Time Consistency and the Duration of Government Debt:
A Signalling Theory of Quantitative Easing
Saroj Bhattarai
University of Texas-Austin
Gauti B. Eggertsson
Brown University and NBER
Bulat Gafarovy
Penn State
Abstract
We present a signalling theory of quantitative easing in which open market operations that change
the duration of outstanding nominal government debt or the size and composition of the central bank’s
balance sheet a¤ect the incentives of the central bank in determining the real interest rate. In a time
consistent (Markov-perfect) equilibrium of a sticky-price model with coordinated monetary and …scal
policy, we show that shortening the duration of outstanding government debt provides an incentive to the
central bank to keep short-term real interest rates low in future in order to avoid higher taxes. Similarly,
in a time consistent (Markov-perfect) equilibrium of a sticky-price model with an independent central
bank, we show that increasing the size of the central bank’s balance sheet or the duration of central
bank’s asset holdings provide it with an incentive to keep short-term real interest rates low in future in
order to avoid capital losses. In both model variations, in a liquidity trap situation, where the current
short-term nominal interest rate is up against the zero lower bound, quantitative easing is e¤ective to
…ght de‡ation and a negative output gap as it leads to lower real long-term interest rates by lowering
future expected real short-term interest rates. We show illustrative numerical examples that suggest
that the bene…ts of quantitative easing in a liquidity trap can be large.
JEL Classi…cation: E31, E52, E58; E61; E63
Keywords: Time-consistent equilibrium; Duration of government debt; Central Bank balance sheet;
Quantitative easing; Liquidity trap; Signalling; Zero lower bound; Sticky prices
We thank Roberto Billi, Jim Bullard, Oli Coibion, Giuseppe Ferrero, Mark Gertler, Marc Giannoni, Andy Levin, Emi
Nakamura, seminar participants at HEC Montreal, Emory University, University of Texas at Austin, and Brown University,
and conference participants at NBER ME spring meeting, HKIMR/New York Fed Conference on Domestic and International
Dimensions of Unconventional Monetary Policy, Society of Economic Dynamics Annual meeting, Mid-west Macro spring meeting, CEPR European Summer Symposium in International Macroeconomics, Annual Conference on Computing and Finance,
NBER Japan Project Meeting, Annual Research Conference at Swiss National Bank, ECB Workshop on Non-Standard Monetary Policy Measures, Annual Research Conference at De Nederlandsche Bank, and Latin American Meetings of Econometric
Society for helpful comments and suggestions. This version: Feb 2015.
y Bhattarai: 2225 Speedway, Stop C3100, Austin, TX 78712; saroj.bhattarai@austin.utexas.edu. Eggertsson: Robinson Hall,
64 Waterman Street, Providence, RI 02912; gauti_eggertsson@brown.edu. Gafarov: Also a¢ liated with National Research
University Higher School of Economics; 605 Kern Building, University Park, PA 16802; bzg134@psu.edu.
1
“The problem with Quantitative Easing (QE) is it works in practice, but it doesn’t work in theory,” Ben
Bernanke, Chairman of the Federal Reserve, Jan 16, 2014 just before leaving o¢ ce.
1
Introduction
Since the onset of the economics crisis that started in 2008, the Federal Reserve has expanded it balance
sheet by large amounts, on the order of 3 trillion mostly under the rubric of Quantitative Easing (QE).
To date the accumulated amount of QE corresponds to about 20% percent of annual GDP. The enormous
scale of this policy has largely been explained by the fact that the Federal Reserve was unable to cut the
Federal Fund rate further, due the zero bound on the short term nominal interest rate. Meanwhile, high
unemployment, slow growth, and low in‡ation desperately called for further stimulus measures.
Many commentators argue that QE in the United States prevented a much stronger contraction, and
that QE is a key reason for why the US has recovered more rapidly from the Great Recession than some its
counterparts. As pointed out by Ben Bernanke, however, one problem is that even if this might be true in
practice, a coherent theoretical rationale has been hard to formulate. This paper contributes to …lling this
gap. We providing an explicit theoretical underpinning for QE: It works because it allows the Central Bank
to credibly commit to expansionary future policy in a zero lower bound situation. We …nd that not only
can one explicitly account for QE in theory, but we can also show some numerical examples in which the
e¤ect is substantial.
What is quantitative easing? Under our interpretation, it is when the central bank buys long-term
government debt with money. We interpret this action though the lenses of two models. First, we treat the
central bank and the treasury as one consolidated agent, i.e. policy is coordinated and budget constraints of
the Treasury and the Central Bank are consolidated. Second, we treat the Central Bank as "independent"
in the sense that it faces its own budget constraint (and thus cares about it own balance sheet) and that its
objective may be di¤erent from the rest of the government and social welfare.
Consider …rst QE under coordinated monetary and …scal policy, a natural and the main benchmark in
the paper. Since the nominal interest rate was zero when QE was implemented in the United States, it makes
no di¤erence if QE was done by printing money (or more precisely bank reserves) or by issuing short-term
government debt: both are government issued papers that yield a zero interest rate. From the perspective
of the government as a whole, quantitative easing at zero nominal interest rates can then simply be thought
of as shortening the maturity of outstanding government debt. The government is simply exchanging long
term bonds in the hands of the public with short term ones.
Consider next quantitative easing from the perspective of the central bank in isolation. Quantitative
easing creates a "duration mismatch" on the balance sheet of the central bank as it is issuing money/reserves
("short-term debt") in exchange for long term treasuries ("long term debt"). This opens up the possibility of
possible future balance sheet losses/gains by the central bank, because the price of its liabilities may fall/rise
relative to its assets. We can interpret quantitative easing as simply increasing the size of the balance sheet
of the central bank, keeping the extent of the duration mismatch on its balance sheet …xed. Alternatively
we can interpret it as only shortening the duration on its asset side, keeping the liability side unchanged.
Either interpretation is valid, and we will look at the data to sort out which interpretation …ts the facts
better for a particular QE episode.
Whether we consider QE from the perspective of a consolidated government budget constraint, or an
independent central bank,we arrive at the same conclusion. In both settings QE operates as a "signal" for
2
lower future short term interest rate in a way we make precise.
The main goal of quantitative easing in the United States was to reduce long-term interest rates, even
when the short-term nominal interest rate could not be reduced further, and thereby, stimulate the economy.
Indeed, several empirical studies …nd evidence of reduction in long-term interest rates following these policy
interventions by the Federal Reserve. For example, Gagnon et al (2011) estimate that the 2009 program
that involved buying various types of debt worth $1:75 trillion reduced long-term interest rates by 58 basis
points while Krishnamurthy and Vissing-Jorgensen (2011) estimate that the 2010 program that involved
buying long-term government debt worth $600 billion reduced long-term interest rates by 33 basis points.
In addition, Hamilton and Wu (2012), Swanson and Williams (2013), and Bauer and Rudebusch (2013) also
…nd similar e¤ects on long-term interest rates.1
From a theoretical perspective however, the e¤ect of such policy is not obvious since open market operations of this kind are neutral (or irrelevant) in standard macroeconomic models, which may have motivated
Bernanke’s quote cited above, holding future interest rate constant. This was pointed out …rst in a wellknown contribution by Wallace (1981) and further extended by Eggertsson and Woodford (2003) to a model
with sticky prices and an explicit zero lower bound on nominal interest rates. These papers showed how
absent some restrictions in asset trade that prevent arbitrage, a change in the relative supplies of various assets in the hands of the private sector has no e¤ect on equilibrium quantities and asset prices in prototypical
macroeconomic models for a given Taylor style monetary policy reaction function.
For this reason, some papers have recently incorporated frictions such as participation constraints due
to “preferred habitat” motives in order to make assets of di¤erent maturities imperfect substitutes. This
in turn negates the neutrality of open market operations as in such an environment, quantitative easing
can reduce long-term interest rates because it decreases the risk-premium. For example, Chen, Curdia, and
Ferrero (2012) augment a quantitative sticky price model with such segmented market frictions and show
that purchases of long-term bonds by the central bank can reduce long-term interest rates by decreasing
the risk-premium. They nevertheless …nd the e¤ects to be fairly modest: based on their estimated model,
they …nd that a $600 billion reduction in outstanding long-term debt, along with a commitment to keep
short-term interest rates at zero for 4 quarters, increases in‡ation by 3 basis points (annualized) and GDP
growth by 0:13% (annualized).
As pointed out by Eggertsson and Woodford (2003) (and further illustrated in Woodford (2012) in the
context of the crisis) quantitative easing need not be e¤ective only because it reduces risk premiums. It
can also reduce long-term interest rates if such policy intervention signals to the private sector that the
central bank will keep the short-term interest rates low once the zero lower bound is no longer a constraint
in the future. This amounts to that QE signals a change in the policy rule taken as given in Eggertsson
and Woodford (2003). In fact, arguably, much of the …ndings of the empirical literature on reduction of
long-term interest rates due to quantitative easing can be attributed to expectations of low future short-term
interest rates. Indeed, Krishnamurthy and Vissing-Jorgensen (2011) and Bauer and Rudebusch (2013) …nd
evidence in support of this channel in their study of the various quantitative easing programs.2
Our contribution is to provide a formal theoretical model of such a “signalling”role of quantitative easing
in a standard general equilibrium sticky price model.3 We consider the two perspectives of the government
1 Note however that empirical studies typically measure nominal interest rates, while theoretically, it is the ability to in‡uence
real interest rates that matter.
2 Woodford (2012) makes a similar argument regarding empirical evidence on most recent balance sheet policies by the
Federal Reserve.
3 Gagnon et al (2011) label this role played by the expected path of future short-term interest rates as a “signalling” role for
quantitative easing. In the literature on central bank intervention in foreign exchange markets, this term has been used often
3
outlined above. In particular, we …rst consider coordinated optimal monetary and …scal policy under discretion and show that in a Markov-perfect (time-consistent) equilibrium, quantitative easing, modeled as
shortening of the duration of outstanding government debt, provides an incentive to the central bank to keep
the short-term real interest rate low in the future. This constitutes optimal policy under discretion because
it avoids capital losses on the government’s balance sheet, which if realized, would entail raising taxes that
are costly in the model. The key intuition for this result is that if the government holds larger amount of
short-term debt then current real short term rate directly a¤ects the cost of rolling the debt over period by
period, while the cost of rolling over long-term debt is not a¤ected as strongly period by period (since the
interest rate on that debt is predetermined at the time the policy is set). This implies that shortening the
maturity of outstanding debt increases the incentive of the government to keep real rates low.
We then consider QE when the central bank is independent, in the sense that it does not internalize the
…scal cost of its actions that is borne by the treasury. We assume, however, that the central bank has a target
remittance to the treasury every period, and that it is costly for the bank to not hit its target of remittances
(for our purposes what is important is if the central bank is below this target). This is an assumed political
economy constraint, but it is worth pointing out that the same type of consideration arises under coordinated
monetary and …scal policy in our model where taxation is costly. Thus the political economy constraint
imposed on our independent central bank – while arti…cial – could directly be motivated by social welfare
considerations. In this second environment we show that if the central bank prints money/reserves and buys
long term bonds, this transaction creates a duration mismatch on its balance sheet. As a consequence, it
becomes more costly for the bank to raise nominal interest rate, as it will depreciate the value of its assets
relative to its liabilities. Thus once again, quantitative easing increases the incentive of the central bank to
keep real interest rates low. Moreover, in this case, simply increasing the size of the balance sheet of the
central bank without changing the composition of the balance sheet, has exactly the same e¤ects on the
incentives of the central bank as long as some duration mismatch exists in the …rst place.
In a liquidity trap situation, where the current short-term nominal interest rate is up against the zero
lower bound and the economy su¤ers from de‡ation and a large negative output gap, it is well known that
signalling about future policy can be very e¤ective. For example, Krugman (1998) emphasizes the importance
of raising in‡ation expectations and Eggertsson and Woodford (2003) emphasize the commitment to lower
future short-term nominal interest rate and allowing output to overshoot its steady state level. But are
these type of commitments credible? In fact, it is well-understood that commitment to future expansionary
policy at the zero lower bound is di¢ cult due to time-inconsistency problems. While the public understands
the bene…ts today of committing to lower future real interest rates, it also appreciates the government’s
incentive in the future to renege on these promises once the economy has recovered (this leads to the so
called “de‡ation bias” developed in Eggertsson (2006)). Our main result, then, is that quantitative easing,
or shortening the duration of government debt, makes promises of expansionary future policy “credible”
because it provides an incentive to the central bank to keep short-term real interest rates low in future to
avoid balance sheet losses. This mitigates the extent of de‡ation and negative output gap that would occur
otherwise in the Markov-perfect equilibrium.
In our calibrated/estimated model, we show numerical experiments in which these e¤ects of quantitative
easing can be quite large. For instance, consider our model. with a consolidated central bank and treasury
where we assess the macroeconomic implications of the Federal Reserve’s policy action starting on Nov 2010,
and appears to have been …rst coined by Mussa (1981) to discuss how foreign exchange interventions might be used to signal
future changes in monetary policy.
4
so called "Quantitative Easing 2." With initial government debt maturity of 16:87 quarters, we calibrate
the size and persistence of the negative demand shock (that makes the zero lower bound binding) such that
output drops by 10%;(annualized) in‡ation by 2%; and the expected duration of the liquidity trap is close to 3
years. We consider a reduction in the duration of government debt to 16:2 quarters, our estimate of the e¤ects
of "Quantitative Easing 2" on the duration of outstanding government debt.4 We then parameterize the
model by matching the e¤ects on future in‡ation and long-term yields of "Quantitative Easing 2" estimated
by Krishnamurthy and Vissing-Jorgensen (2011) In this parameterized model, Quantitative Easing 2 reduces
the extent of de‡ation at the zero lower bound by 23 basis points (annualized) as well as the negative e¤ect
on output by 60 basis points. This suggests that the signalling channel may explain the majority of the
e¤ects of quantitative easing found in the empirical literature.
Finally, while the paper connects with work related to the zero lower bound in models with nominal
rigidities, it also is related to the theoretical literature on how the maturity structure of debt can be
manipulated to eliminate the dynamic inconsistency found in many ‡exible price monetary models (in
particular the “in‡ation bias”) such as Lucas and Stokey (1983), Persson, Persson, and Svensson (1987 and
2006), and Alvarez, Kehoe, and Neumeyer (2004). While the focus of these papers is generally on policies
that eliminates the incentive to in‡ate, the in‡ation bias, (and thus generally imply a lengthening of the
duration of government debt relative to one-period debt), our application can be thought of as precisely the
opposite, that is, the maturity structure of debt is made shorter to solve the “de‡ation bias.” Our paper is
also building on insights of Calvo and Guidotti (1990 and 1992) who studied optimal maturity structure of
nominal government debt in ‡exible price models.
Moreover, the part of our paper with an independent central bank that has balance sheet concerns is
related to Jeanne and Svensson (2007). Jeanne and Svensson (2007) assume direct net worth concerns on
the part of the central bank and show how buying foreign exchange is useful during a liquidity trap as it
commits the central bank to not appreciating the exchange rate in future (since doing so would entail capital
losses on the central bank’s balance sheet). Our paper has a slightly di¤erent formulation of an independent
central bank as well as a di¤erent focus regarding the policy experiment.
There is also a related literature on signalling in which central bank types are …xed (they can either be
"doves" or "hawks", see e.g. Barro (1985)). In these models central banks use nominal interest rates to signal
how much they care about in‡ation. Our signalling mechanism is di¤erent from this literature, because the
central bank’s "type" or preference for in‡ation is endogenous and depends upon the composition of the
asset holdings of the central bank. Furthermore, in contrast to this earlier signalling literature there is full
information about the preferences of the central bank. Thus our interpretation of "signalling" is somewhat
di¤erent, namely, it has to do with credibly changing the central banks’interest rate incentive, or "type" in
the language of this earlier literature.
2
Benchmark model
We start by outlining our benchmark model in which case monetary and …scal policy are coordinated to
maximize social welfare under discretion (i.e. the government cannot commit to future policy). The model
is a standard general equilibrium sticky-price closed economy set-up with an output cost of taxation, along
4 As we detail later in the paper, we construct these measures related to the duration of outstanding government debt in the
consolidated government model using information from the literature on the duration of outstanding treasury debt as well as
reserves issued by the Federal Reserve during quantitative easing. Moreover, the particular exercise we report here corresponds
to our measure of "Quantitative Easing 2."
5
the lines of Eggertsson (2006). The main di¤erence in the model from the literature is the introduction of
long-term government debt. While it may seem like a distraction to write out the fully non-linear model
and de…ne the equilibrium in that context, as we will later on analyze a linear quadratic version of this
model, this is useful for two reasons. First, we will formally show that the linearized …rst-order conditions
of the government’s original non-linear problem are the same as in our linear quadratic model (and this in
general need not be the case, see e.g. Eggertsson (2006)). Second, the non-linear version of the problem will
be important once we allow for fully time-varying duration of government debt, where the linear quadratic
approximation is no longer valid. Laying out the model in this way, also, makes transparent the relationship
between social welfare and the ad-hoc objectives we will assign to the central bank when it is independent.
There, again, we will be working in a linear quadratic framework.
2.1
Private sector
A representative household maximizes expected discounted utility over the in…nite horizon
Et
1
X
t
Ut = Et
t=0
where
1
X
t
[u (Ct ) + g (Gt )
v(ht ))]
(1)
t
t=0
is the discount factor, Ct is household consumption of the composite …nal good, Gt is government
consumption of the composite …nal good, ht is quantity supplied of labor of type i, and
t
is a shock. Et
is the mathematical expectation operator conditional on period-t information, u (:) is concave and strictly
increasing in Ct ; g (:) is concave and strictly increasing in Gt ; and v (:) is increasing and convex in ht :5
i""1
R1h
" 1
;
The composite …nal good is an aggregate of a continuum of varieties indexed by i; Ct = 0 ct (i) " di
where " > 1 is the elasticity of substitution among the varieties. The optimal price index for the composite
hR
i11"
1
…nal good is given by Pt = 0 pt (i)1 " di
; where pt (i) is the price of the variety i: The demand for
the individual varieties is then given by
ct (i)
Ct
=
pt (i)
Pt
"
: Finally, Gt is de…ned analogously to Ct and so
we omit detailed description of government spending.
The household is subject to a sequence of ‡ow budget constraints
Pt Ct + BtS + St Bt + Et fQt;t+1 At+1 g
nt ht + (1 + it
S
1 ) Bt 1
+ (1 + St ) Bt
1
+ At
Pt Tt +
Z
1
Zt (i)di
(2)
0
where nt is nominal wage, Zt (i) is nominal pro…t of …rm i, BtS is the household’s holding of one-period
risk-less nominal government bond at the beginning of period t + 1, Bt is a perpetuity bond, St its price,
and
its decay factor (further described below). At+1 is the value of the complete set of state-contingent
securities at the beginning of period t + 1 and Qt;t+1 is the stochastic discount factor between periods t and
t + 1 that is used to value random nominal income in period t + 1 in monetary units at date t.6 Finally,
it
1
is the nominal interest rate on government bonds at the beginning of period t and Tt is government
taxes.
The way we introduce long term bonds into the model is to assume that government debt not only takes
the form of a one period riskfree debt, BtS ; but that the government also issues a perpetuity in period t
which pays
j
dollars j + 1 periods later, for each j
0 and some decay factor 0
<
1 7
. St is the price
of the perpetuity nominal bond which depends on the decay factor . The main convenience of introducing
5 We
abstract from money in the model and are thus directly considering the “cash-less limit.”
household is subject to a standard no-Ponzi game condition.
7 We follow Woodford (2001).
6 The
6
long term bond in this way is that we can consider government debt of arbitrary duration. For example, a
value of
= 0 implies that this bond is simply a short-term bond while
= 1 corresponds to a classic console
bond. More generally, in an environment with stable prices, the duration of this bond is (1
)
1
. Thus,
this simple assumption allows us to explore a change in the duration of government debt in a transparent
way. The appendix contains details on why the budget constraint takes the form (2). In particular, the
modelling of long-term bond in this way admits a simple recursive formulation of the price of old government
bonds.
For now, observe that we treat
as a constant. We will explore a one-time reduction in this duration as
a main “comparative static” of interest. In other words, a reduction in
answers the question: What does
a permanent reduction in the maturity of government debt do? Toward the end of the paper, however, we
will extend the analysis so that
becomes a time varying choice variable
t:
The main reason for our initial
benchmark assumption is simplicity (and the fact that we get a clean comparative static). But perhaps more
importantly, we will see later that a one-time reduction in
good approximation in our model because
t
(in a liquidity trap) turns out to be a reasonably
is close to a random walk under optimal policy under discretion
at a positive interest rate. When we move towards an independent central bank, the thought experiment
will be somewhat di¤erent, as we soon explain.
The maximization problem of the household is now entirely standard, with the additional feature of the
portfolio choice between long and short term bonds.8 Let us now turn to the …rm side of the model.
There is a continuum of monopolistically competitive …rms indexed by i. Each …rm produces a variety
i according to the production function that is linear in labor yt (i) = ht (i). As in Rotemberg (1983), …rms
. This adjustment cost makes the …rm’s pricing problem
face a cost of changing prices given by d ptp(i)
1 (i)
dynamic. The demand function for variety i is given by
yt (i)
=
Yt
pt (i)
Pt
"
(3)
where Yt is total demand for goods. The …rm maximizes expected discounted pro…ts
Et
1
X
Qt;t+s Zt+s (i)
(4)
s=0
where the period pro…ts Zt (i) are given by
Zt (i) = (1 + s) Yt pt (i)1
"
Pt"
nt (i)Yt pt (i)
"
Pt"
d
pt (i)
pt 1 (i)
Pt
where s is a production subsidy which we will set to eliminate the steady state distortion of monopolistic
competition as is common in the literature.9
We can now write down the necessary conditions for equilibrium that arise from the maximization
problems of the private sector described above. We focus on a symmetric equilibria where all …rms charge
the same price and produce the same amount of output. Note that these conditions hold for any government
policy.
8 The problem of the household is thus to choose {C
S
t+s ; ht+s (i); Bt+s ; Bt+s ; At+s } to maximize (1) subject
to a sequence of ‡ow budget constraints given by (2), while taking as exogenously given initial wealth and
{Pt+s; nt+s (i); it+s ; St+s ( ); Qt;t+s ; t+s ; Zt+s (i); Tt+s }.
9 The problem of the …rm is thus to choose {p
t+s (i)} to maximize (4), while taking as exogenously given
fPt+s; Yt+s ; nt+s (i); Qt;t+s ; t+s g
7
The households optimality conditions are given by
vh (ht )
nt
=
uC (Ct )
Pt
1
= Et
1 + it
St = Et
where
t
=
Pt
Pt 1
uC (Ct+1 )
uC (Ct )
uC (Ct+1 ) t+1
uC Ct ) t
(5)
t+1
1
t+1
(6)
(1 + St+1 )
(7)
t
1
t+1
is gross in‡ation.10 The …rm’s optimality condition from price-setting is given by
"Yt [uC (Ct )
vy (Yt )]
t
+ uC (Ct ) t d0 (
t)
t
= Et [ uC Ct+1 )
t+1 d
0
(
t+1 )
t+1 ]
(8)
where with some abuse of notion we have replaced vh with vy since in a symmetric equilibrium ht (i) =
yt (i) = Yt :
2.2
Government
There is an output cost of taxation (for example, as in Barro (1979)) captured by the function s(Tt
T)
where T is the steady-state level of taxes. Thus, in steady-state, there is no tax cost. Total government
spending is then given by
Ft = Gt + s(Tt
T)
where Gt is aggregate government consumption of the composite …nal good de…ned before.
It remains to write down the (consolidated) ‡ow budget constraint of the government. Note that the
government issues both a one-period bond BtS and the perpetuity Bt . We can write the ‡ow budget constraint
as
BtS + St Bt = (1 + it
1 ) Bt 1
+ (1 + St ) Bt
1
+ Pt (Ft
Tt ) :
Next, we assume that the one-period bond is in net-zero supply (i.e. BtS = 0; which makes clear that we
only introduce this bond explicitly as the risk free short term nominal rate is the key policy instrument of
monetary policy), and write the budget constraint in real terms as
St bt = (1 + St ) bt
where bt =
Bt
Pt .
1
1
t
+ (Ft
Tt )
(9)
We now de…ne …scal policy as the choice of Tt ; Ft ; and bt . For simplicity, we will from now
on suppose that total government spending is constant so that Ft = F: Conventional monetary policy is the
choice of it : We simply impose the zero bound constraint on the setting of monetary policy so that11
it
1 0 We
0:
may also append a standard transversality condition as a part of these conditions or a natural borrowing limit.
bound can be explicitly derived in a variety of environment, see e.g. Eggertsson and Woodford (2003).
1 1 This
8
(10)
2.3
Private sector equilibrium
The goods market clearing condition gives the overall resource constraint as
Yt = Ct + Ft + d (
t) :
(11)
We can then de…ne the private sector equilibrium, that is the set of possible equilibria that are consistent
with household and …rm maximization and the technological constraints of the model. A private sector
equilibrium is a collection of stochastic processes fYt+s; Ct+s ; bt+s ; St+s ;
Gt+s g for s
t+s ;
it+s ; Qt;t+s ; Tt+s ; Ft+s ;
0 that satisfy equations (5)-(7), (8), (9), and (11), for each s
exogenous stochastic process for f
t+s g:
0; given bt
1
and an
To determine the set of possible equilibria in the model, we now
need to be explicit about how policy is determined.
2.4
Markov-perfect Equilibrium
We characterize a Markov-perfect (time-consistent) Equilibrium in which the government cannot commit and
acts with discretion every period.12 A key assumption in a Markov-perfect Equilibrium is that government
policy cannot commit to the actions for the future government. Following Lucas and Stokey (1983), however,
we suppose that the government is able to commit to paying back the nominal value of its debt.13 The only
way the government can in‡uence future governments, then, is via any endogenous state variables that may
enter the private sector equilibrium conditions. Before writing up the problem of the government, it is
therefore necessary to write the system in a way that makes clear what are the endogenous state variables
of the game we study.
De…ne the expectation variables ftE , gtE , and hE
t . The necessary and su¢ cient conditions for a private
sector equilibrium are now that the variables {Yt ; Ct ; bt ; St ;
St ( )bt = (1 + St ( )) bt
1
uC (Ct ) t
; it 0
ftE
1
St ( ) =
gE
uC (Ct ) t t
" 1
uC (Ct )
hE
t = "Yt
"
1
t
+ (F
t ; it ;
Tt } satisfy: (a) the following conditions
Tt )
(12)
1 + it =
Yt = Ct + F + d (
given bt
1
(13)
(14)
t
v~y (Yt )
t
+ uC (Ct ) t d0 (
t)
t
t)
(15)
(16)
and the expectations ftE ; gtE ; and hE
t ; (b) expectations are rational so that
ftE = Et uC (Ct+1 )
gtE
hE
t
= Et uC (Ct+1 )
= Et [uC (Ct+1 )
t+1
1
t+1
t+1
1
t+1
0
t+1 d
(
(17)
(1 + St+1 ( ))
t+1 )
t+1 ] :
(18)
(19)
Note that the possible private sector equilibrium de…ned above depends only on the endogenous state
variable bt
1 2 See
1 3 One
1
and shocks
t:
Given that the government cannot commit to future policy (apart from through
Maskin and Tirole (2001) for a formal de…nition of the Markov-perfect equilibrium.
could model this more explicitly by assuming that the cost of outright default is arbitrarily high.
9
the endogenous state variable), a Markov-perfect Equilibrium then requires that the expectations ftE ; gtE ,and
hE
t are only a function of these two state variables, i.e, we can de…ne the expectation functions
ftE = f E (bt ; t )
(20)
gtE = g E (bt ; t )
(21)
E
hE
t = h (bt ; t ):
(22)
We can now write the discretionary government’s optimization problem as a dynamic programming
problem
V (bt
1; t)
= max[U (:) + Et V (bt ;
it ;Tt
t+1 )]
(23)
subject to the private sector equilibrium conditions (12)-(16) and the expectation functions (20)-(22). Note
that in equilibrium, the expectation functions satisfy the rational expectation restrictions (17)-(19). Here,
U (:) is the utility function of the household in (1) and V (:) is the value function.14 The detailed formulation
of this maximization problem and the associated …rst-order necessary conditions, as well as their linear
approximation, are provided in the appendix.15
3
Linear-quadratic approach
For most of our analysis we take a linear-quadratic approach to the optimal policy problem, which we
will show explicitly is a correct approximation to the original non-linear optimal policy problem of the
government that is maximizing social welfare. This characterization will directly apply when we consider
the consolidated government. In this section, we also consider an independent central bank. The problem
of the independent central bank will be similar to that of the consolidated government, apart from that it
has it own budget constraint and an objective that may deviate from social welfare due to political economy
constraints. We consider …rst the coordinated policy, and then move to an independent central bank.
3.1
Consolidated budget constraint and welfare
We approximate our model around an e¢ cient non-stochastic steady-state with zero in‡ation.16 Moreover,
there are no tax collection costs in steady-state17 . Thus, there is a non-zero steady-state level of debt.18
In steady-state, we assume that there is some …xed total market-value of public debt Sb =
: Then, the
following relationships hold
1+i=
1
; S=
1
; b=
1
1 4 Using
and T = F +
1
:
compact notation, note that we can write the utility function as [u (Ct ) + g (F s(Tt T )) v (Yt )] t :
here that we assume that the government and the private-sector move simultaneously.
1 6 Variables without a t subscript denote a variable in steady state. Note that output is going to be at the e¢ cient level in
steady state because of the assumption of the production subsidy (appropriately chosen) we have made before.
1 7 We can think of this as being due to a limited set of lump sum taxation.
1 8 The steady-state is e¢ cient even with non-zero steady-state debt because of our assumption that taxes do not entail output
loss in steady-state.
1 5 Note
10
We log-linearize the private sector equilibrium conditions around the steady state above to get the relationships
Y^t = Et Y^t+1
t
^bt =
(^{t
= Y^t + Et
1^
bt 1
1
t
S^t = ^{t +
where
and
Et
rte )
t+1
(24)
(25)
t+1
)S^t
(1
T^t
(26)
Et S^t+1
(27)
are a function of structural model parameters that do not depend upon
e¢ cient rate of interest that is a function of the shock
19
t.
The coe¢ cient
T
and rte is the
is also independent of
20
in our experiment.
Here, (24) is the linearized household Euler equation, (25) is the linearized Phillips curve, (26) is the
linearized government budget constraint, and (27) is the linearized forward-looking asset-pricing condition.21
(24) and (25) are standard relationships depicting how current output depends on expected future output
and the current real interest rate gap and how current in‡ation depends on expected future in‡ation and
the current output respectively.22
(26) shows that since debt is nominal, its real value is decreased by in‡ation. Higher taxes also reduce the
debt burden. Moreover, an increase in the price of the perpetuity bond decreases the real value of debt, with
the e¤ect depending on the duration of debt: longer the duration, lower is the e¤ect of the bond price on
debt. Finally, (27) shows that the price of the perpetuity bond is determined by (the negative of) expected
present value of future short-term interest rates. Hence, lower current or future short-term nominal interest
rate will increase the price of the perpetuity bond. Note that when = 0; all debt is of one-period duration
and (26) reduces to the standard linearized government budget constraint while (27) reduces to S^t = ^{t :23
A second-order approximation of household utility around the e¢ cient non-stochastic steady state gives
Ut =
where
and
T
h
2
t
+ Y^t2 +
^2
T Tt
i
(28)
are a function of structural model parameters.24 Compared to the standard loss-function
in models with sticky prices that contains in‡ation and output, (28) features losses that arise from output
costs of taxation outside of steady-state.
To analyze optimal policy under discretion in the linear-quadratic framework we once again maximize
utility, subject the now linear private sector equilibrium conditions, taking into account that the expectation
are functions of the state variables of the game. In the linear system, the exogenous state is now summarized
with re while the endogenous state variable is once again ^bt 1 : Moreover, the expectation variables appearing
t
in the system are now Et Y^t+1 ; Et S^t+1 ; and Et
t+1 .
Accordingly, we will de…ne the game in terms of the state
1 9 The
details of the derivation are in the appendix.
we are thinking of changes in in our experiment as exchanging short bonds with long bonds – e¤ectively reducing/increasing – this interpretation would imply that total value of debt in steady-state – – remains unchanged.
2 1 Variables with hats denote log-deviations from steady state except for the nominal interest rate, which is given as ^
t i
{t = i1+i
:
2 0 Since
Since in the non-stochastic steady state with zero in‡ation, 1 + i = 1 ; this means that the zero lower bound on nominal interest
rates imposes the following bound on ^{t : ^{t
(1
):
2 2 We write directly in terms of output rather than the output gap since we will not be considering shocks that perturb the
e¢ cient level of output in the model.
2 3 It is important to point out one techical detail in this case. The interpretation in this case of ^
bt is that it is the real value
of the debt inclusive of the interest rate payment to be paid next period, that is, if all debt were one period bt = (1 + it )
2 4 The details of the derivation are in the appendix. In particular,
= k" :
11
BtS
Pt
:
variables (rte ; ^bt
and E (^bt ; re ):
1)
and the government now takes as given the expectation functions Y E (^bt ; rte ); S E (^bt ; rte );
t
The discretionary government’s optimization problem can then be written recursively as a linear-quadratic
dynamic programming problem
V (^bt
e
1 ; rt )
2
t
= min[
+ Y^t2 +
^2
T Tt
e
+ Et V (^bt ; rt+1
)]
s.t.
Y^t = Y E (^bt ; rte )
t
^bt =
= Y^t +
1^
bt 1
E
(^{t
1
E
t
(^bt ; rte )
rte )
(^bt ; rte )
(1
)S^t
T^t
S E (^bt ; rte ):
S^t = ^{t +
Observe that once again, in equilibrium, the expectation functions need to satisfy the rational expectations
restrictions that Et Y^t+1 = Y E (^bt ; rte ); Et St+1 = S E (^bt ; rte ); and Et t+1 = E (^bt ; rte ):
We prove in the proposition below that this linear-quadratic approach gives identical linear optimality
conditions as the one obtained by linearizing the non-linear optimality conditions of the original non-linear
government maximization problem that we described above. This provides the formal justi…cation for our
simpli…ed approach.
Proposition 1 The linearized dynamic system of the non-linear Markov Perfect Equilibrium is equivalent
to the linear dynamic system of the linear-quadratic Markov Perfect Equilibrium.
Proof. In Appendix.
3.1.1
Data
Let us now brie‡y review the data we will use when we do numerical experiments with this benchmark model.
What the model is saying, is that under coordinated policy we should only be considering the debt held by
the public (thus, we net out government debt held by the Fed). Consistent with the model, also, we count
reserves issued by the Fed as short-term government debt. The duration of the consolidated government’s
debt is given below in Fig. 1.25 The vertical dashed lines are important events associated with the Federal
Reserve buying long-term treasury bonds: March 2009, November 2010, and September 2011. Around those
dates, the maturity of outstanding government debt declined. The baseline estimation of our model will be
based on the November 2010 or the Quantitative Easing 2 (QE2) program, a common focal point in the
literature. Given the parameter estimates from the QE2 program, we can also assess the macroeconomic
impact of the September 2011 or the Maturity Extension Program (MEP), commonly referred to as QE3.
2 5 In generating this …gure, we …rst use estimates from Chadha, Turner, and Zampoli (2013) on the duration of treasury debt
held outside the Federal Reserve, which we then augment with data on reserves issued by the Federal Reserve that is available
from public sources (FRED).
12
Fig 1: Maturity of outstanding govt debt adjusted for reserves issued by the Federal Reserve
3.2
Central bank balance sheet and political economy
We now present an alternate model where the central bank faces its own budget constraint and minimizes an
ad-hoc loss function that captures directly its balance sheet concerns.26 This model has a political economy
related justi…cation to why the central bank might care about transfers to the treasury. This alternative
formulation, as will become clear, will also require us to view the data through di¤erent lenses than in last
subsection. As we shall see, however, the central insights will remain the same, i.e. QE can have an e¤ect
via signalling.
3.2.1
Optimal policy problem of an independent central bank
We are now interested in studying the problem from the perspective of an independent central bank that
need not act in concert with the rest of the government. The …rst step is to explicitly write down its budget
constraint, before we move on to its objectives. We consider the case where the central bank holds longterm assets (long term government debt). It buys these asset by issuing one-period liabilities (approximating
interest-bearing reserves). We also introduce some "seigniorage net of operations cost" of the centra bank
that is not time-varying (similar to …scal spending that is not time-varying in our previous characterization
when monetary and …scal policy are coordinated). Denote central bank holdings of assets by BtCB and
its liabilities by Lt (with prices St and Qt respectively) and the "seigniorage net of operations cost" by
K: Moreover, let Vt be the transfers to the treasury.
The ‡ow budget constraint of the central bank is then given by
St BtCB + Pt Vt
Qt Lt
K = (1 + St ) BtCB1
Lt
1
2 6 This modelling approach is similar to the one in Jeanne and Svensson (2007), but the precise model details as well as the
balance sheet policy experiments are di¤erent. Berriel and Bhattarai (2009) also uses a related approach, but without studying
issues related to the zero lower bound.
13
where the inverse of Qt is the (gross) short-term nominal interest rate.27 This can be written in real terms
as
St bCB
t
Qt lt = (1 + St ) bCB
t 1
1
lt
t
1
1
t
+K
Vt :
The debt of the treasury is now either held by the central bank or the general public. The market clearing
condition for treasury debt (BtT ) is thus given by
BtT = BtCB + Bt
where Bt is debt held by the public. We assume that the treasury follows “passive …scal policy”that ensures
stable debt dynamics. Thus we completely abstract from …scal policy considerations. We also assume for
simplicity that all central bank reserves are held by the public.
With some algebra outlined in the footnote, and some simplifying assumptions, we can write the linearized
budget constraint as:
h
^bCB
t
where
V
.28
SbCB
V
i
^lt =
1
h
^bCB
t 1
^lt
1
i
^t
) S^t + Q
(1
V
V^t
(29)
Moreover, the price of the long term assets, S^t , is given by the same asset-pricing
condition as before
S^t = ^{t +
Et S^t+1
while the price of the short-term asset, which states that is just the negative of the short-term interest rate,
is given by
^ t = ^{t :
Q
Before we made the critical assumption that in steady state the market value of public debt was given
by some …xed number Sb =
that we linearized around. Here, the most important parameter is the scale
of the balance sheet, measured by
CB
Sb
V
: This number re‡ects how large the asset side of the balance sheet,
is relative to the steady state transfers to the treasury V: Also, note here that the budget constraint
2 7 For some other recent work exploring the implications of the central bank budget constraint, see Hall and Reis (2013) and
Del Negro and Sims (2015).
2 8 The two nonlinear asset pricing conditions will take the form
uC (Ct+1 ) t+1
uC Ct ) t
St = Et
Qt =
In steady-state, like before, we have Q
clearing gives
bCB
1
1; S
(1 + St+1 )
uC (Ct+1 )
uC (Ct )
1
= Et
1 + it
= 1+i =
1
t+1
=
t+1
t
1
t+1
: Moreover, de…ne, as before SbT = ; which from market
1
+ b = : The central bank budget constraint is then given in steady-state by
SbCB
Ql = (1 + S) bCB
We will focus on a steady-state where SbCB = Ql: Since
S
Q
1
l+K
V:
^bCB
t
where
V
=
V
SbCB
Ql ^
lt =
SbCB
1
^bCB
t 1
Ql ^
lt
SbCB
1
1
1
1
; we will have
bCB
l
= (1
) : This then implies
that K = V: We can now linearize the non-linear asset pricing and central bank budget constraint. First, we have S^t =
^ t + Et S^t+1 where in terms of our previous notation the following holds Q
^ t = ^{t : Thus, we have exactly like before the
Q
asset pricing condition for the long-term asset S^t = ^{t + Et S^t+1 : The linearized budget constraint is now given by
=
Ql
SbCB
t
(1
) S^t +
Ql ^
Qt
SbCB
V
V^t
is a parameter. We have to make some assumptions on the steady-state ratio of (mkt value) of liabilities to
assets of the central bank:
Ql
SbCB
: We will assume that it is 1: That gives the linearized buget constraint in the text.
14
is written in terms of the di¤erence between the central banks holding of long term government debt ^bCB
t
which has a …xed duration of and it own issuance of short term debt (interest bearing reserves) ^lt . It can
thus be re-written in terms of the net asset position of the bank, which we de…ne as ^bN;CB
= ^bCB
t
t
^lt . We
will use that formulation later below.
There are several noteworthy features in (29). First notice that up to …rst-order in‡ation has no e¤ect
on the net asset position of the bank. The reason for this is that in‡ation depreciates the value of the
assets (nominal long term bonds) and liabilities (short term nominal debt) by exactly the same extent.
This is a critical di¤erence relative to the consolidated budget constraint. Long term debt does, however,
have an important e¤ect on the net assets of the central bank. This is because while assets are long-term,
the liabilities are short term. There is thus generally a "duration mismatch" in the central bank’s balance
sheet.29
To see fully the implications of this duration mismatch, consider …rst the case in which
= 0: Then, both
the assets and the liabilities have the same duration. Then variations in the short term nominal interest
^ t and S^t cancel out.
rate have no e¤ect on the net worth of the central bank, as Q
Consider now the case in which
> 0. Now the central bank is holding long dated assets and short
dated liabilities. In this case we see that an increase in the short-term nominal interest rate reduces the net
worth of the bank. The reason for this is that an increase in the short rate increases the borrowing cost
of the bank one-to-one. Meanwhile, on the assets side, the increase in the short term nominal interest rate
has a more limited e¤ect , since they are long-dated securities (so the nominal interest rate is multiplier by
1
in expression (29)). Thus, increasing the short-term nominal interest rate sharply will lead to balance
sheet losses for the central bank.
What does quantitative easing mean in the context of this budget constraint? For the consolidated
budget constraint we interpreted it as a one time decline in : That interpretation is not correct in the
current context. Instead, if the central bank increases its holding of long term assets this means an increase
in , i.e. the degree of duration mismatch is enhanced the longer the duration is on the asset side of the
central bank balance sheet.
While an increase in
is one way of interpreting quantitative easing, there is another complementary
interpretation. Note that an increase in
in the context of the CB in isolation means that it is replacing
shorter term bonds on its asset side with bonds of longer duration. But the total value of bonds on the
asset side is constant. Thus, it was simply a change in the composition of the balance sheet. Now consider
the following experiment: Suppose all bonds on the asset side have some …xed
(say, corresponding to …ve
years). Now imagine the central bank increases the purchases of these these bonds by printing interestbearing reserves. While this is not a¤ecting the average duration of bonds on the asset side of the balance
sheet (or average duration mismatch since
is …xed), it is increases the scale of the central bank balance
sheet. The scale of the balance sheet in steady state was evaluated by the parameter
permanent expansion in the balance sheet is therefore measured as a drop in
V
V
=
V
:
SbCB
A
: Hence an alternative way
of interpreting QE is a permanent increase in the balance sheet of the Fed, or a drop in
V:
This then is
a change in the size of the balance sheet of the central bank. We will use this interpretation to study the
implications of some episodes of quantitative easing.
We can write the central bank budget constraint in terms of one state variable, ^bN;CB
= ^bCB
t 1
t 1
^bN;CB =
t
1^N;CB
bt 1
(1
2 9 Jeanne
^t
) S^t + Q
V
V^t :
^lt
1;
as
(30)
and Svensson (2007) in an open economy environment focus on a exchange rate mismatch in the central bank’s
balance sheet.
15
Observe that all the private-sector equilibrium conditions remain the same as when we studied the consolidated budget constraint. We can now conduct the experiment of an increase in the central bank’s balance
sheet via decrease in
V
; holding
assets as an increase in ; holding
…xed. We can also conduct the experiment of larger holding of long-term
V
…xed.
In terms of the central bank’s objective, we take here an ad-hoc loss-function approach, which has a rich
and long history in monetary economics. We posit that the central bank directly cares about transfers to
the treasury for political economy reasons. This means that its period loss function now incorporates a term
related to target transfers to the Treasury, Vt , in addition to the usual terms related to in‡ation and output.
It is thus given by
h
+ Y^t2 +
2
t
V
i
V^t2 :
Does this objective make sense? There is some evidence that central banks care about the transfers to the
treasury (Central bank governors that incur large balance sheet losses — e.g. the Central Bank of Iceland
in 2008 –corresponding to 30% of GDP found themselves without a job shortly thereafter).30 Moreover, as
we have seen, to the extent that these transfers a¤ect tax collection, the central bank should care also from
a social welfare point of view.
The discretionary central bank’s optimization problem can then be written recursively as a linearquadratic dynamic programming problem
e
V (^bN;CB
t 1 ; rt ) = min[
2
t
+ Y^t2 +
^2
T Tt
e
+ Et V (^bN:CB
; rt+1
)]
t
s.t.
Y^t = Y E (^bN;CB
; rte )
t
t
^bN;CB =
t
= Y^t +
1^N;CB
bt 1
S^t = ^{t +
(^{t
E
(1
E
(^bN;CB
; rte )
t
rte )
(^bN;CB
; rte )
t
^t
) S^t + Q
V
V^t
S E (^bN;CB
; rte ):
t
Observe that once again, in equilibrium, the expectation functions need to satisfy the rational expectations
; rte ):
; rte ); and Et t+1 = E (^bN;CB
restrictions that Et Y^t+1 = Y E (^bN;CB
; rte ); Et St+1 = S E (^bN;CB
t
t
t
3.2.2
Data
Ratio of the assets of the Federal Reserve (holdings of treasuries) to pre-crisis remittances to Treasury is
given below in Fig. 2. This is the counterpart to the parameter
1
V
in the model above. Average maturity of
treasury holdings by the Federal Reserve is given below in Fig. 3. This is the counterpart to the parameter
in the model above.31 Again, the vertical dashed lines are important events associated with the Federal
Reserve buying long-term treasury bonds: March 2009, November 2010, and September 2011. The baseline
parameterization of our model will be based on the November 2010 or the Quantitative Easing 2 (QE2)
program. Fig. 2 shows that the de…ning feature of that program was an increase in the size of the Federal
Reserve’s balance sheet with the average maturity not changing by much. Then, given the parameter
estimates from the QE2 program, we also assess the macroeconomic impact of the September 2011 or the
3 0 Jeanne and Svensson (2007) include the central bank’s net worth directly in the loss function. Thus, our modelling approach
is not identical in that sense.
3 1 We used data from publicly available sources to construct these …gures (FRED and the Federal Reserve Board of Governors
website).
16
Maturity Extension Program (MEP). Fig. 3 shows that the de…ning feature of that program was an increase
in the maturity of treasury holdings, with the size of the balance sheet not changing by much.
Fig 2: Ratio of Federal Reserve’s holdings of Treasuries to remittances to Treasury
Fig 3: Maturity of Federal Reserve’s holdings of Treasuries
4
4.1
Results
Consolidated budget constraint and welfare
Let us start with our baseline model where the budget constraints of the treasury and the central bank are
consolidated and the government maximizes welfare of the representative household, subject to its inability
of commit directly to future actions. We discuss the calibration of this model next. As we mentioned before,
17
the baseline calibration of this model will be based on the e¤ects of the QE2 program. For the steady-state
level of debt-to-taxes,
bS
T
=
T
; we use data from the Federal Reserve Bank of Dallas to get the long-run
average of market value of debt over output ( bS
Y ) and NIPA data to estimate the ratio of taxes over output
( YT ): This gives us the value
bS
T
=
= 7:2: Then, as a measure of the e¤ects on the average maturity of
T
outstanding government debt from QE2, we take the di¤erence in Fig. 1 between QE2 and the MEP (the
second and third dashed vertical lines). That is, according to our measure, the reduction in maturity was
from 16:87 q to 16:2 q, with a baseline maturity of 16:87 q.
For the other structural parameters of the model we use a mix of calibration/estimation based on the
e¤ects of QE2 on in‡ation and yields calculated by Krishnamurthy and Vissing-Jorgensen (2011) as well as
the expected duration of the zero lower bound episode. In particular, we pick the quarterly discount factor
of
= 0:99. We allow for debt while at the liquidity trap to be 30% above its steady-state value, which is
in line with the Federal Reserve Bank of Dallas data.32
Then, we choose ( ,
T,
", ) to match the reduction in 8 quarters ahead yield of
and an increase in expected cumulative in‡ation over 10 years
i (8) =
16 b.p.
(40) = 5 b.p. as a result of the QE2
program when the economy is initially in a ZLB situation. These were the estimates of Krishnamurthy
and Vissing-Jorgensen (2011) of the e¤ects of the QE2 program. In addition, we also try to match a 3year average duration of the ZLB period. To model the case of a liquidity trap, we follow Eggertsson and
Woodford (2003) and suppose that a large enough negative shock to the e¢ cient rate of interest rte ; driven
by an increase in the desire to save, makes the zero lower bound binding.33 Moreover, we also assume that
rte follows a two-state Markov process with an absorbing state: every period, with probability , rte takes
e
on a negative value of rL
while with probability 1
; it goes back to steady-state and stays there forever
after. This means that the economy will exit the liquidity trap with a constant probability of 1
every
period and that once it exits, it does not get into the trap again. The appendix contains details about the
computation algorithm. For an average ZLB duration of about 3 years, we target
= 0:91. Our criterion
then is the mean squared weighted relative error given by
Loss =
s
(40)
(40)
(40)
subject to the inequality constraint "
b.p.,
i (8) =
5:5 b.p., and
2
i (8)
+
i (8)
i (8)
2
2
+ 1000
12:The values of the targets for our best match are
(40) = 4:7
= 0:91 (about 3 years). Our estimated parameter values for this match are
given in Table 1:
e
Finally, we pick rL
to get a drop in output of 10% and a 2% percent drop in in‡ation during the ZLB
e
episode to make the experiment relevant for the recent “Great Recession” in the United States. So, rL
is
an implicit function of the model parameters and these two targets. The parameterized values are given in
Table 2:
3 2 We also adjust the quantity of debt level after QE2 to keep the market value of the debt …xed during the QE2 intervention
(it has to be adjusted to 0.297).
3 3 For an alternate way of generating a liquidity trap in monetary models, based on an exogenous drop in the borrowing limit,
see Eggertsson and Krugman (2012).
18
Table 1: Calibration/estimation of model parameters
Parameter
Value
0:99
0:463
0:0066
"
12
0:384
T
bS
T
7:2
Table 2: Calibration/estimation of model parameters for the ZLB experiment
Parameter
Value
e
rL
0:0087
0:91
bL
4.1.1
0:30
Solution at positive interest rates
We start by showing the solution at positive interest rates as it clearly illustrates how the incentives of the
government in setting the real interest rate (through manipulation of the nominal interest rate) get a¤ected
by duration of government debt. The complication in solving a Markov-perfect Equilibrium is that we do not
know the unknown expectation functions
E
; Y E ; and S E : To solve this, we use the method of undetermined
coe¢ cients. All the details of the derivations are provided in the appendix. Let us …rst consider the most
basic exercise to clarify the logic of the government’s problem. How do the dynamics of the model look like
in the absence of shocks when the only di¤erence from steady state is that there is some initial value of debt
with some …xed value of debt duration?
Fig. 4 shows the dynamics of the endogenous variables in the model for an initial value of debt that
is 30 percent above the steady state debt in the model. We see that if debt is above steady state, it is
paid over time back to steady state. For our baseline duration of 16:87 quarters (solid line), the half-life
of debt repayment is about 12 quarters. Note that in the transition for this parameterization, in‡ation is
about 0:75 percent above steady state. Importantly, in the transition phase, we see that the real interest
rate is below its steady state. As a consequence, output is also above its steady state value. Note that this
result is in contrast to the classic Barro tax smoothing result whereby debt follows a random walk. The
reason for this is that debt creates an incentive to create in‡ation for a discretionary government as further
described below. By paying down debt back to steady state, the government thus eliminates this incentive
and achieves better outcomes.
19
Fig 4: Transition dynamics to an increase in debt outstanding at di¤erent levels of duration of debt
The …gure illustrates that for a given maturity of government debt, debt is in‡ationary and implies lower
future real interest rate until a new steady state is reached. What is the logic for this result? Perhaps the
best way to understand the logic is by inspecting the government budget constraint (26). Recall that debt
Bt
b
issued in nominal terms, although in the budget constraint we have rewritten it in terms of ^bt = Pt :
This implies that for a given outstanding debt ^bt
outstanding debt. Accordingly we have the term
b
1;
any actual in‡ation will reduce the real value of the
1
t
term in the budget constraint which re‡ects this
in‡ation incentive. As the literature has stressed in the past (see e.g. Calvo and Guidotti (1990 and 1992)),
if prices are ‡exible then this will reduce actual debt in equilibrium only if the in‡ation is unanticipated.
The reason for this is that otherwise anticipated in‡ation will be re‡ected one-to-one in the interest rate
paid on the debt.
Apart from the incentive to depreciate the real value of the deb via in‡ation, there is a second force
at work in our model. In our model, the government is not only able to a¤ect the price level, it can also
have an e¤ect on the real interest rate. Hence, we see that in Fig. 4 the real interest rate is below steady
state during the entire transition path back to steady state. This will reduce the real interest rate payment
the government needs to pay on debt – in contrast to the classic literature with ‡exible prices where the
(ex-ante) real interest rate is exogenous. We refer to this as the “rollover incentive” of the government.
Intuitively, it may be most straight forward to see this force at work by simplifying the model down to
the case in which
= 0: In that case, the budget constraint of the government can be written as
^bt =
1^
bt 1
1
t
+ ^{t
T^t
and now ^bt is the real value of one period risk-free nominal debt in period t which is inclusive of interest
paid (to relate to our prevision notation in (2) then when
= 0 we have bt =
Bt
Pt
=
(1+it )BtS
Pt
where BtS was
the one period government debt that did not include interest payment). This expression makes clear that
while
t
has a direct e¤ect by depreciating the real value of government debt, the government has another
important margin by which it can in‡uence its debt burden. The term ^{t re‡ects the cost or rolling-over-cost
20
of the one-period debt. In particular, we see that if the interest rate is low, then the cost of rolling over
debt is smaller. This latter mechanism will be critical when considering the e¤ects of varying debt maturity
since its force is highly depending on the value of :
We now consider how reducing the value of
a¤ects the transition dynamics from above. As noted
before, our main interest for this is that a natural interpretation of quantitative easing is that it corresponds
to a reduction in
as in our model, as the duration of debt is given by (1
)
1
: As Fig. 4 reveals, where
we consider a reduction in duration from 16:8 years to 16:2 and 15:6 quarters, this increases in‡ation in
equilibrium considerably, but also reduce the real rate further. Similarly, we see that the debt is now paid
down at a faster clip as higher output gap and in‡ation causes increased distortions, a point we will return
to.
To obtain some intuition for this result, let us again write out the budget constraint, this time substituting
out for St to obtain
^bt =
1^
bt 1
1
t
+ (1
S^E (^bt ; rte )]
)[^{t
T^t :
We observe here that the rollover interest rate is now multiplied by the term (1
(31)
). Intuitively then, if a
larger part of government debt is held with long maturity, the short-term rollover rate matters less, as the
term of the loans are to a greater extent predetermined. Hence, the incentive of the government to lower
the short-term interest rate is reduced.
Moreover, note that we can consider two polar cases of ^bt =
one is with
1^
bt 1
1
t
(1
)S^t
T^t : The …rst
= 0 (only one-period debt), where we get
^bt =
1^
bt 1
1
t
T^t
+ ^{t
which shows that in this case the short-term interest rate a¤ects one-to-one and directly the rollover incentive
of the government. The second polar case is with
^bt =
= 1 (with classic consols only), where we get
1^
bt 1
1
t
T^t
which shows that the interest rate does not a¤ect debt dynamics directly at all. These cases illustrate how
the “rollover incentive”regarding setting the short-term nominal interest rate of the government get directly
a¤ected by the duration of debt.
Having established intuitively and numerically that at positive interest rates, decreasing the duration of
debt increases the incentives of the government to lower short-term real interest rates, we now move on to
analyzing the case where the nominal interest rate is at the zero lower bound.
4.1.2
At the zero lower bound
With this structure, we next consider the following policy experiment. At the liquidity trap, the level of
debt is constant at bL , while out of the trap, it is optimally determined by the government according to the
Markov-perfect equilibrium described previously. Then, we analyze what would be the e¤ect of changing the
duration of debt once-and-for-all, while the zero lower bound is binding. In other words, we are interested in
the comparative statics of the model as we vary the duration of debt in the liquidity trap. Note in particular
that the steady-state market value of debt to taxes is always kept …xed in our experiments. For now a key
abstraction is that the value of
is …xed, an issue we will come back to in the last section.
21
Initial duration of government debt Figs. 5 and 6 show the response of in‡ation and output to a
negative shock to the e¢ cient rate of interest when the duration of government debt is 16:87 quarters.
The solid line shows the expected path for the variables (impulse response) while the thin lines show each
possible economic contingency (thus the third thin line – for example – corresponds to the case in which
the shock reverts to normal in the third quarter). As is clear, because of the zero lower bound constraint,
the economy su¤ers from de‡ation and because of the increase in the real interest rate that it creates (that
is, a gap between the real interest rate and its e¢ cient counterpart), also from a negative e¤ect on output.
This re‡ects what Eggertsson (2006) labelled the “de‡ation bias”of a discretionary central bank at the zero
lower bound. In such a case, generating expectations of future in‡ation would be very bene…cial as it would
decrease the extent of de‡ation and the rise of the real interest rate, as stressed by Krugman (1998) and
Eggertsson (2006). Observe, also, that because of the presence of nominal debt in the economy, the in‡ation
rate once the shock is over is not zero. Instead it is of the order of about 0:75 percent. This “in‡ation bias”
arises from the governments incentive to in‡ate away the nominal value of the outstanding debt.
0.005
0
10
20
30
40
t
50
0.005
0.010
0.015
0.020
Fig 5: Reponse of in‡ation when the duration of government debt is 16:87 quarters. Each line
represents the response when the e¢ cient rate of interest returns to its steady-state in that period.
22
Y
0.02
0
10
20
30
40
t
50
0.02
0.04
0.06
0.08
0.10
Fig 6: Reponse of output when the duration of government debt is 16:87 quarters. Each line
represents the response when the e¢ cient rate of interest returns to its steady-state in that period.
It would be bene…cial for the central bank to commit to keeping the short-term real interest rates lower
in future, once the zero lower bound is no longer binding. This would decrease real long-term interest rates
today and help spur the economy. Thus, one main goal of monetary policy would be to decrease the extent
of increase in real interest rates that is seen in Fig. 7, which shows the real interest rate gap, the gap between
the real interest rate and the e¢ cient rate of interest. In other words, as Krugman (1998) and Eggertsson
(2006) emphasize, the central bank needs to “commit to being irresponsible.”34
3 4 Adam and Billi (2007), another important study of optimal policy in the New Keynesian model under discretion, emphasizes
how the gains from commitment are much stronger once the zero lower bound on nominal interest rates is taken into account.
23
r gap
0.010
0.005
0
10
20
30
40
t
50
0.005
Fig 7: Reponse of the real interest rate gap when the duration of government debt is 16:87
quarters. Each line represents the response when the e¢ cient rate of interest returns to its
steady-state in that period.
Shorter duration of government debt
A reduction in the duration of government debt outstanding
helps achieve this goal. Figs. 8 and 9 show the change in response of output and in‡ation to a negative
shock to the e¢ cient rate of interest when the duration of government debt is 16:2 quarters. As is clear,
the extent of de‡ation is reduced by 23 basis points (annualized) as well as the negative e¤ect on output
by 60 basis points. Note in particular that once the shock is over and the zero lower bound is no longer a
constraint, the change in response of in‡ation and output is positive.
24
dY
0.005
0.004
0.003
0.002
0.001
0
10
20
30
40
t
50
Fig 8: Change in output when the duration of government debt is reduced from 16:87 to 16:2
quarters. Each line represents the response when the e¢ cient rate of interest returns to its
steady-state in that period.
d
0.0020
0.0015
0.0010
0.0005
0
10
20
30
40
t
50
Fig 9: Change in in‡ation when the duration of government debt is reduced from 16:87 to 16:2
quarters. Each line represents the response when the e¢ cient rate of interest returns to its
steady-state in that period.
The main reason why this is achieved is that because the government’s balance sheet has more long-term
bonds (or its debt is more short-term), the central bank keeps the short-term real interest rates lower in
future, especially once the zero lower bound is not binding, in order to keep the real interest rate low on
25
the debt it is rolling over. Thus, quantitative easing indeed provides a signal about the future conduct of
monetary policy and in particular, the future path of short-term interest rates. This then enables it to have
e¤ect on macroeconomic prices and quantities at the zero lower bound, as real interest rates also decline
at the liquidity trap because of higher in‡ationary expectations once the trap is over. The change in the
response of the real interest rate comparing higher to lower duration of debt is given in Fig. 10, where one
can see that the real interest rate is lower through out the horizon. This, then, is the central result of our
paper: quantitative easing acts as a commitment device during a liquidity trap situation.35
dr
0
10
20
30
40
t
50
0.0005
0.0010
0.0015
0.0020
0.0025
Fig 10: Change in the real interest rate when the duration of government debt is reduced from
16:87 to 16:2 quarters. Each line represents the response when the e¢ cient rate of interest returns
to its steady-state in that period.
4.1.3
Capital losses from reneging on optimal policy
We have emphasized and illustrated so far that the reason why lowering the duration of debt during a
liquidity trap situation is bene…cial is that it provides incentives for the government to keep the real interest
rate low in future as it is now rolling over more short-term debt. We have shown these results by comparing
the path of the real interest rate under optimal policy at a baseline and lower duration of debt.
Another way of framing this is that otherwise it would su¤er capital losses on its balance sheet. These
losses then would have to be accounted for by raising costly taxes. One way to illustrate the mechanism
behind this result is to conduct the following thought experiment: suppose that once the liquidity trap is
over, the government reneges on the path for in‡ation and output dictated by optimal policy under discretion
and instead perfectly stabilizes them at zero. In such a situation, how large are capital losses, or equivalently,
how high do taxes have to rise out of zero lower bound compared to if the government had continued to
follow optimal policy? In particular, is this increase in taxes more when debt is of shorter duration ? We
show in Fig. 11 the change in taxes if the government were to renege on optimal policy at a duration of
3 5 This result thus connects our paper with Persson, Persson, and Svensson (1987 and 2006), who show in a ‡exible price
environment that a manipulation of the maturity structure of both nominal and indexed debt can generate an equivalence
between discretion and commitment outcomes.
26
16:87; 16:2; and 15:6 quarters respectively. As is clear, the increase in taxes out of zero lower bound are
higher at a shorter duration of outstanding debt. Thus indeed, lowering the duration of government debt
provides the government with more an incentive to keep the real interest rate low in future in order to avoid
having to raise costly taxes.
Fig 11: Increase in taxes from reneging on optimal policy
4.1.4
Quantitative assessment of MEP
Given the parameter estimates based on QE2 and its e¤ects on expected in‡ation and future short-term
interest rates given in Tables 1 and 2, we now conduct an assessment of the macroeconomic e¤ects of MEP.
This involved a reduction in duration of outstanding government debt as well, as is clear from Fig. 1. Our
experiment is based on the di¤erence is duration between September 2011 and September 2013, which is
equal to 1:8 q and bigger than the e¤ect from QE2. Accordingly, the macroeconomic e¤ects, as shown below
in Figs.12-13 are larger as well as the drop in the real interest rate, as shown in Fig. 14 below, is now bigger.
27
dY
0.015
0.010
0.005
0
10
20
30
40
t
50
Fig 12: Change in output when the duration of government debt is reduced to 14:6 quarters. Each
line represents the response when the e¢ cient rate of interest returns to its steady-state in that
period.
d
0.006
0.005
0.004
0.003
0.002
0.001
0
10
20
30
40
t
50
Fig 13: Change in in‡ation when the duration of government debt is reduced to 14:6 quarters.
Each line represents the response when the e¢ cient rate of interest returns to its steady-state in
that period.
28
dr
0
10
20
30
40
t
50
0.002
0.004
0.006
0.008
Fig 14: Change in the real interest rate when the duration of government debt is reduced to 14:6
quarters. Each line represents the response when the e¢ cient rate of interest returns to its
steady-state in that period.
4.2
Central bank balance sheet and political economy
We now model the e¤ects of quantitative easing using the political economy based model where the central
bank directly cares about transfers to the treasury. We discuss the calibration of this model next. As we
mentioned before, the baseline calibration of this model will be based on the e¤ects of the QE2 program. We
use quarterly average from 2003-2008 of remittances to Treasury from the Federal Reserve of 6:48 as a steady
state value of V . Our baseline value
Ql
V
= 111:47 corresponds to the average ratio of holdings of Treasuries
by the Federal Reserve to remittances to Treasury over the period of 2003-2008. Then, as a measure of the
e¤ects on the size of the Federal Reserve’s balance sheet from QE2, we take the di¤erence in Fig. 2 between
QE2 and the MEP (the second and third dashed vertical lines). That is, according to our measure, the
change in size of the balance sheet was from 111:47 to 231:38, with a di¤erence of 119:91. Finally, we pick
the value of
based on the average maturity of Treasury holdings of the FED during 2003-2008, which is
equal to 7:57 quarters.
For the other structural parameters of the model, like before for the consolidated budget constraint
case, we use a mix of calibration/estimation based on the e¤ects of QE2 on in‡ation and yields calculated
by Krishnamurthy and Vissing-Jorgensen (2011) as well as the expected duration of the zero lower bound
episode. In particular, we pick the quarterly discount factor of
= 0:99. Unlike before for the consolidated
government debt, since there is no clear metric on the net asset position of the Federal Reserve during the
crisis (in terms of deviation from steady-state), we choose to estimate that parameter bN;CB
.36
L
Thus, we choose (bN;CB
; ,
L
T,
",
) to match the reduction in 8 quarters ahead yield of
16 b.p. and an increase in expected cumulative in‡ation over 10 years
i (8) =
(40) = 5 b.p. as a result of the
QE2 program when the economy is initially in a ZLB situation. These were the estimates of Krishnamurthy
3 6 We
also adjust this to keep the market value of the net asset position constant as a result of the QE intervention.
29
and Vissing-Jorgensen (2011) of the e¤ects of the QE2 program. In addition, we also try to match a 3year average duration of the ZLB period. To model the case of a liquidity trap, we follow Eggertsson and
Woodford (2003) and suppose that a large enough negative shock to the e¢ cient rate of interest rte ; driven
by an increase in the desire to save, makes the zero lower bound binding.37 Moreover, we also assume that
rte follows a two-state Markov process with an absorbing state: every period, with probability , rte takes
e
a (large enough) negative value of rL
while with probability 1
; it goes back to steady-state and stays
there forever after. This means that the economy will exit the liquidity trap with a constant probability of
1
every period and that once it exits, it does not get into the trap again. The appendix contains details
about the computation algorithm. For an average ZLB duration of 3 years, we therefore target
= 0:91.
Our criterion then is the mean squared weighted relative error given by
Loss =
s
(40)
(40)
(40)
subject to the inequality constraint "
b.p.,
i (8) =
16:5 b.p., and
2
i (8)
+
i (8)
i (8)
2
2
+ 1000
12:The values of the targets for our best match are
(40) = 5:1
= 0:85 (6.7 quarters). Our estimated parameter values are given in Tables
3 and 4:
e
to get a drop in output of 10% and a 2% percent drop in in‡ation during the zero
Finally, we pick rL
lower bound episode to make the experiment relevant for the recent “Great Recession”in the United States.
e
is an implicit function of the model parameters and these two targets. The estimated value is given
So, rL
in Table 4:
Table 3: Calibration/estimation of model parameters
Parameter
Value
0:99
0:308
0:008
"
8
1:27
T
Ql
V
10
6
111:47
Table 4: Calibration/estimation of model parameters for the ZLB experiment
Parameter
Value
e
rL
0:035
0:85
bN;CB
L
4.2.1
0:15
Solution at positive interest rates
Let us again …rst consider the most basic exercise to clarify the logic of the central bank’s problem. How
do the dynamics of the model look like in the absence of shocks when the only di¤erence from steady state
3 7 For an alternate way of generating a liquidity trap in monetary models, based on an exogenous drop in the borrowing limit,
see Eggertsson and Krugman (2012).
30
is that there is some initial negative value of net asset of the central bank (thus, net assets is below steadystate) with some …xed value of asset duration? We consider transition dynamics when net asset is below
steady-state initially, while in the consolidated budget constraint case we had considered a case where debt
is above steady-state initially. This are exactly the same experiments in essence however, because in both
cases, there is asset accumulation along the transition. Moreover, in both cases, the real interest rate is
rising along the transition, which is the main mechanism behind our paper.38
We will be looking at comparative statics of those transition dynamics with respect to the size of the
central bank balance sheet (holding the duration of the assets …xed) as well as the duration of the assets
on the central bank’s balance sheet (holding the size of the balance sheet …xed). These, as we mentioned
above, is how we interpret QE2 and MEP in the context of our central bank balance sheet concerns model.
Fig. 15 shows the dynamics of the endogenous variables in the model for an initial value of net asset of the
central bank that is 15 percent below the steady state net asset in the model. Analogous to the consolidated
government budget constraint model, we see that if net asset is below steady state, it is accumulated over
time back to steady state. Moreover, note that in this transition, in‡ation is above steady state. Importantly,
in the transition phase, we see that the real interest rate is below its steady state. As a consequence, output
is also above its steady state value. The explanation for these dynamics are the same as for the consolidated
budget constraint.
Fig 15: Transition dynamics to a decrease in net assets of the central bank at di¤erent sizes of the central
bank balance sheet
We now consider how increasing the size of the balance sheet, as given by the ratio of assets to transfers
to the treasury, while holding the duration of the assets on the balance sheet …xed, a¤ects the transition
dynamics. In Fig. 15 we show that bigger the size of the balance sheet (our baseline calibration is 111),
bigger is the drop in the real interest rate. What is the intuition for this result? Consider the linearized ‡ow
budget constraint of the central bank (30) again
^bN;CB =
t
1^N;CB
bt 1
(1
^t
) S^t + Q
V
V^t
3 8 Basically our assumption is saying that the Fed would like to have higher net worth, which is not an unreasonable assumption
under QE when it exposed itself to possible balance sheet losses.
31
1
where
V
is our measure of the size of the central bank’s balance sheet. Now multiply both sides of the
equation above with
1
V
to get
V
1^N;CB
bt
=
This shows clearly that even if we hold
1
V
1^N;CB
bt 1
1
V
h
(1
…xed, an increase in
) S^t
1
V
^t
Q
i
V^t :
makes the e¤ect of the duration mismatch
between assets and liabilities of the central bank worse. Thus, when assets are long-term and liabilities
short-term, for a higher
1
V
; the central bank’s balance sheet is more exposed to interest-rate risk along
the transition where assets are being accumulated and the interest rate is rising, like in Fig. 15. Thus, in
order to avoid capital losses on its balance, the central bank has incentives to keep interest rates lower with
a higher
1
V
:
We next consider how increasing the duration of the assets of the central bank, while holding the size
of the balance sheet …xed, a¤ects the transition dynamics we described above. Fig. 16 shows that longer
is the duration of assets held by the central bank (our baseline calibration is a duration of 7:44 quarters),
larger is the drop in the real interest rate along the transition. This is a counterpart of our results for the
consolidated government budget constraint model and all the intuition and explanation we developed there
applies directly.
Fig 16: Transition dynamics to a decrease in net assets of the central bank at di¤erent levels of duration of
the net assets
Having established intuitively and numerically that at positive interest rates, increasing the size of the
central bank’s balance sheet or increasing the duration of assets held on the central bank’s balance sheet
increases the incentives of the central bank to lower short-term real interest rates, we now move on to
analyzing the case where the nominal interest rate is at the zero lower bound.
4.2.2
At the zero lower bound
With this structure, we next consider the following policy experiment. At the liquidity trap, the level of
net assets of the central bank is constant at bN;CB
, while out of the trap, it is optimally determined by
L
the central bank according to the Markov-perfect equilibrium described previously. Then, we analyze what
32
would be the e¤ect of increasing the size of the central bank’s balance sheet once-and-for-all, while the zero
lower bound is binding. In other words, we are interested in the comparative statics of the model as we vary
the size of the central bank’s balance sheet at the liquidity trap, while keeping the duration of the assets
…xed. Again, we use this experiment as the baseline as it conforms to the data most directly in terms of
QE2. We will also show results based on increasing the duration when we consider the MEP.
At the initial size of the central bank balance sheet Figs. 17 and 18 show the response of in‡ation
and output to a negative shock to the e¢ cient rate of interest when the size of the central bank’s balance
sheet is 111 and the duration of the central bank’s assets is 7:57 quarters. As before, because of the zero
lower bound constraint, the economy su¤ers from de‡ation and because of the increase in the real interest
rate that it creates (that is, a gap between the real interest rate and its e¢ cient counterpart), also from a
negative e¤ect on output. The real interest rate gap is shown in Fig. 19. Thus again, optimal policy at the
zero lower bound would entail reducing the real interest rate gap at the present and in future.
0
10
20
30
40
t
50
0.005
0.010
0.015
0.020
Fig 17: Reponse of in‡ation at the initial level of central bank balance sheet size and net asset
duration. Each line represents the response when the e¢ cient rate of interest returns to its
steady-state in that period.
33
Y
0
10
20
30
40
t
50
0.02
0.04
0.06
0.08
0.10
Fig 18: Reponse of output at the initial level of central bank balance sheet size and net asset
duration. Each line represents the response when the e¢ cient rate of interest returns to its
steady-state in that period.
r gap
0.12
0.10
0.08
0.06
0.04
0.02
0
10
20
30
40
t
50
Fig 19: Reponse of the real interest rate gap at the initial level of central bank balance sheet size
and net asset duration. Each line represents the response when the e¢ cient rate of interest
returns to its steady-state in that period.
At an increased size of the balance sheet An increase in the size of the central bank’s balance sheet,
given by the parameter
1
V
, helps achieve this goal. Figs. 20 and 21 show the change in response of output
34
and in‡ation to a negative shock to the e¢ cient rate of interest when the size of the balance sheet is now
231 (again the duration of the central bank’s assets is still …xed at 7:57 quarters). As is clear, the extent
of de‡ation is reduced by 14 basis points (annualized) as well as the negative e¤ect on output by 40 basis
points. Note in particular that once the shock is over and the zero lower bound is no longer a constraint,
the change in response of in‡ation and output is positive. Note that these e¤ects are in the same order of
magnitude as the e¤ects obtained with the consolidated government budget constraint.
dY
0.004
0.003
0.002
0.001
0
10
20
30
40
t
50
Fig 20: Change in output with an increased central bank balance sheet size and initial level of net
asset duration. Each line represents the response when the e¢ cient rate of interest returns to its
steady-state in that period.
35
d
0.0014
0.0012
0.0010
0.0008
0.0006
0.0004
0.0002
0
10
20
30
40
t
50
Fig 21: Change in in‡ation with an increased central bank balance sheet size and initial level of
net asset duration. Each line represents the response when the e¢ cient rate of interest returns to
its steady-state in that period.
The main reason why this is achieved is that because the central bank’s balance sheet bigger, it magni…es
the e¤ects of interest rate changes on the balance sheet due to the mismatch in duration between its assets
(which are long-term) and liabilities (which are short-term). The central bank then keeps the short-term real
interest rates lower in future, especially once the zero lower bound is not binding, in order to keep the real
interest rate low. Otherwise, it would su¤er from capital losses which are costly for the central bank. Thus,
quantitative easing indeed provides a signal about the future conduct of monetary policy and in particular,
the future path of short-term interest rates. The change in the response of the real interest rate comparing
smaller to bigger duration of debt is given in Fig. 22, where one can see that the real interest rate is lower
through out the horizon. This, then, is a rea¢ rmation of the central result of our paper: quantitative easing
acts as a commitment device during a liquidity trap situation.
36
dr
0
10
20
30
40
t
50
0.002
0.004
0.006
0.008
Fig 22: Change in real interest rate with an increased central bank balance sheet size and initial
level of net asset duration. Each line represents the response when the e¢ cient rate of interest
returns to its steady-state in that period.
4.2.3
Capital losses from reneging on optimal policy
Like with the consolidated government budget constraint model, we now provide an alternative illustration
of the mechanisms behind our separate central bank balance sheet model. This is done to show capital losses
on the balance sheet on its balance sheet if it were to renege on optimal policy. These balance sheet losses
would be costly for the central bank in the model as it would reduce remittances to the treasury too sharply.
We thus conduct the following thought experiment: suppose that once the liquidity trap is over, the
central bank reneges on the path for in‡ation and output dictated by optimal policy under discretion and
instead perfectly stabilizes them at zero. In such a situation, how large are capital losses, or equivalently,
how much lower are transfers to the treasury out of the zero lower bound compared to if the government had
continued to follow optimal policy? In particular, is this decline in transfers more when the central bank’s
balance sheet is bigger ? We show in Fig. 23 the change in transfers to the treasury if the central bank
were to renege on optimal policy at balance sheet sizes of 111; 231, and 351 respectively. As is clear, the
decrease in transfers out of zero lower bound is higher at a larger balance sheet of the central bank. Thus
indeed, increasing the size of its balance sheet provides the central bank with more an incentive to keep the
real interest rate low in future in order to avoid having to su¤er from costly capital losses.
37
Fig 23: Decrease in transfers to treasury from reneging on optimal policy at di¤erent levels of sizes of the
central bank balance sheet
4.2.4
Quantitative assessment of MEP
Given the parameter estimates in Tables 3 and 4 based on QE2 and its e¤ects on expected in‡ation and
future short-term interest rates, we now conduct an assessment of the macroeconomic e¤ects of MEP. This
involved an increase in the duration of assets held by the central bank. We compute the average maturity of
the treasuries on the Federal Reserve’s balance sheet at September 2011 (13:6 q) and September 2013 (18:95
q). The di¤erence between the maturities is 5:4 q. Figs. 24-25 show the associated increase in output and
in‡ation while Fig. 26 shows the resulting decrease in the real interest rate due to this policy intervention.
38
dY
0.0012
0.0010
0.0008
0.0006
0.0004
0.0002
0
10
20
30
40
t
50
Fig 24: Change in output with an initial size of the central bank balance sheet size and increased
net asset duration. Each line represents the response when the e¢ cient rate of interest returns to
its steady-state in that period.
d
0.00025
0.00020
0.00015
0.00010
0.00005
0
10
20
30
40
t
50
Fig 25: Change in in‡ation with an initial size of the central bank balance sheet size and
increased net asset duration. Each line represents the response when the e¢ cient rate of interest
returns to its steady-state in that period.
39
dr
0
10
20
30
40
t
50
0.001
0.002
0.003
0.004
0.005
0.006
Fig 26: Change in the real interest rate with an initial size of the central bank balance sheet size
and increased net asset duration. Each line represents the response when the e¢ cient rate of
interest returns to its steady-state in that period.
5
Extensions
So far, we have focused on analyzing a situation where the duration of government debt is reduced onceand-for all. That is, we have studied comparative statics experiments with respect to : A natural question
that arises in this context is whether there is an incentive for the government to increase the duration of
debt once the economy has recovered and if by not considering that, we are overstating our results. Another
interesting extension to consider is whether the particular way we have modeled the term-structure, using a
perpetuity bond that pays geometrically declining coupons, might be overly restrictive and driving some of
our results. While this was a very convenient modeling device, in particular while calibrating the model to
various average debt durations using a single parameter, it is natural to explore a model with zero-coupon
nominal government bonds with …nite maturity.
5.1
Time-varying optimal duration
To address the …rst question, we now extend our model to allow the government (we only consider the
consolidated government model here for brevity) to pick the duration of government debt optimally, period
by period. That is, now,
t (Bt ) which pays
j
t
is time-varying. In particular, the government issues a perpetuity bond in period
dollars j +1 periods later. Following very similar manipulations as for the …xed duration
case, the ‡ow budget constraint of the government can be written as
St ( t )bt = (1 +
t 1 Wt ( t 1 )) bt 1
40
1
t
+ (F
Tt )
(32)
j
t
where St ( t ) is the period-t price of the government bond that pays
Wt (
bt =
t 1 ) is the period-t price
Bt
Pt : Given these types of
of the government bond that pays
dollars j + 1 periods later while
dollars j + 1 periods later. Moreover,
government bonds, the asset-pricing conditions then take the form
uC (Ct+1 ; t+1 )
uC (Ct ; t )
St ( t ) = Et
Wt (
j
t 1
t 1)
uC (Ct+1 ; t+1 )
uC (Ct ; t )
= Et
1
t+1
1
t+1
(1 +
(1 +
t St+1 ( t ))
(33)
t 1 Wt+1 ( t 1 ))
:
(34)
The rest of the model is the same as before.
The government’s instruments are now it ; Tt ; and
to bt
1;
now,
t 1
t.
Moreover, it is clear from above that in addition
is also a state variable in the model. Then, we can write the discretionary government’s
problem recursively as
J (bt
1; t 1; t)
= max [U (:) + Et J (bt ;
t ; t+1 )]
subject to the three new constraints, (32)-(34), as well as the other private sector equilibrium conditions
that are common from the model in the previous section. Here, U (:) is the utility function of the household
in (1) and J(:) is the value function.39 The detailed formulation of this maximization problem and the
associated …rst-order necessary conditions are provided in the appendix. We discuss below why we take this
non-linear as opposed to a linear-quadratic approach.
We proceed by computing the non-stochastic steady-state and then taking a …rst-order approximation
of the non-linear government optimality conditions as well as the non-linear private sector equilibrium
conditions around the steady-state. Of particular note is that a …rst-order approximation of (32)-(34) leads
to
1
v^t =
v^t
1
1
^t
^t
)L
(1
T^t
(35)
where
v^t = ^bt +
^t
1
^ t = ^{t +
L
S^t
^ t = ^t
W
(36)
^ t+1
Et L
^t
1:
Thus, after undertaking a transformation of variables as given by (36), (35) takes the same form as (26), the
linearized government budget constraint when there was no time variation in duration. Thus, time-variation
in duration does not play a separate role in government debt dynamics upto …rst order. Therefore, if we had
taken a linear-quadratic approach, like before, then with the quadratic loss-function (28) and the linearized
private sector equilibrium conditions including (35), we would not be able to solve for optimal debt duration
dynamics at all.
Given this appropriate rede…nition of the state variable, we can show that the (bounded) solution of the
model at positive interest rates takes the form
"
3 9 Using
v^t
^t
#
=
"
0
v
1
#"
v^t
1
^t
1
#
compact notation, note that we can write the utility function as u (Ct ;
41
t)
+ g (F
s(Tt
T );
t)
v~ (Yt ;
t) :
2
where,
v;
Yv ;
v;
6
4
Y^t
t
r^t
3
2
7 6
5=4
Yv
v
rv
0
3
7
0 5
0
"
v^t
1
^t
1
#
and rv are functions of the model parameters. Here, for simplicity, we are only focussing
on solution of some of the endogenous model variables and do not consider shocks.
Two results stand out: …rst, ^t follows a random-walk like behavior; second, ^t
1
does not a¤ect directly
other other variables such as output, in‡ation, and the real interest rate. This suggests then that even if
we allow the government to pick the duration optimally period by period, there is no incentive for it to
increase the duration of debt after the economy has recovered following a liquidity trap episode. Therefore,
the simple comparative static analysis that we focussed on in the main part of the paper does not appear
to be overstating our results. We show the transition dynamics at positive interest rates with debt above
steady-state in Fig. 27 below that highlights the solution of this extended model. Given these two properties
of the solution, it is now clear that if we consider the su¢ cient state variable v^t 1 similar to ^bt 1 ; then this
model with time-varying duration would lead to identical results as our baseline model with a …xed duration.
Fig 27: Transition dynamics to an increase in debt outstanding at di¤erent initial levels of duration of debt
in a model with optimal time-varying duration of debt
5.2
Alternate model of maturity structure
We now consider a model where the government issues zero-coupon nominal bonds with …nite maturity
(again for brevity, we only consider the baseline model with a consolidated government). Since the goal of
this section is only qualitative, we consider a simple environment of one-period (BtS ) and two-period BtL
zero-coupon nominal bonds. Total bond supply then is given by the market clearing condition Bt = BtS +BtL
and we use a notation similar to before, where the ratio of the two-period to one-period bonds is given by
= BtL =Bt : Thus, a reduction in
will imply that a larger fraction of government debt is short-term. Again,
similar to the notation above, we will denote the two bond prices by (1 + it )
1
and St respectively.
We can write down the ‡ow budget constraints of the consumer and the government respectively as
42
follows
1
Pt Ct + (1 + it )
(1 + it )
1
BtS + St BtL = BtS
BtS + St BtL = BtS
1
+ (1 + it )
1
+ (1 + it )
1
1
BtL
1
Pt Tt
BtL 1 + Pt (F
Tt ) :
Note here that in writing these ‡ow budget constraints, we already impose some arbitrage restrictions. For
future purpose, using the market clearing condition we can write the government budget constraint as
1
(1 + it )
((1
) Bt ) + St ( Bt ) = (1
) Bt
1
1
+ (1 + it )
Bt
1
+ Pt (F
Tt )
or as
Bt (1
1
) (1 + it )
+ St = B t
1
1
1
+ (1 + it )
+ Pt (F
Tt )
which we can rewrite in real terms as
bt
where bt =
Bt
Pt
and
t
(1 + it )
(1
)
+ St
(1 + it )
=
Pt
Pt 1 :
1
= Et
= bt
1
1
+
1
(1 + it )
t
+ (F
Tt )
(37)
The asset pricing conditions then are given by the two relationships
uC (Ct+1 )
uC (Ct )
1
t+1
uC (Ct+1 )
uC (Ct )
; St = Et
1
t+1 (1
+ it+1 )
1
:
(38)
Like before, we can now proceed with a log-linearization around a zero-in‡ation steady-state, where we
will use as
for the steady-state ratio of taxes to market-value of debt
=
T
+
b (1
:
)
Then, (38) log-linearized gives
S^t =
(^{t + Et^{t+1 )
(39)
which is an illustration of the expectation hypothesis in the model while (37) log-linearized gives
^bt =
1^
bt 1
1
t
1
+
(1
2
)+
it
(1
S^t
)+
T^t :
(40)
Further, using (39), one can then write (40) as
^bt =
1^
bt 1
1
t
+
1
(1
2
)+
t
+
(1
)+
( t + Et
t+1 )
T^t :
To get some insights on how the maturity composition of debt a¤ects the roll-over incentives of the
government in determining the short-term interest rate, denote by
=
Et
t+1
t
; the equilibrium rate of
mean reversion of debt. That is, suppose that bt is the only state variable (as is the case in this simple
model), which in the stationary Markov-perfect equilibrium solution with no shocks, gives ^bt = bb^bt 1 and
t
=
b bt 1 .
Then this rate is constant over time and equal to
= bb < 1. This rate is, in general equilibrium,
an endogenous function of the model parameters (including ). But for now, we will next take it as given to
gain some insights using only the government budget constraint, like we did before to illustrate the intuition
of our mechanism.
43
We can then re-write the …nal log-linearized government budget constraint as
^bt =
where
( ; ; )=
1
1^
bt 1
+
1
(Note that in a stationary equilibrium,
1
t
1
(1 +
+
)
+
t
T^t
< 1 for <
1
and
> 0:
< 1). What we call theroll-overincentive in the paper is captured
by how much the current, short-term interest rate a¤ects debt/tax dynamics in the ‡ow budget constraint.
That is, we are interested in how, in the budget constraint
^bt =
1^
bt 1
1
t
+
t
T^t
is a¤ected by : In particular, for our mechanism to be at work, we need
Here, since we hold
to decrease as
increases.
constant in this thought-experiment, we have directly that depends negatively on
@ ( ; ; )
=
@
((
1
< 0:
1) + 1)2
Thus, longer is the maturity composition of government debt, lower is the e¤ect on debt dynamics of the
current, short-term interest rate. This is the main mechanism behind the results in our paper and it continues
to be in operation in this alternate model of the term structure.
6
Conclusion
We present a theoretical model where open market operations that reduce the duration of outstanding
government debt or change the size/composition of the central bank’s balance sheet, so called “quantitative
easing,” are not neutral because they a¤ect the incentive structure of the central bank. In particular, in a
Markov-perfect equilibrium of our model, reducing the duration of outstanding government or increasing
the duration of assets held by an independent central bank, provides an incentive for the central bank to
keep short-term interest rates low in future in order to avoid balance sheet losses. When the economy is
in a liquidity trap, such a policy is thus e¤ective at generating in‡ationary expectations and lowering longterm interest rates, which in turn, helps mitigate the de‡ation and negative output gap that would ensue
otherwise. In other words, quantitative easing is e¤ective because it provides a “signal”to the private sector
that the central bank will keep the short-term real interest rates low even when the zero lower bound is no
longer a constraint in future.
In future work, it would be of interest to evaluate fully the quantitative importance of our model mechanism in a medium-scale sticky price model, along the lines of Christiano, Eichenbaum, and Evans (2005)
and Smets and Wouters (2007). Computation of Markov-perfect equilibrium under coordinated monetary
and …scal policy at the ZLB appears to not have been investigated for such models in the literature If one
departs from the assumption of an e¢ cient steady-state, which would preclude a linear-quadratic approach,
this extension is likely to involve a substantial computation innovation. Moreover, as a methodological extension, it would be fruitful to allow for time-varying duration of government debt to a¤ect real variables.
To do so, it will be necessary to take a higher order approximation of the equilibrium conditions and modify
the guess-and-verify algorithm to compute the Markov-perfect equilibrium accordingly.
44
References
[1] Adam, Klaus, and Roberto M. Billi (2007), “Discretionary Monetary Policy and the Zero Lower Bound
on Nominal Interest Rates,” Journal of Monetary Economics, 54, 728-752.
[2] Alvarez, Fernando, Patrick J. Kehoe, and Pablo A. Neumeyer (2004), “The Time Consistency of Optimal
Monetary and Fiscal Policies,” Econometrica, 72, 541-567.
[3] Barro, Robert J. (1979), “On the Determination of the Public Debt,” Journal of Political Economy,
87, 940-971.
[4] Bauer, Michael D., and Glenn D. Rudebusch (2013), “The Signaling Channel for Federal Reserve Bond
Purchases,” working paper.
[5] Berriel, Tiago C., and Saroj Bhattarai (2009), “Monetary Policy and Central Bank Balance Sheet
Concerns,” BE Journal of Macroeconomics, 9.
[6] Calvo, Guillermo A., and Pablo E. Guidotti (1990), “Credibility and Nominal Debt: Exploring the Role
of Maturity in Managing In‡ation,” IMF Sta¤ Papers, 37, 612-635.
[7] Calvo, Guillermo A., and Pablo E. Guidotti (1992), “Optimal Maturity of Nominal Government Debt:
An In…nite-Horizon Model,” International Economic Review, 37, 612-635.
[8] Chadha, Jagjit S., Philip Turner, and Fabrizio Zampoli (2013), “The Interest Rate E¤ects of Government Debt Maturity,” working paper.
[9] Chen, Han, Vasco Curdia, and Andrea Ferrero (2012), “The Macroeconomic E¤ects of Large-scale Asset
Purchase Programmes,” Economic Journal, 122, 289-315.
[10] Christiano, Lawrence J., Martin Eichenbaum, and Charles Evans (2005), “Nominal Rigidities and the
Dynamic E¤ects of a Shock to Monetary Policy,” Journal of Political Economy, 113, 1-45.
[11] Del Negro, Marco, and Christopher A. Sims (2015), “When Does a Central Bank Balance Sheet Require
Fiscal Support,” working paper.
[12] Eggertsson, Gauti B. (2006), “The De‡ation Bias and Committing to Being Irresponsible,” Journal of
Money, Credit, and Banking, 38, 283-321.
[13] Eggertsson, Gauti B., and Paul Krugman (2012), “Debt, Deleveraging, and the Liquidity Trap: A
Fisher-Minsky-Koo Approach,” Quarterly Journal of Economics, 127, 1469-1513.
[14] Eggertsson, Gauti B., and Michael Woodford (2003), “The Zero Bound on Interest Rates and Optimal
Monetary Policy,” Brookings Papers on Economic Activity, 34, 139-235.
[15] Gagnon, Joseph, Matthew Raskin, Julie Ramache, and Brian Sack (2011), “The Financial Market
E¤ects of the Federal Reserve’s Large-Scale Asset Purchases,”International Journal of Central Banking,
7, 3-43.
[16] Hall, Robert E., and Ricardo Reis (2013), “Controlling In‡ation and Maintaining Central-Bank Solvency
under New-Style Central Banking,” working paper.
45
[17] Hamilton, James D., and Cynthia Wu (2012), “The E¤ectiveness of Alternative Monetary Policy Tools
in a Zero Lower Bound Environment,” Journal of Money, Credit, and Banking, 44, 3-46.
[18] Jeanne, Olivier, and Lars E. O. Svensson (2007), “Credible Commitment to Optimal Escape from a
Liquidity Trap: The Role of the Balance Sheet of an Independent Central Bank,” American Economic
Review, 97, 474-490.
[19] Krishnamurthy, Arvind, and Annette Vissing-Jorgensen (2011), “The E¤ects of Quantitative Easing
on Interest Rates: Channels and Implications for Policy,” Brookings Papers on Economic Activity, 43,
215-287.
[20] Krugman, Paul (1998), “It’s Baaack: Japan’s Slump and the Return of the Liquidity Trap,”Brookings
Papers on Economic Activity, 29, 137-206.
[21] Lucas, Robert, and Nancy Stokey (1983), “Optimal Fiscal and Monetary Policy in an Economy without
Capital,” Journal of Monetary Economics, 12, 55-93.
[22] Maskin, Eric, and Jean Tirole (2001), “Markov Perfect Equilibrium I: Observable Actions,”Journal of
Economic Theory, 100, 191-219.
[23] Mussa, Michael (1981), “The Role of O¢ cial Intervention,” Group of Thirty Occasional Paper.
[24] Persson, Mats, Torsten Persson, and Lars E. O. Svensson (1987), “Time Consistency of Fiscal and
Monetary Policy,” Econometrica, 55, 1419-1431.
[25] Persson, Mats, Torsten Persson, and Lars E. O. Svensson (2006), “Time Consistency of Fiscal and
Monetary Policy: A Solution,” Econometrica, 74, 193-212.
[26] Rotemberg, Julio (1982), “Sticky Prices in the United States,”Journal of Political Economy, 90, 11871211.
[27] Sims, Christopher A. (2011), “Stepping on a Rake: The Role of Fiscal Policy in the In‡ation of the
1970s,” European Economic Review, 55, 48-56.
[28] Sims, Christopher A. (2013), “Paper Money,” American Economic Review, 103, 563-584.
[29] Smets, Frank, and Raf Wouters (2007), “Shocks and Frictions in US Business Cycles: A Bayesian DSGE
Approach,” American Economic Review, 97, 586-606.
[30] Swanson, Eric T, and John C. Williams (2013), “Measuring the E¤ect of the Zero Lower Bound on
Medium-and Longer-Term Interest Rates,” working paper.
[31] Wallace, Neil (1981), “A Modigliani-Miller Theorem for Open-Market Operations,”American Economic
Review, 71, 267-274.
[32] Werning, Ivan (2012), “Managing a Liquidity Trap: Monetary and Fiscal Policy,” working paper.
[33] Woodford, Michael (2001), “Fiscal Requirements for Price Stability,” Journal of Money, Credit, and
Banking, 33, 669-728.
[34] Woodford, Michael (2003), Interest and Prices. Princeton, NJ: Princeton University Press.
[35] Woodford, Michael (2012), “Methods of Policy Accommodation at the Interest-Rate Lower Bound,”in
The Changing Policy Landscape, Federal Reserve Bank of Kansas City.
46
7
Appendix
7.1
7.1.1
Model
The perpetuity nominal bond and the ‡ow budget constraint
Following Woodford (2001), the perpetuity issued in period t pays j dollars j + 1 periods later, for each j 0 and some decay
1 : The implied steady-state duration of this bond is then (1
factor 0
<
) 1:
Let the price of a newly issued bond in period t be St ( ) : Given the existence of the unique stochastic discount factor
Qt;t+j , we can write this price as
1
X
St ( ) = Et
Qt;t+j j 1 :
j=1
O ( )
Now consider the period t + 1 price of such a bond that was issued in period t: We can then write the price St+1
O
St+1
( ) = Et+1
1
X
Qt+1;t+j
j
1
:
j=2
Note …rst that
O
St+1
( ) = St+1 ( )
since
St+1 ( ) = Et+1
1
X
Qt+1;t+j
(41)
j
2
:
j=2
This is highly convenient since it implies that one needs to keep track, at each point in time, of the equilibrium price of only
one type of bond.
Next, we derive an arbitrage condition between this perpetuity and a one-period bond. By simple expansion of the in…nite
sums above and manipulation of the terms, one gets
h
i
O
St ( ) = [Et Qt;t+1 ] + Et Qt;t+1 St+1
( ) :
Since
1
1 + it
Et Qt;t+1 =
we get
h
i
1
O
+ Et Qt;t+1 St+1
( ) :
1 + it
St ( ) =
O ( )= S
Substituting further for St+1
t+1 ( ); we then derive
1
+ Et [Qt;t+1 St+1 ( )] :
1 + it
St ( ) =
(42)
Finally, consider the ‡ow budget constraint of the government
BtS + St ( )Bt = (1 + it
S
1 ) Bt 1
+
0
+ StO ( ) Bt
1
+ Pt (Ft
Tt ) :
O ( )= S
This can be simpli…ed using St+1
t+1 ( ) to
BtS + St ( )Bt = (1 + it
S
1 ) Bt 1
+ (1 + St ( )) Bt
1
+ Pt (Ft
Tt ) :
This is the form in which we write down the ‡ow budget constraint of the household and the government in the main text.
7.1.2
Functional forms
We make the following functional form assumptions on preferences and technology
u (C; ) = C
v (h(i); ) =
47
1
C1
1
1
1
h(i)1+
1+
g (G; ) = G
1
G1
1
1
1
y(i) = h(i)
d( ) = d1 (
1)2
S T = s1 (T
T )2
where we only consider a discount factor shock . Note that
that Y = 1 as well. This implies that we can derive
v~ (Y; ) =
7.2
= 1 in steady-state and that in steady state, we scale hours such
1
1+
Y
1+
:
E¢ cient equilibrium
As benchmark, we …rst derive the e¢ cient allocation.
Using Gt = Ft s(Tt T ) = F s(Tt T ); the social planner’s problem can be written as
max u (Ct ;
t)
+ g (F
s(Tt
T ))
v~ (Yt )
T ))
v~ (Yt )
st
Yt = Ct + F:
Formulate the Lagrangian
Lt = u (Ct ;
+
1t
t)
(Yt
+ g (F
Ct
s(Tt
F)
FOCs (where all the derivatives are to be equated to zero)
@Lt
= v~Y + 1t
@Yt
@Lt
= uC + 1t [ 1]
@Ct
@Lt
s0 (Tt T )
= gG
@Tt
Eliminating the Lagrange multiplier gives
uC = v~Y
gG
s0 (Tt
T ) = 0:
Note that we make the following functional form assumptions on the tax collection cost
s(0) = 0; s0 (0) = 0:
Thus, when taxes are at steady state, that is, Tt = T; then s(Tt T ) = s0 (Tt
E¢ cient allocation thus requires
uC = v~Y
Tt = T:
In steady state, without aggregate shocks, we have
Y =C+F
uC = v~Y
Tt = T:
48
00
T ) = 0: But note that we will allow for s (0) > 0:
7.3
7.3.1
Non-linear markov equilibrium
Optimal policy under discretion
The policy problem can be written as
J (bt
1; t)
t; t)
= max [U (
+ Et J (bt ;
t+1 )]
st
St ( )bt = (1 + St ( )) bt
1
1
t
uC (Ct ;
1 + it =
fte
Tt ) :
+ (F
t)
it
0
1
St ( ) =
uC (Ct ;
"
"Yt
1
(1 + s) uC (Ct ;
"
v~y (Yt ;
t)
t)
t)
gte
+ uC (Ct ;
0
t) d
(
t)
Yt = Ct + F + d ( t )
h
i
1
= f e (bt ; t )
= Et uC (Ct+1 ; t+1 ) t+1
h
i
1
gte = Et uC (Ct+1 ; t+1 ) t+1
(1 + St+1 ( )) = g e (bt ;
t
= het
fte
het
= Et uC (Ct+1 ;
0
t+1 ) d
(
t+1 )
t+1
e
= h (bt ;
t)
t)
Formulate the period Lagrangian
Lt = u (Ct ;
+
1t
+
2t
+
3t
+
4t
+
5t
t)
+ g (F
St ( )bt
( gte
(1 + St ( )) bt
uC (Ct ;
het
"Yt
Ct
t ) St (
"
1
"
F
d(
1t
fte
f e (bt ;
t)
2t
(gte
g e (bt ;
t ))
+
+
+
e
3t ht
+
1t
(it
T ))
v~ (Yt ) + Et J (bt ;
1
1
t
(F
t+1 )
Tt )
uC (Ct ; t )
1 + it
fte
(Yt
s(Tt
e
h (bt ;
))
(1 + s) uC (Ct ;
t ))
t)
0)
49
t)
v~y (Yt ;
t)
uC (Ct ;
0
t) d
(
t)
t
First-order conditions (where all the derivatives should be equated to zero)
@Ls
@ t
@Ls
@Yt
@Ls
@it
@Ls
@St
@Ls
@Ct
@Ls
@Tt
@Ls
@bt
@Ls
@fte
@Ls
@gte
@Ls
@het
h
(1 + St ( )) bt
=
1t
=
v~Y +
=
2t
=
1t
h
h
uC (1 + it )
bt
= uC +
bt
h
2t
s0 (Tt
= gG
"
"
4t
= Et Jb (bt ;
=
2t
+
1t
=
3t
+
2t
=
4t
+
3t
1
2
1
t
1
"
2
i
1
t
+
i
t+1 )
+
+
4t
h
00
uC d
(1 + s) uC
0
uC d
t
i
+
+ "Yt v~yy + "~
vy +
d0
5t
5t
1t
+
uCC (1 + it )
T) +
i
3t
[ uC ]
1
i
[ uCC St ( )] +
+
3t
[St ( )] +
1t
4t
"Yt
"
1
"
(1 + s) uCC
uCC d0
t
+
5t
[ 1]
1t
1t
fbe +
2t
[ gbe ] +
3t
heb
The complementary slackness conditions are
0; it
1t
While the envelope condition is
Jb (bt
1; t)
=
1t
This also implies that
Et Jb (bt ;
7.3.2
t+1 )
= Et
0;
h
1t+1
1t it
=0
(1 + St ( ))
h
1
t
i
(1 + St+1 ( ))
1
t+1
i
Steady-state
A Markov-perfect steady-state is non-trivial to characterize because generally, we need to take derivatives of an unknown
function, as is clear from the FOCs. Here, we will rely on the fact that given an appropriate production subsidy, the Markovperfect steady-state will be the same as the e¢ cient steady-state derived above.
First, note that this requires no resource loss from price-adjustment costs, which in turn requires
d( ) = 0
and thereby ensures
Y =C+F
This means that we need
= 1:
Also, this implies
d0 ( ) = 0:
Next, note from the Phillips curve that this means, we need
"
1
"
(1 + s) uC
v~y = 0:
Now, since the e¢ cient steady-state has uC = v~Y ; the production subsidy then has to satisfy
"
1
"
(1 + s) = 1:
50
We will be looking at a steady-state with positive interest rates
1
1+i=
which means that
1
=0
2
= 0:
and that from the FOC wrt it we have
Also, given that taxes are at steady-state, gG ( s0 (Tt
T )) = 0; from the FOC wrt Tt
1
= 0:
3
= 0:
Given this, in turn, we have from the FOC wrt St
Then, given that
d0
= 0 in steady-state and
d00
is not, and since
1
= 0; it gives from the FOC wrt to
t
= 0:
4
Note that this is highly convenient since these Lagrange multipliers being zero implies
1
=
=
2
3
= 0:
Thus, we do not need to worry about the derivatives of the unknown functions.
This proposed steady-state is consistent with other FOCs. For example, the FOC wrt Yt is now given by
5
= v~Y
and that the FOC wrt Ct is given by
uC =
5
which implies
v~Y = uC :
Finally, FOC wrt bt implies
Jb =
1
(1 + S( ))
1
=0
which is also consistent with the conjectured guess.
Finally, the guess of the steady-state is also consistent with the other model equilibrium conditions, with S( ) given by
S( ) =
[(1 + S( ))]
that is
S( ) =
1
:
Then b and F are linked by
S( )b = (1 + S( )) b + (F
that is
T =F+
7.3.3
1
1
T)
b:
First-order approximation
We now take a log-linear approximation of the Markov perfect FOCs and the private sector equilibrium conditions around
the steady-state above. Also, lets normalize the scale of the economy (with appropriate scaling of hours) so that Y = 1: This
implies C = 1 F: Also the shock t takes a value of 1 in steady-state.
Private sector equilibrium conditions We …rst start with the private sector equilibrium conditions. We denote
variables that are in log-deviations from their respective steady-states by hats, except for ^{t :We denote variables in steady-state
by bars.
First,
Yt = Ct + F + d ( t )
51
gives
^t
Y^t = C C
Second,
"Yt
"
1
"
(1 + s) uC (Ct ;
t)
v~y (Yt ;
t)
+ uC (Ct ;
t) d
0
(
t)
t
= Et uC (Ct+1 ;
0
t+1 ) d
(
t+1 )
t+1
gives
"vyy Y^t "vy ^t + "uC ^t = uC d00 Et ^t+1
^t
uC d00 ^t + "uCC C C
which can be simpli…ed by making use of the log-linearized resource constraint above to yield
vyy ) Y^t = uC d00 Et ^t+1
uC d00 ^t + " (uCC
^t = Et ^t+1 +
" (vyy uCC ) ^
Yt :
uC d00
Third,
1 + it =
gives
uC (Ct ;
h
Et uC (Ct+1 ;
t)
t+1 )
1
t+1
i
^t+1 + uC Et ^t+1
^t + uC ^t = uC ^{t + uCC Et C C
uCC C C
uC Et ^t+1
which can be simpli…ed by making use of the log-linearized resource constraint above to yield
uCC Y^t + uC ^t = uC ^{t + uCC Et Y^t+1 + uC Et ^t+1
uC Et ^t+1
and
uC h ^
uC
Y^t = Et Y^t+1 +
[^{t Et ^t+1 ] +
Et t+1
uCC
uCC
Note here that this implies that the e¢ cient rate of interest is given by
i
uC h ^
rte =
Et t+1 ^t :
uC
Fourth,
St ( ) =
1
uC (Ct ;
t)
gives
^t + S uC ^t + uC S S^t =
S uCC C C
h
Et uC (Ct+1 ;
1
t+1
t+1 )
i
(1 + St+1 ( ))
1 + S uC Et ^t+1
^t+1 +
1 + S uCC Et C C
^t
i
1 + S uC Et ^t+1 +
S uC Et S^t+1
which can be simpli…es by making use of the log-linearized resource constraint above to yield
S uCC Y^t + S uC ^t + uC S S^t =
1 + S uC Et ^t+1
1 + S uCC Et Y^t+1 +
1 + S uC Et ^t+1 +
Note here that by using the log-linearized Euler equation above, one can further simplify as
h
i
S uC ^{t + uCC Et Y^t+1 + uC Et ^t+1 uC Et ^t+1 + uC S S^t
=
1 + S uCC Et Y^t+1 +
1 + S uC Et ^t+1
1 + S uC Et ^t+1 +
S uC Et S^t+1
or
h
uC ^{t + uCC Et Y^t+1 + uC Et ^t+1
=
1+ S
S
uCC Et Y^t+1 +
i
uC Et ^t+1 + uC S^t
1+ S
S
uC Et ^t+1
1+ S
S
uC Et ^t+1 +
Moreover, since
1+ S
=1
S
we have …nally as the asset-pricing condition
^{t + S^t =
Et S^t+1 :
Fifth,
St ( )bt = (1 + St ( )) bt
52
1
1
t
+ (F
Tt )
uC Et S^t+1 :
S uC Et S^t+1 :
gives
^bt + S^t = S^t + 1 + S ^bt
S
1+ S
1^
bt
) S^t
^t
S
T ^
Tt
Sb
which is simpli…ed further as
^bt =
1
Then, …nally, the expectation functions are given by
h
f^te = Et uC (Ct+1 ; t+1 )
h
g^te = Et uC (Ct+1 ;
t+1 )
1
t+1
1
t+1
i
^t
(1
T ^
Tt :
Sb
= uCC Et Y^t+1 + uC Et ^t+1
uC ^t+1
i
(1 + St+1 ( )) = 1 + S uCC Et Y^t+1 + 1 + S uC Et ^t+1
^ e = Et uC (Ct+1 ;
h
t
0
t+1 ) d
(
t+1 )
1 + S uC Et ^t+1 + S uC Et S^t+1
= uC d00 Et ^t+1
t+1
Markov-perfect FOCs Here, note that since all the Lagrange multipliers except one are zero in steady-state, what we
mean by hats will in fact only be deviations from steady-state for all the lagrange multipliers (as opposed to log-deviations).
First,
h
i
h
i
00
0
2
+ 4t
uC d t uC d + 5t
d0 = 0
1t (1 + St ( )) bt 1 t
gives
00
uC d ^4t
1 + S b ^1t
and since
5
= v~y = uC
00
5 d ^t
00
uC d ^4t
1 + S b ^1t
=0
uC d00 ^t = 0:
Second,
v~Y +
"
"
4t
1
"
+ "Yt v~yy + "~
vy +
(1 + s) uC
5t
=0
gives
vY Y Y Y^t + "Y vyy ^4t + ^5t = 0:
Third,
2t
gives
h
uC (1 + it )
2^
2t
uC
Fourth,
1t
gives
h
bt
bt
1
i
+
1t
=0
+ ^1t = 0:
1
t
i
) b ^1t
(1
2
+
3t
[ uC ] = 0
uC ^3t = 0:
Fifth,
uC +
2t
h
uCC (1 + it )
1
i
+
3t
[ uCC St ( )] +
4t
"Yt
"
1
"
(1 + s) uCC
uCC d0
t
+
5t
gives
uC ^
Y^t +
t
uCC
^2t
S ^3t
1 ^
5t = 0:
uCC
" ^4t
Sixth,
s0 (Tt
gG
T) +
1t
=0
gives
00
gG s T T^t + ^1t = 0:
Seventh (after some replacements),
h
Et 1t+1
(1 + St+1 ( ))
gives
1
t+1
i
+
1t
[St ( )] +
2t
fbe +
3t
[gbe ] +
SEt ^1t+1 + S ^1t + fb ^2t + gb ^3t + hb ^4t = 0:
53
4t
heb = 0
[ 1] = 0
7.4
Linear-quadratic approach
7.4.1
Linear approximation of equilibrium conditions
We approximate around an e¢ cient non-stochastic steady-state where = 1: For simplicity, from here on we will assume that
the only shock that hits the economy is a discount factor shock given by . Standard manipulations that are prevalent in the
literature, for example in Woodford (2003), and as shown above in the Markov perfect equilibrium, give (24) and (25) where
( 1+ )
C
and k = "
. Here we again detail the derivations of (26) and (27). Given
= ~Y
00
d
1
uC (Ct ;
St ( ) =
h
Et uC (Ct+1 ;
t)
and
uC (Ct ;
h
Et uC (Ct+1 ;
1 + it =
1
t+1
t+1 )
i
(1 + St+1 ( )) ;
t)
1
t+1
t+1 )
i
1,
and the functional form assumptions above together with in steady state 1 + i =
S^t =
log-linearization gives immediately
Et S^t+1 :
^{t +
Next, given
St ( )bt = (1 + St ( )) bt
and that we assume Ft = F and have from steady state S =
^bt =
1^
bt 1
t
1
1
b; log-linearization gives immediately
T^t :
)S^t
(1
t
Tt )
+ (Ft
; T =F+
1
1
1
1
Note that also the following relationship holds in steady state F = G: We …nally derive an expression for rte ; the e¢ cient rate
of interest
rte = ~ 1 ( t Et t+1 ) :
7.4.2
Quadratic approximation of household utility
For household utility, we need to approximate the following three components
u (Yt
F
d(
g (F
t ); t ) ;
S(Tt
T );
t) ;
v (Yt ;
t) :
Standard manipulations that are prevalent in the literature, for example in Woodford (2003), give as a second-order approximation to household utility
1
2
1
Y
2
T^t2 F
Y + F ^t + F
Y s00 T + F d00 (1) ^ 2t + 2 Y^t
F
d00 (1) ^ 2t
+ ^t
F Y
2Y^t
F Y
2
1
Y
!
+
2
1
!
1
2
2
+ Y^t2
+
^t + 1 Y^t Y
12
2
^t + 1 Y
+1
+1
1
2
Y^t2 Y
1
+ tip
which is in turn given as
1
2
Y^t2
1
F
Y
Y
1
+ T^t2
2 ^t + 1 Y^t
s00 T
Y
1
d00 (1) ^ 2t
+ tip:
1
~Y ), together with the scaling
Now lets multiply everything by +~
1 and consider e¢ cient equilibrium in steady-state (uC = v
that Y = 1 and Y = C + F = 1; to get
~ T^t2 s00 T
~ d00 (1) ^ 2t
Y^t2
2 ( ~ + 1)
2 ( ~ + 1)
2
So, …nally, we get as approximation
where
h
2
t
T
+ Y^t2 +
=
^2
T Tt
s00 T
+~
54
1
i
d00 (1)
"
= :
( + ~ 1)
=
7.4.3
Markov-perfect equilibrium at positive interest rates
Given the Lagrangian where the expectation functions are substituted and the shocks are suppressed
Lt =
+
1
(
2
2
t
+ Y^t2 +
Y^t
2t [ t
^2
T Tt )
b bt ]
^
e
)+
+ Et V (bt ; rt+1
+
1
3t [bt
bt
1t [Yt
1
+
1
Yb bt + ^{t
)S^t + T^t ] +
+ (1
t
rte ]
b bt
^ + ^{t
4t [St
S b bt ]
the …rst-order necessary conditions are given by
@L
=
@ t
t
+
2t
1
+
3t
=0
@L
= Yt + 1t
2t = 0
@Yt
@L
= T Tt +
3t = 0
@Tt
@L
=
1t + 4t = 0
@it
@L
= 3t (1
) + 4t = 0
@St
@L
e
= Et Vb (bt ; rt+1
)
@bt
while the envelope condition is given by
(Yb +
b ) 1t
e
1 ; rt )
Vb (bt
b 2t
1
=
+
Sb
3t
4t
=0
3t
which implies
e
Et Vb (bt ; rt+1
)=
1
Et
3t+1 :
We can then combine the envelope condition with the last FOC to yield
Et
(Yb +
3t+1
b ) 1t
+
b 2t
3t
Sb
4t
= 0:
Sb
4t
=0
To summarize, we have
+
2t
Yt +
1t
t
T Tt
2t
+
3t (1
(Yb +
3t+1
b ) 1t
Y^t = Et Y^t+1
t
1
bt =
bt
4:t
)+
=0
4t
(^{t
S^t =
=0
Et
= Y^t + Et
1
=0
b 2t
1
=0
3t
=0
3t
1t +
Et
1
+
+
rte )
t+1
t+1
)S^t
(1
t
3t
^{t +
Et S^t+1
1
)
T^t
which can be simpli…ed to get
t
[1
b
1
(1
)
1
+
(Yb +
1
Yt = [
b ) (1
)
Y^t = Et Y^t+1
t
bt =
1
bt
(1
1
+
(^{t
1
= Y^t + Et
1
S^t =
1
t
^{t +
55
]
bS
T
)]T^t =
Sb (1
Et
1
+
t+1
rte )
t+1
(1
)S^t
Et S^t+1 :
T^t
^
T Tt
( )
1
T
b
1
Yt + Et T^t+1
The …nal step is then to match coe¢ cients after replacing the conjectured solutions
b bt 1
[1
b
1
(1
1
)
1
+
(Yb +
(Yb bt
b ) (1
Yb bt
1)
1
)
b bt 1
bb bt
1
1
=
bt
[Yb bt
1
1
1]
(ib bt
1]
1)
+
b
1
+
)] [Tb bt
b bt 1 ]
[
=
1
)
((ib bt
1]
=
1
Sb bt
(1
Sb (1
+
= Yb [bb bt
1
1
=[
[
b
bS
T
=
(Tb bt
T
1
( )
[bb bt
[bb bt
T
1)
1
b
[Yb bt
1]
+ [Tb (bb bt
1 )]
1 ]])
1]
) [Sb bt
(1
Sb [bb bt
1) +
]
[Tb bt
1]
1]
1]
which in turn can be simpli…ed to get
b
[(1
1
)
1
b
1
1
+
1
Yb = [
(Yb +
1
b)
(1
b
bb =
1
Sb =
bS
T
(43)
T Tb
( )
1
T
1
b
(44)
Yb + Tb bb
(45)
b bb )
(46)
b bb
(1
b
]
) Tb =
(ib
= Yb +
1
+
Sb ] (1
+
Yb = Yb bb
1
1
)
ib +
)Sb
(47)
Tb
(48)
S b bb :
We now show some properties of b analytically. First, note that we will be restricting to stationary solutions, that is one
where j bb j< 1: Manipulations of (43)-(48) above lead to the following closed-form expression for b
b
where
=
h
= [
bb )
(1
T
lV
1
[
+
1
(1
1
)+
1
bb )] + [(1
(1
1
]
bS
T
1
)
1
(1
bb ) +
1 ] (1
bb ) (1
bb )
1
1
i
:
T
Since j bb j< 1; it is clear that b > 0 for all < 1; : What happens when > 1? Numerically, we have found that often b > 0
still, but sometimes in fact it can hit zero (and then actually go negative if is increased further). It is in fact possible to
pin-down analytically when b = 0: Note that from (46), if b = 0; then Yb = 0: This then implies that for this to be supported
as a solution for all possible values of Tb , from (43), it must be the case that
[
1
(1
1
)
+
1
]=0
which in turn implies that
1
=1+
In this knife-edge case, note that one needs
:
> 1: Moreover, since the upper bound on
1+
1
1
<
or
<1
is
1;
one needs to ensure that
:
This is a fairly restrictive parameterization (not ful…lled in our baseline, for example). Still, it is instructive to note that in
this case of when b does reach 0; then it is indeed possible to show analytically that b declines as duration is increased while
1
1
comparing = 0 with = 1 +
or = 1 with = 1 +
.
7.5
Equivalence of the two approaches
We now show the equivalence of the linearized dynamic systems for non-linear and linear-quadratic approaches.
Consider the system of equations describing private sector equilibrium and optimal government policy
uC
Y^t = Et Y^t+1 +
[^{t
uCC
^t =
Et ^t+1
rte ]
" (vyy uCC ) ^
Yt + Et ^t+1 :
uC d00
^{t + S^t =
56
Et S^t+1 :
1^
bt
^bt =
1
^t
00
uC d ^4t
1 + S b ^1t
T ^
Tt :
Sb
) S^t
(1
5d
00
^t = 0
vY Y Y Y^t + "Y vyy ^4t + ^5t = 0
uC
+ ^1t = 0
) b ^1t
(1
uC ^
Y^t +
t
uCC
2^
2t
^2t
uC ^3t = 0:
S ^3t
1 ^
5t = 0:
uCC
" ^4t
00
gG s T T^t + ^1t = 0:
SEt ^1t+1 + S ^1t + fb ^2t + gb ^3t + hb ^4t = 0:
Under additional functional assumptions outlined in previous sections we get
1
vY Y = Y
1
uCC =
1
C
=
=
1
~
uC = 1
uC = 1
gG = 1
5
=1
Y =1
The …rst four equations are equivalent to their counterparts in the LQ-approach once one use the functional form assumptions to get
uC
= ~
uCC
and introduce new notation
" (vyy uCC )
= :
uC d00
The latter relation implies
"
+~ 1
= d00 :
Let us guess solutions for all variables for the case when the ZLB is slack as a linear function of ^bt
expectations will take form
f^te = uCC Et Y^t+1 + uC Et ^t+1
g^te = 1 + S uCC Et Y^t+1 + 1 + S uC Et ^t+1
uC Et ^t+1 = f^be b^bt
1
1
^ e r^e
+h
r t
where
f^be =
g^be
=
~
1
b
1
b
1
~
1 + S Yb
1
Yb +
b
1+ S b
57
1
b
and r^te . Then the
+ f^re r^te
1 + S uC Et ^t+1 + S uC Et S^t+1 = g^be b^bt
^ e = uC d00 Et ^t+1 = h
^ e b^bt
h
t
b
1
+ SSb b
1
1
+ g^re r^te
^e = "
h
b
1
+~
1
bb
Also under the assumption about the process ^t we get
Et ^t+1 =
and
uC h ^
t
uC
r^te =
^t
i
^t = ^t (1
):
When the economy is out of ZLB r^te = 0:
Now we can …nd explicit representation for the lagrange multipliers
^1t = s00 T T^t :
^2t = 0
00
)] s T T^t :
^3t = [(1
^4t =
^5t = Y^t
00
1 + S bs T T^t
1)
"( + ~
( +~
^t
00
1 + S s T T^t + " ^t
1)
So the last two equations take the form
Y^t +
"^t =
00
Ss T T^t +
Sb
~
1
Yb
b
b
1
1
( +~
~
1)
1
S (1
00
) S bs T T^t + b
(1
1
00
b S bs
00
bs T T^t :
1+ S
)+
"
T T^t
+~
1
bb
1
00
^t = s T SEt T^t+1 :
which after straightforward manipulations become
"
Sb
1+
~
1
^t + Y^t
1
= ~
1
(1
1
)
+
1
00
Yb
b
(1
)+
Sb s T 2
T^t
T ( + ~ 1)
b
00
bS T 2 s
T ( +~
"
1)
T^t :
00
b ^t
=
s T 2 bS
Et T^t+1 :
( + ~ 1) T
These two equations are equivalent for the last two equations from the dynamic system obtained in LQ–approach once we
use the derived weights from the quadratic approximation of the loss function
00
T
=
s T2
( + ~ 1)
1
=
7.6
":
Details of the solution at positive interest rates
Provided that the expectation functions are di¤erentiable, the solution of the model is of the form
t
=
^
b bt 1
^{t = ib^bt
1
+
+
e
r rt ;
ir rte ;
Y^t = Yb^bt
T^t = Tb^bt
1
1
+ Yr rte ; S^t = Sb^bt
+
Tr rte ;
1
and ^bt = bb^bt
+ Sr rte ;
1
+
(49)
br rte
where b ; Yb ; Sb ; ib ; bb ; Tb ; r ; Yr ; Sr ; ir ; br ; and Tr are unknown coe¢ cients to be determined. We make the assumption that
e
the exogenous process rte satis…es Et rt+1
= r rte where 0 < r < 1: Then (49) implies that the expectations are given by
Et
t+1
=
^ +
b bt
e
r r rt ;
Et Y^t+1 = Yb^bt + Yr
e
r rt ;
and Et S^t+1 = Sb^bt + Sr
e
r rt :
(50)
We can then formulate the Lagrangian of the government problem. We substitute out for the expectation function using
58
(50) and suppress the shock for simplicity
Lt =
1
(
2
2
t
+ Y^t2 +
^2
T Tt )
^
Yb^bt + ^{t
^
1^
bt 1
+
1t [Yt
+
3t [bt
+
e
+ Et V (^bt ; rt+1
)
^ +
b bt ]
1
t
^
Y^t
2t [ t
b bt ]
)S^t + T^t ] +
+ (1
Sb^bt ]:
^ + ^{t
4t [St
For now, we are not analyzing the e¤ects of the shock, and hence not carrying around r ; Yr ; Sr ; ir ; br ; and Tr since our key
area of interest at this state is not the e¤ect of the shock at positive interest rates. Instead, we will start focusing on the shock
once the zero bound becomes binding.
The associated …rst order necessary conditions of the Lagrangian problem above and the envelope condition of the minimization problem of the government above are provided in the appendix. Our …rst substantiative result is that the equilibrium
conditions can be simpli…ed, in particular by eliminating the Lagrange multipliers 1t
4t ; to get
t
[1
b
1
(1
)
1
1^
Yt
+
(Yb +
b ) (1
1
=[
)
(1
1
+
)
1
Sb (1
+
1
]
1
)]T^t =
^
(51)
T Tt
( )
1
T
b
1^
Yt
+ Et T^t+1
(52)
which along with (24)-(27) de…ne the equilibrium in the approximated economy. The …nal step to computing the solution
is then to plug in the conjectured solution and to match coe¢ cients on various variables in (24)-(27), (51), and (52), along
with the requirement that expectation are rational, to determine b ; Yb ; Sb ; ib ; bb ; and Tb : The details of this step are in the
appendix.
The most important relationships emerging from our analysis are captured by (51) and (52). (51) is the so called “targeting
rule” of our model. That represents the equilibrium (static) relationship among the three target variables t ; Y^t ; and T^t that
emerges from the optimization problem of the government. (51) thus captures how target variables are related in equilibrium
as governed both by the weights they are assigned in the loss function (
and T ) as well as the trade-o¤s among them as
1+
1 ] 1 ): Note in particular that
1 represents
given by the private sector equilibrium conditions ( 1 and [ 1 (1
)
1
1
1
1
the trade-o¤ between t and Y^t as given by (25) while [
(1
)
+
] represents the trade-o¤ between t and Y^t
vs. T^t as given by the combination of (24), (25), and (26).
(52) is another optimality condition characterizing the Markov-perfect equilibrium and represents the “tax-smoothing
objective” of the government. In contrast to similar expressions following the work of Barro (1979), which would lead to taxes
being a martingale, output appears in (52) because of sticky-prices, which makes output endogenous. Finally, because of the
dynamic nature of (52), as opposed to (51), the unknown coe¢ cients that are critical for expectations of variables, b ; Yb ; and
Sb ; appear in (52). As we shall see – and this is again in contrast to the classic tax smoothing result in which debt is a random
walk – the government will in general have an incentive to pay down public debt if it is above steady-state due to strategic
reasons.
7.6.1
Polar cases on price rigidities
To clarify what is going on in the model, we …rst …nd it helpful to consider two special cases.
a) Fully ‡exible prices When prices are fully ‡exible, then as is well-known, monetary policy cannot control the
(ex-ante) real interest rate r^t : Then, the only way monetary policy can a¤ect the economy is through surprise in‡ation as debt
is nominal. In fact there is a well-known literature that addresses the issue of how the duration of nominal debt in a ‡exible
price environment a¤ects allocations under time consistent optimal monetary policy. For example, Calvo and Guidotti (1990
and 1992) address optimal maturity of nominal government debt in a ‡exible price environment while Sims (2013) explores
how the response of in‡ation to …scal shocks depends on the maturity of government debt under optimal monetary policy. We
conduct a complementary exercise here and want to characterize how in‡ation incentives depend on the duration of debt under
optimal monetary and …scal policy under discretion. Thus, we are interested in how b depends on duration of debt.
For this exercise, we can think of this special case of fully ‡exible prices as
! 1: Note however, that from the two
optimality conditions (51) and (52), while under ‡exible prices Tb = 0; there is indeterminacy in terms of in‡ation and nominal
interest rate dynamics.4 0 This is a well-known result in monetary economics under discretion in a ‡exible price environment,
and comes about because the government cannot a¤ect output and taxes can be put to zero with various combinations of
in‡ation and interest rate choices. To show our result on the role of the duration of debt, we follow the literature such as Calvo
4 0 Note
that
=
"
:
k
59
and Guidotti (1990 and 1992) and Sims (2013) and include a ( very small) aggregate social cost of in‡ation that is independent
of the level of price stickiness. Then, the objective of the government under ‡exible prices will be given by
h 0
i
2
^2
Ut =
t + T Tt
0
0
= 0:001 and the rest of the
parameterizes the cost of in‡ation that is independent of sticky prices. Using
where
parameter values from Table 1, Fig A1 shows how b depends on duration of debt (quarters). We see clearly that with shorter
duration, there is more of an incentive to use current in‡ation. The intuition for this result is that from (26), we see that
everything else the same, when (and thereby, duration) decreases, then there will be a greater period by period incentive to
increase S^t ; that is, keep nominal interest rates low to manage the debt burden. This, however, will increase current in‡ation
further in equilibrium according to our result. As a consequence –and perhaps a little counterintuitively –equilibrium nominal
interest rate will generally increase as well with lower duration since the real rate is exogenously given under ‡exible prices. In
particular, observe that ^{t = Et t+1 (since Y^t = 0 and there are no shocks) and hence it is increasing one-to-one with expected
in‡ation.
b
1.0
0.8
0.6
0.4
0.2
5
10
15
20
Duration
25
Fig A1: In‡ation response coe¢ cients at di¤erent duration of debt under ‡exible prices
b) Fully rigid prices Next we can consider the other extreme case: that of fully rigid prices. In this case, in‡ation is
zero in equilibrium and hence b = 0: Then, one can directly consider the e¤ects on the ex-ante real interest rate by analyzing
the e¤ect on the nominal interest rate since rb = ib
b bb = ib . Thus, we are interested in how ib depends on the duration
of debt. Using the parameters from Table 1, Fig A2 shows how ib depends on duration of debt (quarters). We see that with
shorter duration, there is more of an incentive to keep the nominal interest rate lower. The intuition for this result is again
that from (26), when (and thereby, duration) decreases, then there will be more of an incentive to increase S^t ; that is keep
interest rates low, to manage the debt burden.
60
ib
10
5
20
15
Duration
25
0.2
0.4
0.6
0.8
1.0
Fig A2: Nominal interest rate response coe¢ cients at di¤erent duration of debt under rigid prices
We now move to discussing in detail properties of the solution for the partially rigid price cases that underlies our baseline
calibration.
7.6.2
Real interest rate incentives and duration of outstanding debt
How does the duration of debt a¤ect the real interest rate incentives of the central bank for the partially rigid cases that
underlies our baseline calibration? The e¤ect on the real interest rate of lower duration is at the heart of the matter because
what in‡uences output is eventually the real interest rate and the aim in a zero lower bound situation is precisely to be able
to decrease the short-term real interest rate (today and in future) even when the current short-term nominal interest rate is
stuck at zero. That is, we are interested in the properties of
r^t = ^{t
Et
t+1
= (ib
^
b bb ) bt 1
= rb^bt
1
where r^t is the short-term real interest rate. This is important because in a liquidity trap situation, as is well-known, decreasing
future real interest rate is key to mitigating negative e¤ects on output, and so, if rb depends negatively on duration, then we
are able to provide a theoretical rationale for quantitative easing actions by the government.
Having established the results in the two special cases above, we now move on to the main mechanism in our paper in
the intermediate and quantitatively relevant case of partially rigid prices. Fig A3 shows how rb depends on duration of debt
(quarters). When the duration of debt in the hands of the public is shorter, it unambiguously provides an incentive to the
government to keep the short-term real interest rate lower in future. The intuition again is that by doing so, it reduces the
cost of rolling over the debt period by period. That is, if the debt is short-term, then current real short rates will more directly
a¤ect the cost of rolling the debt over period by period, while the cost of rolling over long-term debt is not a¤ected in the same
way period by period.
61
rb
5
10
15
20
Duration
25
0.2
0.4
0.6
0.8
1.0
Fig A3: Real interest rate response coe¢ cients at di¤erent duration of debt
7.6.3
In‡ation incentives and duration of outstanding debt
While the dependence of real interest rate incentive of the government on the duration of debt is the main mechanism of
our paper, given the attention in‡ation incentives receive in the literature, we now study how b varies with the duration of
outstanding government debt?4 1 Moreover, it helps us emphasize a point that just focusing on in‡ation incentives might not
be su¢ cient to understand the nature of optimal monetary and …scal policy in a liquidity trap situation.
While an analytical expression for b is available in the appendix and it is possible to show that b > 0 for all < 1, a
full analytical characterization of the comparative statics with respect to the duration of debt is not available and so we rely
on numerical results. Fig A4 below shows how b depends on duration of debt (quarters) at di¤erent levels of : As expected,
4 2 More importantly, note that that at our baseline parameterization of
= 0:02;
b is positive at all durations in the …gure.
4 3 At the same time, however, there is a hump-shaped behavior,
does
decrease
with
duration
over
a
wide
range
of
maturity.
b
with b increasing when duration is increased at very short durations.
4 1 For some recent discussion and analysis of how in‡ation dynamics in sticky-price models could depend on duration of
government debt, see Sims (2011).
4 2 Again, it is possible to show analytically that for all
< 1; b > 0: In a very similar model but one with no steady-state
debt, Eggertsson (2006) proved that b > 0 for = 0 (that is, for one period debt).
4 3 Some limited analytical results on the properties of
are available in the appendix. For example, it can
b with respect to
1
be shown that b is positive for all < 1 and that when = 1 +
(and thus, > 1); b = 0: In this sense, for a speci…c
1
case, one can show that b is declining in by comparing some extreme cases (such as = 0 with = 1 +
): Please
1 : So this case of
see the appendix for details. Note also here that the upper bound on is
b = 0 is not necessarily always
reached.
62
b
0.014
0.012
0.010
0.008
0.006
0.004
0.002
5
10
15
20
Duration
25
Fig A4: In‡ation response coe¢ cients at di¤erent duration of debt
What drives this result? First, note from (26) that everything else the same, when (and thereby, duration) decreases,
then there will be an incentive to decrease S^t ; that is keep interest rates low to manage the debt burden. This will then increase
in‡ation in equilibrium. At the same time however, the government’s incentives on in‡ation are not fully/only captured by this
reasoning. This is because what ultimately matters for the cost of debt is the real interest rate, which because of sticky prices,
is endogenous and under the control of the central bank, and which we have seen before robustly depends negatively on the
duration of debt. Therefore, to understand the overall e¤ect on the government’s in‡ation incentives, it is critical to analyze
1+
1 ] plays a key role as it captures
the targeting rule, as discussed above and given by (51). Here, the term [ 1 (1
)
1 here simply captures the role of surprise in‡ation in reducing the
the trade-o¤ between t and Y^t vs. T^t . Note …rst that
debt burden as government debt is nominal. This term would be present even when prices are completely ‡exible. The term
1 (1
1 however, appears because of sticky prices. This re‡ects how the real interest rate is a¤ected by manipulation
)
of the nominal interest rate and how it in turn a¤ects output.
1;
1 , or (1
Thus, in situations where either
) is high, this term can dominate and it can be the case that decreasing
1 is low (or when prices
the duration (or decreasing ) actually leads to a lower b : This means, for example, that when
are more ‡exible), the hump-shaped behavior of b gets restricted to very short maturities only as the channel coming from
sticky-prices is not that in‡uential. We in fact showed this above in the case of fully ‡exible prices, where in‡ation response
depends negatively throughout on duration of debt.
7.6.4
Debt dynamics and duration of outstanding debt
While the primary focus so far is on properties of rb as it determines the real interest rate incentives of the central bank, it is
also interesting to consider the properties of bb ; the parameter governing the persistence of government debt. This exercise is
interesting in its own right, but more importantly, it is also worth exploring because as explained before, what is critical is the
behavior of the real interest rate, and that gets a¤ected by bb as Et t+1 = b^bt = b bb^bt 1 : Unlike for b ; it is not possible
to show a tractable analytical solution (or any property) for bb as it is generally a root of a fourth-order polynomial equation.
We thus rely fully on numerical solutions.
Fig. A5 shows how bb depends on duration of debt. It is clear that the persistence of debt increases monotonically as
the duration increases.4 4 In fact, for a high enough duration, debt dynamics approach that of a random walk (bb = 1), as in
the analysis of Barro (1979). The persistence of debt increases with duration mainly because the response of the short-term
4 4 There is thus, no hump-shaped pattern, unlike for in‡ation. The reason is that what matters directly for persistence of
debt is the real interest rate and there is no hump-shaped pattern there as we show later. Note also that for one-period debt,
often bb is negative (even here, recall that b is still positive). A negative bb is not very interesting empirically as it implies
oscillatory behavior of debt.
63
real interest rate decreases, as we discussed above. Some of this e¤ect is re‡ected in the response of in‡ation decreasing as
also discussed above. Thus, the existence of long-term nominal debt has an important impact on the dynamics of debt under
optimal policy under discretion.
bb
1.0
0.8
0.6
0.4
0.2
10
5
Duration
20
15
25
Fig A5: Debt response coe¢ cients at di¤erent duration of debt
7.7
Computation at ZLB
1.
In our experiment the debt is kept …xed at the zero lower bound at bL . Moreover, at the zero lower bound, ^{t = 1
Given the speci…c assumptions on the two-state Markov shock process, the equilibrium is described by the system of equations
Y^t =
)^bt
b (1
^t =
)^bt +
b (1
1
^{t = 1
rte
L
L
)^bt Yb + YL ;
+ ^{t + (1
+ Yt ;
;
^bt = bL ;
S^t =
^{t +
)Sb bL + SL ) :
((1
where variables with a b subscript denote the solution we compute at positive interest rates while variables with a L subscript
denote values at the ZLB. The solution to this system is
Y^t = YL =
^t =
L
b(
(1
)
b (1
)2 bL
(bL (
(
b (1
=
)2 bL
(bL (
b(
(1
1
^{t = iL =1
S^t = SL =
1)bL
1)Yb )
1) + (
)(1
rte
))
+ iL )
)
1)Yb )
rte + iL )
;
;
; ^bt = bL ;
)Sb bL
(1
1) + (
)(1
(1
b(
^{t
1
where the solution for variables with a b subscript has already been provided above.
7.8
Non-linear markov equilibrium of extended model
We now consider a model where the duration of government debt is time-varying and chosen optimally by the government.
64
The policy problem can be written as
J (bt
1; t 1; t)
= max [U (
t; t)
+ Et J (bt ;
t ; t+1 )]
st
St ( t )bt = (1 +
t 1 Wt ( t 1 )) bt 1
uC (Ct ;
1 + it =
fte
1
t
Tt ) :
+ (F
t)
it
0
1
St ( t ) =
ge
uC (Ct ; t ) t
1
Wt ( t 1 ) =
je
uC (Ct ; t ) t
"
"Yt
1
(1 + s) uC (Ct ;
"
v~y (Yt ;
t)
+ uC (Ct ;
t)
0
t) d
(
t)
Yt = Ct + F + d ( t )
i
1
fte = Et uC (Ct+1 ; t+1 ) t+1
= f e (bt ; t ; t )
h
i
1
gte = Et uC (Ct+1 ; t+1 ) t+1
(1 + t St+1 ( t )) = g e (bt ;
h
het
= Et uC (Ct+1 ;
h
jte = Et uC (Ct+1 ;
t+1 )
t+1 ) d
1
t+1
0
(
(1 +
t+1 )
t+1
e
= h (bt ;
t 1 Wt+1 ( t 1 ))
Formulate the period Lagrangian
Lt = u (Ct ;
t)
+ g (F
St ( t )bt
s(Tt
T ))
t 1 Wt ( t 1 )) bt 1
+
1t
+
2t
+
3t
( gte
uC (Ct ;
t ) St ( t ))
4t
jte
uC (Ct ;
t ) Wt ( t 1 ))
+
+
5t
+
6t
het
(Yt
"Yt
Ct
"
1
"
F
1t
fte
f e (bt ;
2t
(gte
g e (bt ;
+
+
+
e
3t ht
h (bt ;
+
e
4t (jt
j e (bt ;
+
1t
(it
1
t
= het
t; t)
t; t)
= j e (bt ;
t; t)
t ; t+1 )
Tt )
(F
uC (Ct ; t )
1 + it
fte
(
(1 +
v~ (Yt ) + Et J (bt ;
i
t
e
(1 + s) uC (Ct ;
d(
t;
t ))
t)
t ; t ))
t; t)
t ; t ))
0)
65
t)
v~y (Yt ;
t)
uC (Ct ;
0
t) d
(
t)
t
First-order conditions (where all the derivatives should be equated to zero)
@Ls
@ t
@Ls
@Yt
@Ls
@it
@Ls
@St
@Ls
@Wt
@Ls
@Ct
@Ls
@Tt
@Ls
@bt
@Ls
@ t
@Ls
@fte
@Ls
@gte
@Ls
@jte
@Ls
@het
h
=
1t
=
v~Y +
=
2t
h
=
1t
[bt ] +
=
1t
h
t 1 Wt ( t 1 )) bt 1
(1 +
3t
2t
s0 (Tt
2
i
1
t
i
= Et J (bt ;
t ; t+1 )
+
1t
=
3t
+
2t
=
4t
+
4t
=
5t
+
3t
+
1t
+
4t
T) +
t ; t+1 )
2t
+
(1 + s) uC
uCC (1 + it )
= Et Jb (bt ;
=
i
5t
h
uC d
00
uC d
t
+ "Yt v~yy + "~
vy +
i
0
+
6t
d0
6t
[ uC ]
t 1 bt 1
h
1
"
uC (1 + it )
= uC +
= gG
"
"
5t
2
t
+
[ uC ]
1
i
+
3t
[ uCC St ( t )] +
4t
[ uCC Wt (
t 1 )]
+
"Yt
5t
1t
1t
[St ( t )] +
1t
fbe +
2t
[ gbe ] +
heb +
3t
4t
( jbe )
The complementary slackness conditions are
1t
0; it
0;
1t it
=0
While the envelope conditions are
Jb (bt
1; t 1; t)
J (bt
=
1; t 1; t)
1t
=
h
1t
(1 +
h
66
1
t 1 Wt ( t 1 ))
Wt (
t 1 )bt 1
1
t
t
i
:
i
"
1
"
(1 + s) uCC
uCC d0
t
+
6t
[ 1]