Stephanie Schmitt-Grohé and Martín Uribe are
Professors of Economics at Duke University. Their main
line of interest lies in monetary macroeconomics, in
particular issues of optimal stabilisation policy.
Schmitt-Grohé's RePEc/IDEAS
entry. Uribe's RePEc/IDEAS
entry.
Much of our recent research has been devoted to
developing and applying tools for the evaluation of
macroeconomic stabilization policy. This choice of topic
was motivated by an important development in
business-cycle theory. By the late 1990s, a frictionless
model of the macroeconomy was viewed by many as no
longer providing a satisfactory account of aggregate
fluctuations. As a response, the new Keynesian paradigm
emerged as an alternative framework for understanding
business cycles. A key difference between the
neoclassical and the new Keynesian paradigms is that in
the latter, the presence of various nominal and real
distortions provide a meaningful role for stabilization
policy, opening the door once again, after decades of
dormancy, for policy evaluation.
Developing Tools For Policy Evaluation
An obstacle we encountered early on in executing the
research agenda described here was the lack of
appropriate tools to evaluate stabilization policies in
the context of distorted economies. An important part of
our effort was therefore devoted to developing such
tools.
Most models used in modern macroeconomics are too
complex to allow for exact solutions. For this reason,
researchers have appealed to numerical approximation
techniques. One popular and widely used approximation
technique is a first-order perturbation method
delivering a linear approximation to the policy
function. One reason for the popularity of first-order
perturbation techniques is that they do not suffer from
the `curse of dimensionality.' That is, problems with a
large number of state variables can be handled without
much computational demands. Because models that are
successful in accounting for many aspects of observed
business cycles are bound to be large (e.g., Smets and
Wouters, 2004; and Christiano, Eichenbaum, and Evans,
2003), this advantage of perturbation techniques is of
particular importance for policy evaluation. However, an
important limitation of first-order approximation
techniques is that the solution displays the certainty
equivalence property. In particular, the first-order
approximation to the unconditional means of endogenous
variables coincides with their non-stochastic steady
state values. This limitation restricts the range of
questions that can be addressed in a meaningful way
using first-order perturbation techniques. One such
question that is of particular relevance for our
research agenda is welfare evaluation in stochastic
environments featuring distortions or market failures.
For example, Kim and Kim (2003) show that in a simple
two-agent economy, a welfare comparison based on an
evaluation of the utility function using a linear
approximation to the policy function may yield the
spurious result that welfare is higher under autarky
than under full risk sharing. The problem here is that
some second- and higher-order terms of the equilibrium
welfare function are omitted while others are included.
Consequently, the resulting criterion is inaccurate to
order two or higher. The same problem arises under the
common practice in macroeconomics of evaluating a
second-order approximation to the objective function
using a first-order approximation to the decision rules.
For in this case, too, some second-order terms of the
equilibrium welfare function are ignored while others
are not. See Woodford (2003, chapter 6) for a discussion
of conditions under which it is correct up to second
order to approximate the level of welfare using
first-order approximations to the policy function. In
general, a correct second-order approximation of the
equilibrium welfare function requires a second-order
approximation to the policy function.
This is what we set out to accomplish in
Schmitt-Grohé and Uribe (2004a). Building on previous
work by Collard and Juillard, Sims, and Judd among
others, we derive a second-order approximation to the
solution of a general class of discrete-time rational
expectations models. Specifically, our technique is
applicable to nonlinear models whose equilibrium
conditions can be written as: Et
f(yt+1,yt,xt+1,xt)=0,
where the vector xt is predetermined
and the vector yt is nonpredetermined.
The main theoretical contribution of Schmitt-Grohé
and Uribe (2004a) is to show that for any model
belonging to this general class, the coefficients on the
terms linear and quadratic in the state vector in a
second-order expansion of the decision rule are
independent of the volatility of the exogenous shocks.
In other words, these coefficients must be the same in
the stochastic and the deterministic versions of the
model. Thus, up to second order, the presence of
uncertainty affects only the constant term of the
decision rules. But the fact that only the constant term
is affected by the presence of uncertainty is by no
means inconsequential. For it implies that up to second
order the unconditional mean of endogenous variables can
in general be significantly different from their
non-stochastic steady state values. Thus, second-order
approximation methods can in principle capture important
effects of uncertainty on average rate of return
differentials across assets with different risk
characteristics and on the average level of consumer
welfare. An additional advantage of higher-order
perturbation methods is that like their first-order
counterparts, they do not suffer from the curse of
dimensionality. This is because given the first-order
approximation to the policy function, finding the
coefficients of a second-order approximation simply
entails solving a system of linear equations.
The main practical contribution of Schmitt-Grohé and
Uribe (2004a) is the development of a set of MATLAB
programs that compute the coefficients of the
second-order approximation to the solution to the
general class of models described above. This computer
code is publicly available at the authors' websites. Our
computer code coexists with others that have been
developed recently by Chris Sims and Fabrice Collard and
Michel Juillard to accomplish the same task. We believe
that the availability of this set of independently
developed codes, which have been shown to deliver
identical results for a number of example economies,
helps build confidence across potential users of
higher-order perturbation techniques.
Optimal Operational Monetary Policy for the U.S.
Economy
After the completion of the second-order
approximation toolkit, we felt that we were suitably
equipped to undertake a systematic and rigorous
evaluation of stabilization policy. A contemporaneous
development that highly facilitated our work was the
emergence of estimated medium-scale dynamic general
equilibrium models of the U.S. economy with the ability
to explain the behavior of a relatively large number of
macroeconomic variables at business-cycle frequency
(e.g., Christiano, Eichenbaum, and Evans, 2003; and
Smets and Wouters, 2004).
A central characteristic of the studies on optimal
monetary policy that existed at the time we initiated
our research on policy evaluation, was that they were
conducted in the context of highly stylized
environments. An important drawback of that approach is
that highly simplified models are unlikely to provide a
satisfactory account of cyclical movements for but a few
macroeconomic variables of interest. For this reason,
the usefulness of this strategy to produce policy advise
for the real world is necessarily limited.
In a recent working paper (Schmitt-Grohé and Uribe,
2004b), we depart from the literature extant in that we
conduct policy evaluation within the context of a rich
theoretical framework capable of explaining observed
business cycle fluctuations for a wide range of nominal
and real variables. Following the lead of Kimball
(1995), the model emphasizes the importance of combining
nominal and real rigidities in explaining the
propagation of macroeconomic shocks. Specifically, the
model features four nominal frictions, sticky prices,
sticky wages, money in the utility function, and a
cash-in-advance constraint on the wage bill of firms,
and four sources of real rigidities, investment
adjustment costs, variable capacity utilization, habit
formation, and imperfect competition in product and
factor markets. Aggregate fluctuations are assumed to be
driven by supply shocks, which take the form of
stochastic variations in total factor productivity, and
demand shocks stemming from exogenous innovations to the
level of government purchases. Altig et al. (2003) and
Christiano, Eichenbaum, and Evans (2003) argue that the
model economy for which we seek to design optimal
operational monetary policy can indeed explain the
observed responses of inflation, real wages, nominal
interest rates, money growth, output, investment,
consumption, labor productivity, and real profits to
productivity and monetary shocks in the postwar United
States. In this respect, Schmitt-Grohé and Uribe (2004b)
aspires to be a step ahead in the research program of
generating monetary policy evaluation that is of
relevance for the actual practice of central banking.
In our quest for the optimal monetary policy scheme
we restrict attention to what we call operational
interest rate rules. By an operational interest-rate
rule we mean an interest-rate rule that satisfies three
requirements. First, it prescribes that the nominal
interest rate is set as a function of a few readily
observable macroeconomic variables. In the tradition of
Taylor (1993), we focus on rules whereby the nominal
interest rate depends on measures of inflation,
aggregate activity, and possibly its own lag. Second,
the operational rule must induce an equilibrium
satisfying the zero lower bound on nominal interest
rates. And third, operational rules must render the
rational expectations equilibrium unique. This last
restriction closes the door to expectations driven
aggregate fluctuations.
The object that monetary policy aims to maximize in
our study is the expectation of lifetime utility of the
representative household conditional on a particular
initial state of the economy. Our focus on a conditional
welfare measure represents a fundamental departure from
most existing normative evaluations of monetary policy,
which rank policies based upon unconditional
expectations of utility. Exceptions are Kollmann (2003)
and Schmitt-Grohé and Uribe (2004c). As Kim et al.
(2003) point out, unconditional welfare measures ignore
the welfare effects of transitioning from a particular
initial state to the stochastic steady state induced by
the policy under consideration. Indeed, we document that
under plausible initial conditions, conditional welfare
measures can result in different rankings of policies
than the more commonly used unconditional measure. This
finding highlights the fact that transitional dynamics
matter for policy evaluation. In our welfare
evaluations, we depart from the widespread practice in
the neo-Keynesian literature on optimal monetary policy
of limiting attention to models in which the
nonstochastic steady state is undistorted. Most often,
this approach involves assuming the existence of a
battery of subsidies to production and employment aimed
at eliminating the long-run distortions originating from
monopolistic competition in factor and product markets.
The efficiency of the deterministic steady-state
allocation is assumed for purely computational reasons.
For it allows the use of first-order approximation
techniques to evaluate welfare accurately up to second
order, a simplification that was pioneered by Rotemberg
and Woodford (1999). This practice has two potential
shortcomings. First, the instruments necessary to bring
about an undistorted steady state (e.g., labor and
output subsidies financed by lump-sum taxation) are
empirically uncompelling. Second, it is ex ante not
clear whether a policy that is optimal for an economy
with an efficient steady state will also be so for an
economy where the instruments necessary to engineer the
nondistorted steady state are unavailable. For these
reasons, we refrain from making the
efficient-steady-state assumption and instead work with
a model whose steady state is distorted.
Departing from a model whose steady state is Pareto
efficient has a number of important ramifications. One
is that to obtain a second-order accurate measure of
welfare it no longer suffices to approximate the
equilibrium of the model up to first order. Instead, we
obtain a second-order accurate approximation to welfare
by solving the equilibrium of the model up to second
order. Specifically, we use the methodology and computer
code developed in Schmitt-Grohé and Uribe (2004a).
Our numerical work suggests that in the model economy
we study, the optimal operational interest-rate rule
takes the form of a real-interest-rate targeting rule.
For it features an inflation coefficient close to unity,
a mute response to output, no interest-rate smoothing,
and is forward looking. The optimal rule satisfies the
Taylor principle because the inflation coefficient is
greater than unity albeit very close to 1. Optimal
operational monetary policy calls for significant
inflation volatility. This result stands in contrast to
those obtained in the related literature. The main
element of the model driving the desirability of
inflation volatility is indexation of nominal factor and
product prices to 1-period lagged inflation. Under the
alternative assumption of indexation to long-run
inflation, the conventional result of the optimality of
inflation stability reemerges.
Open Questions
There remain many challenging unanswered questions in
this research program. One is to investigate the
sensitivity of the parameters of the optimal operational
policy rule to changes in the sources of uncertainty
driving business cycles. This question is of importance
in light of the ongoing quest in business-cycle research
to identify the salient sources of aggregate
fluctuations. One alternative would be to incorporate
the rich set of shocks identified in econometric
estimations of the model considered here (e.g., Smets
and Wouters, 2004).
The class of operational rules discussed here is
clearly not exhaustive. It would be of interest to
investigate whether the inclusion of macroeconomic
indicators other than those considered here would
improve the policymaker's ability to stabilize the
economy. In particular, the related literature has
emphasized the use of measures of the output gap that
are different from that used by us. Additionally, it has
been argued that in models with nominal wage and price
rigidities the optimal policy should target an average
of wage and price inflation as opposed to only price
inflation, which is the case we analyze.
The optimal policy problem we analyze takes the
central bank's inflation target as exogenously given. A
natural extension is to endogenize this variable.
However, in our theoretical framework, the optimal
inflation target is the one associated with the Friedman
rule. This is because the assumption of full indexation
to past inflation implies the absence of inefficient
price and wage dispersion in the long run. Thus the only
remaining nominal frictions are the demand for money by
households and firms. These frictions call for driving
the opportunity cost of holding money to zero in the
long run. In other words, the zero bound on nominal
interest rate binds in the non-stochastic steady state.
The perturbation technique we employ is ill suited to
handle this case. Therefore, analyzing the case of an
endogenous inflation target entails either changing the
model so that the Friedman rule is no longer optimal in
the long-run or adopting alternative numerical
techniques for computing welfare accurately up to
second-order or higher.
One of our findings is that the initial state of the
economy plays a role in determining the parameters
defining the optimal interest-rate rule. This finding
suggests that the optimal operational rule identified
here is time inconsistent. In Schmitt-Grohé and Uribe
(2004b), we assume that the government is able to commit
to the policy announcements made at time 0. It would be
of interest to characterize optimal operational rules in
an environment without commitment.
Finally, we limit attention to the special case of
passive fiscal policy, taking the form of a
balanced-budget rule with lump-sum taxation. It is well
known that the set of operational monetary rules depends
on the stance of fiscal policy. For instance, the
determinacy properties of the rational expectations
equilibrium associated with a particular monetary rule
can change as fiscal policy is altered. Therefore, it
would be of interest to introduce operational fiscal
rules as an additional policy instrument.
References
Altig, David, Lawrence J. Christiano, Martin
Eichenbaum, and Jesper Lindé (2003): Technology Shocks
and Aggregate Fluctuations manuscript, Northwestern
University.
Christiano, Lawrence J., Martin Eichenbaum, and
Charles Evans (2003): Nominal
Rigidities and the Dynamic Effects of a Shock to
Monetary Policy. Northwestern University.
Kim, Jinill, and Sunghyun Henry Kim (2003): Spurious
Welfare Reversals in International Business Cycle
Models, Journal of International Economics
vol. 60, pages 471-500.
Kim, Jinill, Sunghyun Henry Kim, Ernst Schaumburg,
and Christopher Sims (2003): Calculating
and Using Second Order Accurate Solutions of Discrete
Time Dynamic Equilibrium Models, Finance and
Economics Discussion Series 2003-61, Board of Governors
of the Federal Reserve System.
Kimball, Miles S. (1995): The
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Kollmann, Robert (2003): Welfare Maximizing Fiscal
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Conference Report series. Chicago and London: University
of Chicago Press, pages 57-119.
Schmitt-Grohé, Stephanie and Martín Uribe (2004a): Solving
Dynamic General Equilibrium Models Using a Second-Order
Approximation to the Policy Function, Journal of
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755-775.
Schmitt-Grohé, Stephanie, and Martín Uribe (2004b):
Optimal
Operational Monetary Policy in the
Christiano-Eichenbaum-Evans Model of the U.S. Business
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Schmitt-Grohé, Stephanie and Martín Uribe (2004c): Optimal
Simple And Implementable Monetary and Fiscal Rules,
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Smets, Frank and Raf Wouters (2004): Comparing
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Woodford, Michael (2003): Interest and Prices:
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