期刊名称：Discussion Paper Series / Department of Economics, New York University
出版年度：2005
卷号：1
出版社：New York University
摘要：This paper considers the appropriate stabilization objectives for monetary
policy in a microfounded model with staggered price-setting. Rotemberg and
Woodford (1997) and Woodford (2002) have shown that under certain con-
ditions, a local approximation to the expected utility of the representative
household in a model of this kind is related inversely to the expected dis-
counted value of a conventional quadratic loss function, in which each period's
loss is a weighted average of squared deviations of in°ation and an output gap
measure from their optimal values (zero). However, those derivations rely on
an assumption of the existence of an output or employment subsidy that o®-
sets the distortion due to the market power of monopolistically-competitive
price-setters, so that the steady state under a zero-in°ation policy involves an
e±cient level of output. Here we show how to dispense with this unappealing
assumption, so that a valid linear-quadratic approximation to the optimal pol-
icy problem is possible even when the steady state is distorted to an arbitrary
extent (allowing for tax distortions as well as market power), and when, as a
consequence, it is necessary to take account of the e®ects of stabilization policy
on the average level of output.
We again obtain a welfare-theoretic loss function that involves both in°ation
and an appropriately de¯ned output gap, though the degree of distortion of the
steady state a®ects both the weights on the two stabilization objectives and
the de¯nition of the welfare-relevant output gap. In the light of these results,
we reconsider the conditions under which complete price stability is optimal,
and ¯nd that they are more restrictive in the case of a distorted steady state.
We also consider the conditions under which pure randomization of monetary
policy can be welfare-improving, and ¯nd that this is possible in the case of
a su±ciently distorted steady state, though the parameter values required are
probably not empirically realistic.
According to a common conception of the goals of monetary stabilization policy,
it is appropriate for the monetary authority to aim to stabilize both some measure
of in°ation and some measure of real activity relative to potential. This is often
represented by supposing that the authority should seek to minimize the expected
discounted value of a quadratic loss function, in which each period's loss consists of
a weighted average of the square of the in°ation rate and the square of the \output
gap." It is furthermore typically argued that the two stabilization goals are not fully
compatible with one another, owing to the occurrence of \cost-push shocks," which
prevent a zero output gap from being consistent with zero in°ation. The problem of
¯nding an optimal tradeo® between the two goals is then non-trivial.1
This familiar framework raises a number of questions, however. Most obvious is
the question of how to de¯ne the \output gap" that policy should seek to stabilize.
Should this be understood to mean output relative to some smooth trend, or should
the target output level vary in response to real disturbances of various sorts? A
closely related question is the de¯nition of the \cost-push shocks": how should these
be identi¯ed in practice, and how often do disturbances of this kind actually occur?
And even supposing that we know how to identify the output gap and the cost-push
disturbances, what relative weight should be placed on output-gap stabilization as
opposed to in°ation stabilization?
Here we propose to answer such questions on welfare-theoretic grounds. The
ultimate aim of monetary policy, in our view, should be the maximization of the
expected utility of households. We show, however (following a method introduced by
Rotemberg and Woodford, 1997, and further expounded in Woodford, 2002; 2003,
chap. 6), that it is possible to derive a quadratic approximation to the expected
utility of the representative household that takes the form of a discounted quadratic
loss function of the kind assumed in the traditional literature on monetary policy
evaluation. In the case that the exogenous disturbances are su±ciently small in
amplitude, the best policy (in terms of expected utility) will also be the one that
minimizes the discounted quadratic loss function. We thus obtain precise answers to
the question of what terms should appear in a quadratic loss function, and with which
relative weights, that depend on the speci¯cation of one's model of the monetary
transmission mechanism.2
An important limitation of the method introduced by Rotemberg and Woodford
(1997) is that it requires that the zero-in°ation steady state of one's model involve an
e±cient level of output.3 (They imagine a model in which this is true by assuming the
existence of an output subsidy that o®sets the distortion resulting from the market
power of monopolistically competitive suppliers, though this is obviously not liter-
ally true in actual economies.) For if one were instead to consider the more realistic
case of an economy in which steady-state output is ine±ciently low, one would ¯nd
that expected utility would depend on the expected level of output. An estimate of
expected utility that is accurate to second order would then require a solution for
output (or at any rate, for the expected discounted level of output) that is accurate
to second order in the amplitude of the exogenous disturbances. A log-linear approx-
imation to the structural equations of one's model will then not su±ce to allow one to
determine the evolution of output under one policy or another to a su±cient degree
of accuracy. As a consequence, a linear-quadratic methodology | in which a linear
policy rule is derived so as to minimize a quadratic approximation to the true welfare
objective subject to linear constraints that are ¯rst-order approximations to the true
structural equations | will not generally yield a correct linear approximation to the
optimal policy rule.4
Here we show how the method of Rotemberg and Woodford can be extended
to deal with the case in which the steady-state level of output is ine±cient (owing
to the existence of distorting taxes on sales revenues or labor income, in addition
to the distortions created by market power). Our approach involves computation
of a second-order approximation to the model structural relations (speci¯cally, to
the aggregate-supply relation in the present application), and using this to solve for
the expected discounted value of output as a function of purely quadratic terms.
This solution can then be used to substitute for the terms proportional to expected
discounted output in the quadratic approximation to expected utility. In this way,
we obtain an approximation to expected utility | that holds regardless of the policy
contemplated (as long as it involves in°ation that is not too extreme) | and that is purely quadratic, in the sense of lacking any linear terms. This alternative quadratic
loss function can then be evaluated to second order using an approximate solution
for the endogenous variables of one's model that is accurate only to ¯rst order. One
is then able to compute a linear approximation to optimal policy using a simple
linear-quadratic methodology.
Our proposal to substitute purely quadratic terms for the discounted linear terms
in the Taylor approximation to expected utility builds upon an idea of Sutherland
(2002), who showed how it was possible to take account of the e®ects of macroeco-
nomic volatility on the average levels of variables in welfare calculations for a model
with Calvo pricing like the baseline model considered here. Sutherland's crucial in-
sight was that it is not necessary to compute a complete second-order solution for
the evolution of the endogenous variables under each of the policies that one wishes
to consider in order to evaluate the discounted linear terms needed for the welfare
calculation. Sutherland's approach, however, still requires that one restrict attention
to a particular parametric family of policy rules before computing the second-order
approximations that are used to substitute for the discounted linear terms in the
welfare criterion. Instead, we show that one can substitute out the linear terms using
only a second-order approximation to the structural equations; one thus obtains a
welfare criterion that applies to arbitrary policies.5
An alternative way of attaining a welfare measure that is accurate to second or-
der even in the case of a distorted steady state, that has recently become popular, is
to solve for a second-order approximation to the complete evolution of the endoge-
nous variables under any given policy rule, and then use this solution to evaluate a
quadratic approximation to expected utility (e.g., Kim et al., 2002). However, the
requirement that a system of quadratic expectational di®erence equations be solved
for each policy rule that is contemplated is much more computationally demanding
than the implementation of our LQ methodology. For we are required to consider the
second-order approximation to our structural equations only once | when deriving
the appropriate quadratic loss function, a calculation undertaken in this paper |
after which the evaluation of individual policies requires only that one solve a system
of linear equations. In addition, the method illustrated by Kim et al. requires that one restrict one's attention to a particular parametric family of policy rules, since
the system of equations that is solved to second order must include a speci¯cation of
the policy rule. Our method, by contrast, allows us to determine what variables it is
desirable for policy to depend on without having to prejudge that issue.
Yet another approach that allows a correct calculation of a linear approximation
to the optimal policy rule even in the case of a distorted steady state is to compute
¯rst-order conditions that characterize optimal policy in the exact model (i.e., with-
out approximating either the welfare measure or the structural equations), and then
log-linearize these optimality conditions in order to obtain an approximate charac-
terization of optimal policy (e.g., King and Wolman, 1999; Khan et al., 2003). A
disadvantage of this approach is that it is only suitable for computing the optimal
policy; our quadratic approximate welfare measure also yields a correct ranking of
alternative sub-optimal policy rules, as long as disturbances are small enough, and
the policies under comparison all involve low in°ation. Furthermore, our LQ ap-
proach makes it straightforward to consider whether the second-order conditions for
a policy to be a local optimum are satis¯ed, and not just the ¯rst-order conditions
that are typically considered in the literature on \Ramsey policy", as we show in
section 3.1 below. Under conditions where the second-order conditions are satis¯ed,
our approach and the one used by Khan et al. yield identical approximate linear char-
acterizations of optimal policy; but we believe that the LQ approach provides useful
insight into the aspects of the policy problem that are responsible for the conclusions
obtained. We illustrate this in sections 3.2 and 3.3 by providing an analytical deriva-
tion of results with the same qualitative features as the numerical results reported by
Khan et al. for a related model.
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