Download Optimal Policy with Endogenous Signal Extraction

Optimal Policy with Endogenous Signal
Extraction
Esther Hauk∗
Andrea Lanteri†
Albert Marcet‡§
February 2014
Preliminary and incomplete
Abstract
This paper studies optimal policy in models with multidimensional
uncertainty and endogenous observables. We first consider a very general setup where the policy-maker does not observe the realisations of
the shocks that hit the economy, but only some aggregate variables
that are endogenous with respect to policy, therefore standard first
order conditions do not hold. We derive first order conditions of optimality from first principles and we illustrate why the estimation of
the state of the economy cannot be separated from the determination
of the optimal policy. In an optimal fiscal policy application with incomplete markets and endogenous Partial Information, we find that
the optimal policy response to aggregate data can be quite non-linear:
it calls for tax smoothing across states in normal times, but in some
cases for a strong adjustment of fiscal positions during a slump. We
show that policies that disregard the endogeneity of the filtering problem and hence these non-linearities can be quite wrong. Finally, our
model can rationalise the fiscal response of some European countries
to the Great Recession: a slow reaction, followed by large deficits and
a delayed sharp fiscal adjustment that protracts the downturn.
∗
IAE-CSIC and BGSE, Campus UAB, 08193 Bellaterra (Barcelona); email: [email protected]
†
Department of Economics and Centre for Macroeconomics, LSE, Houghton Street,
London WC2A 2AE; email: [email protected]
‡
ICREA, IAE-CSIC and BGSE, Campus UAB, 08193 Bellaterra (Barcelona); email:
[email protected]
§
The authors appreciate very helpful comments from Wouter den Haan, Robert
Shimer, Jaume Ventura and other participants at LSE Macro work-in-progress seminar,
IAE-CSIC Macro work-in-progress, Barcelona GSE Winter Workshop 2013.
1
”In the policy world, there is a very strong notion that if we only
knew the state of the economy today, it would be a simple matter
to decide what the policy should be. The notion is that we do not
know the state of the system today, and it is all very uncertain
and very hazy whether the economy is improving or getting worse
or what is happening. Because of that, the notion goes, we are
not sure what the policy setting should be today. [...] In the
research world, it is just the opposite. The typical presumption
is that one knows the state of the system at a point in time.
There is nothing hazy or difficult about inferring the state of the
system in most models.” (James Bullard, interview on Review of
Economic Dynamics, November 2013)
1
Introduction
In this paper we aim to build a bridge between the policy world, that is
characterized by a large amount of uncertainty about the current state of
the economy, and academic research on optimal policy, that has typically
separated the issue of making policy decisions from that of inferring the
state of the economy from the data.
Although the problem of signal extraction played a key role in many research papers in the initial stages of the Rational Expectations revolution
during the ’70s and early ’80s, that interest dwindled considerably, perhaps
because finding an optimal policy under limited information when observables are endogenous is still a problem that has not been solved in general.1
The aim of this paper is to address this problem. We design optimal
policy when some relevant shocks that hit the economy are not observable
and aggregate observable variables are themselves endogenous with respect
to policy decisions, so that policy-makers should take into account the feedback between observables and control variables at the time of making their
decisions. We also draw some conclusions about the importance of taking
Partial Information seriously for welfare.
To give a concrete example of key policy decisions made under uncertainty about the state of the economy and based on endogenous aggregate
variables, consider the recent financial crisis. As it unfolded, the policy discussion often hinged on whether the recession was due to a permanent shock
1
See section 2 for a detailed discussion.
2
(perhaps due to a lower expected productive capacity of the economy) or to
a temporary shock (perhaps due to a fall in demand). All we knew for sure in
November 2008 was that employment and output were low, and yet the G20
decided in its Washington meeting to take aggressive expansionary fiscal policy measures in order to reactivate the economy spending 2% of world GDP,
presumably adhering to the idea that the shock was temporary.2 But from
this relatively optimistic initial estimate, several governments came to the
conclusion that a fiscal adjustment was necessary after observing larger than
expected deficits. Many countries have gone through tax hikes and austerity
in spending after that. All along the decisions on fiscal policy were taken
observing output and employment behaviour. In turn, these variables clearly
depend on whether an expansionary fiscal policy or austerity is adopted. How
to deal with this endogeneity is the main focus of this paper.
In order to address this question, we solve for optimal policy in models
with multidimensional uncertainty and endogenous observables. We first
examine a two-period version of Lucas and Stokey (1983) as we consider it the
most standard fiscal policy dynamic model. To model limited information,
we introduce two shocks (to demand and to supply) and to make the issue
relevant we assume incomplete insurance markets. Then we solve for optimal
Ramsey taxation under the assumption that the government does not observe
the realizations of the shocks, but only some endogenous Partial Information.
We first derive a first order condition (FOC) for the optimal policy relying
on first principles. The difficulty here is that the policy choice affects the
distribution of observables, so that standard first order conditions do not
hold. The optimality condition we find modifies the standard FOC found in
dynamic stochastic models as the probabilities of each state of nature need
to be weighted by a kernel that depends on the effect of policy on the the
observed variable. This illustrates why the so-called ”separation principle” of
Kalman filter models fails: we cannot separate estimation and optimization
since the objective conditional probabilities of underlying shocks are not the
proper weights of the derivative of welfare in each state. The FOC is derived
for a general model so that our technique can be widely applied. We show
some cases where the optimality condition coincides with the standard one.
For example, we show why in a linear framework with additive shocks the
2
Coincidentally, the Spanish Finance Minister at the time, Pedro Solbes, published
an article in El Paı́s on the 10th of November 2013 under the title ”Cuando decidı́ salir
del Gobierno” explaining how the outcome of this meeting eventually lead to the current
Spanish Debt crisis and near-default in summer 2012.
3
endogeneity induced by Partial Information does not really matter as has
already been discussed in the literature.3
The Ramsey optimal fiscal policy derived in the model is of interest on
its own. It dictates optimal tax smoothing across states even in a case where
under Full Information taxes would be volatile. This is because one implication of the optimal policy is to average all the possible contingencies using
weights that depend on how reactive observables are to policy and how the
different shocks interact.
We show a case where the implied policy reaction to fiscal adjustments
to an economic crisis is very non-linear. In particular, if future taxes may be
close to the maximum of the Laffer curve, as in some European economies
in the current crisis, a government may go very quickly from not reacting to
low expected output to reacting very strongly, the reason being that there
is a range of observables that may imply a very large increase in taxes in
the future in order to avoid default. The stark and sudden adjustment only
occurs once policy-makers start fearing that a fiscal crisis may materialize
under the worst possible realizations of the shocks, but importantly this
inference is endogenous to the output observed and the policy chosen. This
may rationalize why some countries reacted slowly to the crisis and went for
austerity with a delay.
Finally, we build an infinite-horizon model that confirms the main results developed in the paper. In some cases, the Ramsey government under
Partial Information reacts slowly to a downturn. Interestingly, the delayed
fiscal adjustment induces a longer recession relative to the Full Information
benchmark.
Our main contribution is to provide a general solution to the endogenous
signal extraction and optimization problem when there is no separation of
any kind between these two problems. Furthermore, we explore the welfare consequences of policies that disregard the endogeneity of the filtering
problem. We show that alternative policies based on averages of the more
standard Full Information policy, which would be correct in linear models
with exogenous filtering problems, turn out to be very incorrect. We present
an example where these policies would induce tax cuts in the region of observables where the optimal policy calls for a strong tax increase. We also
show that linear approximations - which are quite standard in the literature
- can be quite misleading: in our fiscal policy model the correct solution is
3
See section 2 below.
4
highly non-linear in nature.
The remainder of the paper is organized as follows. The related literature
is discussed in Section 2. Section 3 introduces our two-period optimal fiscal policy model with incomplete markets and Partial Information. Section
4 provides the general first order condition, compares the Full Information
solution with the Partial Information solution and discusses the case of ”invertibility” when the two solutions coincide. In section 5 we apply these
first order conditions to our Ramsey fiscal policy problem under different
assumptions on preferences and government expenditure levels. Section 6
is dedicated to robustness. Section 7 presents the infinite-horizon model.
Section 8 concludes.
2
Related literature
Policy making under Partial Information is hardly a new topic in macroeconomics. In the ’70s and ’80s the issue of limited information was central
to the development of Rational Expectations (RE) models. In his seminal
paper on RE, Lucas (1972) formulated the signal extraction problem when
agents do not observe nominal and real shocks but only a combination of
the two. Many other papers used limited or asymmetric information.4 The
issue of endogeneity of observables was often in the literature and many papers discussed the validity of conditioning choices on fundamental shocks and
whether or not prices revealed fundamental information, or whether asymmetric information RE equilibria could be reached.
Although reluctantly at the beginning, nowadays the standard approach
in macroeconomics is to build models where all variables and prices are solved
conditional on some observed fundamentals. In their seminal paper on recursive competitive equilibrium, Mehra and Prescott (1980) assumed that
state variables were either directly observable or an invertible function of observables. From then on, the literature has tended to make this assumption,
disregarding the problem of Partial Information.
Limited information is no longer a central issue, in part because it is well
known that for linear models with additively separable errors a ”separation”
principle holds, guaranteeing that the agent can first solve the signal extraction problem (usually using the Kalman filter) and then solve for the optimal
4
For example all the papers with unobserved transitory and permanent shocks.
5
policy conditional on the estimate. Optimal policy under separation just requires a re-definition of the fundamental variables, so that the Kalman filter
becomes the observed variable.
Other papers address the issue of how price-taking agents solve signal
extraction problems where prices are observed. In Townsend (1983) firms
face a signal extraction problem about shocks influencing prices. Prices are
endogenous to the model, but since these prices are taken as given by firms
there is separation in the firms’ minds. Relatedly, Guerrieri and Shimer
(2013) analyze a competitive market where traders have multidimensional
private information.
In a model with dispersed information, Angeletos and Pavan (2010) look
at a problem of optimal policy and show how its contingency on aggregate
variables can affect the way information is distributed in the economy and
agents respond to it. In our model, the only uninformed agent is the government, so that there are no information externalities. However, the government does not benefit from any type of separation as it fully understands
the endogeneity of the distribution of observable variables.
Mirman et al. (1993) considers the problem of a monopolist who sets
quantities under imperfect information about the demand schedule, which is
hit by two different unobserved shocks. They assume that the monopolist
observes the induced price only after an equilibrium has been realized and
hence they abstract from the issue of conditioning on endogenous variables.
However, they show how current decisions affect the distribution of future observables. Relatedly, Wieland (2000a, 2000b) considers optimal policy when
there is limited information, in a class of armed-bandit problems. In this case
there is separation conditional on past endogenous variables so that the government choice affects the information revealed by the equilibrium variables,
there is no influence between today’s choice variable and the distribution of
today’s observable.
In our setup the signal extraction problem is contemporaneously endogenous to the government’s decision. There is a scarce literature on this topic
and it is restricted to the linear model with additively separable Gaussian
shocks. Pearlman et al.(1986), Pearlman (1992), and Svensson and Woodford (2003) point out that with Partial Information on truly forward looking
variables the separation principle might fail because the forward looking variables depend on what policy is going to be like in the future. However, they
show that under linearity, normality and symmetry of information, a separation principle with a modified Kalman filter continues to hold. Baxter et al.
6
(2007, 2011) derive an ”endogenous Kalman filter” for all these cases which
is equivalent to the solution of a standard Kalman filter of a parallel problem
where all the states and the measurement are fully exogenous.
The only exception to the separation principle is Svensson and Woodford
(2004) where like in our paper the government’s information set is a subset
of the private sector’s information set. This papers shows that, in spite of
the failure of the separation principle, there is a suitable modification of
the standard Kalman filter that works, thanks to linearity and additively
separable shocks. Nimark (2008) applies this technique to a problem of
monetary policy where the central bank uses data from the yield curve while
at the same time understanding that it is affecting them.
Our contribution to this literature is to provide a general solution to the
endogenous signal extraction and optimization problem. In our model there
is no separation of any kind as the shocks are allowed to enter non-linearly in
the equilibrium conditions. In this setup, linear approximations can be quite
misleading: we show a model where the correct solution is highly non-linear
in nature.5
The literature of optimal contracts under private information is perhaps
less directly related to our work. This literature usually assumes revelation
of the private information conditional on the equilibrium actions (”invertibility”) and introduces the assumption that agents react strategically to the
optimal contingent policy set up by the principal (in our case the policy function R chosen by the government). Instead, we consider a setup where this
reaction (endogenously) does not take place because agents are atomistic.
However, our results should be useful to study models of optimal contracts
under private and limited information (without invertibility), as the endogeneity of the observed variable that we address would be present also in
that kind of models.
5
Optimal non-linear policies have been found in the literature but for totally different
reasons. Swanson (2006) obtains a non-linear policy when he relaxes the assumption of
normality in the linear model with separable shocks. He considers a model where the
separation principle applies. The non-linearity results entirely from Bayesian updating
on the a priori non-Gaussian shocks. Orphanides and Wieland (2000) obtain optimal
nonlinear policy response even in a world with perfect certainty by relaxing the standard
assumptions on optimal monetary policies that policy makers’ preferences are quadratic
and the economy is linear.
7
3
A simple model of optimal fiscal policy
Before coming to the general problem of optimal policy with endogenous
signal extraction, we present a simple model that we will use to illustrate
the problem and its solution. We consider a very simple two-period version
of the fiscal policy model in Lucas and Stokey (1983) with incomplete insurance markets. A Ramsey government needs to finance an exogenous and
deterministic stream of expenditure (g1 , g2 ), where subscripts indicate time
periods, using distortionary income taxes (τ1 , τ2 ) and bonds b issued in the
first period that promise a repayment in second period consumption units.
The economy is populated by a continuum of agents. Each agent i ∈ [0, 1]
has utility function
h
U (ci1 , l1i , ci2 , l2i ) = γu(ci1 ) − v(l1i ) + β u(ci2 ) − v(l2i )
i
(1)
where cit and lti for t = 1, 2 are consumption and hours worked respectively,
with u′ > 0, u′′ < 0, v ′ > 0, v ′′ > 0.
γ is a temporary preference shock, which we will refer to as a demand
shock, with probability density function fγ (γ). When the realization of the
demand shock is high, agents like first period consumption relatively more.
As will be clear in the following, this will make them willing to work more in
their intratemporal labour-consumption decision and also more impatient in
their intertemporal allocation of consumption. Viceversa, low γ will lead to
lower labour and more patient agents, ceteris paribus. Given that agents are
identical, in the following we can describe the problem of a representative
agent and drop the subscripts i for notational convenience.
The production function is linear in labour and output and is given by
yt = θt lt . We assume that θ1 is equal to the realization of a random variable θ with probability density function fθ (θ) and we will refer to it as the
productivity shock. As far as θ2 is concerned, we will distinguish two cases,
one where θ2 = θ1 , in which case the productivity shock is permanent, and
one where θ2 = Eθ, that is, second period productivity is a known constant,
equal to the mean of the first period shock, in which case the productivity
shock is temporary. Firms maximization implies that agents receive a wage
equal to θt , so that the period budget constraints of the representative agent
are
c1 + qb = θ1 l1 (1 − τ1 )
(2)
8
and
c2 = θ2 l2 (1 − τ2 ) + b
(3)
where q is the price of the government discount bond b.
Maximization of (1) subject to (2) and (3) implies a standard labourconsumption margin condition for each period and a consumption Euler
equation for bonds:
v ′ (l1 )
= θγ(1 − τ1 )
(4)
u′ (c1 )
v ′ (l2 )
= θ(1 − τ2 )
(5)
u′ (c2 )
u′ (c2 )
(6)
q=β ′
γu (c1 )
As anticipated, the demand shock enters the first period labour supply
decision described by (4) as well as the bond pricing equation (6).
A description of a competitive equilibrium is completed by imposing the
resource constraint at both periods:
c t + g t = θt l t .
(7)
A Ramsey government chooses a sequence of taxes and a bond issuance in
order to maximize (1) subject to the competitive equilibrium conditions (2),
(3), (4), (5) and (7). The chosen policy will be a function of the government’s
information set at t = 1 (note that all the uncertainty is resolved after the
first period and second period taxes will have to balance the budget). Before
specifying our assumptions on this information set, let us proceed to show
how in general this Ramsey problem can be reduced to the choice of a first
period tax rate, subject to a single constraint that summarizes the reaction
of the private sector to government policy.
First, by combining the budget constraints (2) and (3) with the first order
conditions (4), (5) and (6), we can derive the implementability constraint of
the Ramsey problem:
γu′ (c1 ) c1 − v ′ (l1 ) l1 + β [u′ (c2 ) c2 − v ′ (l2 ) l2 ] = 0.
(8)
Now, note that we can use the resource constraints (7) to substitute out the
consumption terms in (8) to obtain an equation of the form
G(l1 , l2 , γ, θ) = 0,
9
(9)
where we have left implicit that θ2 will either be equal to θ or to a known
constant depending on the assumption on the persistence of the productivity
shock. Equation (9) defines implicitly a function that maps a choice of first
period labour l1 into a second period labour l2 . Call this map Limp
2 . The
government and the private agents must understand that given a realization
of the shocks, for any choice of l1 there is only one value of l2 , that is Limp
2 (l1 ),
that is consistent with competitive equilibrium conditions. Under some specific assumptions on u and v, it will be possible to solve for l2 as a function of
l1 in closed form. More in general, it will always be possible to characterize
the marginal effect of l1 on l2 by applying the implicit function theorem to
(9).
With this in mind, we can define a new objective function of the Ramsey
problem as follows:
imp
W (l, θ, γ) ≡ U (θl − g1 , l, θ2 Limp
2 (l) − g2 , L2 (l)).
(10)
Note that by using all the equilibrium conditions, we have expressed utility
as a function of first period labour only, for which we have suppressed the
time subscript.
Finally, let us define the reaction function of the private sector to a first
period tax rate τ1 . Using (7) for t = 1 to substitute out consumption in (4),
we get
v ′ (l1 )
− θγ(1 − τ1 ) = 0,
(11)
u′ (θ1 l1 − g1 )
which implicitly defines first period labour as a function of the first period
tax rate and the realizations of the shocks:
l = h(τ, θ, γ)
(12)
where we have again suppressed the time subscript from first period labour
and tax rate. Hence the Ramsey optimal policy can be characterized as
the choice of τ that maximizes (10) subject to (12). This formulation takes
implicitly into account that the future allocation and tax rate will then be
determined by the equilibrium conditions described above.
The government will choose a tax policy contingent on its information
set I G in order to maximize (10) subject to (12). We will consider different
assumptions on the elements of I G . The standard assumption in the Ramsey taxation literature is that of Full Information (FI), which in our model
implies that both the government and the agent observe the realization of
10
(θ, γ) at t = 1 and are allowed to make their decisions based on this observation, so that I G includes (θ, γ) as well as all the parameters of the model
(preferences and government expenditure). However, the aim of this paper
is to characterize the solution to the Partial Information (PI) problem, when
only one aggregate endogenous variable, say labour or output, is observed,
instead of the two exogenous shocks. Hence the elements of I G will be only
this endogenous variable and all the parameters of the model.
The problem of finding the optimal policy contingent on Partial Information is complicated by a key issue: the distribution of the observable variable
is endogenous to a choice of policy, as is clear from (12). Hence, the problem
cannot be treated as a standard problem of optimization under uncertainty,
where the distribution of the shocks is exogenous and conditioning on these
shocks allows to ignore the uncertainty at the time of taking first order conditions. Here, the government must take into account that a certain policy will
imply a certain distribution for l1 which in turn is going to be the argument
of the policy function. This simultaneity issue cannot be dealt with standard
signal extraction methods that are well suited for exogenous or predetermined
observables, and calls for a new technique. In the next section, we will solve
a slightly more general model under both informational assumptions (FI and
PI) and develop the endogenous filtering technique.
4
General first order conditions
We first derive the general solution to our policy problem for any welfare
function W (τ, l, A) of the government and any reaction function l = h(τ, A)
of the atomistic consumers where A refers to the exogenous shocks. Note
that in the model presented in the previous section, A = (θ, γ) and τ does
not directly affect welfare, but only indirectly through an allocation. In
this more general formulation welfare can directly depend on taxes, e.g. the
government might dislike taxes. We will refer to all first period variable
without a subindex. Hence the shocks are θ and γ, the first period tax rate
is τ and the first period labor supply is l.
Under this formulation the government maximizes welfare by choosing a
function R of the observable variables to set the tax rate.
11
4.1
Full Information
Under FI, both the agent and the government observe the shocks in period
1. This implies that the policy function for τ will have the form τ = RF I (A),
i.e. the government chooses τ contingent on the exogenous shock A. In the
fiscal policy example, we have A = (θ, γ). Denote by Φ the space of possible
values of A. Formally, the government chooses R : Φ → ℜ+ to solve
max
{R:Φ→ℜ+ }
h
E W RF I (A), h(RF I (A), A), A
i
(13)
It is easy to find the FOC for this problem since the argument of the function
R (namely A) has a known exogenous distribution, independent of the choice
for R. This is the standard case in macro models, and it can be determined
by standard methods that the optimal policy function RF I∗ is formed by
choosing RF I,∗ (A) = τ where τ satisfies the FOC
Wτ τ, h(τ, A), A + Wl τ, h(τ, A), A hτ (τ, A) = 0
(14)
for all A ∈ Φ, where Wτ and Wl refer to the first derivative of W with respect
to τ and l respectively while hτ refers to the derivative of h with respect to
τ.
The optimal policy function RF I∗ (A) is found by solving for τ in equation
(14) for each A. Note that uncertainty does not really play a role in this setup,
it just indexes the solutions by the realized values of A.
In the model presented in the previous section, the FI policy solves
Wl (l, θ, γ) = 0
(15)
for all realizations of (θ, γ). The implied tax function τ F I (θ, γ) is then obtained by inverting the reaction function h. This is the so-called primal
approach to the Ramsey optimal taxation problem under FI.
The FI policy is one of tax smoothing over time as the government wants
to spread the distortions equally in the two periods. In the case of CRRA
preferences (both u and v are power functions), tax smoothing will be perfect and the government will choose the constant tax rate τ that solves the
intertemporal budget constraint
τ θ1 l 1 − g 1 + β
u′ (c2 )
(τ θ2 l2 − g2 ) = 0.
γu′ (c1 )
12
(16)
It is clear from (16) that the government needs to know the realization
of both productivity and demand shock in order to implement this policy.
In particular, the realization of θ is a crucial piece of information, as it
determines the revenue that a given tax rate is going to raise, while the
demand shock affects the interest rate the government will have to pay on its
debt (or receive on its assets). Furthermore, both shocks clearly contribute
to the determination of an allocation (c1 , c2 , l1 , l2 ).
4.2
Partial Information
We now consider the more interesting case that arises whenever the government cannot observe the exogenous shocks, but only some endogenous
variables, while the agents observe the shocks that hit their preferences and
wages. In particular, we will make the assumption that the only observable variable for the government is labour (and we will consider output as
observable in the robustness section).
The observability of labour makes the optimal fiscal policy particularly
interesting for a number of reasons. First of all, note that tax revenue in the
first period is τ1 θ1 l1 . This means that at the time of setting taxes, the government is uncertain about the revenue that a given policy will generate, as
θ1 is not learned at t = 1 with certainty. Arguably, this is a crucial feature of
actual fiscal policy decision, as very frequent revisions of the official forecasts
for fiscal deficits seem to confirm. Second, employment data have a higher
frequency than other macroeconomic data, like output and its components.
Hence it seems sensible to ask the question of how the government should
set its policy based on this type of information.
Optimal behavior with PI uncertainty implies that the government chooses
a different τ depending on the observed l. Therefore, the government now
chooses a function R : ℜ+ → ℜ+ and given an observed employment level l
will set the tax rate
τ = R(l).
(17)
Optimal behavior under uncertainty means that the government makes contingent plans, a plan for each realization, therefore it chooses the best from
all ”reaction” functions of the form (17).
Let us call L(R, A) the observable l (random variable) induced by the
shock A and a policy R. This will be the equilibrium value of l as defined
13
implicitly by the zero of a function H defined as follows:
H(l, A, R) ≡ l − h(R(l), A) = 0.
(18)
In words, the government knows that a given R implies that the employment level chosen by agents is the random variable L(R, ·) : Φ → ℜ+ given
by the private sector’s reaction function evaluated at the tax implied by R.
The tax rate is then given by
T (R, A) = R L(R, A)
(19)
Notice the distinction between L, T and R. The latter is a function of l
while L and T are functions of R and the realizations of the shocks.
Let
F(R) ≡E (W (T (R, A), L(R; A)) , A)
(20)
be the objective function for a given choice for R.
We can now re-define the PI problem as
max
{R:ℜ+ →ℜ+ }
6
F(R)
(21)
and denote its solution by R∗ .
4.2.1
Case 1: Invertibility
It turns out that there are cases where even under PI, the government can
still implement the FI policy. This is whenever the information set of the
government is invertible, allowing to learn the true state of the economy at
t = 1.
2
To formally define Invertibility, consider the manifold in R+
that describes
all possible (τ, l) at the solution of the FI. This is the set
n
o
2
M ∗ ≡ ((τ, l)) ∈ R+
: τ = R∗F I A and l = h τ, A f or some A ∈ Φ .
(22)
Definition 1 Invertibility holds if for any l such that (τ, l) ∈ M ∗ for some
τ, there exists a unique τ such that (τ, l) ∈ M ∗
6
Notice that F maps the space of functions into R. The expectation operator integrates
over realizations of A using the true exogenous distribution of A, so that the above objective function is mathematically well defined given the above definitions for T , L and
under standard boundedness conditions.
14
Remark 1 Invertibility is automatically satisfied when
• A is one-dimensional and h RF I∗ (A) , A is a monotonic function of
A
• when Φ is a finite set. Then, the dimensionality of Φ does not matter,
we can expect to be able to map an equilibrium into the shock since
there are finitely many realizations, only by coincidence would the same
equilibrium point (τ, l) occur for two different realizations of A if A can
only have finitely many values.7
We prove in Appendix A that
Proposition 1 Under invertibility R∗ = RF I∗
To illustrate a case of Invertibility in the model presented in the previous
section, assume that γ = 1 with certainty, while θ is random and observed
only by the agent at t = 1 (and by the government at t = 2). Assume also
that the government observes only l1 at t = 1 and hence can make its policy
contingent only on labour. This is only apparently a PI problem. As long as
h(τ F I (θ, 1), θ, 1) is invertible with respect to θ, observing the labour choice is
equivalent to observing the true exogenous state θ and hence the government
can implement the FI policy.
4.2.2
Case 2: The general case
The most interesting case of PI arises when the information set of the government is not invertible, meaning that the endogenous observables are not
sufficient to back out the actual realizations of the shocks. Observe that
Remark 2 Invertibility is generally violated
if A is multi-dimensional and
A has a continuous distribution or if h RF I∗ (·) , · is non-monotonic.
To solve (21) for the general case we need to go back to first principles
and derive a variational argument where we take a deviation from the optimal solution in any possible direction and then derive the optimal policy by
exploiting the fact that the optimal deviation is no deviation.
7
An exception is Wallace (1992) who defines the discrete supports of shocks in order to
get non-invertibility even with a finite set Φ. However, this is clearly a degenerate case.
15
Assume for the optimal choice R∗ there is a unique equilibrium (τ, l) for
each realization. Take any function δ : R+ → R and a constant α ∈ R+ . We
will now consider reaction functions of the form R∗ + αδ. We consider only
δ’s for which R∗ + αδ has a unique equilibrium (τ, l) for any realization A if
α is close enough to zero. Fix δ and consider solving the problem
max F(R∗ + αδ)
(23)
α∈ℜ
in other words, now we maximize over small deviations of the optimal reaction
function in the direction determined by δ.
It is clear that
0 ∈ arg max F(R∗ +αδ)
α∈ℜ
F(R∗ ) = max F(R∗ +αδ)
α∈ℜ
(24)
(25)
because, if 0 would not be an arg maxα∈ℜ , since R∗ + αδ is feasible in the
PI problem, this would contradict the fact that R∗ is optimal for the PI
problem.
Since 0 solves the one-dimensional maximization problem (23) the FOC
of that problem give
dF(R∗ + αδ)
|α=0 = 0
(26)
dα
We now compute this one-dimensional derivative and evaluate it at α = 0.
dE (W (T (R∗ +αδ, A), L(R∗ +αδ, A), A))
dF(R∗ + αδ)
=
(27)
dα
dα
R
d Φ W (T (R∗ +αδ, A), L(R∗ +αδ, A), A) dFA (A)
=
(28)
dα
In Appendix B we show that this derivative evaluated at α = 0 is
Z Φ
[Wτ∗ R∗′
+
Wl∗ ] L′
0,δ
+
Wτ∗ δ
dFA (A) = 0
(29)
where it is understood that Wτ∗ , Wl∗ , δ,R∗′ are evaluated at equilibrium op∗ +αδ,A) timal choices and L′0,δ = dL(R dα
which is the only part that needs
α=0
to be determined in (29). To compute L′0,δ we apply the implicit function
theorem to function H defined in (18) to obtain
δ(L(R, A))hτ
dL(R+αδ, A) =
dα
1 − hτ R′
α=0
16
Plugging this into (29) and rearranging, we can conclude that for any variation δ
Z
δ
dFA (A) = 0
(30)
(Wτ∗ + Wl∗ h∗τ )
1 − h∗τ R∗′
Φ
Now we derive implications of these FOC only in terms of primitives of
the problem. Consider any l which has positive density. Formally, given l
and some ε > 0, define the set of realizations for which equilibrium labour is
within ε of l :
n
B ≡ A ∈ Φ : L(R∗ , A) ∈ l − ε, l + ε
o
Note that B is indexed by R∗ , l and ε.
Now, assume h∗ , R∗′ are differentiable at l for all A. Assume also that
these derivatives are bounded for all realizations in B. Assume that l has
positive density in equilibrium, that is ProbB >0 for any ε > 0. Take δ to be
the indicator function8 of the set (l − ε, l + ε) then (30) becomes
Z
B
1
dFA = 0
1 − h∗τ R∗′
(Wτ∗ + Wl∗ h∗τ )
(31)
and letting ε → 0 we have that
E
!
Wτ∗ + Wl∗ h∗τ L(R∗ , A) = l = 0
∗
∗′
1 − hτ R (32)
This will hold for any l that can be an equilibrium and where the differentiability assumption holds.
This says the following: ideally the government would like to reach the
FI optimum and set Wτ∗ + Wl∗ h∗τ = 0 for all realizations. But this is an
impossible task: due to PI a given choice R∗ is compatible with various A
and therefore with various values of Wτ∗ + Wl∗ h∗τ at l. All these derivatives
cannot be made equal to zero by one single choice of the number R∗ (l).
In the first order condition (32) 1−h1∗ R∗′ acts as a kernel in expectations.
τ
Without this kernel we would get the familiar condition of optimization under
uncertainty where the government would simply set E Wτ∗ + Wl∗ h∗τ | L(R∗ , A) = l =
0. Under Partial Information however, the density fl is endogenous to the
8
Strictly speaking we need a smoothed version of the indicator function since as assumed
differentiability in step (27). This is a standard problem with a standard solution in
variational arguments.
17
function R. Therefore when choosing R the government has to take into
account how marginal changes in R affect fl . This is captured by 1−h1∗ R∗′ .
τ
The FOC given by (32) looks difficult to handle since it involves R∗′ , but
with a final step we can obtain a simpler expression without R∗′ . All we
need is to compute the distribution of the shocks conditional on ¯l, that is,
we need to determine the endogenous filter.
To do this, note that we started by defining our objective function as an
integral over realizations of A, that is, a double integral over the support of
θ’s and γ’s.
However, once we condition on ¯l, there is really only one source of uncertainty left, say θ and the other shock γ is uniquely determined by equilibrium.
To see this let us spell out θ and γ in the definition of function H:
H(l, θ, γ, R) ≡ l − h(R(l), θ, γ).
(33)
For each ¯l, (33) defines an implicit function γ = γe (¯l, θ, R) that maps a
realization of θ into the corresponding γ, for a given policy.
Hence, conditional on ¯l, we are now integrating over a line, the locus
of (θ, γ) consistent with ¯l and the chosen policy. Therefore, in (32) we can
integrate over say θ’s only and we get9
Z
Θ(l̄,R)
Wτ∗ + Wl∗ h∗τ
f dθ = 0
1 − h∗τ R∗′ θ|l̄
(34)
where Θ(¯l, R) is the set of θ’s with positive density conditional on observing
¯l, γ is γe (¯l, θ, R∗ ) for all θ̄ in the integral and W ∗ , W ∗ and h∗ are evaluated
τ
l
τ
at ¯l, θ, γe (¯l, θ, R∗ ) .
To find f θ|l̄ we apply Bayes’ rule
f θ|l =
f l|θ fθ ∀ l, θ
fl
(35)
Now, to derive the density of l conditional on θ̄, consider that in equilibrium
l is a function of γ, namely L(R∗ , θ, γ) and observe that by definition γe is
the inverse of L with respect to γ. Hence we just need to apply the change
of variable rule to get the density of the endogenous random variable l as a
function of the density of an exogenous random variable γ:
9
fl|θ ¯l, θ = fγ γe ¯l, θ
el ¯
l, θ γ
(36)
This is without loss of generality. Clearly, we could equivalently integrate over γ’s and
e ¯l, γ, R)
use H to define an implicit function θ(
18
where γel is the partial derivative of γe with respect to ¯l. Also, we have that
f l|θ ¯l, θ = 0 if θ ∈
/ Θ(¯l, R∗ )
Finally, in order to compute the partial derivative γel , we apply once again
the implicit function theorem to H and get
1 − h∗τ R∗′
γel ¯l, θ, R∗ =
h∗γ
(37)
Plugging (37) into (36), using Bayes’ rule and the fact that the denominator fl (¯l) drops out in the first order condition, we obtain the main result,
contained in the following Proposition
Proposition 2 The first order condition of the PI problem is given by
Z
Θ(l̄,R)
Wτ∗ + Wl∗ h∗τ
fγ (γe ∗ )fθ (θ)dθ = 0 f or all ¯l
h∗γ
(38)
where stars denote that the partial derivatives and γ are evaluated at the
optimal policy for a given ¯l.
Note that in the special cases where shocks enter the reaction function
in an additively separable fashion, this expression simplifies significantly and
we have the following corollary.
Corollary 1 If the second cross derivative of the reaction function with respect to the two shocks hγθ |¯l = 0, then it is optimal to just average the Full
Information FOCs using the prior distribution of the shocks.
This is the case in linear models with additively separable shocks (e.g.
New Keynesian optimal monetary policy model). It is also the case in models
where only one of the shock enters the reaction function linearly, which is
sufficient to have hγ independent of θ. Otherwise, the non-linearities imply
that the prior is reshaped using a kernel which now has the simple form of
1
.
h∗γ
It may be worth emphasizing the failure of the separation principle in this
derivation: As can be seen in equation (37), the derivative of the optimal
policy function enters the expression for the derivative of γe with respect to
the observable, which in turn affects the kernel used to weight contingencies
19
in the determination of the policy function. Also, the set of shock realizations
that are compatible with a ¯l is itself a function of R, so the government affects
the possible contingencies that will give rise to a certain observation and also
the density of that observation. Hence it is clear that there cannot be any
separation between the stage of the estimation of the state conditional on
observables and the stage of the solution to the optimal policy problem.
4.2.3
Algorithm for Partial Information
Given (38) it is easy to calculate the PI solution using the following numerical
algorithm. We first need to discretize the support of shocks (θ, γ) and support
of l. Then at each level of l we solve the first order condition (38). We then
take into account that the support of (θ, γ) is endogenous with respect to the
choice of τ . In other words, we iterate to find a fixed point of the mapping
between (i) a policy R that solves (38) at each l for a given conditional
distribution and (ii) the conditional distribution of the shocks consistent
with R at each l.
5
Computations
Let us now illustrate the solution of the optimal fiscal policy model introduced
above. We will proceed by introducing different assumptions on preferences
and persistence of shocks and show how they affect the optimal policy.
5.1
Linear-quadratic utility
First of all, consider the case u(c) = c and v(l) = B2 l2 , that is linear utility
from consumption and quadratic disutility from labour effort. This allows to
derive simple analytical expressions for the reaction function and its derivative. In particular, it is easy to see from the first order condition (4) that
the reaction function (12) specializes to
l = h(τ, θ, γ) =
γθ
(1 − τ ),
B
(39)
implying that both the productivity shock and the demand shock affect the
slope of labour supply with respect to the tax rate hτ , hence making this
model non-linear (in the sense of not having linearly additive shocks), despite
the reaction function being linear in taxes.
20
The two partial derivative hτ and hγ are also easily obtained:
hτ (τ, θ, γ) = −
γθ
B
(40)
θ
(1 − τ )
(41)
B
Let θ be uniformly distributed on a support [θmin , θmax ], θ2 = θ1 (permanent shock), γ uniformly distributed on [γmin , γmax ] and assume β = .96, B
calibrated to get average hours equal to a third and government expenditure
constant and equal to 25% of average output.
By observing a certain ¯l and imposing a tax rate τ̄ , all the government
can infer is a certain realization of the product of the two shocks, but not
the individual realizations of the shocks. Hence the government is uncertain
whether say productivity is high and demand
low, or viceversa,
and in gen
¯
e
eral there is a continuum of realizations θ̄, γ (θ̄, l; R) consistent with the
observation of ¯l and a policy R.
Figure 1 illustrates the optimal policy for this case, plotting the tax rate
against observed labour. The red line is R∗ , while the yellow region is the
set of all equilibrium pairs (lF I , τ F I ) that could have been realized under
Full Information. For the lowest labour that is realized under FI, the government knows that choosing τ F I (θmin , γmin ) is optimal, as that observation,
combined with this policy, allows full revelation of the state. Hence, PI and
FI coincide. The same is true for the highest admissible labour, which implies the FI equilibrium for (θmax , γmax ). In between these two extremes,
there is no full revelation, and it can be seen that the optimal policy calls
for a tax rate in between the minimum and the maximum FI policies for
each observation (but it is sometimes far from being the average of those
tax rates). In general, R∗ is decreasing as higher observed labour suggests
higher conditional expectation for productivity, hence allowing to balance the
intertemporal budget constraint with a lower distortionary tax. Finally, the
figure compares the optimal policy with a linear policy obtained connecting
the two full revelation points with a straight line. While the optimal policy
is not quite linear, in this example a linear approximation would not be too
wrong. We will see below that this property is not robust, in particular it
will not resist changes in preferences.
Figure 2 illustrates the combinations of shocks consistent with observing
an average realization of labour (l = .33), that is, we plot the function
hγ (τ, θ, γ) =
21
Figure 1: Optimal policy with linear-quadratic utility
0.3
τ
0.28
0.26
0.24
0.22
0.2
0.26
0.28
0.3
0.32
0.34
0.36
0.38
0.4
l
γe (θ, .33; R∗ )). As anticipated, this function is decreasing, as the product of
productivity and demand shock consistent with an observation of l and a tax
rate τ must be constant. This line is contrasted with the same object under
FI for the same level of labour.
5.2
Log-quadratic utility
We now introduce curvature in the utility from consumption. Clearly, this induces both risk aversion and a wealth effect on labour supply as the marginal
utility of consumption now enters the reaction function h. For simplicity we
study another special case that allows for an analytical reaction function h.
Assume u(c) = log(c) and again v(l) = B2 l2 . In order to make the PI problem
more interesting, let us also assume that the productivity shock is temporary
and θ2 is known to be equal to the mean of θ.10
Now the first order condition (4) becomes
Bl1 c1 = γθ1 (1 − τ1 )
10
(42)
A permanent θ combined with these preferences leads to equilibrium labour being
independent of θ under the FI tax policy.
22
Figure 2: Set of admissible shocks
1.06
1.04
PI
FI
γ
1.02
1
0.98
0.96
0.94
2.85
2.9
2.95
3
3.05
θ
3.1
3.15
3.2
and after substituting out consumption using the resource constraint, we
obtain that labour supply is the positive root of a quadratic equation, so
that (12) specializes to
l = h(τ, θ, γ) =
Bg1 +
q
(Bg1 )2 + 4Bθ2 γ(1 − τ )
2Bθ
.
(43)
It is important to note that now, differently from the linear-quadratic
case, θ has two opposing effects: the substitution effect between leisure as
consumption (as before) and the wealth effect, that acts in the opposite
direction. With log-quadratic preferences, the second effect dominates and
hence high realizations of θ will lead to low labour, ceteris paribus.
With this in mind, let us illustrate the optimal policy in figure 3 (red
line), once again contrasted to the set of FI equilibria. For low labour, now
the government learns that productivity must be high, so the tax rate can
be rather low. The lowest labour realization leads to the FI equilibrium for
(θmax , γmin ). Then taxes start to increase: higher l’s signal lower expected
productivity and hence revenue, as the set of admissible θ’s is gradually
including lower and lower realizations. This goes on up to a point where
the set of admissible θ’s conditional on l is the whole set [θmin , θmax ]. From
that point on, going to the right, the tax rate changes slope and becomes
23
decreasing and this is because now, with any θ being possible, increasing l
signals an increasing expected revenue, hence allowing lower tax rates, up
to the point where the highest θ’s start being ruled out, at which point the
policy becomes increasing again, up the full revelation point (θmin , γmax ).
Figure 3: Optimal policy with log-quadratic utility
0.275
0.27
0.265
0.26
τ
0.255
0.25
0.245
0.24
0.235
0.23
0.31
0.315
0.32
0.325
0.33
l
0.335
0.34
0.345
Consistently with the description above, the wealth effect of productivity
makes the locus of admissible realization of shocks for ¯l = .33 an increasing
function in the (θ, γ) space, as shown in figure 4. Now conditional on l,
we can have combinations of high productivity (low wealth effect on labour
supply) and high demand or low productivity and low demand.
Optimal policy with PI calls for a substantial smoothing of taxes across
states. This can be seen in figure 5, where the equilibrium cumulative distribution function of tax rates under PI (red line) is contrasted with the one
obtained under FI (blue dotted line). This result is rather intuitive and it
carries a general lesson for optimal fiscal policy decisions under uncertainty:
When the government is not sure about what type of disturbance is hitting
the economy, it seems sensible to choose a policy that is not too aggressive in
any direction and just aims at keeping the budget under control on average.
In our model, this smoothing of taxes across states will imply a larger
24
Figure 4: Set of admissible shocks
1.03
1.02
1.01
γ
1
0.99
0.98
PI
FI
0.97
0.96
2.7
2.8
2.9
3
θ
3.1
3.2
3.3
3.4
variance of tax rates in the second period with respect to the FI policy. In
the second period, all the uncertainty is resolved and the tax rate will be
whatever is needed to balance the budget constraint. This is of course taken
into account at the time of choosing a policy under uncertainty, so that we
could say that optimal policy is very prudent while the source of the observed
aggregate variables is not known and then responsive after uncertainty has
been resolved. In this sense, this model can rationalize the slow reaction
of some governments to big shocks like the current recession. The Spanish
example in the latest recession is a case in point. In 2008, it was far from
clear how persistent the downturn would be and also whether is was demanddriven or productivity-driven and the government did not adjust its fiscal
stance quickly, only to make large adjustments in the subsequent years.
25
Figure 5: Equilibrium CDF of tax rates
1
0.9
PI
FI
0.8
0.7
Fτ
0.6
0.5
0.4
0.3
0.2
0.1
0.23
5.3
0.235
0.24
0.245
0.25
τ
0.255
0.26
0.265
0.27
0.275
Close to the top of the Laffer curve
Let us now look at the case where government expenditure is very high, equal
to 60% of average output in both periods.11 We will see that this leads to a
very non-linear optimal policy and to an exception to tax-smoothing across
states. The government needs to be able to balance the budget in the second
period and is thus now very concerned about the amount of debt that will
need to be issued for a given tax as high debt, combined with high future
expenditure, may call for very high taxes in the future, getting the economy
closer to the top of the Laffer curve, where taxation is most distortionary
and hence consumption is very low.
Figure 6 shows optimal policy for this case (red line), again contrasted
with the set of FI outcomes (yellow region).
For sufficiently low observed labour, the government learns that only
sufficiently high realizations of θ are admissible. Consistently, the tax rate
can be rather low. However, there is a threshold ¯l at which suddenly the
11
All other assumptions on preferences and shocks are the same as in the previous
section.
26
Figure 6: Optimal policy with high government expenditure
0.66
0.64
τ
0.62
0.6
0.58
0.56
0.31
0.315
0.32
0.325
0.33
l
0.335
0.34
0.345
0.35
worst possible outcome in terms of productivity becomes consistent with the
observation. At that point, the government fears that if tax revenue is not
sufficiently high in the first period, then a high debt will need to be issued
and the economy will reach the top of the Laffer curve in the second period.
This calls for raising high taxes in the first period, and it can be seen that the
optimal policy is very steeply increasing. Then, for high l’s optimal policy
becomes smooth again at a higher level of taxation.
In order to gain further intuition on the reason of this strong non-linearity
in the optimal policy, imagine a case with only two possible values of productivity, high and low. The FI Ramsey policy would call for high taxes if
productivity is low and low taxes if it is high. One can think that for most
realisations of the demand shock, the government would be able to infer the
realisation of the productivity shock, but for a very small intermediate range
of demand shocks both realisations of the productivity shock are possible.
Clearly, the PI governemnt would like to be able to replicate the FI policy,
and hence it will do so when this is possible. Now, take the limit of the intermediate range of γ’s that generate confusion going to just a single point.
At that point, the PI government needs to jump from the FI policy for high
27
θ to the FI policy for low θ, and this generates a sharp non-linearity.
However, this is not the end of the story: now that this PI policy is chosen,
the set of admissible shocks conditional on l becomes itself different from the
one obtained under FI. In fact, this example allows us to see the fixed point
between optimal taxes and conditional distributions in action. When the
observed l is high, the government fears low θ and raises taxes accordingly.
However, at high tax rates the agent would like to work less, and if high l
is actually observed, then this must mean that the wealth effect has been
very strong, that is, productivity is very low, confirming the government’s
belief. In this way an optimal policy and a conditional distribution of shocks
consistent with it confirm each other in equilibrium.
Figure 7 shows the conditional loci of the shocks for a low and a high
level of labour. It can be seen that for high labour there is a wider range of
productivities, including low levels that would bring the economy closer to
the top of the Laffer curve. Figure 8 illustrates two possible Laffer curves,
for high and low productivity scenarios conditional on an average level of
observed labour, showing the potential fall in revenue that the government
would face without an adequate fiscal adjustment in the first period.
Figure 7: Set of admissible shocks with high government expenditure
1.2
low l
high l
1.15
1.1
γ
1.05
1
0.95
0.9
2.7
2.8
2.9
3
θ
28
3.1
3.2
3.3
3.4
Figure 8: Laffer curves
0.8
θ
low
0.75
θ
high
revenue
0.7
0.65
0.6
0.55
0.5
0.45
0.5
0.55
0.6
0.65
0.7
τ
0.75
0.8
0.85
0.9
0.95
Increasing the level of government expenditure shows that optimal policy
with PI can be very non-linear in order to avoid the worst outcomes. In
normal times, policy has to be smooth, but when there are contingencies that
are particularly dangerous for agents, then optimal policy calls for being very
reactive to observables in order to prevent those cases to materialize. This is
exemplified by the optimality of increasing taxes steeply in the first period
to avoid having to distort the economy too heavily in the second period if
realized productivity turn out to be low (and hence the fiscal deficit turns
out to be high). This lesson seems relevant for the understanding of the
fiscal policy reaction to the financial crisis in 2008 and afterwards, especially
in countries like Spain and Italy, that arguably where in danger of getting
close to the top of the Laffer curve, as testified by the fact that significant
increases in taxes after 2009 did not raise the amount of revenue as much as
it was desired by these governments.
29
5.4
Approximations and the endogenous filter
As can be seen from figures 3 and 6, the optimal policy can be very nonlinear and hence very different from a policy that simply connects the two
full revelation points (at lowest and highest l) with a straight line. This
suggests that linearization may be misleading in problems of optimal policy
with endogenous Partial Information.
In this subsection, we explore alternative suboptimal policies and we show
how well they approximate the optimal policy derived above. It turns out
that policies that disregard the endogeneity of the filtering problem can look
very different from the optimal one. However, an approximation of the optimal policy that takes into account the endogeneity of the supports of the
shock but does not weight them properly turns out to be a sufficiently good
approximation in many examples.
First let us consider the following two policies that disregard the endogeneity of the filtering problem:
1. RF I,av (l) = E(RF I (θ, γ)|lF I (θ, γ) = l)
2. RF I,ce (l) = RF I (E(θ, γ)|lF I (θ, γ) = l)
We refer to the first as the average of Full Information taxes for each
realized l and to the second as the certainty equivalence policy that applies
that Full Information policy to the conditional expectation of the state, conditional on each l. Note that while the first directly averages tax rates, the
second applies that tax policy to the average state. Both seem intuitive
policies to implement under Partial Information. In an exogenous Kalman
filtering model, policy 2 would correspond to the optimal policy, where one
can separately estimate the state and then solve a Full Information optimization problem conditional on that estimate.
In fact, as the Full Information policy is roughly linear, policy 1 and 2
turn out to be quite similar to each other. However, both of them disregard
the endogeneity of the distribution of observables under endogenous Partial
Information and hence the fixed point argument between distributions and
policy. Hence, they turn out to look very different from optimal policy, as
illustrated by figure 9 for the log-quadratic utility case with high government
expenditure. While the welfare cost of using these policies rather than the
optimal one is not large in absolute value, it is of the same order of magnitude
of the loss in welfare of going from Full Information to Partial Information
30
under the optimal policy, that is, they would double the utility loss for the
agent if implemented. This is because they are averaging the wrong set of
contingencies, that is, the contingencies that are consistent with each l under
Full Information, disregarding the fact that the loci of admissible shocks are
equilibrium outcomes that depend on policy. For example, these policies do
not raise taxes sufficiently in the first period when very low θ’s are possible,
potentially leading to larger required distortions in the second period. Therefore, these approximations based on naive averaging or methods applicable
to exogenous filtering problems are quite wrong.
Figure 9: Optimal policy and suboptimal alternatives
0.62
optimal
FI, av
R
0.615
FI, ce
R
0.61
0.605
0.6
0.595
0.59
0.585
0.31
0.315
0.32
0.325
0.33
0.335
0.34
0.345
0.35
We will now show that in many cases there exists a ”naive” policy that
takes into account the endogeneity of the admissible support of the shocks
and can well approximate the optimal policy. This policy is therefore conceptually very different from policies 1 and 2, as it fully takes into account
that the set of shocks that are consistent with a given observed labour and
a given tax rate is endogenous to this tax rate. This ”naive” policy solves
Z
θ∈Θ(l1 ,Rn )
Wl (l, θ, γe n )fγ (γe n )fθ (θ)dθ = 0
31
(44)
where γe n ≡ γe (l, θ, Rn ). Hence the ”naive” policy averages the FI first order
conditions (15) using the prior distribution of these shocks , that is, it disregards the fraction hhγτ of the optimal first order condition given by equation
(38).12 In other words, the weights given to the admissible realizations are
not correctly updated.
In particular when the variance of hhγτ is low conditional on l, this policy
is a very good approximation for the optimal one, as illustrated in figure 10
for the log-quadratic utility case with low g. Figure 10 shows that the two
tax rates are very close to each other and that the naive policy always results
in a higher tax rate. The reason why the optimal policy implies a lower tax
Figure 10: Optimal and ’naive’ policy with low g
0.2545
optimal
no weighting
0.254
0.2535
τ
0.253
0.2525
0.252
0.2515
0.251
0.322 0.323 0.324 0.325 0.326 0.327 0.328 0.329
l
0.33
0.331
rate than the suboptimal one is instructive about the role played by the endogenous filter. The numerator of the fraction hhγτ is an increasing function of
productivity θ. Intuitively, this means that taxes are more distortionary, the
higher productivity (note that this an intrinsic non-linearity of the model).
Furthermore, the denominator hγ is decreasing in productivity θ. To see this,
note that the demand shock multiplies the utility from consumption. When
12
Recall that in our two period Lukas and Stokey model Wτ = 0. I
32
productivity is high, consumption is high, marginal utility is low and hence
the demand shock is less effective on labour supply. The effect of both the
numerator and denominator of this fraction hhγτ is hence to put more weight
to points with high productivity, where the tax rate would be lower with FI,
leading to a lower tax than the ”naive” one. Summarizing, the fraction hhγτ instructs the Ramsey government to put more weight on contingencies where
(i) observables are more responsive to policy (high slope of labour supply
with respect to taxes) and (ii) the demand shock is less effective.
However, there are also cases where it is important not to disregard the
optimal weighting. High government expenditure is one of these, as exemplified by figure 11, where the distance between the two taxes rates is of the
order of one percentage point.
Figure 11: Optimal and ’naive’ policy with high g
0.612
optimal
no weighting
0.6115
0.611
0.6105
τ
0.61
0.6095
0.609
0.6085
0.608
0.6075
0.33
0.331
0.332
0.333
0.334
0.335
l
We now show the endogenous filter in action in our model. As we have
seen in section 3, computing the optimal policy with PI implies solving the
joint problem of maximizing the expectation of the objective function and
computing the conditional distribution of the shocks conditional on observables. This distribution is given by (35).
Figure 12 plots γel (¯l, θ, R∗ ) against the support of θ’s consistent with ¯l =
33
.33. While the prior was assumed to be uniform, the posterior obtained by
applying the endogenous filter is not uniform and specifically is increasing
in θ, implying that high productivity contingencies have higher conditional
density. After observing a given l, the government updates its prior by
exploiting the restriction on the realization of the two shocks implied by
(37).
Figure 12: Endogenous filtering
6.94
6.92
6.9
6.88
6.86
6.84
6.82
6.8
6.78
2.7
2.8
2.9
3
θ
3.1
3.2
3.3
3.4
It is worth emphasizing once more that this distribution is affected by the
policy decision. In particular, it depends on the slope of the policy function,
R∗′ , because this slope determines the effect of the shock γ on the equilibrium
distribution of the observable variable.
6
6.1
Robustness
Risk aversion and precautionary fiscal adjustments
As we have seen in the previous section, when the economy is close to the
top of the Laffer curve, optimal policy is very non-linear in the observable
34
variable, creating a region of sharp fiscal adjustments for intermediate realized values of labour. The government raises taxes dramatically in the
first period, in order to prevent the worst scenarios with low productivity,
high taxes and low consumption in the second period. In this section, we
investigate how this policy implication changes with different degrees of the
risk-aversion parameter σ.
Figure 13 illustrates the optimal policy for σ = 1 (baseline case), σ = 1.5
and σ = 2. It can be seen that as risk aversion increases, the area where
policy is more reactive of observables becomes wider, and the government
reacts strongly even for weaker signals of a recession. Intuitively, this is
because the government wants to avoid contingencies with high debt that
would lead to high taxes in the second period. The more risk averse the
agent is, the more painful it is to be in those states, where consumption has
to be cut substantially. However, this larger region of reaction also makes
the policy function less steep, as can be seen from the picture.
Figure 13: Changing risk aversion
0.625
σ=1
σ = 1.5
σ=2
0.62
0.615
0.61
0.605
0.6
0.595
0.59
0.31
0.315
0.32
0.325
0.33
35
0.335
0.34
0.345
0.35
6.2
Observable output
We now consider the case where the observable variable is output, instead of
labour. Now the government knows the value of the product θl, but not the
values of the factors independently. Figure 14 illustrates the optimal policy
for this case, with linear-quadratic preferences and permanent productivity
shock. It can be seen that the result is remarkably similar to that obtained
in section 4.1. However, when output is observed, the government has a
lot more information than when labour is observed. This is because current
revenue τ θl is known, and hence there is no uncertainty about the amount
of debt that needs to be issued. The only uncertainty is about the amount
of revenue that will be collected in the future, as the value of (permanent)
productivity is unknown. As we have seen, uncertainty about the debt is key
to get large fiscal adjustments as in section 4.3.
Figure 14: Optimal Policy with output observed
0.32
0.3
τ
0.28
0.26
0.24
0.22
0.2
0.8
0.9
1
y
1.1
1.2
We also performed a further robustness check assuming a truncated normal distrution for our shocks, rather than a uniform distribution as in the
benchmark examples presented. All qualitative results are robust to this
change in the assumption, highlighting the fact that the non-linearities are
36
induced by endogenous PI rather than by an ad hoc distributional assumption.
7
An infinite horizon model with debt
In this section, we present an infinite horizon model of optimal fiscal policy
with endogenous signal extraction and debt and we discuss its numerical
solution. We will see that some key intuitions developed in the two-period
model are still present. In particular, under PI the government sometimes
reacts slowly to recessions and as a consequence needs to raise taxes for a
longer time endogenously prolonging slumps. We assume linear utility from
consumption in order to abstract from time-consistency issues: it is well
known that if we introduced curvature in utility, the government would have
an incentive to twist the interest rate ex-post. Also, in order to simplify the
solution we assume that the shocks are i.i.d. over time. This allows us to
abstract from incentives for the government to use its policy to experiment
and try to learn about the state of the economy. Hence, we are left with
the simplest infinite horizon model of fiscal policy with endogenous signal
extraction. Future work will address the solution to more general infinitehorizon setups.
7.1
Full Information
We consider an incomplete-markets model inspired by Example 2 of Aiyagari
et al. (2002), with linear utility from consumption and standard convex
disutility from labour effort. Preferences of the representative agent are given
by:
E0
∞
X
β t [γt ct − v(lt )]
(45)
t=0
where γt is a demand shock, i.i.d. over time.
The period budget constraint of the representative agent is
ct + qt bt = θt lt (1 − τt ) + bt−1
(46)
where θt is an i.i.d. productivity shock.
The standard first order conditions for utility maximisation are
v ′ (lt )
= θt (1 − τt )
γt
37
(47)
and
γ̄
.
(48)
γt
where γ̄ is the unconditional expectation of the demand shock γ.
The Ramsey government finances a constant stream of expenditure gt =
g ∀t and chooses taxes and non-contingent one-period debt in order to maximise utility of the agent subject to the above competitive equilibrium conditions as well as the resource constraint ct +g = θt lt . Under FI, the government
can choose a sequence of taxes conditional on a sequence of shocks At , where
At = (θt , γt ).
The period implementability constraint with associate Lagrange multiplier λt is
v ′ (lt )
γ̄
bt−1 = ct −
lt + β bt .
(49)
γt
γt
We now introduce an upper bound on debt, bmax . We will assume that
whenever debt goes above this threshold, the government pays a quadratic
utility cost β χ2 (bt − bmax )2 and we will set the parameter χ to be an arbitrarily high number in order to mimic a model with an occasionally binding
borrowing constraint while still retaining differentiability of the problem.
The first order conditions for Ramsey allocations with respect to hours
and debt are:
qt = β
γt θt − v ′ (lt ) + +λt θt −
and
λt
λt ′
[v (lt ) + v ′′ (lt )lt ] = 0
γt
γ̄
= Et λt+1 + χ(bt − bmax )I[bmax ,∞) (bt ).
γt
(50)
(51)
where we denote by I[bmax ,∞) (b) the indicator function for the event b > bmax .
Thanks to the assumption of linear utility from consumption, the Ramsey
policy is time-consistent and allocations satisfy a Bellman equation that defines a value function W F I (bt−1 , At ). Thus taxes are given by a time-invariant
policy function τt = RF I (bt−1 , At )
7.2
Partial Information
We start the description of the PI problem by specifying its timing. At the
beginning of each period t, the Ramsey government observes the realisation
of the exogenous shocks of last period At−1 , the value of its outstanding debt
38
bt−1 and the realisation of current labour lt . Based on this information it sets
the tax rate τt .
Note that because of the i.i.d. assumption on the shocks, information
about outstanding debt summarises all the information about past realisations that is relevant in terms of the objective function and the constraints
of the Ramsey problem. Hence the optimal policy has a recursive structure
and taxes are given by a policy function τt = R(bt−1 , lt ). In other words,
the government cares about past realisations of the exogenous shocks only
to the extent that they affect the level of current outstanding debt. As a
consequence, debt is a sufficient state variable.
The value function of the problem is defined by the following Bellman
equation
W (b) =
max
R: ℜ2 →ℜ+
Eγ(θl − g) − v(l) + βW (
χ (b + g − θl +
−β (
2
βγ̄
v ′ (l)l
)γ
γ
(b + g − θl +
− bmax )2
βγ̄
v ′ (l)l
)γ
γ
)+
(52)
where l satisfies l − h(R(b, l), θ, γ) = 0. The only difference with respect to
the reaction function in the two-period model is that now debt affects labour
indirectly through the tax rate.
Note that in (52) we have substituted future debt from the budget constraint (49). It is important to highlight a key difference with respect to the
FI problem: while in that case a choice of τt implied a choice of bt , now, a
choice of τt implies a function that maps the realisations of At conditional on
lt into a debt level bt . In other words, just like in the two-period model, the
government is uncertain about how much debt will need to be issued and in
particular must take into account that bad realisations of productivity may
lead to a debt level above bmax , if taxes are not sufficiently high.
In order to solve the model, we exploit its recursive structure, by solving
for the PI first order condition at each point on a grid for debt and iterating
on the value function of the problem. To see how this works, consider the
objective function defined by the right-hand side of (52).
For a given guess for the value function, this is just a function of observed
labour to which we can apply the main theorem of the paper (Proposition
2) and obtain the general first order condition with PI.13
13
The FOC is explicitely shown in Appendix C
39
This first order condition will involve the derivative W ′ (b), which satisfies,
by an envelope condition (derived in Appendix C):
γ
W ′ (b) = E W ′ (b′ ) − χ(b′max )I[bmax ,∞) (b′ ).
γ̄
(53)
Hence by solving the first order condition using (53) and iterating on the
Bellman equation (52), we can approximate the optimal policy. In the next
subsection, we show some numerical results obtained after parametrising the
economy. While the model is not meant to be a quantitative model of fiscal
policy, it can nonetheless rationalise important features of the fiscal response
to the Great Recession, with slow and large fiscal adjustments inducing protracted slumps.
7.3
Simulation results
In order to parametrise the economy, we assume quadratic disutility from
labour, and the discount factor as well as shock distributions as in the benchmark two-period model above. We will now evaluate the fiscal response to a
sudden recession that induces uncertainty in the government’s problem under
PI and we will compare its policy response to the FI policy. To do this we first
hit the economy with the mean value of the shocks for a long sequence and
then (in period 400) we hit it with a one-off negative θ shock combined with a
one-off (smaller) positive γ shock, so that the economy enters a recession, but
the PI government is confused about its source. In figure 15 we can see that
the FI government raises taxes in period 400 and after that taxes very gradually go back towards steady-state. However, the PI government reacts slowly
and needs to raise taxes with a delay (first panel). Interestingly, this delayed
fiscal adjustment induces a longer recession as can be seen by the third panel
(hours): higher taxes discourage work for a longer period. This behaviour
of the economy is qualitatively similar to what happened in some European
countries after the financial crisis (e.g. Spain), where an initial slow reaction,
or even an expansionary policy, has been followed by a necessary large fiscal
adjustment and the recovery has so far been very slow and weak. While this
policy is optimal in our setup where only current income can be taxed, the
above findings suggest that allowing for retrospective taxation could improve
welfare. Society would be better off if the government could adjust taxes on
past income after observing the realisation of past shocks and consumers
knew of this possibility. However, retrospective taxation might not be easily
40
implementable in the real world due to time-consistency issues, as ex-post
surprising taxes on past income are non-distortionary.
Figure 15: Impulse responses
taxes
0.265
FI
PI
0.26
0.255
398
399
400
401
402
403
404
405
406
407
408
404
405
406
407
408
404
405
406
407
408
debt
0.2
0.15
0.1
398
399
400
401
402
403
hours
0.33
0.328
0.326
0.324
0.322
398
399
400
401
402
403
Figure 16 illustrates a long stochastic simulation of the model. It is easy
to see that taxes are very responsive to debt. One interesting question is
whether taxes are smoother or more volatile under PI with respect to FI.
Intuitively, there seem to be two opposing forces. On the one hand, the PI
government does not observe the shocks, and hence smooths its policy across
states for a given debt level. However, this policy induces necessary fiscal
adjustments following the dynamics of debt, so that this pushes towards
higher volatility under PI. The results from long simulations is that this
second effect seems to dominate and the FI government is more successful
than the PI government at smoothing tax rates. It can be seen that often
when debt gets close to the borrowing limit (20% of mean output) the PI
government imposes larger fiscal adjustments. This can be rationalised in
analogy with the example of the two-period economy close to the top of the
Laffer curve. Fear of future large required adjustments in the event of low θ
lead the PI government to raises taxes significantly.
41
Figure 16: Simulation
taxes
0.275
FI
PI
0.27
0.265
0.26
0.255
0.25
0
20
40
60
80
100
120
140
160
180
200
debt
0.2
0.18
0.16
0.14
0.12
0.1
0.08
20
8
40
60
80
100
120
140
160
180
200
Conclusion
We derive a method to solve models of optimal policy with limited information without any separation assumption between the optimization and
signal extraction problem. The method works in general and we show that
designing algorithms to solve these problems is quite easy. We also show
that Partial Information on endogenous variables matters as some revealing
non-linearities appear in very simple models.
Optimal fiscal policy under endogenous Partial Information calls for smooth
tax rates across states when the government budget is under control, and for
regions of large response to aggregate data when the economy is close to the
top of the Laffer curve or to a borrowing limit. Uncertainty about the state
of the economy helps to understand the slow reaction of some European governments to the Great Recession, followed by sharp fiscal adjustments and
prolonged downturns.
Clearly, while we have illustrated the technique in a model of optimal fiscal policy, the methodology can be easily extended to other dynamic models,
for example in the analysis of optimal monetary policy in sticky price models
(e.g. Clarida et al. 1999) under the assumption of Partial Information. Our
optimal policy smoothing result is likely to extend to that setup, potentially
leading to a microfoundation for smooth nominal interest rates.
42
References
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[3] Baxter,Brad, Graham,Liam and Stephen Wright (2007) ”The Endogenous Kalman Filter”, Birkbeck School of Economics, Mathematics and
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non-invertible information sets in linear rational expectations models”
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[6] Guerrieri, Veronica and Robert Shimer (2013) ”Markets with Multidimensional Private Information”, Society for Economic Dynamics Meeting Papers series 2013 n. 210.
Lucas Jr, Robert E., Jr. (1972) ”Expectations and the Neutrality of
Money”, Journal of Economic Theory, 4(2), 103-124.
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monetary policy in an economy without capital”, Journal of Monetary
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[8] Mehra, R.and E.C Prescott.(1980).”Recursive competitive equilibrium:
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the bond market”, Journal of Monetary Economics 55, 1389–1400
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[11] Orphanides, A. and Wieland, V. (2000) ”Inflation Zone Targeting”,
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experimentation”, Journal of Economic Dynamics & Control 24: 501534
44
Appendix A: Proof of Proposition 1
It is clear that the PI problem is equivalent with modifying the FI problem
by adding the following constraints
f (A) = R
f A f or all A, A ∈ Φ such that
R
f A ,A
f (A) , A = h R
h R
(54)
to the feasible set. Therefore it is clear that the max of the FI problem is
higher than or equal to the max of the PI problem. But under invertibility
the optimal valueofthe FI problem satisfies the additional restrictions (54)
f (A) = R
f A only if A = A, therefore the FI solution solves the PI
since R
problem.
Appendix B: Derivation of (29)
We compute
namely
dF (R∗ +αδ)
dα
as given by (28) and evaluate it at α = 0. Recall (28),
dF(R∗ + αδ)
d
=
dα
R
Φ
W (T (R∗ +αδ, A), L(R∗ +αδ, A), A)dFA (A)
dα
Under enough boundedness conditions on the derivative we can pass the
derivative operator inside the integral. Hence using T (R∗ +αδ, A) = R∗ (L(R∗ +αδ, A))+
αδ (L(R∗ +αδ, A))
dF(R∗ + αδ)
dα
R
dW
((R∗ (L(R∗ +αδ, A)) + αδ (L(R∗ +αδ, A)) , L (R∗ +αδ, A) , A)
= Φ
dFA (A)
dα
"
#
Z
dL (R∗ +αδ, A)
′
∗′
∗
∗
∗
=
((R (L(R +αδ, A)) + αδ (L(R +αδ, A)))
+ δ (L (R +αδ, A))
dα
Φ
Wτ ((R∗ (L(R∗ +αδ, A)) , L (R∗ +αδ, A) , A)
!
dL (R∗ +αδ, A)
∗
∗
∗
dFA (A)
+ Wl ((R (L(R +αδ, A)) , L (R +αδ, A) , A)
dα
or writing a slightly more elegant expression, letting
45
∗′
∗
R∗′
(55)
α,δ = R (L(R +αδ, A))
′
′
∗
δα,δ = δ (L(R +αδ, A))
∗
δα,δ
= δ (L (R∗ +αδ, A))
dL (R∗ +αδ, A)
L′α,δ =
dα
∗′
Wτ,α,δ
= Wτ ((R∗ (L(R∗ +αδ, A)) , L (R∗ +αδ, A) , A)
∗′
Wl,α,δ
= Wl ((R∗ (L(R∗ +αδ, A), A) , L (R∗ +αδ, A))
the derivative is
i
dF(R∗ + αδ) Z h ∗′ ∗′
′
∗′
′
∗′
∗
=
Wτ,α,δ Rα,δ + αδ α,δ + Wl,α,δ L + Wτ,α,δ δα,δ dFA (A)
α,δ
dα
Φ
Evaluating at α = 0 expressions at (55) we have
R∗′
0,δ
′
δ0,δ
∗
δ0,δ
∗′
Wτ,0,δ
∗′
Wl,0,δ
=
=
=
=
=
R∗′ (L(R∗ , A))
δ ′ (L(R∗ , A))
δ (L (R∗ , A))
Wτ ((R∗ (L(R∗ , A)) , L (R∗ , A)) = Wτ∗
Wl ((R∗ (L(R∗ , A)) , L (αδ, A)) = Wl∗
(56)
Therefore from (26) we have
Z Φ
[Wτ∗ R∗′
+
Wl∗ ] L′
0,δ
+
Wτ∗ δ
dFA (A) = 0
where it is understood that Wτ∗ , Wl∗ , δ,R∗′ are evaluated at equilibrium optimal choices which is our (29).
Appendix C: Derivation of the Envelope Condition (53)
In this Appendix we derive the Envelope Condition (53). First of all let us
introduce the necessary notation. A tax policy is a function of debt and
labour R(b, l) and labour is a function of a policy R, outstanding debt and
the exogenous shock, L(R, b, A) defined by the zero of
H(l, A, R) ≡ l − h(R(b, l), A),
46
(57)
in analogy with the two-period model. By total differentiation of (57), the
partial derivative of labour with respect to debt, Lb , is given by
Lb (R, b, A) = −
γθRb (b, l)
.
+ γθRL (b, l)
(58)
v ′′ (l)
Now, for simplicity consider a case without borrowing penalty. In order
to derive the envelope condition, we differentiate (52) with respect to b and
get
"
!#
′ γ
′
′ ∗
∗
′
∗
∗′ ∗
W (b) = E (γθ − v (l )) Lb + W b
+ βbL Lb
γ̄
where
−θγ + [v ′′ (L(R∗ , A, b))L(R∗ , A, b) + v ′ (L(R∗ , A, b))]
βγ̄
′
b∗L =
l∗ = L(R∗ ; b, A)
L∗b = Lb (R∗ ; b, A).
Using (58) we can write
"
#
′ γ
(−γθR∗b (l, b))
′
+
W
.
b∗
W ′ (b) = E γθ − v ′ (l∗ ) + βW ′ b bL ′′
v (l) + γθR∗L (l, b)
γ̄
(59)
Using Proposition 2, the FOC of PI Ramsey problem is
E
"
′
∗′
∗
′
∗′
θγ − v (l ) + βW (b
′∗
′
)b∗L
#
h∗τ
|¯l = 0
1 − h∗τ R∗L
(60)
for all ¯l. Furthermore, we have that the partial derivative of the reaction
function h with respect to taxes is
hτ =
−γθ
.
v ′′ (l)
So from (59) we get
′
W (b) = E
"
′
∗
γθ − v (l ) + βW
′
b
∗′
′
b∗L
47
#
′ γ
h∗τ R∗b (l, b)
′
.
+
W
b∗
1 − h∗τ R∗L (L, b)
γ̄
Now, applying the law of iterated expectations, using the fact that Rb (l, b)
is known given L, b and using (60), we obtain
′
"
W (b) = E E
"
′
∗
γθ − v (L ) + βW
= E 0+W
′
b
∗′
γ
γ̄
#
′
b
∗′
′
b∗L
!
(62)
Finally, adding the marginal cost of excessive debt this becomes
W ′ (b) = E
#
′ γ
h∗τ R∗b (L, b) ′
L
+
W
b∗ (61)
1 − h∗τ R∗L (L, b) γ̄
γW ′ (b′ )
− χ(b′max )I[bmax ,∞) (b′ ).
γ
48