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\begin{document}
\author{Christopher A. Sims}
\address{Princeton University, \url{sims@princeton.edu}}%
\date{\today}%
\title[Gal\'i-Chari comments]{Comments on papers by Jordi Gal\'i and by Stefania Albanesi,
V.V. Chari, and Lawrence J. Christiano}
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\maketitle
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The theme of these comments is that we need to remain vigilant against the
possibility that ``standard'' modeling conventions in macroeconomics,
originally introduced as experimental or tentative, start to be used
unquestioningly, despite serious drawbacks.  It is in a sense unfair to focus
such comments on the papers presented in this session, since the conventions to
be criticized are not at all special to these papers, but there must be some
point at which we step back and consider where the conventions in our
literature are headed, and a quinquennial World Congress seems as appropriate
an occasion as any for doing this.

I will consider the papers by Gal\'i and by Albanesi, Chari and Christiano
(henceforth ACC) that appear in this volume, and also the empirical paper
\citep{GaliGertlerInflStruct} that forms the foundation for much of the paper
in this volume that Jordi Gal\'i presented.

\section{Optimal Monetary and Fiscal Policy}
Both papers discuss optimal monetary policy in the context of general
equilibrium models.  General equilibrium models, with their own
internally generated, explicit measures of welfare, allow us to avoid
postulating an \textit{ad hoc} objective function for the policy authority,
instead evaluating policy directly in terms of its welfare implications for
private agents.  \G explicitly, and probably ACC also, see this as an
improvement over approaches that use models with a less complete behavioral
interpretation.  Such models must either postulate plausible measure of policy
effectiveness directly, or else present multidimensional measures of policy
performance --- variances and means of a variety of important variables,
usually --- and leave the reader to draw conclusions about ranking the results.

But these papers only confirm that reliable conclusions about rankings of
policies from general equilibrium macro models are, for the time being at
least, completely unavailable.  ACC focus most of their attention on two
welfare effects of inflation --- the decline in demand it generates for ``cash
goods'', under their particular assumptions about timing in discrete time
cash-in-advance style models, and the increase in output toward the efficient
level it can produce under sticky prices.  Output tends to be below the
efficient level in their models because of the existence of monopolistic
competition.  Even within their framework, their conclusions are more fragile
and dubious than a casual reading of the paper might suggest.

\G and related literature he cites, on the other hand, assume both these
effects of monetary policy away at the start.  The assumption that the costs of
squeezing down real balances are small is justified by no more than an appeal
to intuition.  The assumption that there are no gains from pushing output above
its non-inflationary level, despite the same monopolistic competition
assumption as in the ACC model, requires postulating that the government knows
and sets exactly the efficient level of employment subsidy, financing it by
lump sum taxes.  The costs that are left are only those due to
inefficient, but mean-zero, fluctuations around the efficient level of output.

That there is no overlap in the effects of monetary policy considered in these
two papers, since they are representative of much other work in the recent
literature, suggests that we have no professional consensus on how to set about
this task.  Furthermore, a good case can be made that each is considering
potentially important effects, and thus also that each is ignoring potentially
important effects.

It is common to dismiss as unimportant ``shoe-leather costs'' imposed by inducing people to
economize on real balances, so in doing so \G is far from alone.  If we think
of these as the costs imposed on us when inflation induces us to carry less
cash in our wallets, it certainly seems justified to treat them as small.  From
this point of view, the fact that cash-in-advance and limited-participation
models are capable of implying these costs are significant is a flaw in those
models.  ACC do not provide much discussion of why they think we should take
these costs to be quantitatively important.  They also appear to be an
artifact of the models' reliance on discrete time, and the timing ambiguities
that introduces.  As ACC point out, they are using the Svensson, rather than
the Lucas and Stokey, timing assumptions.  But these differences in timing
assumptions are difficult to evaluate for plausibility, since time is in fact not
discrete.

The way the timing assumptions operate to create different conclusions is as
follows.  If $M_t/P_t$ (where the time subscript indicates the date a variable
is known and/or chosen) enters a transactions cost term or the utility
function, then surprise monetary expansion, with flexible prices, is neutral,
though there are still usually real effects from anticipated monetary
expansion.  Expansion in $M$ is accompanied by an expansion in $P$ that offsets
its effects on transactions balances.  The simple version of the natural rate
hypothesis is thus stood on its head.  This result was present in
\citet{LucasStokeyFandM} and has been underlined since, for example in a paper
of mine (\citeyear{CASMandF}).  If instead it is $M_{t-1}/P_t$ that provides
utility or transactions services, then surprise monetary expansions have real
effects, but they are purely contractionary in a flexible price model.  This is
because, with $M_{t-1}$ fixed, any inflationary effects of expansion contract
the availability of transactions services.  It is this type of contractionary
effect on which ACC rely in generating their results.

But the time unit over which people are stuck with cash, and hence subject to
surprise inflation taxes on them, is short on average.  It is also endogenous.
Thus an increase in the price level can be offset by slightly more frequent
trips to the bank, so that the effects on spending of the shrinkage in real
balances are slight.  Cash in advance models, being in their simple form tied
to a unit delay between putting aside cash and spending it, cannot account for
these possibilities.  If one has to choose between $M_t/P_t$ and $M_{t-1}/P_t$
on these grounds, I think the former makes more sense.

In a paper that ACC cite, \citet{NicoliniTConsist} gives a more extensive
discussion of the motivation for this modeling style.  He argues that the unit
time interval is misleading here, that in a realistic model with heterogeneous
agents that make randomly timed visits to the bank, there will be at any given
time some with large balances.  He prefers the $M_{t-1}/P_t$ timing (but really, as a
long run modeling strategy, models with heterogeneous agents) because it seems
to him a stand-in for missing model realism.

My own view is that Nicolini's argument does not go far enough, because it
remains too much focused on cash and transactions technology.  The economy is
laced with incomplete, nominally denominated contracts, both formal and
informal.  Surprise inflation has real effects on the terms of these contracts,
and a lot of surprise inflation is likely to make people move away from simple
nominal contracts.  We do in fact see the nature of some contracts, e.g. labor
contracts, shift with the level and variability of inflation.  Since nominal
contracts reemerge when inflation stabilizes, they apparently are
cost-effective in some sense.  We have little understanding of why these
contracts take the form they do, or of the costs of distorting existing
contracts via inflation, or of inducing people, via surprise inflation, to
contract differently.  These costs may not be very important, but they may also
be large.  We have no quantitative handle on this issue, and none of the
existing approaches to general equilibrium modeling confront the issue.

Nicolini's arguments do not, then, suggest that cash-in-advance or limited
participation models are quantitatively reliable for evaluating the effects of
monetary policy.  They do, by recognizing the importance of heterogeneity and
redistributive effects of inflation,  point toward a broader critique of all
current approaches to using general equilibrium models for monetary policy
evaluation.

\G's approach justifies its measure of welfare effects by a now-widespread
idea, originally due to Rotemberg and Woodford.  Under the monopolistic competition
assumption, and with the Calvo story about the reason for price stickiness,
inflation creates a distribution of prices, with firms that have not for a long
time been hit by the exogenous random variable that allows price changes likely
to have prices far from the desired level.  This leads to firms to have
dispersed output levels, which is inefficient.  But the Calvo story, while a
clever modeling idea, has little claim to be grounded in empirical fact.  In
particular, the costs of inflation it implies rest on firms having no power to
affect the timing of their price changes.  While for some purposes this may be
an acceptable simplification, for evaluating welfare this restriction on firm
behavior becomes central.  If firms can decide to change prices when their
prices get far enough out of line, then the distribution of prices, and thereby
this source of inefficiency, may be quite insensitive to inflation.

The New Keynesian literature of course ends up with a criterion that is a
weighted average of variance in inflation and output.  Usually, as in \G's
paper, we see tables that show the underlying variance numbers as well as the
weighted average that defines welfare.  When this practice is followed, we can
see how robust are the rankings of policies to changes in the weights.
Attempts to translate differences in variances into utility-equivalent
differences in mean consumption, though, depend on our taking seriously the
derivation of the utility function, and should be treated as speculation.

There is no good argument that the non-inflationary level of output would be
socially optimal in the absence of random disturbances.  One might think that
so long as we assume that policy-makers can make commitments, so that the
possibility of short-run employment gains from surprise inflation does not lead
to an inflation bias, it would not make much difference to conclusions if the
noninflationary level of output were not optimal.  But this is not true.  The
effects on welfare of variance in inflation and output are second-order in the
scale of the randomness in the model.  In a nonlinear equilibrium model like
those underlying the \G paper model, randomness affects the mean as well as the
variance of the variables in the model.  The effects on means are also
second-order in the scale of the randomness.  Unless the effects of changing
the means are small, as they are in the neighborhood of the optimal
equilibrium, policies cannot be correctly ranked without considering their
second-order effects on means.  In other words, the log-linearized models
standard in the literature cannot produce correct rankings of policies.
\footnote{This was pointed out in a simple example by \citet{KimKimSpurious},
and there are now a number of individuals or groups preparing and using
software that allows second-order expansion of equilibrium models
\citep{CollardJuillard,SGUribeStickyMF,CASOrder2}}

The defects in the New Keynesian story about welfare costs of inflation and in
the ACC story are both cases of a relatively simple, ad hoc, rigidity having
been suggested as a starting point for modeling non-neutrality, then
propagating in the literature because of its convenience as a modeling
technique.  There is no harm in this process itself.  But we need to remain
aware that there are many potential ways to generate price stickiness and
non-neutrality.  Similar qualitative aggregate observations may be accounted
for by mechanisms with contradictory implications for welfare evaluation of
monetary policy. We therefore need to remain modest in our claims for
conclusions about welfare effects of monetary policy.  Indeed it might be best
to limit our claims to assertions about the effects of policies on the behavior of
aggregate variables.  For such claims we have firmer checks against historical
data.

Both sets of authors deal in the papers at hand with policy evaluation in
abstract models, at best calibrated roughly to data.  In the background is
related literature
\citep[e.g.]{AlbCharChrtExpTrapMPol,ClarGalGertStab,TJSConquest,CogleySargent}
some of it by these same authors, which has examined the last half century or
so of US monetary policy, evaluating how good it has been, whether it has been
improving, and what forces have contributed to its evolution.  While my
comments at the conference included some discussion of this literature,
particularly of the ACC paper (which is what V.V. Chari actually presented at
the conference), here I will simply note that the apparent lack of consensus on
the sources of monetary non-neutrality, the dependence of conclusions on
assumptions about these sources, and the limited attention to statistical fit
in this literature make its conclusions on these issues at best tentative.  Despite a
widespread belief among economists that the inflation of the 70's and the
return to low inflation in the 80's and 90's reflect bad early monetary policy
and better later policy, there is little solid empirical support for this view.
In fact, there is considerable empirical support for precisely the opposite
view:  that the inflation and its end were not generated by monetary policy
decisions, that monetary policy in fact has had roughly the same systematic
component throughout the period.\footnote{\citet{HansonStable,LeeperZhaModest,CASDriftBreak,OrphanidesTrenches}.}

\section{Existence, Uniqueness, and Fiscal Policy}
The first model in the ACC paper, with flexible prices, arrives at the
conclusion that even a monetary authority that cannot make commitments will
choose optimal policy, and that that policy is the Friedman rule --- contract
the stock of non-interest-bearing high-powered money at a rate equal to the
real rate of interest.  ACC observe that only the real allocation in this
equilibrium is unique --- there is a continuum of equilibria indexed by the
initial price level.  But this observation is given no further discussion.

As in most such models, the non-uniqueness of the initial price level
implies the existence of sunspot equilibria, in which the price level is
randomly non-unique at every date.  There are two real variables that are not
uniquely invariant across equilibria:  The level of real balances and the level
of real lump-sum taxation.  Since in reality taxes are not non-distorting,
especially if they must be imposed at high rates, it is clear that the policies
ACC characterize as ``optimal'' in this model could not be recommended in
practice as good policies.  They would require a commitment to tax at
unboundedly high rates if realized prices turn out to be very low.

The non-uniqueness ACC find here is essentially the same as that analyzed by
\citet{BenhabibMultEq}, and rests on the same somewhat strange type of
specification of fiscal policy.  \citeauthor{BenhabibMultEq} postulate that
fiscal policy makes the real primary surplus react positively to the real value
of total outstanding government liabilities, including high powered money as
well as interest-bearing debt.  The ACC fiscal policy, which commits to taxing
at a fixed ratio to the real value of the stock of outstanding money (here the
only government liability), is a special case. We do not see historical
examples of persistent taxation for the sole purpose of contracting the money
supply, and current political budget rhetoric does not seem to leave room for
this reason to tax. Why should a legislature feel an obligation to increase
taxes, with no outstanding debt, when the price level has emerged as lower than
expected?

It is more reasonable to suppose that the fiscal authorities make taxes
respond, if at all, only to the real value of interest-bearing debt.
Furthermore, if we are imagining that the monetary authority is capable of
exactly measuring the real rate of return so as to contract $M$ at that rate,
we might imagine instead that the fiscal authority (another name for the monetary
authority in this model) knows the level of real taxation that is
required to contract $M$ at the desired rate when real balances $M/P$ are
exactly at the satiation level.  If they then formulate policy as a commitment
to tax to produce exactly that real primary surplus, regardless of what happens
to prices, and the monetary authority commits to maintaining a zero nominal
interest rate, then the ACC model will have a Friedman-rule equilibrium with a
uniquely determined price level.

The Friedman rule is still a questionable policy, though.  If the tax authority
sets the primary surplus too low, the policy is unsustainable, and while the
uncertainty about how the policy will become sustainable persists, high
inflation is likely to result.  The surplus could be set high enough to allow
a margin of error --- high enough to contract a volume of real balances well
exceeding the level that satiates transactions demand at a rate above any value
the real discount factor is likely to take on.  But if we recognize a cost to
high taxation, this policy choice is unattractive.

If these transactions costs considerations are the primary reason for worrying
about inflation, then the best policy in practice is likely to be setting a low
positive nominal rate. This, by fixing velocity uniquely, will, in combination
with any primary surplus above some minimum level, guarantee a unique
equilibrium.  Such policies do require fiscal support.  A steady flow of tax
revenue is devoted to contracting the money stock.  However, widespread
existing conventions, in which central banks ``own'' some of the government's
debt and can use the flow of interest income for policy purposes without
legislative second-guessing, would support the necessary flow of fiscal
resources.\footnote{Actually, the situation in the US, where the central bank
backs its high-powered money mainly with holdings of nominal domestic
government debt, is not common.  Where the central bank balance sheet includes
a large component of real assets or of foreign currency denominated securities,
the current conventions of central bank independence would not suffice to
provide fiscal support for such a mildly deflationary policy.  Explicit fiscal
backing would be required. \citep{CASCBInd}}

Specifying monetary policy not as a fixed nominal rate but instead as
contraction of the nominal money stock at a fixed rate $g>\beta$ in this model
is not compatible with commitment to a fixed real primary surplus.  In the
multiple equilibria that ACC find for this case, in all but one real balances
grow at the rate $g/\beta$.  This requires, with their specification of fiscal
policy, that real primary surpluses grow without bound.  With the primary
surplus instead fixed in real terms, the shortfall has to be made up by
borrowing, and the borrowing would have to produce real interest-bearing debt
growth at the rate $\beta^{-1}$, which is incompatible with private
transversality.  The equilibrium in which prices and $M$ both grow at rate $g$,
so $M/P$ remains constant, determines a unique nominal rate, thus a unique
initial $M/P$, and thus a unique initial $P$.  But in general this initial $P$
will not be the value that matches initial $(B+M)/P$ to the discounted present
value of primary surpluses, so there is no corresponding equilibrium.  In other
models, where transactions demand is highly interest-sensitive and a barter
equilibrium exists, the fixed $M$, fixed surplus combination is actually
consistent with equilibrium --- an indeterminate continuum of them, all of
which involve inflationary dissipation of real balances and convergence to
barter equilibrium.

The uniqueness issues I raise here in connection with the first of the three
ACC models in the paper in this volume are more complicated to analyze in the
sticky price and limited participation models, and ACC do not fully unravel
them.  They consider only recursive equilibria with certain natural candidate
state vectors, which leaves open the question of whether there may be
equilibria indexed by, e.g. lagged prices.  Uniqueness issues are brought out,
but in the same sense also incompletely analyzed, in the paper by the same
authors presented at the conference \citep{AlbCharChrtExpTrapMPol}.  There they
suggest that the non-uniqueness that can arise in a model that allows more
elastic response of transactions technology to interest rates might explain
postwar US monetary history.  In that paper it would be even more important to
recognize that jumps between equilibria may imply substantial jumps in the
primary surplus with their assumptions about fiscal policy, and also the
possibility that commitment to a real primary surplus can eliminate
indeterminacy.

The \G paper also ignores the fiscal aspect of monetary policy.  This leads to
its assertion that a fixed nominal interest rate implies indeterminacy of the
price level if interpreted literally.  This is not true if the fixed nominal
interest rate is accompanied by a commitment to a real primary surplus.  It
would be simpler to describe the fixed interest rate policy this way.  As the
paper now describes it, it sounds difficult to implement, whereas in fact it
would be easy.

In a model that recognized the revenue from seignorage, the fixed interest rate
policy implemented as \G describes it here (via a restricted variant on a
Taylor rule, with fiscal policy implicitly making the primary surplus
responsive to the real debt) would probably not deliver the same equilibrium as
an nominal interest rate peg accompanied by a primary surplus peg.  However, in
the model as laid out here, the two versions of a fixed nominal rate policy
probably do deliver equivalent equilibria.  My own experiments suggest that, if
a disturbance is introduced into the Phillips curve as \G does later in the
paper, an interest rate peg can deliver lower inflation variance (though still
higher output variance) than the Taylor rule in this model.  It would be a good
idea, before taking the policy evaluations displayed here as the last word even
for this type of model, to consider the effects of such shocks on the rankings
of policy rules.

\section{The New Phillips Curve}

Simple New Keynesian models use a purely forward-looking ``Phillips Curve''
derived from Calvo-style price adjustment.  These models imply that inflation
``leads'' the output gap, and \citet{FuhrerMooreInfPers} in a careful
two-equation analysis showed that in what was then their usual form these
models are qualitatively at variance with the time series facts.  In another
paper, working with a formally similar inventory model,
\citet{FuhrerMooreSchuh} show that for these models likelihood-based multiple
equation methods are much more reliable than inference based on instrumental
variables.

The literature \G cites on the newest formulation of the
New Phillips curve \citep{GaliGertlerInflStruct,SbordonePandULC}, retreats from
the standard of care in empirical evaluation set by the Fuhrer-Moore paper,
using instrumental variables methods alone for estimation and using
informal measures of fit that can easily be misleading.

To understand the problems associated with the approach in these papers to
assessing fit, consider the New Phillips Curve model laid out the
Gal\'i-Gertler (GG) paper. It specifies the same Phillips Curve that appears in this
volume's \G paper:
\begin{equation}
   \pi_t = \beta E_t\pi_{t+1} + \kappa x_t \label{eq:NPC}\:,
\end{equation}
with $x_t$ interpreted as being the log of the labor income share in non-farm
business sector.  This is, on the face of it, an equation that can be estimated
by an OLS regression of $\pi_{t+1}$ on $\pi_t$ and $x_t$.  Without
explaining why, GG estimate it instead by instrumental variables, normalized as
in \eqref{eq:NPC}, and omit $\pi_t$ from the instrument set.  \footnote{The normalization
would not have mattered if they had kept $\pi_t$ in the instrument set, as in
that case instrumental variables estimates would have been identical to those
from OLS.}

Solving this equation forward in isolation, they obtain an implied relation
between $\pi_t$ and a discounted sum of expected future values of $x$,
\begin{equation}
   \pi_t=\kappa\sum_{t=0}^\infty\beta^sE_tx_{t+s}\label{eq:forward}\:.
\end{equation}
Then they apparently\footnote{The nature of the forecasting rule used to form
$E_tx_{t+s}$ is not completely clear from the paper.  All that is stated
explicitly is that it is based on data on $\pi$ and $x$ dated $t$ and earlier.}
use an unrestricted bivariate VAR to form expected future values of $x$,
substitute them into the right-hand side of \eqref{eq:forward}, and call the
result ``fundamental inflation''.  The theory predicts that fundamental
inflation defined this way is exactly observed inflation.  This is the
intuition behind plotting the two, noting how closely they track each other,
and counting what appears to the eye a close tracking as support for the theory.

But we are not here testing a theory in which an explanatory variable $x$ is
claimed to be important, while under an alternative view $x$ is expected to be
unrelated to $\pi$.  When the gap is measured as the labor share, it is being
measured as a variable that any reasonable macro theory would treat as highly
endogenous, affected by all kinds of structural disturbances.  It is not
surprising that linear combinations of it with $\pi$ should turn out to be
highly correlated with $\pi$.  The theory claims that fundamental inflation
should be identical to actual inflation, and this is a simple linear
restriction on the coefficients of the VAR.  The theory should be tested
directly as such a restriction.  To understand this point, it may help to write
out the unrestricted VAR as
\begin{equation}
   \begin{bmatrix}
      \vec{\pi}_t\\\vec{x}_t
   \end{bmatrix}=z_t=Az_{t-1}+\eps_t\:,
\end{equation}
where $\vec\pi_t$ and $\vec x_t$ are $k\times 1$ vectors containing current and lagged
values of $\pi$ and $x$.  In this form, the system allows us to write
\begin{align}
   E_tz_{t+s}&=A^sz_t\quad\text{and therefore, according to theory,}\\
   \pi_t=e_1z_t&=  e_{k+1}\kappa E_t\left[\sum_{s=0}^\infty \beta^s A^sz_t\right]
   =(I-\beta A)^{-1}z_t\:,
\end{align}
where $e_i$ denotes the unit vector with a one in its $i$'th position. This
implies
\begin{align}
   e_1&=e_{k+1}\kappa (I-\beta A)^{-1}\quad \text{or, equivalently}\label{eq:invrestrict}\\
    e_1\beta A &= e_1-\kappa e_{k+1}\:.
\end{align}
Note that this is simply the assertion that the first equation of the VAR
should be exactly \eqref{eq:NPC}, normalized to have $\pi_{t+1}$ on the left
with a unit coefficient, and with the $E_t$ operator dropped.
The theory should be evaluated by examining, using likelihood-based measures,
the validity of this straightforward linear restriction on $A$.\footnote{Comparing the time
series behavior of the left and right hand sides of \eqref{eq:forward} is
widespread in applications of what is known as the Campbell-Shiller
methodology. Such time series plots can be useful in assessing a model, but
they should not be treated as measures of fit.  Many authors construct formal
tests of \eqref{eq:invrestrict} as a nonlinear restriction on $A$.  As has
recently been pointed out by \citet{MercereauCAStock}, there is good reason to
think that inference about $(I-A)^{-1}$ in these models, based on asymptotics, will be much
less reliable than inference about $A$ itself.}

Even if all the inference were done more carefully, however, we should bear in
mind that we are not interested directly in restrictions within a model
incorporating a single gap measure.  We are interested in comparing models.  We
are interested in how a full-fledged model works in providing forecasts and
policy analysis when incorporating one or the other measure of the ``gap'',
forward or backward-looking specifications, and so on. As yet the literature
arguing for a Phillips Curve with labor share standing in for the gap has given
us no evidence on these questions, even on the simple one of whether forecasts
of a multivariate model are improved by a theoretically constrained addition of
a labor share variable.
\section{Conclusion}
In emphasizing how tenuous and loosely linked to real data are the monetary
policy evaluations in these papers, I do not intend to imply that this is
uninteresting work.  Precisely because we still understand so little in this
area, work with stylized, unrealistic models can be interesting and worthwhile.
Yet in central banks around the world there is increasing recognition of the
value of being more explicit and quantitative about planned time paths of
target variables like inflation, output and employment and about how current
policy actions are expected to influence these time paths.  Accomplishing this
does not require models that explain why people hold money and why money is
non-neutral, but it does require multivariate models grounded in data as well
as in theory.  While these two papers did not set out to provide or improve
such models, they are representative of a great deal of academic macroeconomic
research.  It seems to me that work on improving models that contribute to
current problems in implementing monetary policy deserves relatively more
attention from academic macroeconomists.
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