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Working Paper Series
Tales of Transition Paths: Policy
Uncertainty and Random Walks
WP 15-11
Josef Hollmayr
Deutsche Bundesbank, Frankfurt am
Main
Christian Matthes
Federal Reserve Bank of Richmond
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Tales of Transition Paths:
Policy Uncertainty and Random Walks
Josef Hollmayr∗and Christian Matthes∗∗
September 2015
Working Paper No.15-11
Abstract
What happens when fiscal and/or monetary policy changes systematically?
We construct a DSGE model in which agents have to estimate fiscal and monetary
policy rules and assess how uncertainty surrounding the conduct of policymakers
influences transition paths after policy changes. We find that policy changes
of the magnitude often considered in the literature can lead private agents to
hold substantially different views about the nature of equilibrium than would
be predicted by a full information analysis. In particular, random walk-like
behavior can be observed for a large number of periods in equilibrium, even
though the models we use admit stationary dynamics under full-information
rational expectations.
JEL codes: E32, D83, E62
Keywords: DSGE, Monetary-Fiscal Policy Interaction, Learning
∗
Deutsche Bundesbank, Frankfurt am Main (e-mail: josef.hollmayr@bundesbank.de)
Federal Reserve Bank of Richmond (email: christian.matthes@rich.frb.org). For their useful comments we would like to thank seminar and conference participants at the Midwest Macro meetings
(Columbia, MO), the CEF meetings (Oslo), the Verein für Socialpolitik (Hamburg), the Dynare Conference (Paris), ECB, the T2M conference in Berlin as well as Richard Clarida, Tim Cogley, Eric
Leeper, and Mario Pietrunti. The views expressed in this paper are those of the authors and do not
necessarily reflect the opinions of the Deutsche Bundesbank, the Federal Reserve Bank of Richmond
or the Federal Reserve System. This project was started while Christian Matthes was visiting the
Bundesbank, whose hospitality is gratefully acknowledged.
∗∗
2
1
Introduction
An economic theory asserts how reduced form parameters change when
government policies change.
Thomas J. Sargent, Points of Departure
What happens when fiscal or monetary policy rules change dramatically? Large
changes in fiscal and monetary policy rules are routinely evaluated in micro-founded
dynamic equilibrium models (see for example Curdia and Finocchiaro (2013)). Two
strong assumptions commonly underlie such exercises: Agents are initially unaware of
possible policy changes, and agents become immediately aware of the new policy rule
once it is implemented. Work on Markov-switching DSGE models, such as Bianchi
(2013) and Liu et al. (2011), dispenses of the first assumption but retains the second.
As Sargent (2015) highlights, these assumptions represent different economic theories
about how firms and households in our models react to policy changes. We put forth
another economic theory and model agents as econometricians that have to estimate
coefficients of policy rules to remove the second assumption. We borrow the assumption
of ’anticipated utility’ decision-making (Kreps (1998)) that is common in the learning
literature (Sargent et al. (2006), Primiceri (2005), and Milani (2007) are three examples) and thus keep the first assumption of the standard approach in play.1
A key theme of this paper is how random walk-like behavior (i.e. the VAR representation of the linearized equilibrium dynamics has eigenvalues equal to or slightly larger
than 1) can endogenously arise in models that under rational expectations would only
feature stationary dynamics. This behavior arises even though agents in our model
have substantial knowledge of the economy and act as sophisticated econometricians
to uncover those features of the economy that they do not know about. It is the dynamics of the data in equilibrium that make the learning problem hard, leading to
differences between beliefs of agents and the true policy rule, which in turn fuels the
random walk-like behavior that arises in equilibrium.
We study two stylized examples of policy transitions: One is motivated by the Volcker
disinflation and studies the change of a monetary policy rule from parameter values
that would imply indeterminacy under full-information rational expectations to values that imply determinacy. We thus explicitly model the transition that is absent in
papers that study separately indeterminate and determinate outcomes for the Unites
1
Bianchi and Melosi (2013) introduce a very specific type of learning into a Markov-switching
DSGE model: Their agents do observe the policy rule coefficients currently in play, but are uncertain
how persistent the current regime is.
3
States, such as Lubik and Schorfheide (2004). Throughout this example we keep fiscal
policy passive in the language of Leeper (1991) and Clarida et al. (1999). We revisit
findings by Cogley et al. (2015), namely that even if agents think the economy is stable, unstable behavior can arise if beliefs are different enough from the actual policy
rule coefficients. We build on their findings, but we use a model that allows for fiscal
and monetary policy interaction along the lines of Leeper (1991) and show that the
economy in this scenario will most likely feature random walk-like behavior rather than
outright explosive behavior that would be rejected by the data. We also show that this
behavior can persist for many periods. It is important to emphasize that our focus
here is on unusual (when considered against the backdrop of recent policy behavior)
policy changes. We are not studying small (or ’modest’) policy changes such as those
emphasized by Leeper and Zha (2003).
The second example we study is motivated by the large changes in fiscal and monetary
policy that occurred following the recent financial crisis: a transition from an active
monetary policy/passive fiscal policy regime to a situation where fiscal policy is active
and monetary policy is passive. Not only is this an a priori reasonable description of
current policy in many developed economies, we also show that this policy scenario implies behavior of inflation and debt that is in line with observations since the financial
crisis: Random walk-like behavior for debt, but also low and stable inflation. These
outcomes make learning about the true policy coefficients for the monetary policy rule
hard for the agents in our model, and they lead to substantial periods of confusion
about the nature of equilibrium.
The agents in our model do not place any restrictions on the kind of equilibria they
consider (their ‘perceived law of motion’, or PLM): Both determinate and indeterminate as well as temporarily explosive equilibria are considered as possible outcomes by
our agents.
Once we remove arbitrary restrictions on the kind of equilibria considered by agents,
they can easily find themselves in situations where they misperceive the nature of the
equilibrium: In the first example, they might believe that equilibrium indeterminacy
persists for substantial periods and, in the second example, agents are led to believe
that the economy is temporarily explosive. Temporarily explosive dynamics can also be
a feature of a Markov-switching rational expectations model, an outcome highlighted
by Bianchi and Ilut (2013).
We endow our agents with substantial knowledge of the economy. They are only uncertain about the finite dimensional policy rule parameter vector. Furthermore, we endow
them with the same knowledge of the timing of the structural change that agents in
the standard approach have: Once policy changes, firms and households suspect that
4
policy has indeed changed. In the standard approach, agents immediately learn the
new policy rule coefficients, so their suspicions are instantaneously confirmed. In our
model, agents instead have to estimate the new policy rule coefficients.2 We do so to
minimize the differences between our approach and the standard approach outlined
above. Notwithstanding, even with this substantial knowledge of the economy, equilibrium outcomes in our environment are substantially different from those determined
using the standard approach. We assume that private agents do not have a suspicion
about the direction of the change in the policy rule coefficients. Any announcement
about the magnitude or direction of the policy change is thus viewed as incredible by
our agents.
Our focus on agents that only need to learn about the coefficients of policy rules sets
this paper apart from the earlier literature that studied monetary-fiscal policy interaction under learning such as Eusepi and Preston (2011) and Eusepi and Preston (2013),
who instead endow their agents with less knowledge about the structure of the economy.
The interaction of fiscal and monetary policy under rational expectations in DSGE
models was pioneered by Leeper (1991). Empirical analyses include Traum and Yang
(2011) and Bhattarai et al. (2012). We use similar models and borrow parameter
estimates from that literature, but refrain from using full-information rational expectations.
The effects of regime changes in monetary policy on beliefs and economic dynamics
has been studied in Andolfatto and Gomme (2003). Changes in fiscal policy and their
effects in an RBC economy with distortionary taxation has been studied in Hollmayr
and Matthes (2013). Finally, policy uncertainty more generally has recently received
substantially more attention due to the work of Baker et al. (2012), who quantify policy
uncertainty using various measures and show its impact using VARs.
Our approach is different from the previous literature in that we explicitly try to only
use minimal departure from a full information rational expectations approach, restrict
the private agents’ model uncertainty to the policy rule parameters and focus on the
transition dynamics rather than asymptotic results (i.e. whether equilibria are ’learnable’). The agents in our model use the Kalman Filter to estimate coefficients. We
thus make them sophisticated econometricians who could not easily improve upon their
estimation routine.
The next section uses the model in Leeper (1991) and its analytical solution to shed
some light on how changes in beliefs alter equilibrium dynamics and can lead to random
2
We model this suspicion by exogenously increasing the uncertainty about policy rule coefficients
in the period in which policy actually changes. Section 4 describes this in detail.
5
walk-like behavior. We then turn to our benchmark DSGE model before describing the
learning methodology in detail and discussing the differences between outcomes under
learning and using the standard approach.
2
An Analytical Example: Beliefs and Equilibrium
Outcomes in Leeper (1991)
In this section we highlight the key forces underlying our main results in a simpler
context that allows for analytical results - the model in Leeper (1991). We do not
explicitly model learning in this section, but instead we ask a simpler question - what
happens if beliefs about policy rule coefficients held by private agents happen to be
wrong? In the language of the learning literature, the agents’ perceived law of motion
(PLM) differs from the actual law of motion (ALM). Since agents in our benchmark
DSGE model will be wrong about the policy rules while they are learning about policy
coefficients after a policy change, this exercise will shed some light on the forces driving
our result in the learning model described in later sections. Here we focus on two
examples: one in which agents know that monetary policy is active and fiscal policy
is passive but do not know the exact value of the policy rule coefficients; and a second
situation in which agents still believe that monetary policy is active and fiscal policy
is passive, but the actual policy rule coefficients imply active fiscal policy and passive
monetary policy. For the first example, we will focus on how changing beliefs can alter
equilibrium dynamics more broadly, while, for the second example, we explicitly show
how and in what variables random walk-like behavior can occur. We will revisit the
second example in this section later for our benchmark DSGE model. For analytical
results that speak to the existence of non-stationary dynamics in the first example
(in which fiscal policy stays passive throughout), we refer the reader to Cogley et al.
(2015).3
To start, it is useful to first revisit the nomenclature introduced by Leeper (1991) in
order to discriminate between different classes of equilibria under rational expectations.
3
Cogley et al. (2015) do not explicitly model fiscal policy in their analytical example, but the
underlying assumption is passive fiscal policy, as is common in textbook New Keynesian models.
The form of the monetary policy rule in Leeper (1991) (where monetary policy reacts contemporaneously to inflation) means that changes in beliefs about monetary policy (while keeping fiscal policy
fixed) will not affect the matrix multiplying lagged state variables in the equilibrium law of motion.
Thus, Leeper’s model is less suited to showing how random walk-like behavior can arise in the example
inspired by the Volcker disinflation for our benchmark model.
6
active monetary policy
passive monetary policy
passive fiscal policy
unique equilibrium
multiple equilibria
active fiscal policy
no stable equilibrium
unique equilibrium
Table 1: The nomenclature of equilibria and policies in Leeper (1991).
Table 1 does this. We describe below exactly what parameter values constitute active or
passive policy, but, broadly speaking, passive fiscal policy describes a situation in which
the fiscal authority acts to stabilize debt, whereas active monetary policy describes a
situation in which monetary policy tries to counteract inflation sufficiently.
We now turn to the economy studied in Leeper (1991). The underlying nonlinear
model is described in that paper. To summarize, it is an endowment economy that
features lump-sum taxation, money in the utility function and monetary policy that
follows a Taylor rule. For the sake of brevity, we will directly focus on the linearized
system in inflation πt and debt bt :4
Et πt+1 = αβπt + βθt
(1)
ϕ1 πt + bt + ϕ2 πt−1 − (β −1 − γ)bt−1 + ϕ3 θt + ψt + ϕ4 θt−1 = 0
(2)
θt = ρθ θt−1 + εθt
(3)
ψt = ρψ ψt−1 + εψt
(4)
The ϕ parameters are convolutions of parameters and steady states and are defined in
Leeper (1991). Equation (1) is the result of combining the representative households
consumption Euler equation and the monetary policy rule. Equation (2) is the government’s budget constraint, where we have plugged in the optimality condition for money
holdings and the fiscal and monetary policy rules. A key parameter for our results is
the discount factor β. θt and ψt are exogenous policy shocks - the policy rules used in
deriving these equations are
Rt = απt + θt
(5)
τt = γbt−1 + ψt
(6)
where Rt is the nominal interest rate and τt are lump-sum taxes.
To analyze the dynamics of the system, we can stack the first two equations above:5
4
We will present the non-linear model for our benchmark case later.
For the sake of simplicity, we do not stack the exogenous laws of motion for the shocks in the
system. Our results hold for generic laws of motion of the exogenous process as long as those processes are stationary. Also, to determine the properties of the equilibrium dynamics, we focus on the
5
7
1
0
ϕ1 1
!
πt
bt
!
=
+
0
0
!
πt−1
!
−β/(αβ)
0
+
−ϕ2 β −1 − γ
bt−1
−ϕ3
−1
!
!
!
0 0
θt−1
1/(αβ)
+
Eπt+1
−ϕ4 0
ψt−1
0
!
θt
ψt
!
(7)
To solve the model, we have to plug in above an expression for the one-step ahead
expectations of inflation as a function of pre-determined exogenous variables or exogenous shocks. Once we do that, the system above gives the equilibrium dynamics.
The dynamics of the system and the nature of the prevailing equilibrium under fullinformation rational expectations are governed by αβ and β −1 − γ. If the first term
is larger than one in absolute value, while the second one is not, we are in a world
of monetary activism and passive fiscal policy, whereas, in the reverse case, we are in
a world of active fiscal policy (fiscal policy does not use taxes to stabilize debt) and
passive monetary policy.
Leeper gives the full-information rational expectations solution for inflation and expected inflation in the ’standard’ case of active monetary policy and passive fiscal
policy:
βρθ
θt
αβ − ρθ
β
πt = −
θt
αβ − ρθ
Et πt+1 = −
(8)
(9)
Note that fiscal shocks or the coefficient γ in the fiscal policy rule do not enter either
the expected or actual inflation. This happens precisely because fiscal policy is passive.
Under rational expectations and active fiscal policy, the fiscal shocks would play a role
in determining inflation, as documented by Leeper.
autoregressive matrix in the equilibrium law of motion. That matrix determines stability properties
as long as there is no root cancellation with the roots of the moving average polynomial. This turns
out to be the case here.
8
2.1
The Dynamics Under a PLM When Agents Know Which
Policy is Active
We now turn to a situation where agents have incorrect beliefs about the policy rule
coefficients. As mentioned before, for simplicity we assume here that beliefs are not
updated over time, an assumption we drop later in our learning model. Just as in
the larger benchmark model that we use for our numerical work later, we assume here
that agents observe all past endogenous variables but not the policy shocks, which the
agents have to infer based on the observable data and their beliefs about policy rule
coefficients.
In this section we assume that the true policy rule coefficients indeed imply that fiscal
policy is passive and monetary policy is active, but private agents believe that the
policy rule parameters take on values that are not equal to the true values of those
parameters. Agents have a degenerate prior on the policy rule coefficients given by α
b
and γ
b such that α
bβ > 1 and β −1 − γ
b < 1, but α
b 6= α and γ
b 6= γ, so agents know that
fiscal policy is passive and monetary policy is active, but they do not know the correct
parameter values.6 Given that agents think monetary policy is governed by a policy
rule with coefficient α
b, their perceived policy shock is (we assume that θ is not directly
observable) θbt = (α − α
b)πt + θt . Their conditional expectations of inflation next period
are then given by
Et∗ πt+1 = −
βρθ b
βρθ
θt = −
((α − α
b)πt + θt )
α
b β − ρθ
α
bβ − ρθ
(10)
If we plug these expectations into the matrix equation above, we can see that, in
this case, there are two differences relative to rational expectations: First, the effect
of a shock θ on expectations and thus contemporaneous inflation and debt can be
different since now α
b is in the denominator of the fraction in the equation determining
expectations. Second, there is an additional feedback effect governed by the differences
between beliefs and actual coefficients (α − α
b)πt . At this point, it is instructive to
mention how beliefs about fiscal policy enter the equilibrium dynamics. Equation (7)
uses the actual policy rules and the government budget constraint to solve for debt
dynamics. In the true government budget constraint, beliefs do not directly enter.
Agents use a perceived version of the budget constraint (equation (2), with the true
6
We make this assumption because it simplifies the exposition. This example shares many properties of the first we will use for the DSGE model, but in that case initial beliefs of agents will be
such that they believe they live in an indeterminate equilibrium. To show how changes in beliefs alter
equilibrium dynamics in this analytical example, we do not need to incorporate indeterminacy.
9
parameter values replaced by the agents’ beliefs and the true policy shocks replaced by
the perceived policy shocks)7 to forecast equilibrium outcomes. The perceived fiscal
shock will make their perceived government budget constraint hold. As long as agents
know that monetary policy is active and fiscal policy is passive, beliefs about fiscal
policy only enter equilibrium dynamics insofar as they influence agents’ estimates of
fiscal policy shocks that make the perceived government budget constraint hold. This
is because, under the perception of active monetary policy and passive fiscal policy,
agents do not find fiscal shocks useful to forecast inflation, as can be seen from equation
(10).
2.2
The Dynamics Under a PLM When Agents Are Confused
About Which Policy is Active
Now let us assume that agents hold the same beliefs as before, but the ALM satisfies
αβ < 1 and β −1 − γ > 1. Plugging the expectations derived in the previous section
into the matrix equation that describes the endogenous dynamics of the system, we
get
1+
|
+
1 βρθ
αβ α
bβ−ρθ (α
−α
b) 0
ϕ1
{z
!
−β/(αβ) −
−ϕ3
!
=
bt
1
}
A
1 βρθ
αβ α
bβ−ρθ
πt
0
−1
!
πt−1
−ϕ2 β −1 − γ
{z
}
|
bt−1
0
0
!
(11)
B
!
θt
ψt
!
+
0
0
−ϕ4 0
!
θt−1
!
ψt−1
The key insight to take away here is that πt is still a stationary process (it is perfectly
correlated with the stationary shock θt ). bt , on the other hand, is now non-stationary
because inflation expectations and thus actual inflation do not move to offset the effects
of |β −1 − γ| > 1 since inflation expectations are formed according to the dynamics for
the case in which monetary policy is active and fiscal policy is passive. To see this
more clearly, we can invert the matrix on the left-hand side of the equation above (call
this matrix A) and multiply by the matrix multiplying the lagged endogenous variables
(call this matrix B). This gives us the matrix governing the endogenous dynamics in
the actual law of motion. The inverse of A will also be lower triangular. To see this,
we can apply the formula for the inverse of a 2-by-2 matrix to A. Doing so gives us
7
The perceived government budget constraint also includes perceived versions of ϕ1 and ϕ2 since
those are functions of the perceived monetary policy rule parameter - see the definition of the ϕ
parameters in Leeper (1991).
10
the result that the (2, 2) element of A−1 is 1 and the (1, 2) element is 0. Thus, we get
A−1 B =
0
0
!
−ϕ2 β −1 − γ
(12)
The eigenvalues of that matrix are 0 and β −1 − γ. The second eigenvalue is thus larger
than 1 in absolute value by our assumption. In general, we assume that γ ≥ 0, so the
second eigenvalue is bounded from above by
1
,
β
which is around 1.01 with standard
quarterly calibrations of β. This hints at the possibility that even if agents learn, in
finite samples debt can behave much like a random walk. We will see this in our learning economy.
This example, while abstracting from many important channels through which monetary and fiscal policy can influence economic outcomes, already shows that beliefs of
private agents can have a large impact on economic outcomes and substantially alter the properties of the equilibrium. In particular, it will turn out that the random
walk-like behavior of debt coupled with stationary inflation dynamics will also be an
outcome of our larger model that we discuss next for the second policy experiment.
Even though both Leeper’s model and our larger DSGE model that we turn to next
abstract from issues such as the zero-lower bound on nominal interest rates, the behavior of debt and inflation coming out of both models when studying the second policy
experiment resembles actual data for developed economies since the global financial
crises and the subsequent changes in fiscal policies.
While Leeper’s model is very useful for opening the hood and obtaining further insights
into the issues we want to study, it also has its limitations: An insight that we can only
infer from our benchmark economy (as we will see later) and not from Leeper’s model
is that random walk-like behavior can also occur in the first scenario where agents
know that monetary policy is active and fiscal policy passive.8
Whenever eigenvalues of absolute value 1 or larger appear in arguments based on local
approximations (as is the case here and will be the case in our benchmark model)
we know that we can not use simulations to approximate unconditional moments.
Nonetheless, for simulations or calculations based on finite time periods such as those
presented in this paper, local approximation methods can be reliably used to approximate equilibrium dynamics when there are eigenvalues larger than 1 in play (especially
when these eigenvalues are not much larger than 1 in absolute value). This point has
8
One can use other models that can be solved analytically to show that random walk-like behavior
can occur in the related case where fiscal policy is not explicitly modelled - see Cogley et al. (2015).
11
been made by Kim et al. (2003).9
3
Model
Our model is a standard a medium-scale New-Keynesian model along the lines of,
for example, Smets and Wouters (2007) and Christiano et al. (2005). It incorporates
nominal frictions, habits, capital utilization and, additionally, a fiscal sector. The government accumulates debt if its income from distortionary labor and capital taxation
does not match outlays for government spending, transfers, and debt repayments and
interest payments. First-order conditions and the complete log-linearized model may
be found in the Appendix. The calibration for all parameters is standard in the literature and is mostly taken from Traum and Yang (2011), who estimate a similar model
using U.S. data. We thus relegate the numerical values of the parameters to Table 7
in the Appendix.
3.1
Households
The economy is populated by a continuum of households that each have access to a
full set of state-contingent securities, which leads them to make the same investment
and consumption decisions, even though they each supply a different labor input to the
labor market. Households maximize their expected utility,
10
where the instantaneous
utility function of household j takes the following form:
"
Utj
9
= Utb
(Ct − hCt−1 )1−σ L(j)1+φ
t
−
1−σ
1+φ
#
(13)
The fact that the behavior of a very persistent, but stationary, process can be very similar to
the behavior of a unit root process in finite samples is at the heart of many issues connected with
unit-root test, for example.
10
Agents treat the coefficients both in the fiscal and in the monetary policy rule as fixed when
making their decisions. We thus use an anticipated utility assumption along the lines of Kreps (1998),
which is common in the literature on adaptive learning. A more thorough description follows in the
section where the learning algorithm is described in more detail.
12
The household derives utility from consumption Ct and disutility from hours worked
L(j)t . Utb is a preference shock that follows in its log-linearized form an AR(1) process11 :
b
log(ubt ) = Constbu + ρub log(ubt−1 ) + ut
(14)
Each period, households can choose either to consume, invest (It ), or save in the form of
government bonds (Bt ). Households face the following budget constraint each period:
Ct + Bt + It = Wt (l)Lt (j, l)(1 − τt ) + (1 − τt )RtK Vt K̄t−1 + ψ(Vt )K̄t−1 +
Rt−1 Bt−1
+
+ Zt + P rt
πt
(15)
and the law of motion for private capital:
K̄t = (1 − δ)K̄t−1 +
Uti
It
1−s
It
It−1
(16)
The household’s income stems from working at the wage Wt and interest payments
on their savings at the rate Rt . Zt represents lump-sum transfers and P rt are the
profits households obtain from the intermediate firm. τt denotes the distortionary tax
rate that the government levies equally on labor and capital
rented by households to firms at the rate
RtK
12
. Effective capital K̄t is
and is related to physical capital by its
utilization rate Vt along the lines of
Kt = Vt K̄t−1
(17)
The cost of the utilization rate is denoted by ψ(Vt ) where the functional form follows
standard assumptions in the literature: V is 1 in steady state and ψ(1) = 0. Additionally ψ[0, 1] is defined, so that the following equation is satisfied:
ψ 00 ()
ψ 0 (1)
=
ψ
.
1−ψ
Capital
is subject to a certain depreciation rate δ, but is accumulated over time via investment
It . Investment is subject to adjustment costs s(.) and to the shock Uti ), which captures
an exogenous disturbance as to how efficiently investment can be turned into effective
capital. It also follows a simple AR(1) process in its log-linearized form:
i
log(uit ) = Constiu + ρui log(uit−1 ) + ut
11
(18)
Note that, throughout, lower-case letters are used for log-linearized variables.
In using only one tax rate for both input factors we follow Traum and Yang (2011). For our
application, this assumption allows us to keep the learning problem of the private agents more parsimonious. The forces that drive our results would also be in play in a model with more than one tax
rate.
12
13
3.2
Wages
A composite labor service Lt is produced by labor packers and is given by
Z
1
Lt =
lt (l)
1
1+ηtw
1+ηtw
dl
(19)
0
The demand function of the labor packers stems from the profit maximization problem,
which yields (with Ldt as the composite demand for labor services).
lt (l) = Ldt
Wt (l)
Wt
w
− 1+η
t
ηw
t
(20)
where ηtw is an exogenous markup shock to wages. It follows in its log-linear form an
AR(1) process:
w
log(ηtw ) = Constηw + ρηw log(ηt−1
) + ηt
w
(21)
The nominal aggregate wage evolution is then given by
− η1w
Wt = (1 − θw )W̃t
t
+ θw π
1−χw
1
χw − η w
t
πt−1
− η1w
−ηtw
Wt−1
t
(22)
The fraction θω of households that cannot re-optimize index their wages to past inflation
by the rule:
χω 1−χω
Wt (j) = Wt−1 (j)(πt−1
πss )
3.3
(23)
Firms
The production function of firm i is linear in technology and labor:
Yt (i) = exp(At )Kt−1 (i)α Lt (i)1−α
(24)
where Yt denotes the output produced with a certain level of technology At , labor input
Lt (i), and capital Kt . The exogenous process for technology is given by an AR(1):
log(At ) = ρa log(At−1 ) + A
t
(25)
In terms of price setting, we assume that retailers set their prices according to the Calvo
(1983) mechanism, i.e. each period the fraction (1 − θi ) of all firms are able to reset
14
their prices optimally. Furthermore, we assume that firms that cannot reoptimize their
prices in period t index their prices to the past inflation rate following the equation:
χ
1−χp
Pt (i) = Pt−1 (i)χp πt p πss
(26)
Profits of firm j (in nominal terms) are then equal to
Πt (i) = (Pt (j) − M Ct (i))
Pt (i)
Pt
−ηtp
Yt (i)
(27)
with real marginal costs given by
M Ci,t = (1 − α)α−1 α−α (Rtk )α Wtα A−1
t
and the demand for good i Yt (i) as
Yt (i) = Yt
pt (i)
Pt
− 1+ηp tp
ηt
(28)
where ηtp is an exogenous markup shock to the intermediate good’s price that also
follows an AR(1) process in its log-linear form:
p
log(ηtp ) = Constηp + ρηp log(ηt−1
) + ηt
3.4
p
(29)
Government
The government budget constraint takes the following form:
Bt = Bt−1
Rt−1
− RtK Kt τt − Wt Lt τt + Gt + Zt
πt
(30)
We let the labor tax rule react to past levels of debt. For simplicity, all other fiscal
rules follow Gaussian AR(1) processes. We assume that agents know all fiscal policy
rules except for the tax rule. Government spending is given by
log(Gt ) = ConstG + ρg log(Gt−1 ) + G
t
(31)
15
Zt denotes transfers, which behave as follows:
log(Zt ) = ConstZ + ρz log(Zt−1 ) + Zt ,
(32)
As mentioned above, the tax rate is modeled as a rule with the feedback coefficient ρb ,
which the agents do not know for sure and have to infer:
log(τt ) = Constτ + ρb log(Bt−1 ) + τt
(33)
Using the debt-to-GDP ratio as the right-hand side variable instead does not alter our
results. Monetary policy is conducted via a simple Taylor-type rule, which is only
reacting to lagged inflation.
log(Rt ) = ConstR + φπ log(πt−1 ) + R
t
(34)
The firms and households in our model know the form of the labor tax rule and the
monetary policy rule as described above, but they do not know the coefficients, which
they have to estimate. They also know that the government budget constraint has
to hold in every period. We model the government and the central bank as reacting
to lagged endogenous variables so as to circumvent endogeneity issues in the learning
problem of private agents that would occur if the government and the central bank
were to react to contemporaneous endogenous variables. Given the information and
implementation lags present in economic policymaking, we do not think that this is an
overly strong assumption.
3.5
Market Clearing
Demand on the part of the government and households in the form of investment and
consumption in addition to the adjustment costs related to the utilization costs must
fully absorb the output of the firms:
Yt = Ct + It + Gt + ψ(Vt )Kt−1
Market clearing in the bond market implies that all bonds issued by the government
are bought by the households in the economy.
16
4
Learning Mechanism
Our approach to modeling learning is borrowed from our earlier work, Hollmayr and
Matthes (2013), which in turn builds on Cogley et al. (2015).
The private agents in our model observe all state variables in the economy and all
exogenous innovations except for the true policy shocks - instead of the true policy
shocks in the tax and monetary policy rules, they observe the perceived policy shocks,
which are the residuals in their corresponding estimated policy rules. They use those
observations to estimate the coefficients of the policy rules. Firms and households know
all other aspects of the model. All private agents share the same beliefs and carry out
inference by using the Kalman filter. The choice of the Kalman filter as the agents’
estimation procedure is motivated by the fact that we want to stay as close as possible
to a (limited information) rational expectations setup, and the Kalman filter returns the
posterior distribution for conditionally linear models with Gaussian innovations such
as ours.13 The one departure from limited information rational expectations in our
setup is the use of the anticipated utility assumption (Kreps (1998)), which is common
in the literature on learning in macroeconomics (see, for example, Milani (2007)). This
assumption amounts to the private agents using a point estimate each period to form
their beliefs (rather than integrating over the posterior) and not contemplating future
changes in beliefs when making decisions.
We first describe the model that agents use to form estimates of policy rule coefficients.
Then, we will describe what those estimates imply for the agents’ view about the
dynamics of the economy - we derive their perceived law of motion (PLM). Finally,
we ask what those perceptions imply for actual equilibrium dynamics - we derive the
actual law of motion (ALM) for the model.
We denote by Ωt the vector of policy rule coefficients φπ and ρb that agents want to
estimate. In order for agents to be able to use the Kalman Filter for inference, we need
to build a state-space system that encompasses our assumptions about the learning
behavior of agents. The observation equation is obtained by stacking the monetary
policy rule and the fiscal policy rule for taxes, whereas the state equation represents
the perceived dynamics in policy rule coefficients.
13
For a comparison of learning based on the Kalman Filter and learning based on recursive least
squares algorithms that are also common in the literature, see Sargent and Williams (2005).
17
The vector of observables ξt is given by
"
ξt =
14
cc
log(Rt ) − R
#
log(τt ) − τbc
(35)
The observation equation is then:
ξt = Xt−1 Ωt + ηt
(36)
where ηt collects the iid disturbances in the policy rules. Xt−1 collects the right-hand
side variables in the two policy rules, lagged inflation and lagged debt, respectively.
What is left to specify then is the perceived law of motion for Ωt - how do firms
and households in the economy think policy rule coefficients change over time? We
study two assumptions: Agents either know when the policy rule changes and take
into account that policy rule coefficients before and after the break date are fixed,
or they suspect that policy changes every period. The following law of motion for
the coefficients encodes these assumptions, inspired by the literature on time-varying
coefficient models in empirical macroeconomics (such as Cogley and Sargent (2005) or
Primiceri (2005))
15
:
Ωt = Ωt−1 + 1t νt
(37)
If we set the variance of νt to a conformable matrix of zeroes, then the private agents
in our model believe that policy rule coefficients do not change and they estimate unknown constant coefficients. The indicator function 1t selects in what periods agents
perceive there to be a change in policy. We will assume that this indicator function is
0 unless the policy rule actually changes.
Given beliefs for Ωt , agents in our model will adhere to the anticipated utility theory of
decision-making: They will act as if Ωt is going to be fixed at the currently estimated
level forever onwards. This is a common assumption in the literature on learning, see for
example Milani (2007). We use the posterior mean forecast Et−1 (Ωt ) = Ωt|t−1 calculated via the Kalman Filter as a point estimate that the agents in the model condition
on when forming expectations. By the random walk assumption on the parameters
that agents use in the Kalman filter, this also implies that Ωt|t−1 = Et−1 (Ωt−1 ).
14
For simplicity, we assume that the steady states are known to the private agents (Cogley et al.
(2015) highlight the fact that the differences between dynamics under learning and the full information case emerge mainly from different views held by agents on policy rule response coefficients, not
intercepts). Given their knowledge of the steady state and last period’s estimates for the policy rule
coefficients, agents can back out an estimate of the intercepts in the policy rule, which they use in
their state space system. Those estimates are denoted below with a hat.
15
This assumption has been applied in the learning literature by Sargent et al. (2006), for example.
18
If we denote the vector of all variables (plus a constant intercept) in the model economy by Yt , then we can stack the log-linearized equilibrium conditions (approximated
around the steady state, which we assume is known to agents) and the estimated policy
rules to get the log-linearized perceived law of motion in the economy:
A(Ωt|t−1 )Yt = B(Ωt|t−1 )E∗t Yt+1 + C(Ωt|t−1 )Yt−1 + Dε∗t
(38)
ε∗t contains the standardized (i.e. variance 1) actual shocks that agents observe as
well as the standardized perceived policy shocks (the residuals in the estimated policy
rules). Those residuals are standardized because we choose to include the standard
deviations of the shocks in the system matrices. Because we use the anticipated utility
assumption, agents act as if their beliefs will not change in the future and this system
can be solved using a number of standard algorithms for the solution of linearized
rational expectations models such as Gensys (Sims (1994)).16 The resulting reduced
form perceived law of motion is given by
Yt = S(Ωt|t−1 )Yt−1 + G(Ωt|t−1 )ε∗t
(39)
S(Ωt|t−1 ) solves the following matrix quadratic equation17 :
S(Ωt|t−1 ) = (A(Ωt|t−1 ) − B(Ωt|t−1 )S(Ωt|t−1 ))−1 C(Ωt|t−1 )
(40)
and G(Ωt|t−1 ) is given by
G(Ωt|t−1 ) = (A(Ωt|t−1 ))−1 D
(41)
To derive the ALM, we replace the perceived policy rule coefficients in C(Ωt|t−1 )
with the actual policy rule coefficients and use the actual innovation vector εt :
actual
A(Ωt|t−1 )Yt = B(Ωt|t−1 )E∗t Yt+1 + C
16
(Ωt|t−1 )Yt−1 + Dεt
(42)
If the estimated policy rule coefficients imply an indeterminate equilibrium, we use the equilibrium
returned by Gensys.
17
The perceived law of motion can be derived by assuming a VAR perceived law of motion of order
1 and then using the method of undetermined coefficients.
19
To solve the model, we can plug the PLM into the ALM twice to get
actual
A(Ωt|t−1 )Yt = B(Ωt|t−1 )(S(Ωt|t−1 )2 Yt−1 + S(Ωt|t−1 )G(Ωt|t−1 )ε∗t ) + C
(Ωt|t−1 )Yt−1 + Dεt
(43)
Note that there are two types of shocks appearing in the last equation: the true and
the perceived shocks. We can solve for the dynamics of Yt by only inverting A(Ωt|t−1 )
as long as we can derive an expression for the perceived shocks that only depends on
pre-determined and exogenous variables. Fortunately enough, this is true in our case:
ε∗t = εt + IP (C
actual
(Ωt|t−1 ) − C(Ωt|t−1 ))Yt−1
(44)
where IP denotes a selection matrix that selects those rows of the vector it multiplies,
which are associated with the policy instruments about whose dynamics the agents
are learning and rescales those rows by the inverse of the standard deviation of the
corresponding policy shock.
Plugging that expression into equation (43), we can derive the reduced form actual law
of motion:
Yt = F(Ωt|t−1 )Yt−1 + R(Ωt|t−1 )ε∗t
F(Ωt|t−1 ) = A
+ A
R(Ωt|t−1 ) = A
−1
−1
−1
(45)
2
(Ωt|t−1 )(C(Ωt|t−1 ) + B(Ωt|t−1 )F (Ωt|t−1 ))
actual
(Ωt|t−1 )(B(Ωt|t−1 )F(Ωt|t−1 )G(Ωt|t−1 )IP (C
(Ωt|t−1 ) − C(Ωt|t−1
(46)
)))
(Ωt|t−1 )(B(Ωt|t−1 )F(Ωt|t−1 ))G(Ωt|t−1 ) + D
(47)
This derivation departs from the derivation used in Cogley et al. (2015) because we
found our approach of solving for the equilibrium dynamics to be more numerically
stable (once we have solved for the ALM, our approach only requires invertibility of
A(Ωt|t−1 )).
5
Equilibrium Outcomes Under Learning
We now turn to simulating our learning economy. We consider the following two scenarios described: First, we consider a scenario where we leave fiscal policy constant at
all times. It is passive in the sense that it stabilizes debt. Monetary policy is assumed
to switch from passive to active: απ changes from 0.8 to 1.5. This scenario is introduced to broadly mimic the disinflation period during Paul Volcker’s chairmanship of
20
the Federal Reserve in the early 1980s in the Unites States.
We then consider a second scenario in which monetary policy becomes passive and
distortionary taxes react less to the level of debt. In particular, we consider a one-time
switch in the policy rule coefficients απ and ρb from 1.5 and .1 to .8 and .04. Put differently, the economy undergoes a switch from monetary dominance to fiscal dominance.
This scenario is meant to approximate the situation in many developed countries after
the recent financial crisis. The numerical values for the policy rule parameters pre- and
post-policy change for both scenarios are in line with parameter estimates for the U.S.
economy (Traum and Yang (2011)).
We run 500 simulations of 100 periods with the policy switch occurring in period
10.18 To keep the number of additional parameters manageable and to keep those
parameters interpretable, we assume agents use a covariance matrix of the innovations
in the perceived policy rule coefficients of the following form19 :
"
E(νt νt0 ) =
(scale ∗ (1.5 − 0.8))2
0
0
(scale ∗ (0.1 − 0.04))2
#
(48)
For simplicity, we will use the same calibration for both scenarios. The initial values
of the parameters are given by the pre-policy change true values - we can think of this
as the agents initially living in a world where any learning process has converged. For
scale, we use 1 as a benchmark value, but we also show selected results for 2 and 4.
Agents thus experience a shock that is either a 1 standard deviation shock, a shock of
size half a standard deviation, or a one-fourth standard deviation shock.
18
We choose not to put the policy switch at the beginning of the simulations in order to minimize
the effect of the choice of the initial covariance matrix for the Kalman Filter.
19
We have used covariance matrices of this form in our previous work and found them useful for
interpreting the perceived amount of time variation.
21
6
6.1
Results
A Switch From Passive to Active Monetary Policy - Revisiting Volcker
Figure 1 shows the probability that the perceived equilibrium of the private agents is
determinate. The black line gives the actual equilibrium behavior and the colored lines
give the probabilities for the different sizes of perceived policy changes. We see that,
depending on the prior of the size of the policy change, initially the probability that
agents know that the policy rules actually imply a unique equilibrium after the policy
change is only between 20% and 50%. Furthermore, learning about the nature of the
equilibrium is slow. Before we turn to figuring out why this is, we want to point out that
agents in our simulations for this scenario never think that they are in a situation in
which the equilibrium is unstable - therefore the complement of the probability plotted
in Figure 1 is indeed the probability that the agents think the equilibrium is still
indeterminate. Does the fact that agents never think that the equilibrium is unstable
imply that the actual equilibrium dynamics are stable? Figure 2 answers this. It plots
for each simulation the largest eigenvalue in absolute value of F(Ωt ) encountered in
that simulation versus the number of periods for which an eigenvalue of that matrix
is larger than 1 in absolute value. First, we see that large eigenvalues are pervasive in
this scenario. All simulations have at least one period with eigenvalues larger than 1.
Second, the eigenvalues, while larger than 1, are not substantially larger than 1, so the
economy does not explode within a few periods. Rather, eigenvalues of this magnitude
imply random walk-like behavior. Third, there are many simulations that feature
large eigenvalues for substantial periods of time - the probability of random walk-like
behavior is substantial. And finally, there is a negative relationship between the size
of the largest eigenvalue during a simulation and the number of periods for which the
simulation feature random walk-like behavior. This last feature can be explained by
simple econometrics: The larger the eigenvalue is, the more volatile the economy tends
to be. If there is more volatility, then the agents can more easily identify the postpolicy change parameters since the right-hand side variables in their models become
more volatile. But how can it be that the economy features random walk-like behavior
when the perceived law of motion is stable? The logic behind this result is given in
Cogley et al. (2015). (This paper does not focus on the size of the eigenvalues, however,
and instead focuses on solving for an optimal monetary policy rule.) The logic can also
22
be seen by revisiting the equation determining F(Ωt|t−1 ):
F(Ωt|t−1 ) = A
+ A
−1
−1
2
(Ωt|t−1 )(C(Ωt|t−1 ) + B(Ωt|t−1 )F (Ωt|t−1 ))
(Ωt|t−1 )(B(Ωt|t−1 )F(Ωt|t−1 )G(Ωt|t−1 )IP (C
actual
(Ωt|t−1 ) − C(Ωt|t−1 )))
The first term on the right-hand side gives the equilibrium dynamics under the perceived law of motion, which are always stable in this scenario. However, the second
term on the right-hand side features the difference between actual and perceived laws
of motion. If that difference is large enough, the eigenvalues of the entire matrix can
become larger than 1 in absolute value, even though the perceived law of motion is
stable. In the standard approach to modeling policy changes, we do not see random
walk-like behavior - the economy switches from one stable equilibrium to another. To
see what impact this has on equilibrium outcomes, Figure 3 plots the difference between the outcomes under the standard approach and the learning approach. To obtain
these results, we use the same shock series for both approaches, calculate the levels of
the relevant variables and derive the ratio of the median under one approach to the
median under the other approach, normalized by the steady state. For any variable at
time j, we thus plot
Dif fjW
(Wjlearning − Wjstandard )
=
W
(49)
where Wjlearning is the median outcome in levels under learning and Wjstandard the
corresponding outcome for the standard approach. To make the differences in GDP
more visible, we plot cumulative differences for GDP instead:
Dif fjW
=
j
X
(Wtlearning − W RE )
t
t=1
W
(50)
We also plot the average parameter estimates.
We can see that sizeable differences arise for GDP, where the differences are most visible. Agents learn about the monetary policy coefficient slowly - inflation is not volatile
enough to make them learn faster. Nonetheless, differences are -maybe surprisinglysmall. Even under rational expectations, our model features substantial persistence in
many variables, so the emergence of random walk-like behavior does not lead to vastly
different outcomes.20 This does not mean that our findings are not important for outcomes - we will see larger differences in our second experiment that is motivated by
20
This is also evidenced by the standard deviations of the variables (which we omit here for the sake
of brevity) under the two approaches - they are not very different from each other for this scenario.
23
1
1/4 std
1/2 std
1std
0.9
Prob. of determinate Solution
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
40
50
time
60
70
80
90
100
Figure 1: Perceived equilibria in the ‘Volcker’ experiment
Scatter Plot
90
80
Periods of Explosiveness
70
60
50
40
30
20
10
0
1.02
1.04
1.06
1.08
1.1
Eigenvalues
1.12
1.14
1.16
Figure 2: Persistence in the ‘Volcker’ experiment
24
Changes (π)in %
Cum. Changes (GDP)in %
5
0.05
0
−5
0
0
50
time
φπ
100
−0.05
2
0.1002
1
0.1
0
Changes (K)in %
0.2
0.0998
0
50
time
100
0
0
50
time
ρb
100
0
50
time
100
−0.2
Changes (B)in %
2
0
0
50
time
100
−2
0
50
time
100
Figure 3: Summary of outcomes for the ‘Volcker’ experiment
recent economic events. Also, even though we do not explicitly model credible policy
communication in this paper, our findings highlight that policy communication could
be useful if it speeds up learning of the agents.
6.2
The Dawn of Active Fiscal Policy
For our second experiment, the analytical results from Leeper’s model already hint at
the strong possibility of random walk-like behavior when fiscal policy becomes active,
but private agents have incomplete knowledge of this. Note that, even if agents learn
quickly about changes in the fiscal policy rule, they can be misled if learning about
the monetary policy coefficient is slow. This is exactly what happens in our second
scenario. Figure 4 plots the probability of a stable perceived equilibrium. While, under
rational expectations, agents would know that the equilibrium is always stable, here
agents are indeed misled and believe they are in an unstable equilibrium. In contrast to
the first example, here even the perceived law of motion is unstable. This was not the
case in our analysis of Leeper’s model, where we assumed a stable PLM for simplicity.
An unstable PLM can lead to instability in the ALM because of the first term on the
right-hand side in the equation for F(Ωt|t−1 ). The instability of the PLM creeps into
the ALM in our case, as can be seen in Figure 5, which plots the largest eigenvalue
in absolute value in the ALM versus the number of periods for which an eigenvalue
larger than 1 in absolute value is present in the equilibrium dynamics. Are all variables
affected by this random walk-like behavior? Figure 6 shows some randomly selected
sample paths from our simulation for inflation and debt. In spite of the difference in
the nature of the PLM, we see the same results as for Leeper’s model: Inflation is low
25
1
1/4 std
1/2 std
1std
0.9
0.8
Prob. of stable PLM
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
40
50
time
60
70
80
90
100
Figure 4: Perceived equilibria in the second experiment
and stable, whereas debt is very persistent and displays random walk-like behavior.
It is the stable behavior of inflation that makes learning about the monetary policy
coefficient hard, as can be seen by the average estimate in Figure 7. While debt is
volatile enough in our economy so that the average coefficient for the fiscal policy
rule is learned almost immediately,
21
it takes a long time to learn about the change
in monetary policy. In many simulations, agents thus think that both monetary and
fiscal policy are active, leading them to believe that the equilibrium is unstable. In this
scenario, we see larger differences in (cumulated) GDP and debt between the learning
economy and the standard approach to modeling policy changes.
Even though our model abstracts from many issues that are important in the current
economic situation, this experiment still reproduces the broad patterns of low and
stable inflation and substantial persistent movements in debt. If we were to model a
fiscal stimulus along the lines of Drautzburg and Uhlig (2011), Cogan et al. (2010),
or Hollmayr and Matthes (2013), we would see a larger initial jump in debt. The
learning problem that agents face in this scenario is hard because inflation does not
move enough to identify the changes in monetary policy. Incorporating the zero-lower
bound on nominal interest rates would only make this identification issue harder. The
issues we highlight here therefore seem very much relevant for the current economic
environment.
To check whether or not our results hinge on the specific signal-to-noise structure
21
Not all simulations feature fiscal policy coefficient that directly collapse to the true value, but they
all move in the right direction and the variation is quite small, so focusing on the average estimate
across simulations is not misleading.
26
Scatter Plot
90
80
Periods of Explosiveness
70
60
50
40
30
20
10
0
1.002
1.003
1.004
1.005
1.006
Eigenvalues
1.007
1.008
1.009
1.01
Figure 5: Persistence in the second experiment
−3
3
Outcomes Inflation
x 10
2
1
0
−1
−2
0
10
20
30
40
50
60
70
80
90
100
60
70
80
90
100
Outcomes Debt
0.55
0.5
0.45
0.4
0.35
0
10
20
30
40
50
Figure 6: Inflation and debt sample paths in the second experiment
Changes (π)in %
Cum. Changes (GDP)in %
5
0.1
0
0
0
50
time
φπ
100
−0.1
1.5
0.2
1
0.1
0.5
0
50
time
100
0
0
0
50
time
ρb
50
time
100
Changes (K)in %
Changes (B)in %
1
2
0
0
−1
0
50
time
100
−2
0
50
time
100
Figure 7: Summary of outcomes for the second experiment
100
27
1
1/4 std
1/2 std
1std
0.9
0.8
Prob. of stable PLM
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
40
50
time
60
70
80
90
100
Figure 8: Perceived equilibria in the second experiment when monetary policy error
standard deviation is reduced by 50 %
inherent in our calibration, we next turn to a calibration where we set the standard
deviation of the monetary policy error R
t to half of the value in our benchmark calibration. This will increase the information value of observations. Nonetheless, as can
be seen in Figure 9, the nature of perceived equilibria is basically unchanged. Agents
learn somewhat faster that the true equilibrium is stable, but the difference from the
standard calibration is not large. These similarities carry over to other equilibrium
outcomes, which we omit here for the sake of brevity. Thus, even with a substantially
smaller noise term in the policy rule, we still get the same results - inflation in this
experiment just does not vary enough to make agents update their beliefs faster.22
7
Conclusion
This paper analyzes different modeling strategies for the analysis of discrete changes
in economic policy. We show that removing immediate knowledge of the new policy
rule from agents can alter the nature of the perceived equilibrium. We show that persistent, random walk-like behavior naturally occurs when agents have to learn about
policy changes and that the resulting equilibrium dynamics can make learning the
true policy rule coefficients difficult. It is important to remember that the agents
22
In the appendix we show that only once we almost let the noise term vanish do we get much
faster learning. In that case, the standard deviation of the noise term in the monetary policy equation
is 1/20th of our benchmark number. Even with such a small noise term, learning is not immediate,
though.
28
in our models are sophisticated econometricians - the properties of observed data in
equilibrium make learning hard, not the use of unsophisticated econometric techniques.
29
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32
Appendix
FOCs and Log-linearized Equation
A First-Order Conditions
Households
λt = Utb (Ct − hCt−1 )−σ
λt+1
λt = βRt Et
πt+1
(1 − τt )Rtk = ψ 0 (Vt )
i
λt+1 h
k
(1 − τt+1 )Rt+1
Vt+1 − ψ(Vt+1 ) + (1 − δ)Qt+1
Qt = βEt
λ
t
It
It
It
0
1 = Qt 1 − Γ
−Γ
It−1
It−1 It−1
"
#
λt+1 0 It+1
It+1 2
+ βEt Qt+1
Γ
λt
It
It
33
Firms
Yt M Ct
Lt
Y
M
C
t
t
Rtk = α
K
"
##
" ∞t
χp
s
Y
X
π
π
ss
t+k−1
P
(βωp )s λt+s ȳt+s p̄t
− (1 + ηt+s
)M Ct+s
0 = Et
πss
πt+k
Wt = (1 − α)
s=0
k=1
P
ȳt+s =
s
Y
p̄t
k=1
"
πt+k−1
πss
1
ηtP
(1 − ωP )p̄t
1 =
"
0 = Et
χp
"
+ ωP
πss
πt+k
πt−1
πss
"
t+s
χP
πss
πt
1
ηtP
##ηtP
∞
s
X
Y
(1 + ηtw )ψ L̄ξt+s
πss
s
−
(βθw ) λt+s L̄t+s w̃t
πt+k
(1 − τt+s )λt+s
t=0
##
k=1
s
Y
πss
w̃t
πt+k
"
L̄t+s =
1+η
!− ηPt+s
w
t
#− 1+η
ηtw
Lt+s
k=1
1
ηtw
wt
1
ηtw
= (1 − θw )w̃t
+ θw
πss
πt
wt−1
1
ηtw
B Log-Linearized Model
Households
σ
σh
log(ct ) +
log(ct−1 )
1−h
1−h
log(λt ) = Constλ + log(Rt ) + Et log(λt+1 ) − Et log(πt+1 )
τss
ψ
log(rtk ) = ConstV +
log(vt ) +
log(τt )
1−ψ
1 − τss
k
k
K
log(qt ) = ConstQ + Et log(λt+1 ) − log(λt ) + β(1 − τss )Rss
Et log(rt+1
) − βτss Rss
Et log(τt+1 )
log(λt ) = ConstC + log(ubt ) −
+β(1 − δ)Et log(qt+1 )
log(kt ) = ConstK + log(vt ) + log(kt−1 )
log(k̄t ) = ConstK̄ + (1 − δ)log(k̄t−1 ) + δ(log(uit ) + log(it ))
1
(log(qt ) + log(uit )) − βEt log(it+1 ) = log(it−1 ) + ConstI
(1 + β)log(it ) −
s
34
Firms
Css
Iss
Gss
ψ 0 (1)Kss
log(ct ) +
log(it ) +
log(gt ) +
log(vt )
Yss
Yss
Yss
Yss
log(yt ) = ConstY + log(at ) + αlog(kt ) + (1 − α)log(lt )
β
χp
log(πt ) = Constπ +
E
log(π
)
+
log(πt−1 ) + κp log(mct ) + κp log(ηtp )
t
t+1
1 + χp β
1 + χp β
log(mct ) = ConstM C + αlog(rtk ) + (1 − α)log(wt ) − log(at )
log(yt ) = ConstAgg +
log(rtk ) = ConstRK + log(lt ) − log(kt ) + log(wt )
β
1
log(wt−1 ) +
Et log(wt+1 )
log(wt ) = ConstW +
1+β
1+β
τss
χw
b
log(τt ) +
− κw log(wt ) − νlog(lt ) − log(ut ) + log(λt ) −
log(πt−1 )
1 − τss
1+β
1 + χw β
β
−
log(πt ) +
Et log(πt+1 ) + κw log(ηtw )
1+β
1+β
Policy Rules and Shocks
Wss Lss
Rk Kss
log(τt ) + log(rtk ) + log(kt )
(log(τt ) + log(wt ) + log(lt )) + τss ss
Bss
Bss
1
1
1
Gss
Zss
= ConstB + log(Rt−1 ) + log(bt−1 ) − log(πt ) +
log(gt ) +
log(zt )
β
β
β
Bss
Bss
log(gt ) = ConstG + ρG log(gt−1 ) + G
t
log(bt ) + τss
log(zt ) = ConstZ + ρZ log(zt−1 ) + Z
t
log(τt ) = Constτ + ρb log(bt−1 ) + τt
log(Rt ) = ConstR + αlog(πt−1 ) + R
t
log(at ) = ConstA + ρA log(at−1 ) + A
t
i
log(uit ) = Constu i + ρui log(uit−1 ) + ut
b
log(ubt ) = Constu b + ρub log(ubt−1 ) + ut
p
log(ηtp ) = Constηp + ρηp log(ηt−1
) + ηt
p
w
log(ηtw ) = Constηw + ρηw log(ηt−1
) + ηt
with the constants given by
w
35
Constant
Expression
ConstG
log(Gss )(1 − ρG )
ConstZ
log(Zss )(1 − ρZ )
Constτ
L ) − ρ log(B )
log(τss
ss
b
ConstR
log(Rss ) − φπ log(πss )
Constui
i )(1 − ρ )
log(Uss
ui
Constub
b )(1 − ρ )
log(Uss
ub
Constηw
w )(1 − ρ w )
log(ηss
η
Constηp
p
log(ηss
)(1 − ρηp )
ConstB
Rss
k
ss
log(Bss )(1 − β1 ) + τss LssBW
(log(τss ) + log(Wss ) + log(Lss )) + τss Kss
Bss (log(τss ) + log(Rss )+
ss
k
log(Kss )) − β1 log(Rss ) + β1 log(πss ) −
Gss
Bss log(Gss )
−
Zss
Bss log(Zss )
ConstY
log(Yss ) − log(Ass ) − (1 − α)log(Lss ) − αlog(Kss )
ConstA
log(Ass )(1 − ρA )
ConstAgg
log(Yss ) −
Constπ
(1 −
ConstLam
−log(Rss ) + log(πss )
ConstQ
K log(RK ) + βτ RK log(τ )
(1 − β(1 − δ))log(Qss ) − β(1 − τss )Rss
ss ss
ss
ss
ConstRK
K ) − log(W ) − log(L ) + log(K )
log(Rss
ss
ss
ss
ConstW
(1 + κw −
k (1−τ )K
Rss
ss
ss
Css
Gss
Iss
log(vt )
Yss log(Css ) − Yss log(Gss ) − Yss log(Iss ) −
Yss
p
β
p
p
(1+χp beta) − χ (1 + βχ ))log(πss ) − κp log(mcss ) − κp log(ηss )
1
(1+β)
b ) + κ log(λ )−
− β(1 + β))log(Wss ) − κw νlog(Lss ) − κw log(Uss
w
ss
w
τss
w) − ( χ
−κw (1−τ
log(τss ) − κw log(ηss
(1+β) −
ss )
(1+βχw )
(1+β)
+
ConstI
i )
− 1s log(Qss ) − 1s log(Uss
ConstM C
log(mcss ) − αlog(RsK s) − (1 − α)log(Wss ) + log(Ass )
ConstL
(1 − α)log(Lss ) + αlog(Kss ) + log(Ass ) − log(Yss )
ConstC
σ
( 1−h
−
ConstV
σh
b
1−h )log(Css ) + log(λss ) − log(Uss )
ψ
τss
K
1−ψ log(Vss ) − log(Rss ) + (1−τss ) log(τss )
ConstK
log(Kss ) − log(Vss ) − log(K̄ss )
ConstK̄
i ) − δlog(I )
δlog(K̄ss ) − δlog(Uss
ss
β
(1+β) )log(πss )
36
C Parameters
Calibrated parameters of benchmark DSGE model
Description
impatience
CES utility consumption
CES utility labor
Level shifter labor
habits
Capital intensity
Depreciation rate
Price indexation
Wage indexation
Calvo prices
Calvo wages
Inv. adjustment cost parameter
Capital utilization cost param.
Steady state tax rate
coeff. on inflation in TR
coeff. on B in tax rules
AR parameter transfer rule
AR parameter gov. spending
AR parameter technology
AR parameter price mark-up
AR parameter wage mark-up
AR parameter preference
AR parameter investment
Std.deviation technology
Std.deviation gov. spending
Std.deviation transfers
Std.deviation tax
Std.deviation interest rate
Std.deviation investment
Std.deviation preference
Std.deviation price mark-up
Std.deviation wage mark-up
Parameter
β
σ
φ
ν
h
α
δ
χp
χw
θp
θw
γ
ψ
τss
φπ
ρb
ρz
ρg
ρa
ρηp
ρηw
ρub
ρui
σa
σg
σz
στ
σr
σui
σub
ση p
ση w
Value
0.99
1
1.4
3.19
0.8
0.33
0.025
0.26
0.35
0.9
0.79
4
0.34
0.32
1.5
0.1
0.34
0.97
0.35
0.69
0.42
0.86
0.87
0.69
0.15
0.91
0.24
0.25
0.35
0.38
0.065
0.21
Table 2: Calibrated parameters of the model
where κp = [(1 − βθp )(1 − θp )]/[θp (1 + βχp )] and
κw = [(1 − βθw )(1 − θw )]/[θw (1 + β)(1 +
1+η w ν
)]
ηw
37
D Substantially smaller noise in the monetary policy
rule
1
1/4 std
1/2 std
1std
0.95
0.9
Prob. of stable PLM
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0
10
20
30
40
50
time
60
70
80
90
100
Figure 9: Perceived equilibria in the second experiment when monetary policy error
standard deviation is 1/20th of the original value.
@FedFRASER