Full text of Working Papers (Federal Reserve Bank of Richmond) : Learning About Fiscal Policy and the Effects of Policy Uncertainty, Working Paper 13-15

The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.

Working Paper Series

Learning about Fiscal Policy and the
Effects of Policy Uncertainty

WP 13-15

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/

Learning about Fiscal Policy and the
Effects of Policy Uncertainty
Josef Hollmayr∗and Christian Matthes∗∗

September 30, 2013
Working Paper No. 13-15

Abstract
The recent crisis in the United States has often been associated with substantial amounts of policy uncertainty. In this paper we ask how uncertainty about
fiscal policy affects the impact of fiscal policy changes on the economy when the
government tries to counteract a deep recession. The agents in our model act
as econometricians by estimating the policy rules for the different fiscal policy
instruments, which include distortionary tax rates.
Comparing the outcomes in our model to those under full-information rational
expectations, we find that assuming that agents are not instantaneously aware of
the new fiscal policy regime (or policy rule) in place leads to substantially more
volatility in the short run and persistent differences in average outcomes.

JEL codes: E32, D83, E62
Keywords: DSGE, Fiscal Policy, Learning
∗

Deutsche Bundesbank, Frankfurt am Main (e-mail: josef.hollmayr@bundesbank.de).
Federal Reserve Bank of Richmond (email: christian.matthes@rich.frb.org). The views expressed
in this paper are those of the authors and do not necessarily reflect those of the Bundesbank, the Federal Reserve Bank of Richmond, or the Federal Reserve System. We would like to thank Klaus Adam,
Tim Cogley, Michael Krause, Geert Langenus, Stéphane Moyen, Albert Marcet, Taisuke Nakata, Johannes Pfeifer, Ctirad Slavik, and Nikolai Stähler, seminar participants at the University of Hamburg
and the Bundesbank as well workshop participants in Mannheim and Bucharest for helpful comments.
This project started while Matthes was visiting the Bundesbank, whose hospitality is gratefully acknowledged.
∗∗

2

1

Introduction

Partly motivated by the recent financial crisis and the subsequent recession, economists
have recently placed greater emphasis on identifying uncertainty about monetary and
fiscal policy as a potentially important factor determining economic outcomes, as highlighted by Baker et al. (2012). Natural questions seem to be how this uncertainty
arises, what the exact transmission mechanism is and how this uncertainty affects
equilibrium outcomes. In this paper we propose one model of fiscal policy uncertainty:
an RBC-type model with distortionary taxation and government debt, in which agents
act as econometricians and update their beliefs about fiscal policy every period. In our
model, agents use past realizations of fiscal variables to learn what actual policy rules
are in place and thus whether changes in those fiscal variables are temporary (driven
by exogenous shocks) or permanent (driven by changes in the parameters of the fiscal
policy rules). In our model uncertainty about fiscal policy is partly endogenous since
the properties of the estimators of the fiscal policy rule coefficients employed by private
agents change as the private sector’s behavior changes. This behavior occurs because
choice variables of the representative private agent enter the fiscal policy rules.
The task of disentangling permanent from temporary changes in fiscal policy is identified as a major source of fiscal policy uncertainty by Baker et al. (2012), who use
an index of tax code expiration data to measure fiscal policy uncertainty1 . Figure 1
plots their index of fiscal uncertainty. Uncertainty increases substantially during the
recent American Recovery and Reinvestment Act (ARRA) program, a major period
of policy change, but it is very small beforehand and decreases afterward. We will use
these patterns in this measure of objective uncertainty to inform our (learning) model
of subjective uncertainty2 .
We analyze a one-time permanent change in the government spending policy rule and
use Monte Carlo simulations of our model to assess how beliefs evolve and how these beliefs affect allocations. Learning leads to substantially different outcomes even though
learning is quite fast: There is a substantial temporary spike in volatility under learning that is absent under full information. In addition, there are persistent average
differences between the outcomes under learning and under full information. We show
that investment plays a big role in creating the average differences - temporary dif1

They state on the associated website www.policyuncertainty.com that ”Temporary tax measures are a source of uncertainty for businesses and households because Congress often extends them
at the last minute, undermining stability in and certainty about the tax code.”.
2
The subjective measure of fiscal policy uncertainty used in Baker et al. (2012), a measure of
disagreement among professional forecasts of fiscal spending, shows a similar pattern around the
introduction of the ARRA program.

3
ferences in investment between the learning and full information environments have
long-lasting effects via the capital stock. The uncertainty about government spending
induces uncertainty about the steady state of other variables such as GDP and debt,
which in turn influences uncertainty about the steady state of other fiscal policy instruments, even though the coefficients of those policy rules are tightly (and correctly)
estimated. Thus, even though we only consider changing a small subset of the fiscal
policy coefficients, this uncertainty creeps into other fiscal variables.
To check for robustness, we consider various assumptions about the agents’ information set and their preferences as well as an alternative change in fiscal policy. Our
qualitative results remain unchanged throughout.
We are far from being the first to model fiscal policy in an environment in which agents
adaptively learn about the economy. Papers such as that of Eusepi and Preston (2011)
encompass both monetary and fiscal policy, but have a smaller set of fiscal policy instruments (in particular no distortionary taxation). We instead choose to focus on
fiscal policy alone, leaving the interesting issue of fiscal and monetary policy interaction for future work. We do, however, have a larger set of fiscal policy instruments.
Giannitsarou (2006) does feature distortionary taxation and is interested in issues similar to ours, but does not feature government debt, which we include in order to be able
to view the current policy debate in the United States through the lens of our model.
Mitra et al. (2012) focus on the question of anticipated versus unanticipated changes
in fiscal policy when agents are learning, but they only study the case of lump-sum
taxation.
What sets our model apart is the way agents form their beliefs about the stance of fiscal
policy. In contrast to the previously mentioned papers, our agents know the structure
of the economy, except for the policy rules followed by the fiscal authority. Our paper thus follows the approach laid out in Cogley et al. (2011), who study a model of
monetary policy. Firms and households in our model estimate the coefficients of the
policy rules using the Kalman filter and incorporate both these beliefs and all crossequation restrictions coming from knowledge of the structure of the economy when
making their decisions. Knowledge of the timing of the policy change is incorporated
by agents into their estimation problem by using a time-varying covariance matrix for
the parameters. Furthermore, the agents in our model are aware that the government
budget constraint has to hold. Thus they estimate policy rules for all but one fiscal
policy instrument, with the beliefs about the last policy instrument being determined
by the period-by-period government budget constraint.
In our model, agents are uncertain not only about future fiscal policy, but also about
the policy rules currently in place. Papers such as Davig et al. (2010) instead model

4
the fiscal policy rule coefficients as being governed by a discrete state Markov chain,
which is observable to private agents. Thus agents in those environments know the
policy rule coefficients in place in the current period.
Another strand of the literature that studies uncertainty3 (or risk) about future fiscal
policy is represented by Born and Pfeifer (2013) and Fernandez-Villaverde et al. (2011),
who study stochastic volatility in the innovations of otherwise standard fiscal policy
rules. The view of uncertainty encoded in the latter two papers is quite different from
both our approach as well as the approach used by Davig et al. (2010) and similar papers: In our model, agents are uncertain as to how the government systematically sets
its fiscal policy instruments (both currently and in the future), whereas in Born and
Pfeifer (2013) and Fernandez-Villaverde et al. (2011) agents are uncertain as to how
important (i.e. volatile) the random component of fiscal policy will be in the future.
Davig et al. (2010), Born and Pfeifer (2013) and Fernandez-Villaverde et al. (2011) use
full-information rational expectations models, whereas our approach encodes a form
of bounded rationality common in the learning literature (the anticipated utility approach of Kreps (1998)), which sets the approaches further apart. The anticipated
utility approach we use abstracts from precautionary behavior driven by model uncertainty on behalf of the private agents. Our results can thus be viewed as representing
a bound on the difference between full information and limited information approaches.

2

Model

Our model is a simplified version of Leeper et al. (2010). It is a real model of a closed
economy without habits and other frictions. The only deviation from the simplest
possible RBC model (Kydland and Prescott (1982)) is the rich fiscal sector with distortionary taxation, government spending, and transfers. First-order conditions and
the complete log-linearized model may be found in the Appendix.
3

When we talk about uncertainty, we do not mean Knightian uncertainty. For a study of (optimal) fiscal policy when agents face Knightian uncertainty, see Karantounias (2013). A corresponding
analysis of optimal fiscal policy when agents are learning is provided by Caprioli (2010). Both papers
use a smaller set of fiscal policy instruments than we do.

5

2.1

Households

Households are expected utility maximizers4 with the instantaneous utility function of
the representative household taking the following form:
Ct1−σ
L1+φ
Ut =
− t
1−σ 1+φ

(1)

with consumption Ct and labor Lt . Each period households can choose to consume,
save in the form of government bonds (Bt ) or invest (It ) in the capital stock (Kt ) that
they hold. Therefore they maximize the infinite sum of discounted utility under the
following constraints:
Ct (1 + τtC ) + Bt + It = Wt Lt (1 − τtL ) + (1 − τtK )RtK Kt−1 + Rt−1 Bt−1 + Zt

(2)

Kt = (1 − δ)Kt−1 + It

(3)

where the first constraint is the budget constraint of the household and the latter is
the law of motion for capital. The household’s income stems from working at the wage
Wt , gains from renting out capital RtK and interest payments on their savings at the
rate Rt . Zt represents lump-sum transfers or taxes. τti with i = K, L, C denotes the
various tax rates that the government levies on capital, labor and consumption.

2.2

Firms

The production function is of the standard Cobb-Douglas type:
α
Yt = exp(At )Kt−1
Lt1−α

(4)

where Yt denotes the output produced with a certain level of technology At , capital
Kt and labor Lt . Technology follows an AR(1) process. The exogenous process for
technology is an AR(1):
At = ρa At−1 + A
t
4

(5)

This statement extends up to their beliefs about fiscal policy rule coefficients, which they treat as
fixed when making their decisions. We use an anticipated utility assumption, which is common in the
literature on adaptive learning. It is described in detail in the section that elaborates on our learning
algorithm.

6

2.3

Government

The government in this setup only consists of the fiscal branch. The government budget
constraint is given by:
Bt = Bt−1 Rt−1 − RtK Kt τtK − Wt Lt τtL − Ct τtC + Gt + Zt

(6)

We follow Leeper et al. (2010) in the choice of right-hand side variables for the policy
rules, except that we make time t fiscal policy instruments functions of time t − 1
endogenous variables. This assumption simplifies our learning algorithm, which we
discuss later. Given the lags in fiscal policy decision-making, this assumption does not
seem overly strong5 .
Government Expenditure:
log(Gt ) = Gc − ρg,y log(Yt−1 ) − ρg,b log(Bt−1 ) + G
t

(7)

log(Zt ) = Zc − ρz,y log(Yt−1 ) − ρz,b log(Bt−1 ) + Zt ,

(8)

Transfers:

Consumption Tax Rate Rule:
log(τtC ) = τcc + C
t

(9)

log(τtL ) = τcl + ρL,y log(Yt−1 ) + ρL,b log(Bt−1 ) + Lt

(10)

Labor Tax Rate Rule:

Capital Tax Rate Rule:
log(τtK ) = τck + ρK,y log(Yt−1 ) + ρK,b log(Bt−1 ) + K
t

(11)

In contrast to Leeper et al. (2010) we simplify the model and do not assume that the
innovations to the tax rates are contemporaneously correlated.
The firms and households in our model know the form of the policy rules 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.
5

For a discussion of the link between simple fiscal policy rules like the ones employed here and
optimal fiscal policy, see Kliem and Kriwoluzky (2013).

7

2.4

Market Clearing

Demand on the part of the government and households must fully absorb the output
of the competitive firm:
Yt = Ct + It + Gt
The bond market in our model is simple and market clearing in this market implies
that all bonds issued by the government are bought by the households in the economy.

3

Calibration

The model is calibrated to the U.S. economy at a quarterly frequency. All parameters of the model are chosen to be consistent with other dynamic stochastic general
equilibrium models in the literature. Therefore, the discount factor, the parameter
which indicates the impatience of households, β, is set to 0.99. This value yields a
steady state real interest rate of 3.6 percent in annual terms. The capital share in the
Cobb-Douglas function α is one-third

6

and the depreciation rate of capital is set at

0.025, which is equivalent to a total annual depreciation of 10 percent. These values
are in line with accounting standards. The CES parameters σ and φ govern the utility
function, which takes as its input consumption and labor. Both parameters are fixed
at 2.
Lastly, all coefficients in the fiscal rules come from the estimation of the DSGE model
in Leeper et al. (2010). Although their model includes more frictions such as consumption habits and a capital utilization rate, we think that it is reasonable to adopt their
estimation results for these parameters.
To obtain the same steady state values as Leeper et al. (2010) for tax rates, government spending over GDP, and debt capital over GDP, we set the respective constants
accordingly. The steady state values for the consumption tax, the capital tax, and
the labor tax are therefore 0.0287, 0.2452, and 0.1886, respectively. The ratio for the
shares of government spending and capital to GDP are 0.09 and 7.10. The volatilities of all shock processes are also taken from the estimation in Leeper et al. (2010).
We discuss the parameters governing initial beliefs and learning when we present the
learning algorithm in the next section.
All parameters and steady state values are shown in tables 1 and 2, respectively.
6

This value is within the band that is implied by the prior mean by Smets and Wouters (2007)(0.3)
and the calibrated parameter by Bernanke et al. (1999) (0.35)

8

4

A Change in Fiscal Policy

We want to ask how beliefs and economic outcomes evolve during a recession when
fiscal policy acts to counteract the recession. This section lays out the main policy
experiment we consider. As initial values for the policy rule coefficients we use the
estimates from Leeper et al. (2010), which we reproduce in table 1. The analysis is
carried out via a Monte Carlo simulation - 1000 simulations of 100 periods each. In
period 9, a negative technology shock hits that puts the technology level 5 percent
below its steady state level. In the next period, the fiscal policy authority changes the
process for government spending. We consider a permanent policy change in which only
the intercept in the policy rule changes to reflect an average increase of government
spending across the board. All other coefficients of the fiscal policy rules remain fixed
at the original levels (including the intercepts in the respective policy rules)7 .
We pick the size of the change in Gc using the following thought experiment: Given the
original steady state values for debt and GDP, by how much would we have to change
Gc to increase the steady state level of government spending by 1 percent of GDP?
The ’1 percent of GDP’ number is in line with the maximum increase in Gt used by
Cogan et al. (2010), who calibrate their Gt sequence to the ARRA spending program.
To illustrate our choice of the change in Gc , it is useful to look at equation (7) in levels
at the original steady state:
G = exp(Gc )Y −ρg,y B −ρg,b

(12)

Uppercase letters without a subscript denote the original steady state in this case. We
solve for the new value of the intercept in the log version of the government spending
rule G∗c using the following equation:
G + 0.01Y = exp(G∗c )Y −ρg,y B −ρg,b

(13)

This is a back-of-the-envelope calculation since it does not take into account that a
change in Gc will affect the steady state values of GDP and debt, and thus it will not
lead to an increase of 1 percent of GDP. In our benchmark case the actual increase
in G due to this policy change is 0.81 percent of original GDP, so the back-of-theenvelope calculation is not far off. We use this calculation because it is a calculation a
7

This implies that we do not change how the government raises revenues - the way government
spending is paid for is still encoded in the policy rule coefficients we have borrowed from Leeper et al.
(2010). The endogenous variables in our model will adjust to make sure that those policy rules imply
that the increase in government spending is paid for.

9
government might carry out without knowledge of the entire model as long as precise
estimates of the original steady state values are available.

5

Learning about Fiscal Policy

The agents in our model act as Bayesian econometricians. They observe all relevant
economic outcomes and use those observations to estimate the coefficients of the policy
rules (7)-(11). Firms and households know all other aspects of the model.
We first describe how agents update their estimates of fiscal policy coefficients, then
go on to derive the beliefs about the equilibrium dynamics induced by those estimates
and finally derive expressions for the equilibrium dynamics in our model.
All private agents share the same beliefs and carry out inference by using the Kalman
filter8 , which means that they recursively apply Bayes’ law. If we denote by Ωt the
vector of coefficients of all fiscal policy rules (which is exactly what the agents have to
estimate) and by τt the vector of fiscal policy instruments at time t (i.e., the left-hand
side variables of equations (7)-(11)), then the observation equation for the state space
system used by the Kalman filter is given by:
τt = Xt−1 Ωt + ηt

(14)

where ηt collects the iid disturbances in the fiscal policy rules. Xt−1 collects the righthand side variables in the fiscal policy rules. In a previous section we have laid out
how policy actually changes. Now we have to specify the perceived law of motion for
Ωt - how do firms and households in the economy think policy rule coefficients change
over time? While we move away from the assumption of full-information rational
expectations, the agents’ views on policy changes are very close to the actual law of
motion of the policy rule coefficients (i.e. the actual policy change we consider). In
particular, our agents know at what time the policy rule coefficients change - they
just do not know what coefficients change and the magnitude of the change. To be
clear, agents also update their beliefs about fiscal policy in the periods in which the
policy does not change. The following law of motion for the coefficients encodes these
assumptions:
Ωt = Ωt−1 + 1t νt
8

(15)

For a comparison of learning when using the Kalman filter versus learning when using the common
recursive least squares approach, see Sargent and Williams (2005).

10
1t is an indicator function that equals 1 in the period in which fiscal policy changes9
and νt is a Gaussian vector with mean 0 for each element. This law of motion is inspired
by the literature on time-varying coefficient models in empirical macroeconomics (such
as Cogley and Sargent (2005) or Primiceri (2005))

10

. The perceived law of motion for

the coefficients makes agents realize that fiscal policy changes infrequently. A similar
modeling device has been introduced in time-varying parameter VAR models by Koop
et al. (2009), who replace 1t with a random variable that can take on only the values
0 or 1. In the literature on learning in macroeconomic models, Marcet and Nicolini
(2003) propose a learning mechanism in a similar spirit: Agents place greater weight
on recent data if they suspect that there has been a structural change (i.e., whenever
the estimated coefficients fit the data poorly). Introducing 1t into the agents’ learning
algorithm helps us to match the pattern of uncertainty displayed in figure 1.
If we were to set the variance of νt to a conformable matrix of zeros, then the private
agents in our model would believe that fiscal policy rule coefficients do not change and
they would estimate unknown constant coefficients. A non-zero covariance matrix for
νt implies the belief that fiscal policy rule coefficients change when the actual policy
change happens. This begs the question of how we calibrate the covariance matrix
for νt , Σν . We set this matrix to a scaling factor s times a diagonal matrix with the
ith element on the diagonal being equal to the square of the ith element of Ω0 . Ω0 is
the initial estimate of the policy rule coefficients, which we set to the true pre-policychange values. This assumption makes any calibration for s easily interpretable - if
s = 1, then a 1-standard-deviation shock can double the parameter, for example. We
choose different values for s that endow the agents with different views on how likely
or unlikely the actual policy change is - we calibrate s such that the policy changes we
consider in our subsequent simulations represent either a 1, 2, or 3-standard-deviation
shock according to Σν . In order to be able to use the Kalman filter for the agents’
inference problem, we have to assume that agents know the variance of the shocks in
the policy rules.
Next, we move on to describe how the private agents in our model view the world what is their perceived law of motion?
Given beliefs for Ωt , agents in our model will adhere to the anticipated utility theory
of decision-making (Kreps (1998)): they will act as if Ωt is going to be fixed at the cur9
We thus implicitly assume that the government can credibly announce that there is a change in
fiscal policy, but it cannot credibly communicate in what way fiscal policy changes.
10
An assumption of this kind (with 1t = 1∀t) has been applied in the learning literature by Sargent
et al. (2006), for example.

11
rently estimated level forever more

11

. This is a common assumption in the literature

on learning, see for example Milani (2007) or Sargent et al. (2006). Cogley et al. (2007)
show that in a model of monetary policy the differences between anticipated-utility decision making and fully Bayesian learning are not large. They succinctly summarize the
relationship between uncertainty and anticipated-utility decision making: ”Although
an anticipated-utility decision maker learns and takes account of model uncertainty, he
does not design his decisions intentionally to refine future estimates.”
A change in beliefs about fiscal policy will also induce a change in the beliefs about
the steady state of the economy (see the description of the perceived steady state in
the Appendix for details). 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 perceived steady state) and the estimated fiscal
policy rules to get the log-linearized perceived law of motion in the economy12 :
A(Ωt−1 )Yt = B(Ωt−1 )Et∗ Yt+1 + C(Ωt−1 )Yt−1 + Dε∗t

(16)

The asterisked expectations operator denotes expectations conditional on private sector
beliefs about the economy. The asterisked vector of shocks ε∗t includes the perceived
fiscal policy shocks as well as the technology shock that agents can observe perfectly.
This expectational difference equation can be solved using standard algorithms to yield
the perceived law of motion for the economy at time t:
Yt = S(Ωt−1 )Yt−1 + G(Ωt−1 )ε∗t

(17)

S(Ωt−1 ) solves the following matrix quadratic equation13 :
S(Ωt−1 ) = (A(Ωt−1 ) − B(Ωt−1 )S(Ωt−1 ))−1 C(Ωt−1 )

(18)

and G(Ωt−1 ) is given by
G(Ωt−1 ) = (A(Ωt−1 ) − B(Ωt−1 )S(Ωt−1 ))−1 D

(19)

The beliefs in those equations are dated t−1 because of our timing assumption: Agents
enter the current period (and make decisions in that period) with beliefs updated at
11

We use the posterior mean produced by the Kalman filter as a point estimate that the agents in
the model condition on when forming expectations.
12
This derivation follows Cogley et al. (2011). We also borrow their use of a projection facility: If
no stable perceived law of motion exists, agents use the previous period’s estimates.
13
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.

12
the end of the previous period. This makes the solution method recursive, otherwise
we would have to jointly solve for outcomes and beliefs in every period.
Having described how agents update their estimates and their views on the dynamics of
the variables in the model, we are now in a position to derive the equilibrium dynamics
- the actual law of motion of the economy. This actual law of motion can be derived
as follows: we modify C(Ωt−1 ) to now include the true policy coefficients. We call this
matrix C true (Ωt−1 ). Then the actual law of motion solves:
A(Ωt−1 )Yt = B(Ωt−1 )Et∗ Yt+1 + C true (Ωt−1 )Yt−1 + Dεt

(20)

where we now use the actual shock vector εt . Using the perceived law of motion to
solve out for the expectations gives
Yt = H(Ωt−1 )Yt−1 + G(Ωt−1 )εt

(21)

As can be seen from this derivation, actual economic outcomes will depend on both
perceived and actual policy rule coefficients. H is given by:
H(Ωt−1 ) = S(Ωt−1 ) + (A(Ωt−1 ) − B(Ωt−1 )S(Ωt−1 ))−1 (C true (Ωt−1 ) − C(Ωt−1 ))

(22)

We calibrate the initial covariance matrix of the estimators so that the initial standard
deviation for each parameter is equal to 10 percent of its original value (which is also
the true pre-policy-change value). We want agents to be reasonably confident about
the pre-policy-change fiscal policy rules (so that before the policy change our agents
behave very similarly to agents who know the fiscal policy rules perfectly). Since the
policy change in our simulations only happens in period 10 and the agents update
their estimates as well as the associated covariance matrix in the first 9 periods of the
simulations, the exact calibration of the initial covariance matrix is not critical.

13

6
6.1

Results
A Roadmap

We will first present results for the full-information rational expectations benchmark14 .
We will then show how learning affects equilibrium outcomes by first discussing results
in our benchmark specification, in which agents think that the true policy change is a
2-standard-deviation shock. We then go on to show how our different beliefs about the
possible size of the policy change affect outcomes. After that we ask if learning would
have any effects if there were no actual policy change.
Next, we ask how different information structures affect our results: Does it matter
if agents know that only one specific coefficient changes or if agents think that other
variables could affect fiscal policy?
We also assess the robustness of our result with respect to the specification of preferences: As we will see below, the behavior of labor supply seems to play an important
role in the dynamics of our model. We thus check to see if our results hold under
two preference specifications that imply very different behavior of labor supply: the
preferences of King et al. (1988) and Greenwood et al. (1988), respectively. Finally, we
show that our findings are robust to the choice of policy instrument that is changed:
We consider a decrease in the intercept of the policy rule for the capital tax rate.

6.2

Rational Expectations

Figure 2 plots the median of the logarithm of the outcomes for our experiment under
full-information rational expectations15 . We see that there are very persistent effects
on output, but ultimately output returns to a level very close to the initial steady
state. The steady state of other variables is very much affected by the policy change
though: Debt and the capital tax rate are permanently higher, leading to a permanently
lower capital stock. The long-run level of the labor tax, on the other hand, remains
basically unchanged, stemming from the parameter values of the policy rule for that
instrument. Consumption shows very persistent effects and converges toward a lower
14

Full-information rational expectations might be a misnomer since the agents in this economy do
not anticipate the policy change - a common assumption when analyzing structural change in rational
expectations models. When the change in fiscal policy happens, the agents are fully aware of the new
policy, though.
15
Mean outcomes are very similar.

14
steady state. Households raise their labor supply to partially offset the drop in capital.
Overall, the effects of the policy change are a short-term small increase in output
relative to a scenario in which the policy rule does not change (shown in figure 13 in
the Appendix), coming at the cost of changes in the long-run behavior of the economy.
As mentioned above, we will later check how robust our outcomes are to different
preference specifications that lead to different behavior of the labor supply.

6.3

Benchmark Results

Now we turn to the economy under learning. First, we ask to what extent outcomes
are different under learning relative to rational expectations when agents’ beliefs about
time variation are calibrated in such a way that the actual policy represents a 2standard-deviation shock under the beliefs of the agents in the economy. Figure 3
shows a summary of the outcomes in that environment. The bottom panel shows
the distribution of point estimates (median as well as 5th and 95th percentile bands)
across simulations for the parameters in the government spending policy rule16 . Agents
quickly pick up on the change in Gc . Before the policy change, the uncertainty surrounding policy rule parameters is very small. There is a substantial increase in that
uncertainty, as measured by the difference of the percentile bands, as policy changes.
The uncertainty decreases again after the policy change for Gc . These patterns are
consistent with the uncertainty index constructed by Baker et al. (2012)17 . The uncertainty surrounding the response coefficients grows over time, but is very small in
magnitude. There is also a slight bias in the estimation of these coefficients, but by
inspecting the y-axis of these graphs one can see that the bias is small, too18 . Thus,
agents in this setup learn fast and the largest uncertainty in quantitative terms (that
around Gc ) disappears reasonably quickly. Does learning have any effect on outcomes
then?
The top panel shows how average outcomes change relative to full-information rational expectations19 : We plot the cumulated difference between median outcomes under
16

Agents estimate the coefficients in all policy rules, but since the policy change occurs in the
government spending policy rule we focus on those parameters.
17
If we were to set 1t = 1∀t we would not get this strong reduction in uncertainty.
18
The uncertainty in these response coefficients does not make a substantial difference for our results.
This will become clear in the robustness check below in which agents only have to estimate Gc . The
qualitative results in this case are the same as in our benchmark case.
19
Note that the results under learning up to any period t are the same under our assumption of a
permanent change in fiscal policy as they would be under the assumption of a temporary change that
ends in period t + 1. This is not true under full-information rational expectations.

15
learning and under rational expectations relative to the original steady state. We thus
plot
Dif fjW

=

j
X
(Wtlearning − W RE )
t

t=1

W

(23)

where Wt is the median of the variable of interest in levels, W is the associated original
steady state, and the superscripts denote outcomes under learning and rational expectations20 . We see that before the negative technology shock and the associated policy
change the cumulative differences are basically zero - there is no difference in average
outcomes between learning and the full-information case. After the technology shock
and the fiscal policy change in period 10 differences emerge - for a while consumption
is higher under learning and hours worked lower . In those periods the agents in the
learning model are actually better off on average. After a few periods the cumulative difference in consumption decreases again and ultimately becomes negative. The
cumulative difference for GDP stays negative throughout. These effects are quantitatively significant: 40 periods (10 years) after the policy change the cumulative loss in
GDP is 2 percent of the original steady state. The cumulative difference in the capital
stock is persistently negative, which explains the differences in GDP given that the
cumulative difference in hours is small. When it comes to fiscal policy instruments, we
see that the cumulative difference in capital tax rates is basically zero, but that there
are huge differences when it comes to debt. To summarize, not taking into account
learning can have sizable effects on average outcomes in the economy. This is only one
side of the coin though - the middle panel of figure 3 shows the standard deviation of
(the log of) each variable relative to the volatility across the simulations under rational
expectations. Consumption is substantially more volatile under learning at the time
of the policy change (a 20 percent increase). Volatility also increases for GDP (around
2 percent) and other variables. These increases in volatility are smaller than those for
GDP, but they are still significant. The changes in standard deviations are short-lived
though, which is consistent with our observations that the estimated coefficients converge quickly. Why then are average outcomes affected so much? The sudden large fall
in average investment under learning has very persistent effects via the capital stock.
Thus, even though agents pick up quickly on changes, the short period of ’confusion’
has persistent effects. This in turns stems from the underestimation of the persistence
of the increase in government spending by agents - it takes them a few periods to fully
grasp that the increase in government spending comes from an increase in Gc rather
than a sequence of large shocks. The belief that part of the changes in government
20

In this calculation the outcomes under rational expectations and learning are calculated using the
same shock sequences.

16
spending are temporary leads agents to believe that permanent increases in debt and
capital taxes are not as large as they actually are, which substantially affects their
investment decisions. Further evidence for this can be gathered by looking at figure
12. The figure plots the actual median path of the capital rate in levels under learning
(this path is very similar under learning and rational expectations), the steady state
capital tax rate associated with the original policy, the steady state capital tax rate
associated with the new policy rule and the median perceived steady state across simulations. As the policy change happens, the rational expectations agents immediately
realize that the new steady state of capital taxes is the green line, whereas agents
under learning think the steady state is given by the perceived steady state. Thus,
relative to steady state rational expectations agents find it more profitable to invest
even at the time of the crisis because they know that the capital tax will be higher
on average than the learning agents think. In more technical terms, the log-linearized
equilibrium conditions we use will give investment as a negative function of (among
other things) log(τtK ) − log(τ K ), which will be larger in absolute value for the rational
expectations agents because they know that the steady state is larger. This is only
a partial explanation because the coefficients multiplying the log difference term are
also a function of the (perceived or actual) steady state. Nonetheless, the dynamics of
the perceived steady state of capital taxes seem to be one factor contributing to the
difference in investment. This also sheds light on an interesting feature of our model:
The agents are very much certain about the coefficients of the capital tax policy rule
(they estimate them, but the associated estimates do not move significantly), but they
are still very uncertain about the steady state value of that policy instrument. This is
due to their uncertainty about the steady state of debt and GDP owing to the uncertainty surrounding government spending. GDP and debt enter the right-hand side of
the capital tax policy rule and thus influence the steady state of the capital tax rate.
In at least one direction we are underestimating the average effects of learning: If the
policy were autocorrelated, it would take the agents longer to figure out that a change
in Gc drives the policy change, rather than a sequence of shocks.

6.4

The Effect of Agents’ Beliefs

Next we ask to what extent outcomes under learning would be different if agents either
think that the same policy change is more likely than before (it represents a 1-standarddeviation shock) or less likely (it represents a 3-standard-deviation shock). The shape
of the plotted objects remains the same as before. However, the magnitudes do change

17
substantially and there is a clear pattern: The less likely agents find a large change
in policy, the bigger the differences in average outcomes between learning and rational
expectations - it takes agents longer to learn. This longer transition has the effect of
substantially decreasing volatility. Thus it is not clear if a policymaker contemplating
a policy change would want agents to be uncertain about policy and consider large
changes, or if that policymaker would want agents to believe that there will be only
small policy changes. Ultimately this will depend on the preferences and the decision
horizon of the policymaker.

6.5

Learning When There is no Policy Change

An important question is what drives the differences between learning and rational
expectations: Is it the change in policy or would learning also lead to different outcomes when there is no policy change? The pre-policy-change part of the results above
strongly indicates that if agents did not contemplate a policy change (i.e., 1t = 0∀t),
then there would be no noticeable difference between learning and rational expectations. But what would happen if the agents did contemplate a policy change just as
above, but there was none? Figure 6 tackles that question. Comparing this figure with
figure 3, we see that the mere suspicion of a policy change on the part of the agents
already leads to substantial increases in volatility (which are smaller than in the case
with changes to fiscal policy, though), but average effects are substantially smaller.

6.6

Information Structure

Does it matter whether agents know exactly what parameter in the fiscal policy rule
changes or what variables enter into the fiscal policy rules? We turn to these questions
next. Both of these experiments use the benchmark calibration for the agents’ beliefs.
First, we endow agents with the knowledge that only Gc changes. The results of this
exercise are given in figure 7. In this case volatilities are dampened relative to our
benchmark case depicted in figure 3, but average outcomes behave very similarly.
Next we ask what would happen if the agents thought that another variable (in our
case consumption) would enter the right-hand side of the policy rule for government
spending. We initialize the beliefs about the coefficient on consumption at zero. Figure
8 shows the relevant outcomes. The parameter estimates for the other coefficients are
very similar to our benchmark case (the estimate for the coefficient on consumption

18
stays centered on zero throughout). Average outcomes and volatilities are very similar
to the benchmark case as well - it seems that agents entertaining more general models
(within certain bounds) does not substantially change our conclusions.

6.7

Preferences

Do our results hold when agents have different preferences? To address this issue with
a particular focus on the behavior of labor supply, we redo our benchmark analysis for
two classes of preferences that imply very different wealth effects on labor supply: the
preferences of Greenwood et al. (1988) and those of King et al. (1988). The equations
for both cases are laid out in the Appendix. Figures 9 and 10 show the results for these
two cases. While the dynamics differ from our benchmark case for both preferences,
the big picture remains the same: We see substantial differences in average outcomes
and increases in volatility relative to rational expectations.

6.8

Capital Tax Change

After a negative shock hits the economy, government spending is not the only instrument the fiscal sector can change to boost the economy. In figure 11 we study a capital
tax decrease equivalent to 1 percent of GDP. This is calculated along the lines of Leeper
et al. (2010) and our own calculations for the government spending case, so that the
decrease of total capital tax revenues approximately equals one percent of overall prepolicy-change steady state GDP. Qualitatively the results are the same as under the
scenario of an increase of government spending. Cumulated GDP is lower by about
5 percent after the end of our simulation horizon while cumulated debt is around 15
percent higher in the case of learning compared to the rational expectations outcome.
Investment and therefore also capital are decreasing constantly throughout. Volatility
increases are quite small for all variables.

7

Conclusion

Our experiments point to the conclusion that we should be cautious when evaluating fiscal policy proposals solely on the basis of a full-information analysis. We have

19
endowed agents with substantial knowledge of the structure of the economy and the
timing of the policy change, thus focusing the uncertainty agents face on a very specific
aspect - the post-policy-change values of the policy rule coefficients. Yet we still find
meaningful differences between a rational expectations model and our learning model.
The views that agents hold about the magnitude of possible policy changes has a significant impact on outcomes, pointing toward a possible role for communicating policy
changes. However, a policymaker would have to be sure of the effects of their communication on the public’s views to avoid undesired outcomes - if that communication
only increases the probability that private agents assign to large policy changes then
communication would lead to substantially more volatility after the policy change.

20

References
Baker, S. R., Bloom, N., and Davis, S. J. (2012). Has economic policy uncertainty
hampered the recovery? Technical report.
Bernanke, B., Gertler, M., and Gilchrist, S. (1999). The financial accelerator in a quantitative business cycle framework , volume 1, chapter 21, pages 1341–1393. Handbook
of Macroeconomics.
Born, B. and Pfeifer, J. (2013). Policy risk and the business cycle. CESifo Working
Paper Series 4336, CESifo Group Munich.
Caprioli, F. (2010). Optimal fiscal policy when agents are learning. Working paper.
Cogan, J. F., Cwik, T., Taylor, J. B., and Wieland, V. (2010). New Keynesian versus
old Keynesian government spending multipliers. Journal of Economic Dynamics and
Control , 34, 281–295.
Cogley, T. and Sargent, T. J. (2005). Drift and volatilities: Monetary policies and
outcomes in the post WWII U.S. Review of Economic Dynamics, 8(2), 262–302.
Cogley, T., Colacito, R., and Sargent, T. J. (2007). Benefits from U.S. monetary policy
experimentation in the days of Samuleson and Solow and Lucas. Journal of Money,
Credit & Banking, 39, 67–99.
Cogley, T., Matthes, C., and Sbordone, A. M. (2011). Optimal disinflation under
learning. Technical report.
Davig, T., Leeper, E. M., and Walker, T. B. (2010). Unfunded liabilities" and
uncertain fiscal financing. Journal of Monetary Economics, 57(5), 600–619.
Eusepi, S. and Preston, B. (2011). Learning the fiscal theory of the price level: Some
consequences of debt-management policy. Journal of the Japanese and International
Economies, forthcoming.
Fernandez-Villaverde, J., Guerron-Quintana, P., Kuester, K., and Rubio-Ramirez, J.
(2011). Fiscal volatility shocks and economic activity. PIER Working Paper Archive
11-022, Penn Institute for Economic Research, Department of Economics, University
of Pennsylvania.
Giannitsarou, C. (2006). Supply-side reforms and learning dynamics. Journal of Monetary Economics, 53(2), 291–309.
Greenwood, J., Hercowitz, Z., and Huffman, G. W. (1988). Investment, capacity utilization, and the real business cycle. American Economic Review , 78(3), 402–17.

21
Jaimovich, N. and Rebelo, S. (2009). Can news about the future drive the business
cycle? American Economic Review , 99(4), 1097–1118.
Karantounias, A. G. (2013). Managing pessimistic expectations and fiscal policy. Theoretical Economics, 8, 193–231.
King, R. G., Plosser, C. I., and Rebelo, S. T. (1988). Production, growth and business
cycles : I. the basic neoclassical model. Journal of Monetary Economics, 21(2-3),
195–232.
Kliem, M. and Kriwoluzky, A. (2013). Toward a Taylor rule for fiscal policy. Working
paper, Deutsche Bundesbank and Rheinische Friedrich-Wilhelms Universität Bonn.
Koop, G., Leon-Gonzalez, R., and Strachan, R. (2009). On the evolution of monetary
policy. Journal of Economic Dynamics and Control , 33, 997–1017.
Kreps, D. (1998). Anticipated Utility and Dynamic Choice, pages 242–274. Frontiers
of Research in Economic Theory. Cambridge University Press.
Kydland, F. E. and Prescott, E. C. (1982). Time to build and aggregate fluctuations.
Econometrica, 50(6), 1345–70.
Leeper, E. M., Plante, M., and Traum, N. (2010). Dynamics of fiscal financing in the
United States. Journal of Econometrics, 156, 304–321.
Marcet, A. and Nicolini, J.-P. (2003). Recurrent hyperinflations and learning. American
Economic Review , 93, 1476–1498.
Milani, F. (2007). Expectations, learning and macroeconomic persistence. Journal of
Monetary Economics, 54(7), 2065–2082.
Mitra, K., Evans, G. W., and Honkapohja, S. (2012). Fiscal policy and learning.
Research Discussion Papers 5/2012, Bank of Finland.
Primiceri, G. (2005). Time varying structural vector autoregressions and monetary
policy. Review of Economic Studies, 72(3), 821–852.
Sargent, T., Williams, N., and Zha, T. (2006). Shocks and government beliefs: The
rise and fall of American inflation. American Economic Review , 96(4), 1193–1224.
Sargent, T. J. and Williams, N. (2005). Impacts of priors on convergence and escapes
from Nash inflation. Review of Economic Dynamics, 8, 360–391.
Sims, C. A. (2001). Solving linear rational expectations models. Computational Economics, 20, 1–20.
Smets, F. and Wouters, R. (2007). Shocks and Frictions in US Business Cycles: A
Bayesian DSGE Approach. American Economic Review , 97(3), 586–606.

22

Appendix
A First-Order Conditions
Households:
Ct−σ
1 + τtC

= Et

−σ
βRt Ct+1
C
1 + τt+1

L1+φ
(1 + τtC ) = Ct−σ (1 − τtL )(1 − α)Yt
t


−σ
Ct+1
(1 + τtC )
K αYt+1
(1 − τt+1 )
1 = βEt −σ
+ (1 − δ)
C )
Kt
Ct (1 + τt+1

Firms:
Wt =
RtK

=

(1 − α)Yt
Lt
αYt
Kt−1

B Log-Linearized Model
Households:

τcL
(1 + φ)log(Lt ) +
= Const + log(Yt ) −
log(τtL ) − σlog(Ct )
1 − τcL




1
τcC
1
τcC
1
C
C
log(Ct ) = ConstC −
log(τ
)
+
log(τt+1
) + log(Ct+1 ) − log(Rt )
t
σ 1 + τcC
σ 1 + τcC
σ


τcC
1 + τcC



log(τtC )

L



log(Kt ) = ConstLoM + (1 − δ)log(Kt−1 ) + δlog(It )
log(Yt ) = ConstY + log(At ) + αlog(Kt−1 ) + (1 − α)log(Lt )
τcC
τcC
C
σEt log(Ct+1 ) = ConstK + σlog(Ct ) −
E
log(τ
)
+
log(τtC )
t
t+1
(1 + τcC )
(1 + τcC )
Yss
Yss
Yss
K
+ β(1 − τcK )α
Et log(Yt+1 ) − β(1 − τcK )α
log(Kt ) − βτcK α
Et log(τt+1
)
Kss
Kss
Kss

23

Firms:
Css
Iss
Gss
log(Ct ) +
log(It ) +
log(Gt )
Yss
Yss
Yss
log(At ) = ConstA + ρa log(At−1 ) + tA
log(Yt ) = ConstAgg +

Policy Rules:
Yss
Yss
Css
(log(τtK ) + log(Yt )) + τcL (1 − α)
(log(τtL ) + log(Yt )) + τcC
(log(τtC ) + log(Ct ))
Bss
Bss
Bss
1
1
Gss
Zss
= ConstB + log(Rt−1 ) + log(Bt−1 ) +
log(Gt ) +
log(Zt )
β
β
Bss
Bss
log(Gt ) = Gc − ρg,y log(Yt−1 ) − ρg,b log(Bt−1 ) + G
t
log(Bt ) + τcK α

log(Zt ) = Zc − ρz,y log(Yt−1 ) − ρz,b log(Bt−1 ) + Z
t
log(τtC ) = τcc + C
t
log(τtL ) = τcl + ρL,y log(Yt−1 ) + ρL,b log(Bt−1 ) + L
t
log(τtK ) = τcl + ρK,y log(Yt−1 ) + ρK,b log(Bt−1 ) + K
t

with the constants given by:
Constant

Expression

Gc

log(Gc ) + ρg,y log(Yss ) + ρg,b log(Bss )

Zc

log(Zc ) + ρz,y log(Yss ) + ρz,b log(Bss )

τcl
τck
τcc

log(τcL ) − ρL,y log(Yss ) − ρL,b log(Bss )

ConstB

Yss
Yss
log(Bss )(1 − β1 ) + τcK α B
(log(τcK ) + log(Yss )) + τcL (1 − α) B
(log(Yss ) + log(τcl ))
ss
ss

log(τcK ) − ρK,y log(Yss ) − ρK,b log(Bss )
log(τcC )
Css
+τcC B
(log(τcC ) + log(Css )) − β1 log(Rss ) −
ss

Gss
Bss log(Gss )

ConstLoM

δ(log(Kss ) − log(Iss ))

ConstL

(1 + φ)log(Lss ) +

ConstC

1
σ log(Rss )

ConstY

log(Yss ) − log(Ass ) − αlog(Kss ) − (1 − α)log(Lss )

ConstA

log(Ass )

ConstAgg

log(Yss ) −

ConstK

−β(1 −

τcC
log(τcC )
1+τcC

− log(Yss ) +

−

Zss
Bss log(Zss )

τcL
log(τcL )
1+τcL

Css
Gss
Iss
Yss log(Css ) − Yss log(Gss ) − Yss log(Iss )
Yss
Yss
τcK )α K
log(Yss ) + β(1 − τcK )α K
log(Kss )
ss
ss

Yss
+ β(τcK )α K
log(τcK )
ss

24

C Parameters
Calibrated Parameters
Description
Impatience
Capital share
Depreciation rate
CES utility consumption
CES utility labor
Coeff. on Y in gov. exp. rule
Coeff. on B in gov. exp. rule
Coeff. on Y in transfer rule
Coeff. on B in transfer rule
Coeff. on Y labor tax rule
Coeff. on B labor tax rule
Coeff. on Y capital tax rule
Coeff. on B capital tax rule
AR parameter technology
Std. deviation technology
Std. deviation gov. spending
Std. deviation transfers
Std. deviation cons.tax
Std. deviation labor tax
Std. deviation capital tax

Parameter
β
α
δ
σ
φ
ρg,y
ρg,b
ρz,y
ρz,b
ρL,y
ρL,b
ρK,y
ρK,b
ρa
σa
σg
σz
σc
σl
σk

Value
0.99
0.33
0.025
2
2
0.034
0.23
0.13
0.5
0.36
0.049
1.7
0.39
0.9
0.0062
0.031
0.034
0.04
0.03
0.044

Table 1: Calibrated parameters of the model

Initial Steady State Values of the Actual Law of Motion

25
Description
Output
Consumption
Cons. tax rate
Capital tax rate
Labor
Investment
Capital
Debt
Labor tax rate
Government spending
Transfers
Technology
Interest rate

Parameter
Yss
Css
τcC
τcK
Lss
Iss
Kss
Bss
τcL
Gc
Zc
Ass
Rss

Value
2.0601
1.5010
0.0287
0.2452
0.7847
0.3655
14.6195
0.5623
0.1886
0.1936
0.2709
1
1.01

Table 2: Calibrated parameters of the model

Perceived Steady States
The perceived steady states in the updating algorithm of the agents are given by the
following twelve equations:

R =

1
β
1

− (1 − δ)
αY
β
=
K
1 − τK
1+φ
C
−σ
L (1 + τ ) = C (1 − τ L )(1 − α)Y
Y

= AK α L1−α

Y

= C +I +G

I = δK
1
B = B − τ K αY − τ L (1 − α)Y − τ C C + G + Z
β
G
Const = log(G) + ρg,y log(Y ) + ρg,b log(B)
ConstZ = log(Z) + ρz,y log(Y ) + ρz,b log(B)
L

= log(τcL ) − ρL,y log(Y ) − ρL,b log(B)

Constτ

K

= log(τcK ) − ρK,y log(Y ) − ρK,b log(B)

Constτ

C

= log(τcC )

Constτ

26
for the twelve variables: Y, K, L, C, G, Z, τ L , τ K , τ C , B, I, R, which are solved numerically.

27

D Figures
1600

Tax Expiration

1400

1200

1000

800

600

400

200

0
1985

1990

1995

2000

2005

2010

2015

Figure 1: Fiscal uncertainty index by Baker et al. (2012)
GDP

Consumption

0.75

0.42

0.7

0.4

Cons. Tax
−3.52
−3.54
−3.56
−3.58

0.38

0.65
20

40 60 80
time
Cap. Tax

100

20

40 60
time
Labor

80

100

20

40 60 80
time
Investment

100

20

40 60 80
time
Labor Tax

100

20

40 60 80
time
Interest Rate

100

−1

−1.35
−1.4
−1.45
−1.5
−1.55

−0.24
−1.2
−0.26
20

40 60
time
Capital

80

100

20

2.7

−0.4

2.65

−0.5

2.6

40 60
time
Debt

80

100

−1.65

−1.7

−0.6
20

40 60 80
time
Gov. Spending

100

20

40 60 80
time
Transfers

100

−3

x 10

−1.25
−1.3
−1.35
−1.4
−1.45

−1.5
−1.6
−1.7
20

40 60
time

80

100

10

20

40 60
time

80

100

9

20

40 60
time

Figure 2: Log outcomes under rational expectations

80

100

28

Cum. Changes in %
1

Cum. Changes in %
5

Cum. Changes in %
20

Cum. Changes in %
5

0

0

10

0

Cons.

−1
−2

GDP

0

50
time

Hours

−5
−10

100

Captax

Investment

0

50
time

0
−10

100

0

50
time

Capital

−5

Debt

−10

100

Gov.Spending

0

50
time

100

LE std / RE std, log C
LE std / RE std, log GDP LE std / RE std, log debt LE std / RE std, log capital
1.05
1.5
1.02
1.01

1

0.5

1

0

50
time
Gc

0.98

100

−1.6

1

0

50
time
ρgy

0.95

100

1

0

50
time
ρgb

100

0

50
time

100

0.99

0

50
time

100

0.24

0.036

0.235
−1.7

Median

0.034
0.23

Upper
Lower

−1.8

0

50
time

Actual

100

0.032

0

50
time

100

0.225

Figure 3: Summary of outcomes under learning
Cum. Changes in %
1

Cum. Changes in %
5

0
Cons.

−1
−2

0

50
time

100

Investment

−5

Cum. Changes in %
2

10

0

Hours

GDP

0

Cum. Changes in %
20

0

50
time

100

Captax

0
−10

Debt

0

50
time

100

Capital

−2
−4

Gov.Spending

0

50
time

100

LE std / RE std, log GDP LE std / RE std, log debt LE std / RE std, log capital
LE std / RE std, log C
1.05
1.05
1.5
1.02

1

0.5

1

0

50
time
Gc

100

−1.6

0.95

1

0

50
time
ρgy

100

0.95

1

0

50
time
ρgb

100

0

50
time

100

0.98

0

50
time

100

0.24

0.038
0.036

−1.7

0.23

Median
Upper

0.034

Lower

−1.8

0

50
time

Actual

100

0.032

0

50
time

100

0.22

Figure 4: Summary of outcomes under learning, 1-standard-deviation case

29

Cum. Changes in %
2

Cum. Changes in %
5

Cum. Changes in %
20

Cum. Changes in %
5

0

0

10

0

Cons.

−2
−4

Hours

−5

GDP

0

50
time

−10

100

LE std / RE std, log C
1.2

0

Investment

0

50
time

−10

100

1

0

50
time
Gc

−1.7

0.98

100

−1.6

Median

Capital

−5

Debt

0

50
time

−10

100

Gov.Spending

0

50
time

100

LE std / RE std, log GDP LE std / RE std, log debt LE std / RE std, log capital
1.05
1.02
1.005

1

0.8

Captax

1

0

50
time
ρgy

0.95

100

0.035

0.235

0.034

0.23

1

0

50
time
ρgb

100

0

50
time

100

0.995

0

50
time

100

Upper
Lower

−1.8

0

50
time

Actual

100

0.033

0

50
time

100

0.225

Figure 5: Summary of outcomes under learning, 3-standard-deviations case
Cum. Changes in %
0.5

Cum. Changes in %
1
0.5

0

−0.5

0

50
time

100

Hours

0
−0.5

Cum. Changes in %
0.5

1

Cons.
GDP

Cum. Changes in %
2

Investment

0

50
time

100

Captax

0
−1

0

Capital

Debt

0

50
time

100

Gov.Spending

−0.5

0

50
time

100

LE std / RE std, log C
LE std / RE std, log GDP LE std / RE std, log debt LE std / RE std, log capital
1.5
1.02
1.01
1.02

1

0.5

1

0

50
time
Gc

100

−1.65
−1.7

0.98

1

0

50
time
ρgy

100

0.99

0.038

0.26

0.036

0.24

0.034

0.22

1

0

50
time
ρgb

100

0

50
time

100

0.98

0

50
time

100

Median

−1.75

Upper
Lower

−1.8

0

50
time

Actual

100

0.032

0

50
time

100

0.2

Figure 6: Summary of outcomes under learning when there is no fiscal policy change

30

Cum. Changes in %
2

Cum. Changes in %
5

Cum. Changes in %
20

Cum. Changes in %
5

0

0

10

0

Cons.

−2
−4

GDP

0

50
time

100

Hours

−5
−10

Investment

0

50
time

100

Captax

0
−10

Debt

0

50
time

100

Capital

−5
−10

Gov.Spending

0

50
time

100

LE std / RE std, log C
LE std / RE std, log GDP LE std / RE std, log debt LE std / RE std, log capital
1.05
1.004
1.05
1.005
1.002
1

1

1

1
0.95

0

50
time
Gc

100

−1.6

−1.7

0.998

0

50
time
ρgy

100

0.95

2

2

1

1

0

0

0

50
time
ρgb

100

0

50
time

100

0.995

0

50
time

100

Median
Upper
Lower

−1.8

0

50
time

Actual

100

−1

0

50
time

100

−1

Figure 7: Summary of outcomes under learning when agents only need to learn about
Gc

31
Cum. Changes in %
2

Cum. Changes in %
5

Cum. Changes in %
20

Cum. Changes in %
5

0

0

10

0

Cons.

−2
−4

0

50
time

Hours

−5

GDP

−10

100

Investment

0

50
time

100

Captax

0
−10

0

50
time

Capital

−5

Debt

−10

100

Gov.Spending

0

50
time

100

LE std / RE std, log C
LE std / RE std, log GDP LE std / RE std, log debt LE std / RE std, log capital
1.5
1.02
1.02
1.02

1

0.5

1

0

50
time
Gc

−1.6

−1.7

0.98

100

1

0

50
time
ρgy

100

0.98

0.038

0.25

0.036

0.24

0.034

0.23

1

0

50
time
ρgb

100

0

50
time

100

0.98

0

50
time

100

Median
Upper
Lower

−1.8

0

50
time

Actual

100

0.032

0

50
time

100

0.22

Figure 8: Summary of outcomes under learning when agents think that consumption
enters the policy rule for government spending
Cum. Changes in %
20

Cum. Changes in %
20

0

0

−40

GDP

0

500
time

1000

0

Hours

−20
−40

Cum. Changes in %
20

100

Cons.

−20

Cum. Changes in %
200

Investment

0

500
time

1000

Captax

0
−100

Debt

0

500
time

1000

Capital

−20
−40

Gov.Spending

0

500
time

1000

LE std / RE std, log C
LE std / RE std, log GDP LE std / RE std, log debt LE std / RE std, log capital
1.1
1.05
1.05
1.005

1

0.9

1

0

500
time
Gc

1000

−1.6

0.95

1

0

500
time
ρgy

1000

0.95

1

0

500
time
ρgb

1000

0

500
time

1000

0.995

0

500
time

1000

0.24

0.038
0.036

−1.7

0.23

Median
Upper

0.034

Lower

−1.8

0

500
time

Actual

1000

0.032

0

500
time

1000

0.22

Figure 9: Summary of outcomes under learning when agents have GHH preferences

32
Cum. Changes in %

Cum. Changes in %

1

Cum. Changes in %

5

0
Cons.

−1
−2

0

50
time

−5

0

50
time

100

1

50
time
Gc

0

10
0

0

50
time

Capital

−5

Debt

100

−10

Gov.Spending

0

50
time

100

LE std / RE std, log GDP LE std / RE std, log debt LE std / RE std, log capital
1.1
1.05
1.04

1.5

0

20
Captax

Investment

100

5

Hours

GDP

LE std / RE std, log C
2

1

0

Cum. Changes in %

30

0.9

100

−1.4

1

0

50
time
ρgy

100

0.06

0.95

1.02

0

50
time
ρgb

100

0

50
time

100

1

0

50
time

100

0.4

0.05
−1.6

0.2

Median

0.04

Upper
Lower

−1.8

0

50
time

Actual

100

0.03

0

50
time

100

0

Figure 10: Summary of outcomes under learning when agents have KPR preferences
Cum. Changes in %
5

Cum. Changes in %
10

Cum. Changes in %
20

0
0

−5

10

Cons.
GDP

0

50
time

100

0

Hours

−10
−20

Cum. Changes in %
10

Captax

Investment

0

50
time

100

0
−10

Debt

0

50
time

100

Capital

−10
−20

Gov.Spending

0

50
time

100

LE std / RE std, log C
LE std / RE std, log GDP LE std / RE std, log debt LE std / RE std, log capital
1.01
1.005
1.005
1.005

1

0.99

1

0

50
time
Taxkc

100

−2.38

0.995

1

0

50
time
ρky

100

1.72

0.995

1

0

50
time
ρkb

100

0

50
time

100

0.995

0

50
time

100

0.4

−2.4

0.395
Median

−2.42

1.7
0.39

Upper
Lower

−2.44

0

50
time

Actual

100

1.68

0

50
time

100

0.385

Figure 11: Summary of outcomes under learning when the capital tax policy rule
changes

33

0.255

0.25

0.245

0.24

0.235

0.23

0.225

perceived SS
median learning
original SS
new SS
0

10

20

30

40

50

60

70

80

90

100

Figure 12: The capital tax rate

34

E Robustness Checks: Different Utility Function Specifications
A: First-order conditions of households: As robustness checks we consider the following
utility function (compare Jaimovich and Rebelo (2009)):
U = E0

∞
X
t=0


1−σ
Ct − ψNtθ Xt
−1
β
1−σ
t

(24)

1−γ
with Xt = Ctγ Xt−1
which nests both the King et al. (1988) Preferences (γ = 1) and
the Greenwood et al. (1988) preferences (γ = 0).

−σ
Ct − ψNtθ Xt
+ µt γCtγ−1 Xt1−γ = λt (1 + τtc )
i

−σ
h
γ
Xt−γ
Ct − ψNtθ Xt
ψNtθ + µt = βEt µt+1 (1 − γ)Ct+1

−σ
Ct − ψNtθ Xt
ψθNtθ−1 Xt = λt (1 − τtl )Wt


1 = βEt



λt+1
K
K
(1 − τt+1
)Rt+1
+ (1 − δ)
λt

B: First-order conditions in the GHH case:

−σ
Ct − ψNtθ
(1 + τtC )

θ
Rt Ct+1 − ψNt+1
+ βEt
c )
(1 + τt+1


−σ

ψθNtθ (1 + τtC ) = (1 − τtL )(1 − α)Yt

−σ


θ
Ct+1 − ψNt+1
(1 + τtC )
Yt+1
K
1 = βEt
(1 − τt+1 )α
+ (1 − δ)

−σ
C )
Kt
Ct − ψNtθ
(1 + τt+1

35
C: Log-linearized conditions in the GHH case



τcC
τcL
C
L
log(τ
)
=
Const
+
log(Y
)
−
log(τtL )
t
t
1 + τcC
1 − τcL




θ
τcC
σψθNss
τcC
C
C
C
log(τt ) = Const + Rt −
log(τt+1
)
log(Nt ) −
θ
1 + τcC
1 + τcC
Css − ψNss
θ
σCss
σψθNss
log(C
)
+
log(Nt+1 )
t+1
θ
θ
Css − ψNss
Css − ψNss
θ
σψθNss
τcC
σCss
log(N
)
−
log(τtC ) −
log(Ct )
t+1
C
θ
θ
(1 + τc )
Css − ψNss
Css − ψNss
θ
σψθNss
τcC
C
log(N
)
+
Et log(τt+1
) = ConstK
t
θ
(1 + τcC )
Css − ψNss
Yss
Yss
Et log(Yt+1 ) − β(1 − τcK )α
log(Kt )
β(1 − τcK )α
Kss
Kss
Yss
K
Et log(τt+1
)
βτcK α
Kss


θlog(Lt ) +
−

σCss
log(Ct ) +
θ
Css − ψNss
−

σCss
Et log(Ct+1 ) −
θ
Css − ψNss
+
+
−

D: First-order conditions in the KPR case:

−σ


Ct − ψNtθ Ct
1 − ψNtθ
(1 + τtC )


−σ


θ C
θ
Rt Ct+1 − ψNt+1
1 − ψNt+1
t+1
= βEt
c )
(1 + τt+1

ψθCt Ntθ (1 + τtC ) = (1 − ψNtθ )(1 − τtL )(1 − α)Yt

−σ
θ C
θ )(1 + τ C )
Ct+1 − ψNt+1
(1 − ψNt+1
t+1
t
1 = βEt


−σ
C )
Ct − ψNtθ Ct
(1 − ψNtθ )(1 + τt+1


Yt+1
K
+ (1 − δ)
(1 − τt+1 )α
Kt

36
E: Log-linearized conditions in the KPR case



τcC
τcL
C
L
log(τ
)
+
log(C
)
=
Const
+
log(Y
)
−
log(τtL )
t
t
t
1 + τcC
1 − τcL
θ
ψθNss
log(Nt )
θ
1 − ψNss




θ
(1 − σ)ψθNss
τcC
τcC
C
C
C
log(Nt ) −
log(τt ) = Const + Rt −
log(τt+1
)
θ
1 + τcC
1 + τcC
1 − ψNss
θ
(1 − σ)ψθNss
σlog(Ct+1 ) −
log(Nt+1 )
θ
1 − ψNss
θ
θ
(1 − σ)ψθNss
τcC
(1 − σ)ψθNss
C
log(N
)
−
log(τ
)
−
σlog(C
)
−
log(Nt )
t+1
t
t
θ
θ
(1 + τcC )
1 − ψNss
1 − ψNss
Yss
τcC
Yss
C
Et log(τt+1
) = ConstK + β(1 − τcK )α
Et log(Yt+1 ) − β(1 − τcK )α
log(Kt )
(1 + τcC )
Kss
Kss
Yss
K
βτcK α
Et log(τt+1
)
Kss


θlog(Nt ) +
−
−σlog(Ct ) −
−
σEt log(Ct+1 ) +
+
−

F Simulation
The simulation of our learning economy is carried out via the following steps:
1. We endow agents with initial beliefs Ω0 , which coincide with the true pre-policychange parameter values.
2. Given the beliefs Ωt−1 , the perceived steady states are calculated and then used
to log-linearize the equilibrium conditions, which together with the estimated policy
rules gives the following expectational difference equation:
A(Ωt−1 )Yt = B(Ωt−1 )Et∗ Yt+1 + C(Ωt−1 )Yt−1 + Dε∗t
which yields the perceived law of motion (using the RE solution algorithm Gensys by
Sims (2001))
Yt = S(Ωt−1 )Yt−1 + G(Ωt−1 )ε∗t .
3. The actual law of motion takes the perceived steady states but uses the true policy
parameters C true (Ωt ) to arrive at the system:
A(Ωt−1 )Yt = B(Ωt−1 )Et∗ Yt+1 + C true (Ωt−1 )Yt + Dεt

37
with the actual shock vector εt . To solve out for the expectations we use the perceived
law of motion to obtain
Yt = H(Ωt−1 )Yt−1 + G(Ωt−1 )εt
4. Shocks are realized by drawing from a multivariate Gaussian distribution, which
together with the transition matrices produced by step 3 determine the macroeconomic outcomes for period t.
5. Observing these outcomes, beliefs are updated via the Kalman filter, which gives Ωt .
We simulate the economy for each setting 1000 times with a sample length of T = 100.

38

G Additional Figures

−3

3

x 10

−3

GDP
5

x 10

Consumption

Cons. Tax
1

2

0.5
0

1

0
−5

0
−1

−0.5

Both minus only Tech

0

50
time

100

−10

0

50
time
−3

Cap. Tax
0.04

6

0.03

x 10

100

−1

0

50
time

Labor

100

Investment
0
−0.005

4

0.02

−0.01
2

0.01

−0.015
0

0
−0.01

0

50
time

100

−2

−0.02
0

Capital

50
time

100

−0.025

Debt

0

0.1

−0.005

0.05

−0.01

0

−0.015

−0.05

6

0

50
time
−3

Labor Tax

−4

Interest Rate

x 10

100

4
2
0

0

50
time

100

0

Gov. Spending

50
time

100

−2

Transfers

0.15

0.02

0.1

0

0.05

−0.02

2

0

50
time

x 10

100

1
0
−1

0

−0.04

−0.05

−0.06

0

50
time

100

−2
0

50
time

100

−3

0

50
time

100

Figure 13: Difference in median (log) outcomes between the RE cases with and without
fiscal policy change