation rate beyond the corresponding increase in
the expected inflation rate that would materialize when this channel is absent. But, since
the nominal interest rate rises by less one-for-one, the actual decline in the expected real
interest rate is larger than when the cost channel is not operative, which in turn leads to a
higher government spending multiplier.
19
The opposite holds for the active central bank. The decline in the expected real interest
rate with the cost channel is higher than otherwise, which makes the cost-channel multiplier
lower. Moreover, the impact of the cost channel on the size of the multiplier is more significant
for a passive policy that for an active policy. This observation may reflect the fact that the
government spending multiplier with the active central bank is already (relatively) low and
the cost channel has a relatively less ability to lower it even further.
The result about the active central bank is conceptually in line with the benchmark
analyses (section 4). In the benchmark analyses, the central bank increases the nominal
interest rate by more than one-for-one with the increase in the inflation rate. In particular,
φπ = 1.5, which implies that the response to the expected inflation rate is φρπ = 1.875. In
both cases, the rise in the expected real interest rate with the cost channel is larger than
otherwise, and the spending multiplier is lower accordingly.
The gaps between the multipliers with the and without the cost channel are larger for
higher values of ψ for an active central bank and for more negative values of ψ for a passive
central bank. This is particularly true for the passive central bank; for example, with
ψ = −0.7, the multiplier without the cost channel is roughly 6 and the multiplier with δ = 3
is above 11. Furthermore, the effects are stronger for more flexible prices. These results can
be provided upon request.
More generally, the analyses of this section support the findings of the benchmark model
by experiementing on the model’s specification and the interest-rate rule. The cost channel
has implications for the strength of the expected inflation channel, and accounting for it
provides a more accurate estimation of the strength of the expected inflation channel and
its implications for the government spending multiplier.
6. The Constant Interest-Rate Government Spending Multiplier
So far, the analyses assumed that the nominal interest rate is free to adjust and that it is
not at the zero lower bound. In what follows, I relax this assumption by letting the nominal
(policy) interest rate be fixed. More specifically, it is assumed that the ZLB constraint on the
nominal policy interest rate binds. As discussed in the introduction, it is well known by now
that this class of models predicts a large spending multiplier when conventional monetary
policy becomes powerless.
Another assumption to be relaxed in this section is that the loan interest rate and the
policy interest rate are exactly the same. If they were equal, then the cost channel would not
affect the size of the multiplier at the ZLB, and the results of this analysis would converge
to those of Christiano et al. (2011). Furthermore, if the loan rate equals the policy rate,
then the loan rate is zero as well, which goes against the evidence that I will present in this
section.
The first subsection presents the fiscal multiplier in the model with a constant policy
interest rate. The second subsection shows evidence regarding the size of the corporate
credit spread and the effects of government spending multiplier on the this spread. The
20
third subsection then reports the numerical findings. I close by a short description of the
results with the separable utility function.
6.1. Analytical Analyses
To account for possible deviations between the nominal loan and policy interest rates, I
assume that there is a spread (ηt ) between these two rates. As discussed in the introduction,
this could result from market power in the intermediation sector and costs that are associated
with changes in the scale of operations of intermediaries. Expanding the size of deposits or
loans may require further hiring and training (e.g. of loan officers) by the bank, increasing
advertisements and possibly constructing new branches to cope with the rise in the scale of
operation. This assumption follows multiple existing studies (e.g. Klein, 1971; Monti, 1972;
Elyasiani et al. (1995) and Freixas and Rochet, 1997 and Kopecky and VanHoose, 2012 ). I,
therefore, assume that:
RL,t = Rt + ηt
(30)
which, as shown in Appendix A.3, can be derived from the solution to the problem of
an intermediary that has market power in the loan market and/or faces cost functions as
outlined above. In reality, however, deviations between the loan rate and the policy rate may
arise because of other reasons and we shall assume that ηt captures all possible factors. In
the context of the cost channel, a similar form of condition (30) appears also in Chowdhury
et al. (2006).
̂t = 0), condition (30) can be rewritten in log deviations from
At a constant policy rate (R
the steady state as:
η
̂
R
η̂t
(31)
L,t =
R+η
and, since the interest rate that is relevant for consumers is Rt , condition (2) remains unaltered. On the other hand, the Phillips Curve is now given by:
π̂t = β Et π̂
ct +
t+1 + κ (̂
n
̂
n̂t + δ R
L,t )
1−n
(32)
The response of the credit spread to changes in government spending is governed by:
η̂t = Z ĝt
(33)
Since we have no endogenous state variables and ĝt = 0 outside the ZLB state, all endogenous control variables jump on impact, but they revert back to their steady state values
when the shock expires (i.e. when the nominal interest rate is no longer at the ZLB). For
̂t and Et ĉ
this reason, following the literature, we can write Et π̂
ct , where p is
t+1 = pπ
t+1 = p̂
the probability that the ZLB remains in effect next period, and thus the expected duration
1
of the ZLB is T = 1−p
. The solution then yields:
A=
g(1 − βp)(1 − p)[γ(σ − 1) + 1] − gpκ + (1 − g)pκδλZ
1
n
(1 − βp)(1 − p) − pκ(1 − g) [ 1−g
+ 1−n
]
21
(34)
η
A
t
and the government spending multiplier is given by dy
dgt = g , with λ = R+η . For δ = 0 or
when the borrowing costs of firms do not respond to government spending shocks (Z = 0),
the constant interest-rate government spending multiplier restores that of Christiano et al.
(2011). The cost channel alters the size of the multiplier, but the exact effect depends on the
response of the spread to government spending shocks and the strength of the cost channel.
Crucially, it also depends on the size of the denominator (which will be referred to as ∆).
With a positive ∆, if the rise in government spending raises firm’s borrowing costs, then
the cost channel actually strengthens the multiplier. The intuition behind this result is the
following: the rise in government spending raises the inflation rate and inflation expectations.
If the borrowing costs of firms rise in response to the expansionary fiscal policy, then the
marginal cost will increase. The overall result will be a bigger rise in inflation and inflation
expectations, which in turn implies a stronger expected inflation channel. As a result, the
spending multiplier will be larger than in the model without the cost channel. The opposite
holds if, with a constant nominal policy interest rate, the borrowing costs of firms actually
decline following a rise in government spending. With a negative ∆, the opposite occurs; if
firms’ borrowing costs rise following an expansionary fiscal policy, then the multiplier will
be lower than in an otherwise setup without the cost channel.
If we restrict the analyses to the case for which government spending leads to a rise in
the credit spreads, then the effects of the cost channel on the multiplier will crucially depend
on the sign of ∆. The analyses will be conducted under the standard assumption of ∆ > 0
that was maintained in Christiano et al. (2011) and Carlstrom et al. (2014), among others.
The size of ∆, in turn, depends on the probability that the policy interest rate will remain
constant (p), and thus the duration of the interest rate peg (T ). Therefore, this parameter
is set so that ∆ remains positive, in line with these studies.
For comparison, the government spending multiplier of normal times when the model
allows for deviations between the loan rate and the policy rate is given by:
dyt κ(φπ − ρ) + (1 − ρ)(1 − βρ − δωκφπ )[γ(σ − 1) + 1] − (1 − g)(φπ − ρ)κδλZ
=
dgt (1 − βρ − δωκφπ )[1 + (1 − g)φy − ρ] + (1 − g)(φπ − ρ)κ [ 1 + n + δωφy ]
1−g
1−n
(35)
R
with ω = R+η
. The last term in the numerator clearly shows that a rise in the spread (i.e.
Z > 0) following government spending shocks leads to a decline in the government spending
multiplier. This effect is magnified by the strength of the cost channel as captured by δ.
6.2. Empirical Evidence
This subsection studies the effects of government spending shocks on the credit spread and
the estimation of Z. Two quarterly series for government spending are considered: First, the
“Real Government Consumption Expenditures and Gross Investment”.3 Second, the “Gross
3
Ramey and Zubairy (2015) use nominal Government Consumption Expenditures and Gross Investment
deflated by the GDP deflator.
22
Domestic Product by Expenditure in Constant Prices: Government Final Consumption
Expenditure for the United States”, which does not account for government investment.
The credit spread is measured as the difference between Moody’s Seasoned Baa Corporate
Bond Yield and a 3-Month Treasury Bill Rate. All data are available in the FRED database
of the Federal Reserve Bank of St. Louis.
Figure B.3 shows the credit spread for the period 1960:Q4-2014:Q3, for which data have
been available at the time of writing this version. The key observation for our analyses is
that, during the ZLB episode in the U.S., the spread has been clearly positive and volatile,
and it remained so even after the end of the Great Recession. Indeed, the average spread has
been 5.7%, which is almost twice the average spread of the pre-Great Recession period (see
Table B.3 for more details). Interestingly, this has also been a period with higher government spending, which raises interest in formally exploring the linkages between government
spending and the credit spread, to which I turn next.
To study the effects of government spending on the spread, I proceed in two ways. First,
I estimate a Victor Auto-Regression (VAR) that includes the credit spread, output gap,
tax receipts-GDP ratio and the 10-year minus 3-month U.S. government bond spread. The
results of the VAR, which are summarized in Figure B.4, indicate that the spread positively
responds to government spending shocks and that it is (very) countercyclical, which is in
line with expectations. The other two variables mostly do not have significant effects on the
credit spread at the 95% confidence level.
Description
Govt. spending: Series 1
J-Test (p-value)
Govt. spending: Series 2
J-Test (p-value)
1960:Q1-2007:Q3
1984:Q1-2007:Q3
0.724
(0.424, 1.024)
0.696
(0.379, 1.014)
0.130
0.482
0.902
(0.531, 1.272)
0.741
(0.376, 1.105)
0.138
0.474
191
95
Obs.
Table 3: Empirical results- GMM estimation of condition (33). The list of instruments includes four lags of
the output gap, 10-year minus 3-month U.S. government bond spread, defense spending and government tax
receipts as a percentage of GDP. 95% confidence intervals in parentheses. The J-Test tests the hypothesis
that the over-identifying restrictions are satisfied. Series 1: Real Government Consumption Expenditures
and Gross Investment. Series 2: Gross Domestic Product by Expenditure in Constant Prices: Government
Final Consumption Expenditures. Defense spending: Federal defense consumption expenditures.
I next estimate the value of Z using GMM by running the credit spread on government
spending and the output gap (to control for the cyclical behavior of the spread). The list of
instruments and the estimation results are summarized underneath Table 3. The elasticity
of the credit spread with respect to government spending is positive and mostly around 1.
23
Indeed, the hypothesis that the coefficient is equal to one cannot be rejected for the full
sample or the sample that starts in 1984. I thus set Z = 1 in the benchmark analyses and
then report the results using another value of this parameter.
6.3. Results
Figure 6 illustrates how the cost channel alters the results regarding the cost channel at
the ZLB. In this figure, I show the results using both my benchmark value of κ and using
that of Christiano et al. (2011) for two reasons: first, as is well known, small changes in κ
can make significant differences regarding the size of the multiplier. Second, to enable close
comparisons with the results of the aforementioned study. The analysis is conducted using
p = 0.80, in line with Christiano et al. (2011), which implies a duration of fixed interest
rate of 5 quarters. Under this parameterization, ∆ is positive. I set η = 0.057 to match the
average credit spread over the period 2008:Q4-2014:Q3. This value implies λ = 0.054δ. The
rest of the parameters remain as in Table 2.
The multiplier is increasing linearly in the size of δ (which can be easily seen in condition
(34)), and the effects can be very large. With the benchmark value of κ, the spending
multiplier jumps from 8 to nearly 19 in the range of plausible values of δ. Even a small value
of δ = 1 makes a significant difference as the multiplier increases from 8 to roughly 10.
20
8
7.5
18
7
6.5
Multiplier
Multiplier
16
14
12
6
5.5
5
4.5
10
4
8
0
1
2
3

4
5
3.5
6
0
1
2
3

4
5
6
Figure 6: The government spending multiplier for various values of δ. Left panel: benchmark calibration
(κ = 0.034). Right panel: Christiano et al. (2011) calibration (κ = 0.030).
With the value of κ of Christiano et al. (2011), the multiplier starts from a value of 3.7,
which is exactly the value in their study, and then more than doubles at δ = 6. Therefore,
while there are differences in terms of the size of the multiplier for both values of κ, the effects
of the cost channel are quite clear and they support the expectations: the cost channel plays
24
a significant role in strengthening the “inflation channel” and renders the spending multiplier
considerably larger during periods of liquidity trap. Furthermore, the comparison between
the two panels reveals, as in the model with variable interest rates, that the degree of price
rigidity matters a lot for the size of the multiplier and that it becomes more important when
it is coupled with the cost channel.
When the nominal policy interest rate is fixed, the expected real interest rate can by
expressed as (see Appendix A.4 for details):
Et r̂
gt ,
t+1 = −pQ̂
(36)
which indicates that the decline in the expected real interest rate following a government
spending shock is bigger with a stronger response of the inflation rate to this shock (e.g.
with a higher Q). With the cost channel, the response of inflation is stronger and the
corresponding decline in the expected real interest rate is bigger, leading to a larger multiplier
in the presence of this channel than otherwise. This is numerically illustrated in Figure B.5
where the response of the expected real interest rate is reported for the plausible range of δ
(and holding p fixed).
This figure also illustrates the importance of price rigidity: with more flexible prices, the
rise in the inflation rate (hence, the expected inflation rate) following government spending
shocks is bigger and the decline in the expected real interest rate is bigger. In turn, this
explains the differences between the fiscal multiplier under the benchmark calibration of this
paper and the calibration of Christiano et al. (2011). Similar arguments can be made about
the actual real interest rate as can be seen in this Appendix. In fact, the actual real interest
rate in this case is only a scale up of the expected real interest rate.
Furthermore, Figure B.6 shows that the size of Z matters markedly for the spending
multiplier. I show the results for a range of Zs between 1 and 2 since that is the range in
which most point estimates lie and to be conservative regarding the value of this parameter.
Consistent with condition (34), the multiplier rises linearly in Z. The cost channel makes
a clear difference even for relatively small values of both Z and δ, and it becomes more
significant for higher values of both parameters.
The figure also presents the corresponding spending multipliers of normal times that is
implied by condition (35). In this case, the government spending multiplier with positve
values of Z is lower for any δ, suggesting that if, in addition to a rise in the nominal policy
interest rate, the spread rises, then the spending multiplier will be even lower. Notice,
however, that the effects in this case are smaller. The main reason for this observation is
that the rise in the policy interest rate in itself leads to a rise in firms’ borrowing costs even
if the spread remains the same and, therefore, most of the rise in the costs of borrowing of
firms will be already factorized in the rise in the policy interest rate.
Clearly, the differences between the fiscal multipliers of fixed and non-fixed nominal policy
interest rates can be very large and they increase considerably as the cost channel becomes
stronger. These differences can also be well seen in the impulse responses of output, con25
sumption and the inflation rate to government spending shocks (Figure B.7). For example,
the cost channel makes the decline in consumption outside the ZLB bigger, but it leads to a
larger increase in consumption at the ZLB. The spending multiplier depends on the response
of consumption to the government spending shock, which in turn depends on the response
of inflation along the lines that have been outlined above.
The differences between the government spending multipliers of liquidity trap and normal
times are underestimated when the cost channel is not accounted for. Therefore, while
these findings support the fact that the government spending multiplier is larger during
liquidity trap times, they intensify the debate between the empirical findings about the size
of the multiplier (e.g. Ramey and Zubairy, 2015) and the results that are obtained from
quantitative models, particularly since the work of Christiano et al. (2011).
6.4. Separable Utility Function
With separable preferences, as in (23), we have:
A=
gσ[(1 − βp)(1 − p) − pκ] + (1 − g)pκδλZ
σ(1 − βp)(1 − p) − pκ[σ + ν(1 − g)]
(37)
A
t
and the multiplier will be dy
dgt = g . With δ = 0 or Z = 0, the constant interest-rate government
spending multiplier restores that of Carlstrom et al. (2014). Overall, the effects of the cost
channel on the government spending multiplier are as in the benchmark model. In particular,
under the assumption ∆ > 0, the cost channel leads to a rise in the spending multiplier and
the effect is bigger for higher values of κ.
2.3
4
2.2
3.5
2.1
3
Multiplier
Multiplier
2
2.5
1.9
1.8
1.7
1.6
1.5
2
1.4
1.5
0
1
2
3

4
5
1.3
6
0
1
2
3

4
5
6
Figure 7: The government spending multiplier for various values of δ. Left panel: benchmark calibration
(κ = 0.034). Right panel: Carlstrom et al. (2014) calibration (κ = 0.028).
26
The analyses with the parameterization of Carlstrom et al. (2014) starts from their benchmark parameterization of T = 5 and κ = 0.028, which delivers a multiplier of 1.3 (Figure 7).
With the cost channel, the multiplier can reach 2.3. The multiplier rises dramatically under
a bigger T and/or a bigger κ. For example, by only increasing T to 6, the multiplier jumps
from 5 with no cost channel to 7.16 with δ = 1 and 17.94 with δ = 6. The sensitivity of
the multiplier to changes in κ and T have been discussed in Carlstrom et al. (2014) and
further details can be found in that study. The key insight of the analyses of this section
is that, holding κ and T fixed, the cost channel matters a great deal for the size of the
multiplier during periods of constant nominal interest rates. Models that abstract from the
cost channel underestimate the size of the multiplier during this type of periods.
Also, since with a constant nominal policy interest rate the central bank is passive (in
̂t = 0), the only source of variations in firms’ borrowing costs is through
the sense that R
variations in the credit spread ηt . And since the model that has been used by Dupor and Li
(2015) is exactly as in Carlstrom et al. (2014), the government spending multiplier for the
constant interest rate policy for Dupor and Li (2015) is exactly as in condition (37). The
only differences will be numerical that result from differences in the parameterization.
The separable utility function provides a good case to compare the spending multiplier of
normal times with the liquidity-trap multiplier. Using the Euler condition (24), the response
of consumption to government spending shocks in normal times can be written as:
ĉt
(φπ − ρ)
=−
Q
ĝt
σ(1 − ρ)
(38)
which, given that Q > 0 and φπ > ρ, implies that consumption will decline in response to
a government spending shock. This corresponds to the standard crowding out result of
consumption, and it explains why the spending multiplier is less than one with separable
preferences. The cost channel makes Q larger, thus amplifying the (negative) response of
consumption to government spending shocks, leading to even a lower multiplier than in a
model that does not account for this channel.
When the nominal interest rate is fixed, the response of consumption to government
sending shocks reads:
ĉt
p
=
Q
(39)
ĝt σ(1 − p)
which is positive. In this case, the cost channel leads to a higher multiplier as Q is larger.
This case, thus, illustrates how does the cost channel generates a bigger gap between the
multipliers of normal times and periods of liquidity trap.
Appendix A.5 presents an alternative model with the following features. First, the deposit
interest rate is not necessarily equal to the policy rate. Second, in line with the analyses of
this section, the loan rate is different from the policy rate. Third, government spending is
financed through taxation and issuing bonds. The key conditions that yield the government
spending multiplier do not change, and therefore, the multipliers in this section continue to
be given by conditions (34) and (37).
27
Finally, Figure B.8 presents an alternative calculation of the government spending multiplier. Following the literature (e.g. Mountford and Uhlig, 2009 and Zubairy, 2014), we show
the “impact spending multiplier”, which is the change in output k periods ahead (∆yt+k )
in response to a change in government spending at the current period (∆gt ). Formally, the
impact multiplier is given by:
∆yt+k
Impact Multipliert,t+k =
(40)
∆gt
and it is calculated based on the impulse responses that are shown in Figure B.7. The impact
multiplier is highest on impact, but it declines over time and reaches zero after nearly 5 years.
When the nominal interest rate is free to adjust following spending shocks (“normal times”),
the cost channel reduces the size of the spending multiplier on impact, but the differences
shrink as the economy reverts back to the initial steady state. On the other hand, the
cost channel markedly raises the impact multiplier of liquidity-trap periods, and significant
differences persist for nearly four years. These observations confirm the main findings of the
paper and suggest that the exact measurement of the spending multiplier does not affect the
conclusions regarding the implications of the cost channel for the effectiveness of government
spending.
7. Conclusions
The government spending multiplier in a model with the cost channel of the nominal interest
rate is studied in this paper. It is found that the cost channel matters a great deal for the
size of the spending multiplier. In normal times, where the nominal interest rate is free to
adjust, the cost channel reduces the multiplier considerably as it leads to a bigger rise in
the expected real interest rate following a surge in government spending, which crowds out
consumption. When the nominal interest rate is fixed, as may happen with this interest
rate being at the zero lower bound, the cost channel leads to a bigger drop in the expected
real interest rate and, consequently, to a larger government spending multiplier. Therefore,
by ignoring the cost channel, the spending multiplier is overestimated in normal times and
underestimated in periods with fixed interest rates. In this respect, the cost channel provides
even a better case for raising government spending at the zero lower bound. On the other
hand, because ZLB episodes are uncommon, the spending multiplier with the cost channel
will mostly be lower than in previous estimates.
The paper also provides a more updated evidence about the existence of the cost channel
and the response of corporate credit spreads to government spending shocks, suggesting that
firms’ borrowing costs are not fixed during zero lower-bound episodes. In turn, this allows
the cost channel to matter even when the policy nominal interest rate is fixed. Furthermore,
it is shown that since 1984, the cost channel is having a bigger impact on the dynamics of
inflation than it previously had.
This study adds to the voluminous literature on the government spending multiplier and
to the literature on the cost channel. By so doing, the paper provides first analysis regarding
28
the implications of the cost channel for the effectiveness of fiscal policy as previous literature
on the cost channel naturally focused on monetary policy and the dynamics of inflation.
The recent debates about the 2009 American Recovery and Reinvestment Act as well as
the size of the government spending multiplier, particulary at the zero lower bound, make
this topic very important to investigate. By finding that the fiscal multiplier is even larger
at the zero lower bound, the paper adds to the debate between empirical and model-based
quantitative studies regarding the size of the fiscal multiplier during periods of liquidity trap.
The analyses can be enriched by allowing for unconventional monetary policy to be operative
in addition to the cost channel, but this is left for a future work.
References
Auerbach, A. J. and Y. Gorodnichenko (2012): “Measuring the Output Responses to Fiscal Policy,”
American Economic Journal: Economic Policy, 4, 1–27.
Bils, M. and P. J. Klenow (2004): “Some Evidence on the Importance of Sticky Prices.” Journal of
Political Economy, 112, 947–985.
Bouakez, H., M. Guillard, and J. Roulleau-Pasdeloup (2015): “Public Investment, Time to Build,
and the Zero Lower Bound,” Mimeo.
Caggiano, G., E. Castelnuovo, V. Colombo, and G. Nodari (2015): “Estimating Fiscal Multipliers:
News From A Non-linear World,” Economic Journal, 125, 746–776.
Calvo, G. A. (1983): “Staggered Prices in a Utility-Maximizing Framework,” Journal of Monetary Economics, 12, 383–398.
Carlstrom, C. T., T. S. Fuerst, and M. Paustian (2014): “Fiscal Multipliers under an Interest Rate
Peg of Deterministic versus Stochastic Duration,” Journal of Money, Credit and Banking, 46, 37–70.
Carrillo, J. A. and C. Poilly (2013): “How Do Financial Frictions Affect the Spending Multiplier
During a Liquidity Trap?” Review of Economic Dynamics, 16, 296–311.
Chowdhury, I., M. Hoffmann, and A. Schabert (2006): “Inflation Dynamics and the Cost Channel
of Monetary Transmission,” European Economic Review, 50, 995–1016.
Christiano, L. J., M. Eichenbaum, and C. Evans (2005): “Nominal Rigidities and the Dynamic Effects
of Shock to Monetary Policy,” Journal of Political Economy, 113, 1–45.
Christiano, L. J., M. Eichenbaum, and S. Rebelo (2011): “When Is the Government Spending
Multiplier Large?” Journal of Political Economy, 119, 78–121.
Dupor, B. and R. Li (2015): “The Expected Inflation Channel of Government Spending in the Postwar
U.S.” European Economic Review, 74, 36–56.
Elyasiani, E., K. J. Kopecky, and D. D. VanHoose (1995): “Costs of Adjustment, Portfolio Separation,
and the Dynamic Behavior of Bank Loans and Deposits,” Journal of Money, Credit and Banking, 27, 955–
974.
Freixas, X. and J.-C. Rochet (1997): “Microeconomics of Banking,” The MIT Press.
Gali, J., D. Lopez-Salido, and J. Valles (2007): “Understanding the Effects of Government Spending
on Consumption,” Journal of the European Economic Association, 5, 227–270.
Klein, M. A. (1971): “A Theory of the Banking Firm,” Journal of Money, Credit and Banking, 3, 205–218.
Kopecky, K. and D. D. VanHoose (2012): “Imperfect Competition in Bank Retail Markets, Deposit
and Loan Rate Dynamics, and Incomplete Pass Through,” Journal of Money, Credit and Banking, 44,
1185–1205.
Leeper, E. M., N. Traum, and T. B. Walker (2011): “Clearing Up the Fiscal Multiplier Morass,”
Mimeo, Indiana University.
29
McCallum, B. T. (1983): “On Non-Uniqueness in Rational Expectations Models: An Attempt at Perspective,” Journal of Monetary Economics, 11, 139–168.
——— (1999): “ORole of the Minimal State Variable Criterion in Rational Expectations Models,” International Tax and Public Finance, 6, 621–639.
Monacelli, T. and R. Perotti (2008): “Fiscal Policy, Wealth Effects, and Markups,” NBER Working
Paper No. 14584.
Monti, M. (1972): “Deposit, Credit, and Interest Rate Determination under Alternative Bank Objectives,”
In: Karl Shell and Giorgio Szego, editors, Mathematical Methods in Investment and Finance, NorthHolland, Amsterdam, 431–454.
Mountford, A. and H. Uhlig (2009): “What Are the Effects of Fiscal Policy Shocks?” Journal of
Applied Econometrics, 24, 960–992.
Nakamura, E. and J. Steinsson (2008): “Five Facts About Prices: A Reevaluation of Menu Cost
Models,” Quarterly Journal of Economics, 123, 1415–1464.
Ramey, V. A. and S. Zubairy (2015): “Government Spending Multipliers in Good Times and in Bad:
Evidence from U.S. Historical Data,” Mimeo, University of California, San Diego.
Ravenna, F. and C. Walsh (2006): “Optimal Monetary Policy with the Cost Channel,” Journal of
Monetary Economics, 53, 199–216.
Smets, F. and R. Wouters (2007): “Shocks and Frictions in US Business Cycles: A Bayesian DSGE
Approach,” American Economic Review, 97, 586–606.
Tillmann, P. (2008): “Do Interest Rates Drive Inflation Dynamics? An Analysis of the Cost Channel of
Monetary Transmission,” Journal of Economic Dynamics and Control, 32, 2723–2744.
Zubairy, S. (2014): “On Fiscal Multipliers: Estimates from a Medium Scale DSGE Model,” International
Economic Review, 55, 169–195.
30
A. Mathematical Appendix
I start by presenting the households’ problem and then move to presenting the expected real
interest rate and the response of inflation to spending shocks with both fixed and non fixed
nominal policy interest rates. I also show the problem of the intermediation sector and an
alternative approach to introducing the cost channel in the model.
A.1. The Households’ Problem
The problem of the households is to
∞
[cγt (1 − nt )1−γ ]1−σ − 1
max
)
E0 ∑ β (
{ct ,Dt ,Mt ,nt }∞
1−σ
t=0
t=0
t
(A.1)
subject to the sequence of budget constraints
Pt ct + Dt + Mt+1 = Mt + Rt Dt + Wt nt + Tt + Πt
(A.2)
and the cash-in-advance (CIA) constraint:
Pt ct = Mt + Wt nt − Dt
(A.3)
where Wt denotes the nominal wage, Pt is the price level, Rt is the gross nominal policy
(or market) interest rate, Tt are nominal net transfers and Πt are nominal profits from the
ownership of firms and the financial intermediary.
Denote the Lagrange multipliers on conditions (A.2) and (A.3) by λt and µt , respectively.
Then, the first-order conditions with respect to ct , Dt , Mt and nt are, respectively, given by:
1−γ
[cγt (1 − nt )1−γ ]−σ γcγ−1
= Pt (λt + µt )
t (1 − nt )
(A.4)
λt Rt = λt + µt
(A.5)
λt = βEt (λt+1 + µt+1 )
(A.6)
[cγt (1 − nt )1−γ ]−σ (1 − γ)cγt (1 − nt )−γ = Wt (λt + µt )
(A.7)
Combining conditions (A.4)-(A.6) gives the Euler equation:
[cγt (1 − nt )1−γ ]−σ γctγ−1 (1 − nt )1−γ = βRt Et (
1−γ
[cγt+1 (1 − nt+1 )1−γ ]−σ γcγ−1
t+1 (1 − nt+1 )
)
πt+1
(A.8)
Similarly, combining (A.4) and (A.7) gives the labor supply condition:
(1 − γ)ct
= wt
γ(1 − nt )
(A.9)
Log Linearization of conditions (A.8) and (A.9) around the deterministic steady state
yields conditions (2) and (3) in the text, respectively.
31
A.2. Variable Nominal Policy Interest Rate
This appendix outlines the effects of the cost channel on the response of the expected real
interest rate and, consequently, the spending multiplier. First, let the response of inflation
to government spending shocks be governed by:
π̂t = Q̂
gt
(A.10)
where:
Q=
1
n
(1 − ρ)gκ[γ(σ − 1) + 1](1 − g) [ 1−g
+ 1−n
+ δφy ] − κg[1 − ρ + (1 − g)φy ]
1
n
(1 − g)(φπ − ρ)κ(1 − g) [ 1−g
+ 1−n
+ δφy ] + (1 − g)(1 − βρ − δκφπ )[1 − ρ + (1 − g)φy ]
(A.11)
which can be derived in a similar manner to the derivation of A.
The expected inflation rate then reads:
̂
Et π̂
t+1 = Q Et g
t+1
(A.12)
or, using the AR(1) process of government spending:
gt
Et π̂
t+1 = ρQ̂
(A.13)
With no response to output in the interest-rate rule, which is the assumption in Christiano
et al. (2011) among others, we have:
̂t = φπ π̂t
(A.14)
R
And the expected real interest rate:
̂
̂
Et r̂
t+1
t+1 = Rt − Et π
(A.15)
which, using, (A.10), (A.13)-(A.14) gives condition (21) in the text:
Et r̂
gt ,
t+1 = (φπ − ρ)Q̂
(A.16)
The actual real interest rate is given by:
̂t − π̂t
r̂t = R
(A.17)
r̂t = (φπ − 1)Q̂
gt
(A.18)
or, using conditions (A.10) and (A.14):
Therefore, to the extent that φπ > 1 and Q > 0, the actual real interest rate will rise
following government spending shocks.
32
̂t = φπ π̂t + φy ŷt , the corresponding expected real interest rate reads:
With R
̂t
Et r̂
t+1 = [(φπ − ρ)Q + φy A] g
(A.19)
which is condition (22) in the text.
A.3. The Intermediation Sector
There is a continuum of measure 1 of identical intermediaries (banks) in the (0,1) interval.
Let Di,t and Li,t be the time-t amounts of deposits in bank i and the amount of loans
extended by this bank, respectively. Let also D−i,t and L−i,t be the amounts of deposits
and loans of all other banks. The total amount of deposits and loans are then given by:
Dt = Di,t + D−i,t and Lt = Li,t + L−i,t .
Banks operate in an environment with Cournot competition with each bank i taking
D−i,t and L−i,t as given. The deposit rate RD,t and the loan rate RL,t are determined by
the total supply of deposits and the total demand for loans. Formally, RD,t = RD (Dt ) and
RL,t = RL (Lt ), which are the inverse demand for deposits and supply of loans, respectively.
The deposit interest rate is increasing in the total amount of deposits and the loan interest
∂R
∂R
> 0 and ∂LL,t
< 0.
rate is decreasing in the total amount of loans: ∂DD,t
t
t
The problem of bank i is to choose Di,t and Li,t to maximize its expected present discounted stream of real profits. The period nominal profit function of bank i is given by:
Πbi,t = RL,t Li,t − RD,t Di,t − Rt Hi,t − Γi,t
(A.20)
with Γi,t = Γ(Di,t , Di,t−1 , Li,t , Li,t−1 ) being the adjustment cost of deposits and loans and Hi,t
being the nominal balance (position) of bank i in the interbank market. The latter can be
either positive (if bank i is a net lender), negative (if the bank is a net borrower) or zero.
The nominal policy interest Rt is the interest rate at which interbank lending takes place,
which similar to the Federal Funds Rate in the U.S. Furthermore, the adjustment cost is
∂Γ
∂Γ
> 0 and ∂Li,t
> 0.
increasing in deposits and loans: ∂Di,t
i,t
i,t
Profit maximization by bank i is subject to its balance sheet:
Li,t = Di,t + Hi,t
(A.21)
where, for simplicity, I abstract from reserve requirements.
Substituting condition (A.21) into the present discounted stream of profits and taking
first-order conditions with respect to Di,t and Li,t , respectively, yield:
RD,t = Rt −
∂RD,t
∂Γi,t
Di,t −
∂Dt
∂Di,t
(A.22)
RL,t = Rt −
∂RL,t
∂Γi,t
Li,t +
.
∂Lt
∂Li,t
(A.23)
33
Condition (A.22) suggests that the deposit interest rate RD,t is lower than the market
interest rate Rt . The gap between both interest rates is due to the market power of banks (as
∂Γ
∂RD
captured by ∂Dtt ) and the adjustment cost of deposits (which is captured by ∂Di,t
). On the
i,t
other hand, the loan interest rate is higher than the market interest rate due to the market
power of banks and the adjustment cost of loans. As a result, we have RL,t > Rt > RD,t ,
suggesting that banks generate profits by paying a below-market interest rate on deposits
and charging an above-market interest rate on the loans that they extend.
Focusing on the loan market, we get condition (30) in the text from condition (A.24):
RL,t = Rt + ηt
(A.24)
where ηt measure the spread, driven in this analyses by market power and the adjustment
costs. In the real world, of course, this spread may result from other factors that are not
accounted for in this illustrative case. The key goal of this analysis is to demonstrate how
the loan market can differ from the policy interest rate, implying that even at the ZLB, the
loan rate is not zero (or not constant).
Since all banks are symmetric in equilibrium, the interbank lending of each bank is zero
and, therefore, Ht = 0. In addition, loans equal the part of the wage bill that is paid in
advance. Therefore, in aggregate, Lt = Wt nt . Substituting these two results in the real
version of condition (A.21) gives:
dt = wt nt
(A.25)
which is the clearing condition of the intermediation sector.
Furthermore, in the current setup, we have the special case of the deposit rate being
equal to the policy rate. Therefore:
RD,t = Rt
(A.26)
This result does not change the key finding of the paper regarding the role of the cost
channel in affecting the size of the multiplier as condition (A.24) involves only the spread
between the loan rate and the policy rate, ηt .
A.4. Constant Nominal Policy Interest Rate
When the nominal interest rate is fixed, the expected inflation rate is given by:
̂t
Et π̂
t+1 = pπ
(A.27)
Et π̂
gt
t+1 = pQ̂
(A.28)
or:
with:
Q=
κg
n
1
κg [ 1−g
+ 1−n
] [γ(σ − 1) + 1] − 1−g
+ κδλZ
1
n
1 − βp − κp(1−g)
1−p [ 1−g + 1−n ]
34
(A.29)
which is clearly larger in the presence of the cost channel.
̂t = 0, the expected real interest rate will be given by:
Moreover, since R
Et r̂
gt
t+1 = −pQ̂
which is condition (36) in the text. Similarly, the real interest rate is given by:
(A.30)
̂t − π̂t
r̂t = R
(A.31)
r̂t = −Q̂
gt .
(A.32)
or:
and, thus, the behavior of the actual real interest rate is similar to the behavior of the
expected real interest rate, but with differences in the magnitudes because p < 1.
A.5. Alternative Model
In this appendix, I consider an alternative model that yields the same conditions. Households
have access to government bonds (denoted by Bt ) in addition to money and deposits, supply
labor and consume. In addition, households derive utility from money, which is in the form
of cash and deposits. The problem of the representative household is to:
∞
max
{ct ,Bt ,Dt ,Mt ,nt }∞
t=0
E0 ∑ β t u (ct , nt ,
t=0
Xt
)
Pt
(A.33)
where Xt is a money composite that consists of cash and deposits in order to capture the
∂u
t
utility from holding money in both forms. The utility function u(ct , nt , X
Pt ) satisfies: ∂c > 0,
Xt
∂u
∂2u
∂u
∂2u
∂2u
∂c2 < 0, ∂x > 0, ∂x2 < 0, ∂n < 0 and ∂n2 < 0, where xt = Pt stands for real money balances.
∂xt
t
Also, xt is increasing in real cash and deposits: ∂m
> 0 and ∂x
∂dt > 0.
t
The sequence of households’ budget constraints is given by:
Pt ct + Mt + Bt + Dt = Mt−1 + RD,t−1 Dt−1 + Rt−1 Bt−1 + Wt nt + Tt + Πt
(A.34)
where Rt is the gross nominal policy (or market) interest rate and RD,t is the gross nominal
interest rate on deposits. All other variables are defined as before.
Optimization by households leads to the following first-order conditions:
uc,t = βRt Et (
−
un,t
= wt
uc,t
uc,t+1
)
πt+1
(A.35)
(A.36)
uc,t+1
∂xt
) + ux,t
πt+1
∂dt
uc,t+1
∂xt
uc,t = β Et (
) + ux,t
πt+1
∂mt
uc,t = βRD,t Et (
(A.37)
(A.38)
where uk,t denotes the derivative of the utility function with respect to argument k
35
(k =1,2,3). Equations (A.35) and (A.36) are the standard consumption Euler equation and
the labor supply condition, respectively. Condition (A.37) governs the choice of deposits by
households and condition (A.38) governs the choice of cash.
The conditions that matter for the spending multiplier are (A.35) and (A.36). With nonseparable preferences, we get conditions (2) and (3). Separable preferences yield conditions
(24) and (25).
The intermediation sector is described in Appendix (A.3), and it yields deviations between
the three nominal interest rates. Importantly, we have:
RL,t = Rt + ηt .
(A.39)
The production sector does not change; therefore, the Phillips Curve condition continues
to be given by:
̂
st + δ R
(A.40)
π̂t = β Et π̂
t+1 + κ (̂
L,t )
which, with no spread (ηt = 0), gives condition (11) in the text. With non-separable preferences, we get condition (16), and with separable preference we get condition (26). When ηt
is not zero, we have the same conditions but with RL,t replacing Rt .
Government expenditures are financed by lump-sum taxes and the issuance of bonds:
Pt gt + Rt−1 Bt−1 = Bt + Tt
(A.41)
The resource constraint of the economy remains unchanged:
ŷt = (1 − g)̂
ct + g ĝt
(A.42)
with g = yg is the steady-state government spending-output ratio, and government spending
evolving according to the following AR(1) process:
ĝt = ρ̂
gt−1 + ut
(A.43)
with ρ being the persistence parameter and ut ∼ N (0, σu2 ).
The Euler equation, labor supply condition, resource constraint and the condition that
governs the deviation between the nominal loan rate and the nominal policy rate are not
affected by allowing for the deposit rate to differ from the policy rate or allowing for government spending to be financed via bonds alongside taxes. Therefore, the expressions for the
multipliers are not affected either.
36
B. Tables and Figures
Description
BK Filter
With Baa
κ
0.014
(0.005, 0.024)
0.031
(0.012, 0.050)
δ
2.477
(1.936, 5.250)
1.409
(1.100, 1.482)
Price Duration
9.198
(7.168, 16.186)
6.365
(5.089, 10.172)
J-Test (p-value)
Obs.
0.335
191
0.695
191
Table B.1: Empirical results- estimation of condition (11). The list of instruments is as in Table 1. For the
specification with Baa: Moody’s Seasoned Baa Corporate Bond Yield replaces the 3-month T-bill rate. 95%
confidence intervals in parentheses. The J-Test tests the hypothesis that the over-identifying restrictions are
satisfied.
Description
With Cost Channel
No Cost Channel
κ
0.042
(0.009, 0.076)
0.046
(0.012, 0.080)
δ
1.646
(1.391, 3.819)
Price Duration
5.488
(4.221, 11.639)
5.286
(4.122, 10.070)
LR-Stat
LR-Test (p-value)
16.194
0.000
Obs.
191
191
Table B.2: Empirical results- estimation of condition (11). Results of Maximum Likelihood estimation.
95% confidence intervals in parentheses. LR-Stat: the statistic of the likelihood ratio test for rejecting the
hypothesis that the restricted model (without the cost channel) is true.
Description
Mean
Std. Dev.
Obs.
1960:Q1-2007:Q3
1960:Q1-2014:Q3
2008:Q4-2014:Q3
0.032
0.016
0.035
0.018
0.057
0.010
191
219
24
Table B.3: Descriptive statistics of the corporate credit spread, measured as the difference between Moody’s
Seasoned Baa Corporate Bond Yield and a 3-Month Treasury Bill Rate.
37
0.05
0.045
0.04
Response of Et(rt+1)
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0
1
2
3

4
5
6
Figure B.1: The response of the expected real interest rate to
government spending shocks for various values of δ.
1
0.8
Multiplier
0.6
0.4
0.2
0
-0.2
0
1
2
3

4
5
6
Figure B.2: The government spending multiplier for various values
of δ and φy = 0.5/4. All other parameters are as in Table 2.
38
.10
.08
.06
.04
.02
.00
-.02
60
65
70
75
80
85
90
95
00
05
10
Figure B.3: The corporate credit spread, measured as the difference between Moody’s Seasoned
Baa Corporate Bond Yield and a 3-Month Treasury Bill Rate. The shaded area indicates the
period 2008:Q4-2014:Q3.
Government Spending
GDP
.0030
.001
.0025
.000
.0020
.0015
-.001
.0010
-.002
.0005
.0000
-.003
-.0005
-.0010
-.004
2
4
6
8
10
12
14
16
18
20
2
4
6
8
Tax-GDP Ratio
10
12
14
16
18
20
14
16
18
20
TB3M10Y
.0012
.001
.0008
.000
.0004
.0000
-.001
-.0004
-.002
-.0008
-.003
-.0012
-.0016
-.004
2
4
6
8
10
12
14
16
18
20
2
4
6
8
10
12
Figure B.4: VAR-based impulse responses of the corporate credit spread to a positive one standard deviation. Shaded areas indicate the 95% confidence intervals. Government spending series:
Real Government Consumption Expenditures and Gross Investment. TB3M10Y: 10-year minus
3-month U.S. government bond spread.
39
-0.12
-0.2
-0.14
-0.3
-0.16
-0.18
Response of Et (rt+1)
Response of Et (rt+1)
-0.4
-0.5
-0.6
-0.2
-0.22
-0.24
-0.26
-0.7
-0.28
-0.8
-0.3
-0.9
0
1
2
3

4
5
-0.32
6
0
1
2
3

4
5
6
Figure B.5: The response of the expected real interest rate to government spending shocks for various values
̂t = 0. is Left panel: benchmark calibration (κ = 0.034). Right panel: Christiano et al. (2011)
of δ with R
calibration (κ = 0.030).
1.1
20
Z=0
Z=1
Z=2
1
18
16
Multiplier
Multiplier
0.9
0.8
14
12
0.7
10
0.6
0.5
8
0
0.5
1
1.5

2
2.5
6
3
0
0.5
1
1.5

2
2.5
Figure B.6: The government spending multiplier for various values of Z and δ. Left panel: multiplier of
normal times. Right panel: the liquidity-trap multiplier.
40
3
y

c
0.18
0
No Cost Channel
With Cost Channel
0.16
-0.01
0.14
-0.02
0.12
-0.03
0.1
-0.04
0.08
-0.05
0.06
-0.06
0.04
-0.07
0.02
-0.08
0
5
10
15
0.025
-0.09
20
0.02
0.015
0.01
0.005
5
10
y
15
0
20
5
2
15
20
15
20

c
2.5
10
3
0.7
2.5
0.6
0.5
2
1.5
0.4
1.5
0.3
1
1
0.2
0.5
0
0.5
5
10
15
0
20
0.1
5
10
15
0
20
5
10
Figure B.7: Impulse responses to a positive shock to government spending. Top panel: variable nominal
interest rate (φπ = 1.5,φy = 0.5/4.). Bottom panel: constant nominal interest rate. Percentage deviations
from the deterministic steady state. Time unit: quarters.
Normal Times
1
Liquidity Trap
14
No Cost Channel
W ith Cost Channel
0.9
12
0.8
10
0.7
0.6
8
0.5
6
0.4
0.3
4
0.2
2
0.1
0
5
10
15
0
20
5
10
15
20
Figure B.8: The impact government spending multiplier for various values of δ. Time unit: quarters.
41