Download Time-Consistent Management of a Liquidity Trap

Time-Consistent Management of a Liquidity Trap:
Monetary and Fiscal policy with Debt∗
Dmitry Matveev†
Job market paper
October 2014
Abstract
We study optimal monetary and fiscal policy in the New Keynesian economy with
occasionally binding zero lower bound, which leads to liquidity trap episodes. We
compare capacity of discretionary government spending and labor income tax use for
the purpose of economy stabilization at the liquidity trap. Reliance on either of these
instruments critically depends on whether government budget can be relaxed or has
to be balanced. Tying hands of the government with balanced budget one makes it
optimal for the government to rely more on spending instrument to the extent of order
of magnitude larger response to binding zero lower bound than in the case with debt
dynamics. Use of government debt makes the government to rely more on extracting
stabilization benefits from effect of labor tax on aggregate economic activity. Moreover,
we show that risk of falling into liquidity trap leads to accumulation of optimal long-run
government debt buffer set to reduce frequency of reaching zero lower bound.
JEL: E31, E52, E63, H21, H63
Keywords: Zero lower bound, Government debt, Markov-perfect equilibria
∗ For useful discussions during different stages of this project I thank Francesc Obiols, Albert Marcet,
Stefano Gnocchi, Jordi Galí, Rigas Oikonomou, Sarolta Laczo, Jeffrey Campbell, Pontus Rendahl, Klaus
Adam, Hannes Mueller, Arnau Valladares-Esteban, Yuliya Kulikova, Guillem Pons Rabat. I also benefited
from discussions with participants of the Barcelona GSE PhD Jamboree 2014 and UAB SoW workshop. All
remaining errors are mine.
† Universitat Autònoma de Barcelona, Barcelona GSE. Email: [email protected]
1
Introduction
Shortfall of demand in an economy calls for a firm policy response. Conventional wisdom
until Great Recession was for monetary policy to bear the burden of economy stabilization.1
At the onset of the Great Recession in December 2008 Federal Reserve pushed federal funds
rate down next to zero. Situation when zero bound prevents monetary authority from
providing enough stimuli by means of the interest rate policy is often referred to as a
liquidity trap. So far it has been the reason for conventional monetary policy in the United
States being constrained for more than five years. This led academics to revisit stabilization
merits of a number of alternatives including fiscal policy.
Large boost in studying fiscal stabilization at the zero lower bound came along with
policy actions implemented during the Great Recession. February 2009 saw enactment of
unprecedented stimulus package “the American Recovery and Reinvestment Act” in the
United States. Rationalized with Keynesian ideas, major part of the package came in the
form of government spending and payroll tax adjustments. Discretionary use of government
spending for stabilization purpose at the liquidity trap has been proven to be optimal in
a number of studies with New Keynesian models by Nakata (2013), Schmidt (2013), and
Werning (2011). However, the payroll tax, or any distortionary tax for that matter, is missing in these studies, for taxes there are assumed to be lump-sum and Ricardian equivalence
is in place. It is therefore instructive to consider the case of the joint use of the two fiscal
instruments and understand if there are reasons to rely more on either of them.
This paper contributes to the line of research that studies optimal time-consistent monetary and fiscal policy with occasionally binding zero lower bound constraint on nominal
interest rates. Framework of our analysis is a standard New Keynesian model, with monopolistic competition and costly price adjustment, augmented with fiscal sector. We consider
setup where benevolent government acts discretionary in a sequential manner. The government jointly chooses optimal (Markov-perfect) policy for short-term nominal interest rate,
level of government spending that has an intrinsic value, distortionary linear labor income
tax rate, and government debt supply.2 Model economy is subject to uncertain demand
stemming from ad hoc variation in the time preference of the households. High enough
preference shock may bring demand low enough to make “natural” rate of interest negative
and for a full stabilization to require negative nominal interest rate, which effectively makes
zero lower bound binding and brings economy into the liquidity trap.
Our focus on discretionary policy is dictated by the evidence in Bodenstein et al. (2012)
on lower credibility of Federal Reserve following economic turmoil in 2008, and by discretionary nature of the fiscal measures implemented at the same time.3 Zero lower bound
forces particular demand stabilization challenge on monetary policy of dicretionary nature.
In this paper we wish to examine the scope of solutions fiscal policy can offer to remedy this
problem while also being discretionary. In order to ponder this problem we eliminate usual
static inflationary bias and do not take into account first order fiscal effects of monetary
policy by studying the economy that in its deterministic version would settle at an efficient
steady state with zero public debt. To solve stochastic version of the model with demand
variation we put to use global nonlinear solution method.
Both government spending and labor tax take on stabilization role upon entering the
1 See
Kirsanova et al. (2009) discussing consensus assignments of monetary and fiscal policy.
for a larger set of tax instruments one could use them to circumvent relevance of the zero lower
bound along the lines of unconventional fiscal policy discussed in Correia et al. (2013).
3 Chen et al. (2013) find that discretionary monetary policy is preferred by the data over 1960-2008.
2 Allowing
2
liquidity trap. We contrast response of these two instruments when the government is using
debt to the case where the government is forced to maintain its budget balanced in every
period. Government with balanced budget relies heavily on the use of government spending.
Relaxing government budget makes the response of spending one order of magnitude smaller
than in the case with balanced budget. Labor tax response on the other hand is more vocal
with debt dynamics. We highlight the role public debt plays in stabilization of non-Ricardian
economy without balanced government budget assumption by means of its effect on private
sector expectations under discretionary policymaking. By bequeathing debt from one period
to another, the government can implicitly influence private expectations through debt effect
on policy in the future. The goal of debt dynamics in response to binding zero lower bound is
to exploit the benefit of creating outlook of future tax policy that insulate against deflation
bias accompanying discretionary policymaking without fiscal stabilization. The use of labor
tax in conjunction with debt also has consequences for optimal policy prescription at the
times when zero lower bound is not binding. We show how the risk of reaching zero bound
on nominal interest rate forces the government to increase and keep outstanding positive
buffer of government debt.
Government debt augments discretionary policy with credible history-dependence, which
is crucial to achieve effective stabilization closer to the case under commitment. Optimal
way to manage the liquidity trap under commitment in a monetary model without fiscal
instruments is well articulated in Eggertsson and Woodford (2003), and boils down to monetary authority engaging into forward guidance promising to keep (nominal) interest rate
at a low level for a prolonged period of time beyond the point in time when natural interest
rate becomes positive.4 Binding zero bound creates intertemporal trade-off between output
and inflation stabilization. Forward guidance prescription is a natural way to cope with this
trade-off by tolerating extra inflation and associated cost in the future in exchange for less
severe recession today. One can think of this as having central bank that targets nominal
GDP. More precisely, as Werning (2011) elaborated, the ultimate goal is to keep real rates
low in the future so as to smooth out current recession by generating extra output boom in
the future. Yet this solution is subject to time-consistency problem and may not be credible,
because monetary authority will be tempted to renege on its promise to tolerate inflationary
output boom once time passes and recession subsides.
Assuming commitment away turns out detrimental for the ability of the monetary authority on its own to stabilize economy with occasionally binding zero lower bound and leads
to substantial decline in welfare as demonstrated by Adam and Billi (2007).5 They also show
that optimal discretionary policy displays a deflationary bias compared to commitment policy. Inability of using forward guidance to create credible beneficial expectations of future
policies leads to suboptimally high deflation and real interest rates during the liquidity trap
episodes. Rational agents away from the liquidity trap then take this risk into account and
adjust their expectations of real interest rate and deflation upward, which makes monetary
policy more accommodative. Overall, this makes liquidity trap episodes under policy discretion both more frequent and costly. Failure to provide optimal (history-dependent) dynamic
response to exogenous shocks at the heart of this problem is an alternative manifestation of
the more general stabilization bias emphasized by Clarida et al. (1999).
In our model economy, with fiscal instruments added into consideration, government
can improve stabilization against large demand fluctuations that would make zero lower
4 That said, monetary forward guidance has its limitations and its effectiveness vanish with severity and
persistence of demand shock as explored in Levin et al. (2010) and Bundick (2013).
5 See also Levine et al. (2008) and Nakov (2008) on assessment of welfare gain from monetary commitment.
3
bound effectively binding. Upon arrival of such shock, it is optimal to respond by temporarily increasing government spending and raising labor tax rate. Both policy actions are
inflationary, the former due to cushioning aggregate demand and the latter due to positive
cost-push effect. The magnitude of both responses depends crucially on whether government
budget is balanced or relaxed. Government spending response is one order of magnitude
larger under balanced budget. On the contrary, labor tax rate raise is roughly three times
larger under relaxed budget constraint. Government spending is helpful in cushioning demand fall, but its use requires deviation from the constant efficient level and therefore brings
direct welfare cost. Using labor tax in conjunction with debt improves its stabilization efficacy and government shifts reliance away from spending instrument to reduce the associated
with it welfare cost. Under relaxed government budget, relative magnitude of government
spending and labor tax response is such that primary surplus of the government budget
increases and only a fraction of the inherited debt is rolled-over. To improve labor tax
stabilization efficacy it is important to influence expectations of tax rate increase to persist
during the liquidity trap for current response to be efficient in restraining recession. Level
of the bequeathed debt determines the extent to which the government in the next period
is willing to tolerate change of price level, because it would affect real debt burden and
roll-over cost. Given mean-reverting nature of the preference shock, it is likely demand
improves next period and therefore rolling over only a fraction of debt is enough to guard
against deflation bias. In both cases of balanced and relaxed government budget, initial
government spending and labor tax response monotonically declines over time and both
instruments revert to pre-crisis level by the time liquidity trap is over. After liquidity trap
is over, gradual accumulation of the debt to its long-run level begins.
In normal times when zero lower bound is not binding, the optimal discretionary policy
dictates the government to accumulate or reduce debt until it reaches positive optimal stock
exceeding efficient deterministic zero level. Government, having a positive long-run level of
nominal debt above efficient one, under discretionary policy is tempted to reduce its debt
burden. Without zero lower bound, Leith and Wren-Lewis (2013) show that it is optimal
for the government to eliminate this temptation by adjusting policy instruments so as to
return debt to the efficient level. Particular mixture of policy instruments implementing
fiscal adjustment depends on the size of the debt stock. Optimal long-run debt level in our
model is low enough to stay in the region where responsibility of debt stabilization mainly
weights on fiscal policy and is implemented via taxation. Keeping moderate positive debt
therefore would require higher labor tax that in turn translates into higher marginal costs.
Agents attach expectations of higher inflation and lower real interest rate to states with
higher debt level, which makes monetary policy tighter. Increasing debt level in the longrun thus produces the opposite to deflationary bias effect. Optimal long-run level of debt
trades-off the cost of associated extra distortionary taxation against the benefit of reaching
zero lower bound at a lower frequency.
Benefits from abandoning tax smoothing and varying government spending gap of the
kind that we observe in our model has been shown to exist under joint monetary and
fiscal commitment in Eggertsson and Woodford (2006) and Nakata (2011). Fiscal part of
the optimal commitment policy mix is history-dependent and features forward guidance
component. For the government currently in the liquidity trap it is optimal to commit itself
to support inflationary output boom due to period of low interest rate with expansionary
lower tax rate and reversal of government spending expansion. Absence of commitment
in our setting provides a major qualification to this result. We show that while debt as a
natural state variable can deliver beneficial history-dependence during the liquidity trap it
4
may not be enough to make fiscal forward guidance credible.
The use of debt as means to shape expectations of future policy outside the liquidity
trap to improve credibility problem was introduced in Eggertsson (2006). Two features are
notably different in our setup. First, the model of taxation there is of costly lump-sum type à
la Barro (1979).6 In that analysis debt is effective only because it provides a way to influence
expectations of monetary authority engaging into keeping rates low after the liquidity trap
for the sake of debt adjustment. Such mechanism critically relies on assumption of policy
coordination, whereas in our setup debt is used to form expectations through supply-side
effects of labor tax on aggregate economic activity and pricing decisions, for which policy
coordination should be of lesser importance. Second, there a great degree of tractability is
achieved by modeling economy that starts in the liquidity trap and is expected to escape
from it each period with a fixed probability without ever revisiting it again. Under such
assumption, the role of uncertainty is limited because it is tied to shock persistence and
concerns only exit from the liquidity trap.
We analyze economy with occasionally binding zero lower bound, thereby studying the
role that risk of emerging liquidity trap plays in design of optimal policies. In our calibrated
model, we set parameters of demand shock process in order for the economy to spend at
the liquidity trap around eight per cent of the time. This number is slightly larger than
estimates of five to six per cent in Reifschneider and Williams (2000) based on the data
for Great Moderation period. The optimal prescription to address stabilization challenge
that such risk entails is for the government to target the amount of outstanding debt equal
to five per cent of annual GDP. Size of the optimal government debt buffer increases with
liquidity trap risk, and we consider debt-to-GDP buffer of five per cent to be still in the
ball park of a lower bound for a number of reasons. First, taking into account that zero
lower bound is expected to be binding in the United states well into the 2015 means that
zero lower bound will have been binding for almost a quarter of time since the beginning of
Great Moderation period, which by far exceeds the five per cent probability estimate. Even
if one considers current events to be extreme tail event, Chung et al. (2012) have employed
a variety of estimation techniques including first two years of the Great Recession into the
sample and concluded that five per cent frequency is an underestimation. Second, recently
revived hypothesis of secular stagnation (see Summers (2013)) and IMF (2014) forecast of
downward trend in the natural real interest rate would imply that the probability of hitting
zero lower bound is increasing given the same variation in demand. Finally, the very chance
of having larger odds of hitting zero lower bound would force the policymaker of Hansen and
Sargent (2007) robust controller type act according worst case scenario and keep a larger
government debt buffer.
The paper is organized as follows. In section 2 we outline our model. Section 3 defines
equilibria under two policy regimes: (1) Ramsey equilibrium with commitment policies, (2)
Markov-perfect equilibrium with time-consistent policies. We then analyze deterministic
steady states of the two equilibria and show analytically that absence of commitment creates
long-run biases in inflation and government spending. Our desire to focus on the role of
policy for stabilization motivates us to close this section discussing design of a labor subsidy
in a way that eliminates long-term biases of discretionary policy. Section 4 presents our
numerical results of the optimal time-consistent policies in the Markov-perfect equilibrium
with occasionally binding zero lower bound. Section 5 concludes.
6 Burgert and Schmidt (2013) study optimal discretionary monetary and fiscal policy with distortionary
taxes, but keep taxes constant in their baseline model.
5
2
The Model
We consider a standard dynamic stochastic general equilibrium New Keynesian model of a
closed economy with monopolistically competitive intermediate goods markets and costly
price adjustment. This section describes the economy, defines competitive equilibrium and
derives first-best allocation as a reference point for policymaker problems defined in the next
section.
2.1
Households
The representative household consumes composite goods, which are produced from a continuum of differentiated products indexed by i ∈ [0, 1] using constant-elasticity-of-substitution
production technology. Total supply of the aggregate good Yt is given by
1
ˆ
θ−1
θ
θ
θ−1
,
Yi,t di
Yt =
0
where Yi,t is the input of differentiated good i, and θ > 1 is the elasticity of substitution.
This composite good is used either for private or public consumption
(1 − κt )Yt = Ct + Gt ,
(2.1)
accounting for resource cost of price adjustment, κt , incurred by producers of the differentiated goods. Private consumption of the aggregated consumption good is denoted by Ct ,
and Gt denotes spending on public good provision by the government.
Representative household values private consumption and public goods, and dislikes
labor. Preferences of the representative household are given by
!
ˆ 1
∞
t−1
X
Y
t
E0
β
ξs
u(Ct ) + g(Gt ) −
v(hi,t )di ,
(2.2)
t=0
0
s=−1
where 0 < β < 1 is a discount factor, and hi,t denotes quantity of labor of type i supplied. We
assume that function v is increasing and convex in labor, function g is increasing and concave
in consumption of public goods, and function u is increasing and concave in consumption
of private goods. Exogenous stochastic disturbance ξt is a preference shock that affects the
marginal rate of substitution between consumption at time t and consumption at time t+1.7
We normalize ξ−1 = 1 and assume it follows an AR(1) process
ln(ξt ) = ρ ln(ξt−1 ) + εt ,
where 0 6 ρ < 1, and ε ∼ N (0, σ 2 ).
Labor of type i is used to produce differentiated good i. Private consumption of a bundle
of differentiated goods is indexed using constant-elasticity-of-substitution aggregator
ˆ
Ct =
1
θ−1
θ
Ci,t di
θ
θ−1
,
0
7 Preference
shock enters preferences analogously to Braun et al. (2013), Nakata (2013) and Ngo (2013).
6
Index of public goods Gt is defined analogously. Given individual goods prices Pi,t , minimum
cost of a unit of the aggregated good is defined as the corresponding price index
ˆ
Pt =
1
1−θ
1
1−θ
Pi,t
di
.
0
The household enters period t holding assets in the form of nominal non-contingent (risk
free, zero coupon) one-period bonds Bt−1 . Supply of bonds in equilibrium is determined by
the government policies discussed later. Under described market arrangement, household’s
flow budget constraint is of the following form
ˆ 1
ˆ 1
−1
Πi,t di − Tt ,
(2.3)
Wi,t hi,t di + Bt−1 +
Pt Ct + Rt Bt 6 (1 − τt )
0
0
where Wi,t is the nominal wage of labor of type i, and Πi,t is the nominal profits from
sales of differentiated good of type i distributed in a lump-sum way, and Tt are lumpsum government transfers. Labor income of the household is taxed at a linear tax rate
τt . To ease the notation we do not explicitly describe market for private claims. Still our
setup is isomorphic to the model with complete set of private state-contingent securities,
because those would not be traded in equilibrium under assumption of the representative
household. Also note that we consider “cashless” limit of the monetary economy in the
spirit of Woodford (2003) and therefore abstract from money holdings.
To have a well-defined intertemporal budget constraint and rule out “Ponzi schemes” we
impose additional constraint on household behavior that has to hols at each contingency
" T
!
#
Y
−1
lim Et
Rk
BT > 0.
(2.4)
T →∞
k=t
Household maximizes (2.2) by choosing consumption, industry-specific labor and bond
purchases {Ct , ht (i), Bt } subject to budget constraint (2.3) and no-Ponzi condition (2.4),
taking as given prices, policies and firms’ profits {Pt , Wt , Rt , τt , Gt , Tt , Πt (i)}∞
t=0 , exogenous
stochastic process of preference shocks {ξt }∞
t=0 , and initial bond holdings B−1 .
The optimal plan of the households has to satisfy (2.3) and (2.4) with equality (the latter
then referred to as transversality condition), and the following first order conditions
wi,t
Rt−1
0
1
v (hi,t )
=
,
(1 − τt ) u0 (Ct )
0
u (Ct+1 )
= βξt Et
,
u0 (Ct )πt+1
(2.5)
(2.6)
where πt+1 ≡ Pt+1/Pt is the gross one period inflation rate, and wi,t ≡ Wi,t/Pt is the real
wage. Equation (2.5) describes intratemporal trade-off between consumption and leisure.
Equation (2.6) is an Euler equation describing intertemporal allocation of consumption and
savings.
2.2
Intermediate goods producers
There is a continuum of firms of unit mass producing imperfectly substitutable differentiated
goods with a linear technology in labor Yi,t = hi,t . Firm producing good i sets the price
7
Pi,t and hires labor of type i necessary to satisfy realized demand on a perfectly competitive
labor market. We assume that the government allocates its spending on the good varieties
identically to the households, the resulting demand for good i is then
−θ
Pi,t
Yt .
(2.7)
Yi,t =
Pt
Firm chooses the price Pi,t so as to maximize its present discounted real value of profits
E0
∞
X
t=0
β t λt
Pi,t
Yi,t − (1 − s)wi,t Yi,t − κi,t Yt ,
Pt
subject to demand function (2.7), where λt is the marginal utility of real income for the representative household, and s is the time-invariant rate of a labor (employment) subsidy that
can be used to eliminate deterministic steady state distortions associated with monopolistic
competition and distortionary labor income tax. Following Rotemberg (1982) we assume
quadratic cost of price adjustment
2
Pi,t
ϕ
−1 .
κi,t ≡
2 Pi,t−1
Our choice of this specification for sticky prices over Calvo (1983) is common in the literature
for solving optimal policy problems without appealing to linear-quadratic approach, because
in that latter case state-space of the model would be enlarged due to the need to track price
dispersion.8
In a symmetric equilibrium where all the firms charge identical price, or equivalently
there exists a representative firm, the first order condition for the firm’s problem is
u0 (Ct+1 ) Yt+1
(θ − 1)
= ϕ (πt − 1) πt − βξt Et 0
(πt+1 − 1) πt+1 , (2.8)
θ (1 − s)wt −
θ
u (Ct ) Yt
Condition (2.8) is a New Keynesian Phillips curve in its nonlinear form stating that current
inflation depends on the marginal cost of production and expected inflation.
2.3
Government
In the “cashless” economy that we consider, the government supplies nominal claims also
known as “money” that do not provide nonpecuniary return and thus only impose a zero
lower bound on the gross nominal interest rates
Rt > 1.
(2.9)
Noting optimizing behavior of firms, we can map resource constraint (2.1) and household’s flow budget constraint (2.3) to nominal flow budget constraint of the government
Rt−1 Bt = Pt (Gt − (τt − s)wt Yt ) + Bt−1 − Tt .
Since we leave money out of consideration we automatically forgo seigniorage revenues
obtained by the government. We assume that lump sum transfers Tt are used for the sole
8 See
exceptions in Anderson et al. (2010) and Ngo (2013) for fully nonlinear solutions with Calvo (1983)
pricing.
8
purpose of transferring resources corresponding to labor subsidy. Furthermore, since the
purpose of the subsidy is to address only permanent distortions in the economy, we fix real
value of the lump-sum transfers over time equal to the steady-state value of the subsidy
level, and the flow budget constraint of the government in real terms is then given by
Rt−1 bt =
bt−1
+ (Gt + ςt − τt wt Yt ) ,
πt
(2.10)
where bt ≡ Bt/Pt is the real value of government bonds, and ςt ≡ swt Yt − swss Y ss is the real
deviation of the subsidy from its steady state level.
The government consists of a central bank and a treasury. The former controls shortterm interest rate Rt by means of open-market operations that vary level of the real money
balances held by the households. The latter decides on the amount of public goods Gt
provided to the households in the form of aggregate consumption good. To cover spending
on public goods, it levies labor income tax and participates in the bond market. No arbitrage logic implies that monetary policy rate equals interest rate on government bonds in
equilibrium. Following Chari and Kehoe (1993) and Lustig et al. (2008) we assume that
households participate at the bond market anonymously, thus bonds issued by households
are unenforceable and the government wouldn’t buy them. This assumption restricts the
range of the government portfolio positions. We capture this assumption with the following
inequality constraint that has to hold at all times and in all states
bt > 0.
(2.11)
Under this constraint the government can borrow from the households but does not provide
loans to the households.9 Note that under (2.11), the no-Ponzi condition (2.4) is automatically satisfied.
Therefore the government, subject to zero lower bound constraint (2.9), chooses {Rt , Gt , τt }
that, at the equilibrium prices, uniquely pin down {bt } as satisfying (2.10) and (2.11). The
government’s problem will be introduced and discussed in the next section.
2.4
Competitive equilibrium
We focus on the symmetric equilibria where producers of intermediate goods charge equivalent price, Pi,t = Pt for all i ∈ [0, 1], and therefore face the same demand Yi,t = Yt , hire
the same amount of labor hi,t = ht and pay the same competitive wage wi,t = wt . Production function for intermediate goods then implies that Yt = ht . Taking this conditions into
account, we now define competitive equilibrium
Definition 1 (Competitive equilibrium). Given exogenous process for household’s preference shocks {ξt }∞
t=0 and initial outstanding government debt b−1 > 0, a rational expectations
symmetric equilibrium is a sequence of stochastic processes {Ct , Yt , πt , wt , bt , Rt , Gt , τt }∞
t=0
satisfying
9 No
government lending constraint is also imposed in Faraglia et al. (2013).
9
u0 (Ct ) = βξt Rt Et
u0 (Ct+1 )
πt+1
,
(2.12)
v 0 (Yt ) = (1 − τt )wt u0 (Ct ),
0
u (Ct+1 ) Yt+1
θ(1 − s)wt = (θ − 1) + ϕ (πt − 1) πt − βξt Et
(π
−
1)
π
,
t+1
t+1
u0 (Ct ) Yt
ϕ
2
0 = Ct + Gt − 1 − (πt − 1) Yt ,
2
b
t−1
+ (Gt + ςt − τt wt Yt ) ,
Rt−1 bt =
πt
Rt > 1,
bt > 0,
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
(2.18)
and transversality condition, for all t > 0; where Euler equation (2.12) and equation (2.13)
are first-order conditions of the household problem, New-Keynesian Phillips curve equation
(2.14) is the first-order condition of the intermediate goods producer, equation (2.15) is the
aggregate resource constraint, equation (2.16) is the government budget constraint, (2.17)
represents zero lower bound constraint on nominal interest rates, (2.18) is the government
no-lending constraint.
2.5
First-best allocation
The first-best allocation is the one maximizing household’s utility (2.2) subject to technology constraints while abstracting from economic distortions in the private markets. Such
allocation is efficient and coincides with the solution of the fictitious social Planner’s problem described in appendix A. Efficiency dictates for all t > 0 to equate marginal utilities of
private and public consumption to marginal disutility of labor
u0 (Ct ) = v 0 (Yt ),
g 0 (Gt ) = v 0 (Yt ).
We use these efficiency conditions as a benchmark when discussing problems of the government in the next section.
3
Policy Regimes
In this section we formulate optimal policy problem. Throughout the paper we assume
full cooperation between fiscal and monetary authorities. The government is assumed to
be benevolent and, hence, have maximization of household’s utility as an objective. Firstbest allocation is in general not attainable due to the number of economic distortions in
our model. Therefore we start by establishing a second-best allocation that takes economic
distortions into account. To do that we define Ramsey equilibrium, which accounts for
nominal rigidities, distortive labor tax and monopoly power of firms, as well as bounds on
bond holdings and nominal interest rate. Ramsey policymaker is assumed to fully commit to
its announced policies. Then we proceed by dropping commitment assumption and studying
10
optimal (Markov-perfect) allocation under discretionary policymaking. Optimal Markovian
policy and corresponding equilibrium is a focus of our paper and is by construction timeconsistent.
Last part of this section is devoted to the analysis of the deterministic steady states of
the two policy regimes. We derive a number of results that point to long-term differences
between Ramsey and Markovian policies that are driven by the discretionary nature of the
latter. These results are then used to derive labor subsidy that is capable of eliminating
the difference between the two by making both of the deterministic steady states efficient.
Using this subsidy allows us to focus on the stabilization role of discretionary policy and
its implications in the Markov-perfect equilibrium when studying numerical results for the
stochastic version of the equilibrium in the next section.
3.1
Ramsey equilibrium
We assume the government can commit to follow through the policies it announces at the
beginning of time. The government then announces policy plan for all future contingencies in
order to implement the best competitive equilibrium that we refer to as Ramsey equilibrium.
We formally capture this idea in the following definition.
Definition 2 (Ramsey equilibrium). Ramsey equilibrium consists of a state-contingent
plan {C(ξ t ), Y (ξ t ), π(ξ t ), w(ξ t ), b(ξ t ), R(ξ t ), G(ξ t ), τ (ξ t )}∞
t=0 chosen at the initial time period t = 0 and for all possible histories ξ t ≡ (ξ0 , . . . , ξt ) of preference shocks that, given
initial outstanding government debt b−1 > 0, maximizes expected discounted sum of future
utilities
!
∞
t−1
X
Y
t
E0
β
ξs [u(Ct ) + g(Gt ) − v(Yt )]
t=0
s=−1
subject to equations (2.12) - (2.18) characterizing competitive equilibrium.
The Lagrangian of the Ramsey problem and first-order conditions characterizing Ramsey
allocation are described in appendix B.
3.2
Markov-perfect equilibrium
Assume that the government is benevolent and credibly repays its debts in the future, but
the rest of its fiscal and monetary policies are credible only within current period.10 Due
to a range of economic distortions embedded in our setup, government has incentives to
renege on its promises in the future. This gives rise to a time inconsistency of the Ramseytype policies. We therefore focus on the policy maker choosing policies from the class of
sequential rules, so that optimal policies are time-consistent by construction.
We model optimal discretionary policy as an outcome of the game between private sector
and a government represented by treasury and monetary authority playing cooperatively to
maximize expected discounted utility (2.2) of the representative household. In particular,
we look for policies that belong to stationary Markov-perfect equilibria of this game.
Each period, the government acts first as a Stackelberg leader and announces current
period policies. When doing so, it internalizes the effect of its decision on actions of the
10 Credible
debt repayment can be supported by a high implicit cost of the outright debt default.
11
private sector, acting as a Stackelberg follower and taking future policies as given. Optimal
policies for each period are chosen by the government sequentially based on the minimal
payoff-relevant state of the economy. Such aggregate state of the economy at time t is
described by realization of the preference shock ξ and inherited government liabilities from
the previous period b−1 . Although the government at a given period can not credibly
commit to its future policies and hence does not directly influence future actions of the
private sector, it can indirectly influence both through her current debt issuance policy
that determines future inherited state of the economy. In the Markov-perfect equilibrium,
sequentially optimal choices are time consistent and recursively determined by stationary
rules, hence so are private actions in equilibrium turn out to be stationary.
We proceed by using recursive formulations to formally develop the concept of optimal
discretionary policy. We drop time indices and denote next period value of a given variable
x by x0 . The government, at a given period in time, correctly anticipates future policies
together with corresponding equilibrium allocation and prices that we denote as governed
by stationary functions C, Y, Π , W, R, B, T , G. When announcing policies for an ongoing
period the government is free to deviate from the anticipated rules by implementing best
possible competitive equilibrium. As it is common in the optimal taxation literature, one
can think of the government as choosing simultaneously its policies, prices and allocation,
provided that they satisfy equilibrium implementability conditions from definition 1.11 We
capture it with the following Markov optimization problem of the discretionary government
max
C,Y,π,w,R,τ,G,b
u(C) + g(G) − v(Y ) + βξE {V (b; ξ 0 )}
subject to the constraints
0 = [θ(1 − s)w − (θ − 1) − ϕ(π − 1)π] Y uc
+ ϕβξE {Y (·) (Π (·) − 1) Π (·) uc (C (·))} ,
ϕ
0 = 1 − (π − 1)2 Y − C − G,
2
0 =R−1 b − π −1 b−1 − (G + ς − τ wY ) ,
uc (C (·))
,
0 =uc − RβξE
Π (·)
0 =w(1 − τ )uc − vy ,
b >0,
R >1.
For optimal policies to be time consistent, the government should find no incentives to
deviate from anticipated rules. We capture this in the formal equilibrium definition.
Definition 3 (Markov-perfect equilibrium). A Markov-perfect equilibrium is a function
V (b−1 ; ξ) and a tuple of allocation rules {C, Y, Π , W, R, B, T , G} each being a function of
b−1 and ξ, such that:
1. Given V (·), tuple of rules solves Markov problem of the government,
11 Strictly
speaking, we also assume that the primitives of the model permit correspondence between
sequential formulation in definition 1 and recursive formulation employed here to be valid.
12
2. V (·) is the value function of the government
V (b−1 ; ξ) = u (C (·)) + g (G (·)) − v (Y (·)) + βξE {V (B (·) ; ξ 0 )} .
We restrict our attention to equilibria with differentiable value and policy functions. Under this assumption, equilibria are characterized by first-order conditions of the government
problem.12 Derivation of these conditions is delegated to the appendix C.
3.3
Deterministic steady state analysis
We set σ = 0, ξt = 1 for all t > 0 and analyze time-invariant deterministic long-run
steady states of both Ramsey and Markov-perfect equilibria. We start with the Ramsey
policy regime. Standard to this class of models, there is in fact a continuum of Ramsey
steady states indexed with initial level of government debt. To put it the other way, a
continuous range of debt levels can be supported in equilibrium, hence the model exhibits
indeterminacy of degree one. We characterize this continuum of steady states with the
following proposition.
Proposition 4 (Ramsey Steady State). Level of debt in the Ramsey steady state is indeterminate. For a given debt level b, Ramsey steady state is characterized by
π = 1 and R = β −1 ,
and marginal utility conditions
G + (1 − β)b
θ−1 1
−
θ (1 − s)
Y
u0 (C) = v 0 (Y ),
(3.1)
g 0 (G) > v 0 (Y ).
(3.2)
Proof. See appendix B.1.
This result is similar to Adam (2011) and Motta and Rossi (2013). All Ramsey steady states
share the same zero inflation condition and identical positive nominal interest rate.
Until further notice, consider the case when there is no employment subsidy, s = 0, and
compare marginal utility conditions for Ramsey steady state with analogous conditions for
the first-best allocation. Conditions (3.1) and (3.2) show that in general there are wedges
between marginal utilities of private and public consumption and marginal disutility from
labor. Wedge in equation (3.1) implies u0 (C) > v 0 (Y ) so that private consumption in Ramsey
steady state falls below first-best optimum. This wedge appears due to firms charging
monopolistic mark-up and distortionary nature of taxes required to finance steady state
government spending and debt interest payments (for positive levels of debt). The fact
that government spending requires collection of distortionary tax provides incentive for the
planner to reduce government spending below first-best optimum as described by condition
(3.2). Clearly the output also appears to be below efficient level.
If no-lending constraint (2.18) was not in place, one could show that there exists a
unique negative level of government debt for which Ramsey steady state is in fact efficient.
In this steady state government uses interest receipts from holding assets in order to finance
12 We
refrain from a formal proof of equilibrium existence (or uniqueness).
13
government spending and offset monopolistic distortion without resorting to distortionary
labor tax. Gnocchi and Lambertini (2014) show that such efficient asset level turns out to
be a large magnitude of GDP.
Now turning to Markov-perfect equilibrium. Steady-state of Markov-perfect equilibrium
in general is hard to characterize, because first-order conditions of the government problem
contain derivatives of unknown functions. One can still devise a limited analytical insight
into properties of Markov steady state allocation. To do so, we use Ramsey steady state
as a reference and see what happens if policymaker were to achieve price stability π = 1
in Markov steady state. Under price stability, a subset of first-order conditions imply the
reaction function
u0 (C) = v 0 (Y ),
g 0 (G) = v 0 (Y ),
under which the government is trying to equate marginal utilities of private and public consumption to marginal disutility of labor just like in the first-best allocation. Such behavior
is counterproductive in the presence of economic distortions and cannot be sustained in
equilibrium, which leads to deviation of Markov steady state from Ramsey steady state. In
particular the following results regarding inflation holds.
Proposition 5 (Markov steady state). Given discount factor β close enough to 1, Markovperfect steady state exhibits positive trend inflation
π > 1.
Proof. See appendix C.1.
It is then straightforward to show that government spending in Markov steady state is
in general different from its counterpart in Ramsey steady state. This result shows that
lack of credibility has consequences for the long-run properties of optimal policies in our
setup. Discretionary policymaker in attempt to boost inefficiently low output in the longrun leads to biases in inflation and government spending. The former is a reminiscence of
the debated outcome due to credibility problem from Kydland and Prescott (1977), and
the latter is an accompaniment that arises due to distortionary financing of endogenous
government spending akin Adam and Billi (2014) where compared to our setup there is no
coordination between monetary and fiscal government authorities.
Focus of our study is on discretionary stabilization policies and we want to study them
in isolation from the long-term effects caused by the absence of commitment in combination
with permanent economic distortions. To do so we use labor subsidy to offset permanent
distortions in our economy. Below we show that one can design such subsidy, which eliminates monopoly power distortion and effect of distortionary taxation in a given Ramsey
steady state. Using same subsidy in the Markov-perfect equilibrium ensures that there is
a corresponding steady state identical to efficient Ramsey steady state, hence this subsidy
eliminates long-term differences between two regimes.
Proposition 6 (Employment subsidy). Given debt level b, there exists a unique employment
subsidy rate s such that
• corresponding Ramsey steady state is efficient,
14
• there is a Markov steady state with debt level b that is also efficient.
Proof. See appendix D.
Therefore in general, in the deterministic setting without uncertainty, one can design subsidy to support any debt-to-GDP ratio with efficient allocation under both policy regimes.
Labor subsidy at the deterministic steady state is financed with lump-sum transfer. Outside
of the deterministic steady state the government has to resort to distortionary labor tax to
adjust the size of the subsidy relative to the steady state level, like for instance in Leith and
Wren-Lewis (2013).
In what follows we pick a particular subsidy rate under which efficient steady states of
both policy regimes features zero government debt. By establishing zero debt as a status
quo in the setting without uncertainty, we make sure that any long-term level of debt that
appears in the stochastic version of the model is due to the risk of binding zero lower bound
and any debt dynamics is for stabilization purposes induced by materialization of this risk.
4
Results
This section presents numerical results for the stochastic Markov-perfect equilibrium of our
model with occasionally binding zero lower bound. We lay our parametrization strategy
and describe fully nonlinear solution method. Then we start by discussing optimal longrun policy of debt accumulation and after that proceed with discussion of optimal policy
subject to variation in preference shock that can bring economy into the liquidity trap when
government budget constraint is relaxed. This section is closed considering the case when
government budget is balanced.
4.1
Parametrization
Each period in the model represents one quarter of the year. The steady state time discount
rate, β, is set to 0.995, corresponding to annual real interest rate of 2 per cent.
Household preferences for private and public goods consumption are described by u(ct ) ≡
c
c1−γ
t
1−γc
1−γg
G
h
1+γh
t
t
, g(Gt ) ≡ νg 1−γ
, and disutility from work is specified as v(ht ) ≡ νh 1+γ
. Curvature
g
h
parameters of the utility functions, γc , γg and γh , are set to 2, 1 and 1 correspondingly. We
choose values for utility weights νh and νg equal to 100 and 1.25 correspondingly so that
in the steady state households spend one quarter of their unitary time endowment working
and government spending amounts to 20 per cent of value added.
Monopoly power of the firms is described by elasticity of substitution between intermediate goods, θ. We choose it equal to 11 in order to match the markup of price over marginal
cost of 10 per cent. The parameter of price adjustment cost, ϕ, is set to 116.505, which is
consistent with a Calvo (1983) price-setting specification where quarter of firms reoptimize
their prices every period.
We set parameter governing persistence of the demand shock, ρ, equal to 0.8. This
value is a common choice in the literature studying zero lower bound and, as FernándezVillaverde et al. (2012) discuss, it implies that demand shock process has a half life of about
three quarters. For the standard deviation of the shock innovations we choose a value of
0.0015. This value results in having a probability of being at the zero lower bound around
5 per cent of the time.
Parameters values are summarized in table 1.
15
Parameter
β
γc
γg
γh
νg
νh
ϕ
θ
ρ
σε
Description
Discount rate
Interpemoral elasticity for C
Intertemporal elasticity for G
Inverse Frish elasticity
Utility weight on G
Utility weight on labor
Price adjustment cost
Elasticity of substitution among goods
Demand shock persistence AR(1)
Demand shock innovation s.d.
Value
0.995
2
1
1
1.25
100
116.505
11
0.8
0.0015
Table 1: Baseline parametrization
4.2
Solution method
Equilibrium policy functions for our model can not be computed in closed form. We solve
the model numerically by a global nonlinear approximation method. We use value function
iteration procedure to search for a fixed point of value function and policy functions. Starting
with a guess of the next period value function and future policy functions, on a discretized
state space, we solve Markov optimization problem of the discretionary government. New
solution is used to update guesses of the value and policy functions, and repeat the procedure
until convergence when value and policy functions from two consecutive iterations become
arbitrarily close.
Off the grid points we use cubic splines to interpolate value and policy functions. Expectations are computed with Gauss-Hermite quadrature. As for the grid, we start with
equidistant points and then augment them with a set of adaptive points. Borrowing idea
from Brumm and Grill (2014), we choose adaptive part of the grid to better capture the
kink due to occasionally binding zero lower bound. Appendix E contains the details of the
algorithm.
4.3
Optimal long-run debt
Under appropriately chosen subsidy rate and without uncertainty our economy features
deterministic steady state without outstanding government debt and efficient allocation of
private consumption, government spending and labor. In economy with demand fluctuations
zero lower bound is known to prevent full stabilization of the economy by means of interest
rate changes. Binding zero lower bound precludes from keeping allocation constant at its
efficient level. In this section we show that presence of liquidity trap risk in the long run
drives model economy away from its efficient deterministic steady state.
Keeping positive debt in the long run is used as an instrument to permanently change
the point in the state space around which economy fluctuates. Figure 1 shows optimal
government debt policy in the absence of realized demand shocks but with the risk of such
taken into account. Let economy start at the point identical to deterministic steady state,
i.e. without outstanding debt b−1 = 0 and ξ = 1. Optimal policy prescription when demand
fluctuations do not occur is then to accumulate government debt until a certain optimal level
b∗ is reached. Conversely, economy starting with debt level above optimal b∗ , conditional
on ξ = 1, should experience debt stock decline until this level is reached. For our baseline
16
parametrization, this optimal level of government debt amounts to ~5 per cent of annual
GDP. If economy today is in the state with outstanding government debt at the optimal level
b∗ and no demand deviation, then optimal policy is to keep economy at this point. Therefore
such point can be referred to as a risky steady state in the terminology of Coeurdacier et al.
(2011).
[Figure 1 about here.]
Existence of the risky steady state with positive outstanding debt around which economy
fluctuates results in a positive unconditional mean of government debt level. Such outcome
is an example of how responses around deterministic steady state are sub-optimal and
short-run dynamics drive the long-run of the economy away from its counterpart without
uncertainty.
Keeping optimal level of debt in the risky steady state, b∗ , is preferred to no debt, for it
opposes discretionary effects of debt stabilization bias against those of demand stabilization
bias introduced by presence of zero lower bound to balance the magnitude of shock at
which zero lower bound is reached. Following high enough preference shock, binding zero
lower bound prevents monetary authority from offsetting negative demand effects of the
shock. Abstracting from fiscal policy and without further actions of monetary authority,
falling demand pulls down prices and output contracts. Deflationary pressure is larger
under discretionary monetary policy because monetary policy can not credibly compromise
stabilization upon exiting the liquidity trap and generate output boom that would effectively
reduce desire of agents to save into the future and improve stabilization at the liquidity
trap. This effect was coined “deflation bias” in Eggertsson (2006). Matters get worse under
occasionally binding zero lower bound, when deflationary pressure and output contraction
spill into rational expectations of agents even when nominal interest rate is positive and
economy is away from the liquidity trap. Contemporaneous inflation and output are then
adjusted downward. Adam and Billi (2007) show that this makes monetary policy more
accommodative leading to the lower bound being reached at a higher frequency. It is this
consequence of the liquidity trap anticipation that can be affected by varying long run level
of debt.
Government with outstanding debt exceeding efficient level has to collect additional revenue via distortionary taxation to support its liabilities. Given incomplete market structure,
it is optimal for the government to commit and spread the associated cost of distortionary
taxation over time. In the setting with commitment debt then would follow a random walk
type of behavior described in Aiyagari et al. (2002) and Schmitt-Grohé and Uribe (2004).
Yet this policy is not credible and the government will be tempted to reduce its debt burden
by using policy instruments at its disposal. Therefore under discretion the government can
not enjoy the benefits of spreading costs over time and has to stabilize debt stock around its
efficient level. Adjustment strategy changes depending on the outstanding amount of debt
and price stickiness. Leith and Wren-Lewis (2013) show that at low debt levels, comparable
to our optimal long run level b∗ , adjustment is done primarily by increasing tax, whereas at
higher debt levels main role is assigned to the use of interest rate that reduces debt service
costs. Hence, at low debt levels although discretionary policy overstabilizes debt (compared
to commitment), it does so with one of the fiscal instruments preserving consensus assignment. Yet low positive levels of debt have indirect effect on the conduct of monetary policy.
They promote not only expectations of higher taxes but also associated with taxes higher
marginal costs. To curb the pass-through of this effect on prices monetary policy should
conduct tighter monetary policy.
17
[Figure 2 about here.]
From the perspective of the zero lower bound analysis, the implication of higher long
run debt is the reduction in frequency of falling into liquidity trap. Figure 2 shows how
optimal interest rate response on impact changes with beginning of the period outstanding
debt. The higher is outstanding debt level the large is the preference shock required for
zero lower bound to be reached. Optimal long-run level of debt balances the benefit from
less frequently binding zero lower bound against the cost of distortionary taxation required
to support this level. Increasing long-run debt level would further reduce occasions of the
liquidity traps, but would lead to larger long-run gaps in output and both private and public
consumption. On the other hand reducing long-run level of debt would make those gaps
lower at a cost of more frequent liquidity traps.
4.4
Optimal responses
We proceed with examination of optimal policy and allocation in Markov equilibrium by
considering impulse responses to varying demand when government budget constraint is
relax starting from the optimal long run debt level b∗ . Demand variation is a result of
exogenous time preference changes. An increase in the preference shock ξ makes agents
more patient, reduces their desire to spend today, and increases their desire to save.
Moderate fluctuations in demand can be successfully offset with variations in nominal
interest rate. Figure 3 shows impulse responses under optimal discretionary policy following
positive preference shock of one unconditional standard deviation magnitude. Lowering
policy rate reduces excess savings and makes allocation of aggregate economic activity intact.
Labor supply and both private and public consumption are stabilized at the values that
are slightly below their deterministic steady state counterparts, because risky steady state
level of government debt is positive compared to zero debt in the deterministic steady
state. Keeping allocation constant also requires keeping government spending and tax rate
unaltered. Lower policy rates raise government bond prices, hence government has to reduce
borrowing to keep its liabilities market value constant. Debt adjustment helps to balance
intertemporal government budget constraint in response to variation in interest rate while
going without variations in primary balance.
[Figure 3 about here.]
If preference shock ξ deviation from its average generates large enough fall in demand,
zero lower bound starts binding. Figure 4 shows impulse responses under optimal discretionary policy following positive preference shock of three unconditional standard deviations
magnitude. When this happens, real interest rate fails to offset increase in saving desire
of households and this leads to a fall of private demand that in equilibrium will lead to a
fall of private consumption below potential. Falling private demand manifests itself into
reduction of firms’ labor demand and downward pressure on real wages. As a result, firms
become more eager to lower their prices even in the presence of price adjustment costs,
which effectively would lead to resource losses. Because of the shock persistence, agents
expect deflationary pressure to continue for some time in the future. Through expectations
this creates a further increase in real interest rates that reinforces savings desire and endogenously aggravates fall in private demand. The larger is the preference shock the more
pronounced are negative private consumption gap and deflationary pressure.
[Figure 4 about here.]
18
In models without fiscal sector where private sector is the only source of demand, falling
private demand is the only force changing aggregate demand. Differently, in the class of
models with endogenous government spending, it is possible for the government to affect
aggregate demand. Optimal policy mix in the liquidity trap features increase in the government spending that cushions decline in aggregate demand due to a fall in private demand,
which improves hiring incentives of the firms. Wages adjust upward in reaction to higher labor demand and so do prices. Initial government spending expansion dies out until reaching
pre-crisis level when zero lower bound stops binding. The fact that optimal discretionary
policy of government spending is used to stabilization purposes when zero lower bound is
binding, which is not the case when demand fluctuations are not strong enough to put economy into the liquidity trap, has been shown in Nakata (2013) and Schmidt (2013) for the
Ricardian case with lump-sum taxes.
Using government spending to pull inflation up is costly, because it opens the gap by
driving public consumption further away from its efficient level. Like government spending
that takes on a new role during the liquidity trap, so does the labor tax responds to binding
zero lower bound when its variation is otherwise avoided. Another part of optimal policy
response mix to a shock that puts economy into the liquidity trap includes temporarily rising
tax rate and setting it back to pre-crisis level once zero lower bound stops binding. Higher
labor tax is optimal due to its supply effect on prices and output determination. Agents are
less willing to work when tax rate is higher, therefore firms are forced to increase wages to
produce a given amount of output. Firms then pass some degree of marginal cost increase
on prices.
Cost push inflation generated by tax variation would not be as beneficial if created
only in the impact period. The important part is to create credible expectations of this
inflationary policy to continue into the future while economy is still in the liquidity trap. If
created, such expectations offset expectations of deflation due to binding zero lower bound,
which reduces real interest rate and effectively mitigates effect of the preference shock. Debt
stock passed from one time period to another is a natural state variable that makes policy
history-dependent. Using this mechanism, current fiscal policy can be used to condition the
degree to which future tax policy is inflationary. Higher tax rates on impact upon entering
the liquidity trap generate more revenue than is required to match expanding government
spending. Flow of extra profit from increase in primary surplus is then spent on reduction
of the debt stock. Underpinning this debt response is the mean-reverting nature of the
preference shock. Conditional on falling into the liquidity trap today, one expects demand
to improve in the future and deflationary pressure to be lower than on impact. Bequeathing
lower stock of outstanding nominal debt is enough to achieve required credibility, for it
implies gradual decline in tax rate. Next section described the consequences for optimal
government spending and labor tax response of turning this channel off in the case of
balanced government budget.
4.5
Comparison with the balanced budget case
We now contrast previous analysis to the case where the government is restrained to keep
flow budget balanced in every period. When doing so, we do not change nature of taxation
instrument. Government spending has to be financed with revenue collected via distortionary tax on labor income. Balanced budget prevents the government from using government debt to shape private sector expectations at the liquidity trap and away from it.
Figure 4 shows impulse responses under optimal discretionary policy without debt fol-
19
lowing positive preference shock of three unconditional standard deviations magnitude. As
before following such a shock in attempt to offset increased savings desire of agents nominal
interest rate is brought down to the zero lower bound. Fiscal responses are qualitatively
similar to the case with government debt. Both government spending and labor income tax
rate temporarily increase reducing deflationary momentum. However, government debt now
can not be used to create credible expectations of optimal path for the two fiscal instruments throughout the liquidity trap. Real rates stay high and economy experiences stronger
demand fall. In response to falling demand wages plummet and we observe larger private
consumption gap and falling prices.
Inability to exploit intertemporal dimension pushes the government towards considerably
more pronounced intratemporal response of the government spending. Increasing government spending in order to cushion fall of the aggregate demand, the government goes much
further to the point where its response is order of magnitude larger than in the case with
government debt. On the other hand, there is less reliance on tax instrument as the rate set
in response is around one percentage point lower than in the case with government debt.
This change of the roles lead to a different output response. Stronger use of expansionary
government spending and milder use of contractionary labor tax do not depress output as
much as with government debt. In fact with our parametrization we observe positive output gap at the liquidity trap with balanced government budget. Notice that even observed
larger than efficient level of output is not enough to bring along positive inflation without
government having ability to affect private expectations. Absolute magnitude of the output
gap is smaller than in the case with government debt, but this comes at a cost of having
considerably larger gap in public consumption.
[Figure 5 about here.]
Absence of government debt not only affects fiscal policy and allocation at the liquidity
trap, but also it affects interest rate policy and allocation when the zero lower bound is not
binding. Without government debt economy fluctuates around its efficient deterministic
steady state, but moderate fluctuations in demand are not precisely offset with variations in
nominal interest rate. Figure 6 shows impulse responses under optimal discretionary policy
following positive preference shock of one unconditional standard deviation magnitude.
[Figure 6 about here.]
Under balanced budget assumption, the government increases nominal rate more aggressively in response to a preference shock before reaching zero lower bound than it would in
the economy without balanced budget. First, absence of positive level of government debt
makes monetary policy more accommodative. Second, different dynamics during the liquidity trap episodes without government debt spill over into dynamics when zero lower bound
is not binding through expectations. Forward-looking agents and firms take into account
higher real rates and lower wages corresponding to the states with binding zero lower bound
and adjust correspondingly their consumption and prices downward even when zero lower
bound is not yet binding. The higher is conditional probability of reaching the liquidity
trap, the stronger is such pass-through. Monetary policy lowering nominal interest rate
more aggressively counteracts this effect. In equilibrium, more accommodative monetary
policy leads to output boom and positive private consumption gap.
20
5
Conclusion
This paper characterizes optimal monetary and fiscal policy under discretion in a New
Keynesian model with a zero lower bound on nominal interest rates. Two fiscal instruments,
government spending and distortionary labor income tax, enter optimal stabilization mix
during liquidity trap episodes due to binding zero lower bound. Government with the
balanced government budget relies more on government spending instrument. Upon falling
into the liquidity trap, spending increase of the government with balanced budget is one
order of magnitude larger than of the government with relaxed budget. On the other hand,
the government whose budget constraint is relaxed places more weight on the use of labor
income tax. Change of the labor tax by the government with relaxed budget constraint is
three times larger than by the government with balanced budget.
Joint use of labor tax and debt when government budget constraint is relaxed also
influences optimal policy at the times when zero lower bound is not binding. Risk of reaching
zero bound on nominal interest rate makes it optimal to accumulate positive long-run level
of debt, albeit a relatively small one.
The analysis in this paper—among a number of simplifying assumptions—omits fiscal
consequences of the monetary policy. Extending analysis by considering consequences of
higher outstanding debt levels and different maturity length is deemed as important direction
for future research.
21
References
Adam, K., “Government debt and optimal monetary and fiscal policy,” European Economic
Review 55 (January 2011), 57–74. 13
Adam, K. and R. M. Billi, “Discretionary monetary policy and the zero lower bound on
nominal interest rates,” Journal of Monetary Economics 54 (April 2007), 728–752. 3, 17
———, “Distortionary fiscal policy and monetary policy goals,” Economic Letters 122
(2014), 1–6. 14
Aiyagari, R., A. Marcet, T. Sargent and J. Seppälä, “Optimal Taxation without
State-Contingent Debt,” Journal of Political Economy 110 (2002), 1220–1254. 17
Anderson, G., J. Kim and T. Yun, “Using a projection method to analyze inflation bias
in a micro-founded model,” Journal of Economic Dynamics and Control (2010). 8
Barro, R., “On the determination of the public debt,” The Journal of Political Economy
87 (1979), 940–971. 5
Bodenstein, M., J. Hebden and R. Nunes, “Imperfect credibility and the zero lower
bound,” Journal of Monetary Economics 59 (March 2012), 135–149. 2
Braun, A., L. Korber and Y. Waki, “Small and orthodox fiscal multipliers at the zero
lower bound,” Federal Reserve Bank of Atlanta, 2013. 6
Brumm, J. and M. Grill, “Computing equilibria in dynamic models with occasionally
binding constraints,” Journal of Economic Dynamics and Control 38 (2014), 142–160. 16
Bundick, B., “Forward Guidance Under Uncertainty,” , 2013. 3
Burgert, M. and S. Schmidt, “Dealing with a Liquidity Trap when Government Debt
Matters: Optimal Time-Consistent Monetary and Fiscal Policy,” Institute for Monetary
and Financial Stability, 2013. 5
Calvo, G., “Staggered prices in a utility-maximizing framework,” Journal of monetary
Economics 12 (1983). 8, 15
Chari, V. and P. Kehoe, “Sustainable plans and mutual default,” The Review of Economic Studies 60 (1993), 175–195. 9
Chen, X., T. Kirsanova and C. Leith, “How Optimal is US Monetary Policy ?,” , 2013.
2
Chung, H., J.-P. Laforte, D. Reifschneider and J. C. Williams, “Have We Underestimated the Likelihood and Severity of Zero Lower Bound Events?,” Journal of Money,
Credit and Banking 44 (February 2012), 47–82. 5
Clarida, R., J. Gali and M. Gertler, “The science of monetary policy: a New Keynesian perspective,” Journal of Economic Literature 37 (1999), 1661–1707. 3
Coeurdacier, N., H. Rey and P. Winant, “The Risky Steady-State,” American Economic Review 101 (2011), 398–401. 17
22
Correia, I., E. Farhi, J. P. Nicolini and P. Teles, “Unconventional Fiscal Policy at
the Zero Bound,” American Economic Review 103 (June 2013), 1172–1211. 2
Eggertsson, G., “The Deflation Bias and Committing to Being Irresponsible,” Journal of
Money, Credit, and Banking 38 (2006), 283–321. 5, 17
Eggertsson, G. and M. Woodford, “The Zero Bound on Interest Rates and Optimal
Monetary Policy,” Brookings Papers on Economic Activity 34 (2003), 139–235. 3
———, “Optimal Monetary and Fiscal Policy in a Liquidity Trap,” , 2006. 4
Faraglia, E., A. Marcet, R. Oikonomou and A. Scott, “The Impact of Debt Levels
and Debt Maturity on Inflation,” The Economic Journal 123 (February 2013), F164–F192.
9
Fernández-Villaverde, J., G. Gordon, P. Guerrón-Quintana and J. F. RubioRamírez, “Nonlinear adventures at the zero lower bound,” Federal Reserve Bank of
Philadelphia, 2012. 15
Gnocchi, S. and L. Lambertini, “Monetary and Fiscal Policy Interactions with Debt
Dynamics,” , 2014. 14
Hansen, L. P. and T. Sargent, Robustness (Princeton University Press, 2007). 5
IMF, “World Economic Outlook - Recovery Strengthens, Remains Uneven,” (2014). 5
Kirsanova, T., C. Leith and S. Wren-Lewis, “Monetary and Fiscal Policy Interaction:
The Current Consensus Assignment in the Light of Recent Developments,” The Economic
Journal 119 (2009), 482–496. 2
Kydland, F. and P. Prescott, “Rules rather than discretion: The inconsistency of
optimal plans,” The Journal of Political Economy 85 (1977), 473–492. 14
Leith, C. and S. Wren-Lewis, “Fiscal Sustainability in a New Keynesian Model,” Journal of Money, Credit and Banking 45 (December 2013), 1477–1516. 4, 15, 17
Levin, A., D. López-Salido, E. Nelson and T. Yun, “Limitations on the effectiveness
of forward guidance at the zero lower bound,” International Journal of Central Banking
(2010), 143–189. 3
Levine, P., P. McAdam and J. Pearlman, “Quantifying and sustaining welfare gains
from monetary commitment,” Journal of Monetary Economics (2008). 3
Lustig, H., C. Sleet and e. Yeltekin, “Fiscal hedging with nominal assets,” Journal
of Monetary Economics 55 (May 2008), 710–727. 9
Motta, G. and R. Rossi, “Ramsey Monetary and Fiscal Policy: the Role of Consumption
Taxation,” , 2013. 13
Nakata, T., “Optimal Government Spending at the Zero Bound: Nonlinear and NonRicardian Analysis,” , 2011. 4
———, “Optimal Fiscal and Monetary Policy With Occasionally Binding Zero Bound Constraints,” , 2013. 2, 6, 19
23
Nakov, A., “Optimal and Simple Monetary Policy Rules with Zero Floor on the Nominal
Interest Rate,” International Journal of Central Banking (2008), 73–127. 3
Ngo, P., “Optimal Discretionary Monetary Policy in a Micro-Founded Model with a Zero
Lower Bound on Nominal Interest Rate,” , 2013. 6, 8
Reifschneider, D. and J. C. Williams, “Three lessons for monetary policy in a lowinflation era,” Journal of Money, Credit and Banking 32 (2000). 5
Rotemberg, J., “Sticky prices in the United States,” The Journal of Political Economy
(1982). 8
Schmidt, S., “Optimal Monetary and Fiscal Policy with a Zero Bound on Nominal Interest
Rates,” Journal of Money, Credit and Banking 45 (October 2013), 1335–1350. 2, 19
Schmitt-Grohé, S. and M. Uribe, “Optimal fiscal and monetary policy under sticky
prices,” Journal of Economic Theory 114 (February 2004), 198–230. 17
Summers, L., “Why Stagnation Might Prove to be the New Normal,” The Financial Times
(2013). 5
Werning, I., “Managing a Liquidity Trap: Monetary and Fiscal Policy,” , 2011. 2, 3
Woodford, M., Interest and Prices: Foundations of a Theory of Monetary Policy (Princeton University Press, 2003). 7
24
A
First-best allocation characterization
Lagrangian corresponding to the Planner’s problem is
!
∞
t−1
X
Y
t
L ≡ E0
β
ξs [u(Ct ) + g(Gt ) − v(Yt ) + γt (Yt − Ct − Gt )] .
t=0
s=−1
First-order conditions are as follows
u0 (Ct ) = γt ,
g 0 (Gt ) = γt ,
v 0 (Yt ) = γt .
Eliminating Lagrange multiplier γt leaves us with two equations
u0 (Ct ) = v 0 (Yt ),
g 0 (Gt ) = v 0 (Yt ),
that together with resource constraint Yt = Ct + Gt characterize first-best allocation as a
solution of the Planner’s problem.
B
Ramsey equilibrium characterization
Lagrangian corresponding to the Ramsey problem is
!
∞
t−1
X
Y
t
L ≡ E0
β
ξs [u(Ct ) + g(Gt ) − v(Yt )
t=0
s=−1
uc,t+1
− λ1,t uc,t − Rt ξt βEt
πt+1
λ1,t−1
uc,t
−
uc,t−1 − Rt−1 ξt−1 β
ξt−1 β
πt
− λ2,t ([θ(1 − s)wt − (θ − 1) − ϕ(πt − 1)πt ] Yt uc,t + ϕξt βEt {(πt+1 − 1)πt+1 Yt+1 uc,t+1 })
λ2,t−1
−
([θ(1 − s)wt−1 − (θ − 1) − ϕ(πt−1 − 1)πt−1 ] Yt−1 uc,t−1 + ϕξt−1 β(πt − 1)πt Yt uc,t )
ξt−1 β
− λ3,t (wt (1 − τt )uc,t − vy,t )
ϕ
− λ4,t 1 − (π − 1)2 Yt − Ct − Gt
2
bt−1
−1
− (Gt + ςt − τt wt Yt )
− λ5,t Rt bt −
πt
bt
−1
− Et λ5,t+1 ξt+1 β Rt+1 bt+1 −
− (Gt+1 − τt+1 wt+1 Yt+1 )
πt+1
+ νt (Rt − 1) + ηt bt )],
First-order conditions
25
uc,t = λ1,t ucc,t + λ2,t [θ(1 − s)wt − (θ − 1) − ϕ(πt − 1)πt ] Yt ucc,t + λ3,t wt (1 − τt )ucc,t − λ4,t
ucc,t
− λ1,t−1 Rt−1
− λ2,t−1 ϕ(πt − 1)πt Yt ucc,t
πt
ϕ
vy,t = −λ2,t [θ(1 − s)wt − (θ − 1) − ϕ(πt − 1)πt ] uc,t + λ3,t vyy,t − λ4,t 1 − (π − 1)2 − λ5,t ((τt − s)wt )
2
− λ2,t−1 ϕ(πt − 1)πt uc,t
bt−1
0 = λ2,t ϕ(2πt − 1)Yt uc,t + λ4,t ϕ(πt − 1)Yt − λ5,t 2
πt
uc,t
− λ1,t−1 Rt−1 2 − λ2,t−1 ϕ(2πt − 1)Yt uc,t
πt
0 = λ2,t θ(1 − s)Yt uc,t + λ3,t (1 − τt )uc,t + λ5,t ((τt − s)Yt )
0 = λ3,t uc,t − λ5,t Yt
gG,t = −λ4,t − λ5,t
bt
uc,t+1
+ λ5,t 2 + νt
0 = λ1,t ξt βEt
πt+1
Rt
λ5,t+1
λ5,t
0 = βEt ξt+1
−
+ ηt
πt+1
Rt
B.1
Ramsey steady state
TBA
C
Markov equilibrium characterization
Lagrangian for the Markov problem of the government is as follows
L ≡ u(C) + g(G) − v(Y ) + βξE {V (b; ξ 0 )}
− λ1 [uc − RβξE {S (b; ξ 0 )}]
− λ2 [[θ(1 − s)w − (θ − 1) − ϕ(π − 1)π] Y uc + ϕβξE {Z (b; ξ 0 )}]
− λ3 [w(1 − τ )uc − vy ]
i
h
ϕ
− λ4 1 − (π − 1)2 Y − C − G
2
b−1
−1
− λ5 R b −
− (G + ς − τ wY )
π
+ ν [R − 1] + ηb,
where
uc (C (b; ξ 0 ))
,
Π (b; ξ 0 )
Z (b; ξ 0 ) ≡Y (b; ξ 0 ) · (Π (b; ξ 0 ) − 1) · Π (b; ξ 0 ) · uc (C (b; ξ 0 )) .
S (b; ξ 0 ) ≡
26
Note that multipliers ν > 0, η > 0 and have to satisfy Kuhn-Tucker conditions
0 = ν (R − 1) ,
0 = ηb.
Ignoring functions’ arguments, the corresponding first-order conditions, apart from equilibrium implementability constraints, include
[C] :
[Y ] :
[π] :
[w] :
[τ ] :
uc = (λ1 + λ2 [θ(1 − s)w − (θ − 1) − ϕ(π − 1)π] Y + λ3 w(1 − τ )) ucc − λ4
(C.1)
ϕ
vy = − λ2 [θ(1 − s)w − (θ − 1) − ϕ(π − 1)π] uc + λ3 vyy − λ4 1 − (π − 1)2 − λ5 (τ − s)w
2
(C.2)
b−1
(C.3)
0 =ϕ (λ2 (2π − 1)uc + λ4 (π − 1)) Y − λ5 2
π
0 = (λ2 θ(1 − s)Y + λ3 (1 − τ )) uc + λ5 (τ − s)Y
(C.4)
0 =λ3 uc − λ5 Y
(C.5)
[G] : gG = − λ4 − λ5
n 0o
[R] :
0 =λ1 βξE S + λ5 bR−2 + ν
n 0o
n 0o
n 0o
[b] :
0 =βξE Vb + βξλ1 RE Sb − λ2 βξϕE Zb − λ5 R−1 + η
(C.6)
(C.7)
(C.8)
where subscripts denote partial derivatives. Envelope theorem implies
0
0
Vb =
λ5
Π0
(C.9)
Simplifying the system
One can work around with the system of first order conditions and simplify it. Using resource
constraint (2.1) we solve for government expenditures
ϕ
G = Ĝ(Y, C, π) ≡ 1 − (π − 1)2 Y − C.
2
Analogously we use Phillips curve (2.8) and consumption leisure trade-off (2.5) to express
wage and labor tax
n 0o

E
Z
(θ − 1)
ϕ
(π − 1)π − β
,
w = ŵ (Y, C, π, b) ≡
+
θ(1 − s) θ(1 − s)
Y uc
τ = τ̂ (Y, C, π, b) ≡ 1 −
1 vy
.
ŵ uc
Private Euler equation (2.6) along with zero-lower bound condition (2.9) deliver
uc
R = R̂ (C, b) ≡ max 1,
,
βξE {S 0 }
27
We proceed by explicitly solving for Lagrange multipliers λ2 , λ3 , λ4 and λ5 from equations
(C.3)-(C.6).
gG τ̂
Y
λ2 = λ̂2 (Y, C, π, b) ≡ Ω
+
,
uc θ
(1 − τ̂ )uc
gG
λ3 = λ̂3 (Y, C, π, b) ≡ −Ω Y,
uc
λ4 = λ̂4 (Y, C, π, b) ≡ (Ω − 1) gG ,
λ5 = λ̂5 (Y, C, π, b) ≡ −ΩgG ,
where Ω < 1 is defined as follows
Ω (Y, C, π, b) ≡
ϕπ 2 (π − 1)Y
ϕπ 2 (π − 1)Y + ϕπ 2 (2π − 1)Y τ̂θ +
Y
(1−τ̂ )uc
.
+ b−1
Non-binding ZLB
If zero-lower bound is not binding, then from (C.7)
λ1 = λ̂1 (Y, C, π, b) ≡ Ω
gG
(uc )
2 bβξE
n 0o
S ,
if η = 0,
and we are left with 4 unknowns (Y, C, π, b) and the same number of equations
0 =uc − λ̂1 + λ̂2 [θŵ − (θ − 1) − ϕ(π − 1)π] Y + λ̂3 ŵ(1 − τ̂ ) ucc + λ̂4 ,
(C.10)
ϕ
0 =vy + λ̂2 [θ(1 − s)ŵ − (θ − 1) − ϕ(π − 1)π] uc − λ̂3 vyy + λ̂4 1 − (π − 1)2 + λ̂5 τ̂ ŵ,
2
(C.11)
b−1
− Ĝ + ςˆ − τ̂ ŵY ,
(C.12)
0 =R̂−1 b −
( 0 π)
n 0o
n 0o
λ̂5
+ βξ λ̂1 R̂E Sb − λ̂2 βξϕE Zb − λ̂5 R̂−1 ,
0 =βξE
(C.13)
0
Π
where first two correspond to first-order conditions (C.1) and (C.2), equation (C.12) is the
government budget constraint, and (C.13) is the so-called generalized Euler equation (GEE)
characterizing optimal bond purchases choice. The GEE equates the discounted expected
utility loss resulting from a tighter budget constraint in the future with the current direct
and indirect gains resulting from a marginal relaxation of the budget constraint today and
the other equilibrium implementability constraints. In our setup there is also ad-hoc lending
constraints (2.11), which means that GEE has to hold only for the interior solution b > 0.
Binding ZLB
For the case of binding zero-lower bound we can not use (C.7) to express λ1 . Instead, the
very fact of binding zero-lower bound delivers extra equation of the form
ξβ
E {S (b; ξ 0 )}
= 1.
uc
28
(C.14)
One can then use equation (C.1) to solve for λ1 as follows
uc − gg
+
λ1 = λ̂1 (Y, C, π, b) ≡
ucc
gG 1 uc
τ̂
Y
ΩY
+ ŵ(1 − τ̂ ) −
+
[θ(1 − s)ŵ − (θ − 1) − ϕ(π − 1)π]
uc Y ucc
θ
(1 − τ̂ )uc
and use system of equations (C.11)-(C.13) along with (C.14) to solve for unknown (Y, C, π, b).
Solution
Described system of equations has to be satisfied for each (b−1 ; ξ), hence it is a system
of functional equations with solution described by a tuple of functions {Y, C, Π , B}. As
it is common, the system characterizing Markov-perfect equilibria appears to contain yet
unknown policy functions and their derivatives.
C.1
Markov steady state
TBA
D
Subsidy rate
TBA
E
Solution method
TBA
We implement solution method in Matlab using open source nonlinear optimization
solver IPOPT. Interface of IPOPT solver for Matlab environment is implemented in a freeware third-party OPTI toolbox. To improve computation speed, solution of the Markov
optimization problem over the grid is done via parallel computing.
29
Figure 1: Optimal debt change policy, (B(b−1 , ξ) − b−1 ), conditional on ξ = 1
Figure 2: Contour lines of optimal nominal interest rate policy, R(b−1 , ξ)
Notes: White state space area corresponds to binding zero lower bound. Dotted line marks
optimal long run debt level b∗ .
Figure 3: Impulse responses when zero lower bound is not reached
Notes: Impulse responses to a +1 unconditional standard deviation preference shock starting
from optimal long-run debt level b∗ . Dashed blue line in the top left panel shows the natural
real interest rate.
Figure 4: Impulse responses when zero lower bound is reached
Notes: Impulse responses to a +3 unconditional standard deviation preference shock starting
from optimal long-run debt level b∗ . Dashed blue line in the top left panel shows the natural
real interest rate. Shaded areas show conditional 68% confidence bands.
Figure 5: Impulse responses when zero lower bound is reached under balanced budget
Notes: Impulse responses to a +3 unconditional standard deviation preference shock starting
from optimal long-run debt level b∗ . Dashed blue line in the top left panel shows the natural
real interest rate. Shaded areas show conditional 68% confidence bands.
Figure 6: Impulse responses when zero lower bound is not reached under balanced budget
Notes: Impulse responses to a +1 unconditional standard deviation preference shock starting
from optimal long-run debt level b∗ . Dashed blue line in the top left panel shows the natural
real interest rate.