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Japanese Monetary Policy,
1991–2001
Bennett T. McCallum

uring recent years, Japanese monetary policy has been the topic of a
great deal of discussion, commentary, and debate. This is not only
because of the great practical importance of the long-lasting slump
of the world’s second largest national economy, but also because the situation
in Japan has raised interesting issues concerning some fundamental topics in
monetary theory. Accordingly, this paper considers issues relating to recent
and prospective policy measures of the Bank of Japan (BOJ).
It is hard to avoid the impression that Bank of Japan (BOJ) policy has
been overly restrictive for approximately a decade. That statement does not
imply that Japan’s poor economic performance during the 1990s was entirely
or even primarily attributable to monetary policy, for structural flaws have
also been very important.1 It does suggest, however, that Japanese economic
performance would have been less undesirable if BOJ policy had been less
restrictive. In the pages that follow, I will attempt to support the foregoing
claim, discuss the difficulty faced by the BOJ because of the zero lower bound
on nominal interest rates, and illustrate this difficulty with a small quantitative
study. Then I will take up some of the nonstandard policy approaches that have
been proposed and will argue that the most promising of these would entail
rapid monetary base growth effected largely through purchases of foreign
H. J. Heinz Professor of Economics, Carnegie Mellon University, and Visiting Scholar, Federal
Reserve Bank of Richmond. This paper has been prepared for the Federal Reserve Bank of
Richmond. It is in part based upon position papers presented at meetings of the Shadow
Open Market Committee held in Washington, D.C., in April and October of 2001. For useful
comments, the author is indebted to Marvin Goodfriend, Bob Hetzel, Tom Humphrey, Allan
Meltzer, Edward Nelson, Roy Webb, and John Weinberg. The views expressed in this article
are not necessarily those of the Federal Reserve Bank of Richmond or the Federal Reserve
System.
1 Major banking-system difficulties are widely recognized, and in addition it is likely that

the growth rate of “potential” or “natural-rate” output has fallen from the level of the 1970s and
1980s. But the severity of the bank-solvency problem has been increased by the deflation of
the past several years, and it is almost certainly the case that actual output has fallen far below
potential.

Federal Reserve Bank of Richmond Economic Quarterly Volume 89/1 Winter 2003

Federal Reserve Bank of Richmond Economic Quarterly

exchange. Such a strategy has faced two major objections, however, so much
of the paper is devoted to counterarguments to these objections. The first
objection is based on legal provisions of the Bank of Japan Law and the second
on the concern that such actions would constitute a “beggar-thy-neighbor”
policy that would reduce Japanese demand for imports. It is argued that
neither of these objections is appropriate. With respect to the former, it is
suggested that the BOJ Law, as written, includes conflicting provisions and
that foreign exchange purchases for the purpose of monetary control could be
conducted if the BOJ were to request permission of the government. In this
regard, the intimate connection between monetary and exchange-rate policies
is emphasized. With respect to the beggar-thy-neighbor issue, it is argued
that in fact an expansionary monetary policy of the type recommended would
increase net Japanese imports. In this regard, a major portion of the paper
is devoted to a quantitative analysis of the trade-balance effects of a policy
of the recommended type. The analysis is carried out in the context of a
dynamic optimizing model of an open economy, which is exposited in some
detail. Policy simulation exercises conducted with this model represent a
major feature of the paper.

HAS BANK OF JAPAN POLICY BEEN TIGHT?

That BOJ policy has been quite tight—low interest rates notwithstanding—
is suggested by the most prominent and widely-respected guideline for the
conduct of monetary policy, i.e., the policy rule developed by John Taylor
(1993a). The Taylor rule can be expressed as
Rt = 3 + pta + 0.5(pta − 2) + 0.5(yt − ȳt ),

(1)

where R is the call rate, pta is the average inflation rate (GDP deflator) over
the previous four quarters, y is real GDP and ȳ is its potential value.2 A chart
contrasting Taylor-rule prescriptions for the overnight call rate3 with actual
values of this rate over the years 1972–1998 appeared in a recent paper in
this journal (McCallum, 2000b) to which the reader is referred for various
details.4 That comparison is reproduced in the top half of Figure 1. There it
is clear that the actual value exceeded the setting prescribed by Taylor’s rule
during almost every quarter beginning with 1993Q1 and continuing through
1998Q4. Of course, the negative values called for by the rule are not feasible,
2 Here the long-run average real rate of interest is taken to be 3 percent per annum (p.a.)
and the inflation target rate to be 2 percent. Some versions of the rule use other values for these
and for the coefficients attached to the target variables.
3 The (uncollateralized) overnight call rate was the BOJ’s operating target or instrument variable through the period of the 1990s. The procedure was changed in March 2001.
4 The most important of these, of course, is the measurement of “potential” output—which
has been especially problematic for Japan in recent years. Its reliance on this inherently difficult
concept is one weakness of the Taylor rule.

B. T. McCallum: Japanese Monetary Policy

Figure 1 Policy Rule Indications

but that does not alter the fact that Taylor’s policy guideline has called for
greater monetary ease through this period.
An alternative rule involving management of the monetary base has been
promoted in several of my papers (e.g., McCallum 1988, 1993, 2000b). It can
be written as
bt = 5 − vta + 0.5(5 − xt−1 ),

(2)

Federal Reserve Bank of Richmond Economic Quarterly

where b and x are logs of the monetary base and nominal GDP, while vta is
the average rate of base velocity growth over the previous four years. Here
5 is the target value for nominal GDP growth, obtained from a 2 percent
inflation target and a 3 percent assumed long-run average growth rate for real
GDP. This rule is much less prominent than Taylor’s, primarily because actual
central banks focus upon interest rates, not monetary base growth rates, in
designing their policy actions. Especially in an environment with near-zero
call rates, however, its prescriptions may be of interest. In any event, the
actual and rule (2) settings for base growth rates are shown in the lower panel
of Figure 1.5 There the indication is that actual BOJ policy has been too tight
virtually all of the time since the middle of 1990!6
Increased base money growth rates have been recommended for several
years by Mr. Nobuyuki Nakahara, a member of the BOJ’s Monetary Policy
Board (MPB).7 But until the change that was announced at the MPB meeting
on March 19, 2001, the BOJ’s position was that additional base growth would
have no stimulative effect since short-term nominal interest rates were close to
zero. With such low rates, base money and short-term government securities
(bills) become almost perfect substitutes, so purchases of the latter by the
BOJ have no effect on asset markets and consequently none on the economy,
according to the BOJ view. That position will be discussed in the next two
sections.

2. THE BANK OF JAPAN’S DIFFICULTY
Over the period 1999–2001, commentary in influential nonacademic publications including the Economist, the Financial Times, and the Wall Street Journal became increasingly critical of the BOJ for not providing more monetary
stimulus to aggregate demand in Japan. The plots presented in the previous
section also suggest that more stimulus is needed and has been needed for
years, but nevertheless I believe that much of the press commentary has failed
to recognize the difficulty of the problem that has faced the BOJ. It is not
just stubbornness that has prevented the BOJ from providing such stimulus,
for the nature of monetary policy actions is sharply different when short-term
interest rates are effectively equal to zero. It is not true that there has been
“nothing more that the BOJ can do,” but what needs to be done is different
than in normal conditions and the policy actions are more difficult to design.
5 The plot is reproduced from the same source as before, which provides details.
6 Some early indication that BOJ policy was too tight during 1990–92 appears in McCallum

(1993, 35–36). Also see McCallum and Hargraves (1995).
7 Mr. Nakahara’s term as an MPB member ended in April 2002. The MPB currently includes
Mr. Shin Nakahara, who is not related to Mr. Nobuyuki Nakahara.

B. T. McCallum: Japanese Monetary Policy

For some years, the BOJ took the position that nothing more could be
done, beyond lowering its overnight call rate well below one percent and
finally almost to zero. These statements were of questionable validity, as we
shall see, and perhaps reflected a fundamentally misguided tendency to think
of levels of nominal interest rates as direct indicators of monetary conditions,
with low rates representing expansionary policy. In fact, nominal rates will
be low (for given real rates) when expected inflation is low; thus low rates are
in large part an indication that monetary policy has been tight in the past, not
that it is loose in the present. Recognizing this last point, several critics have
argued that the BOJ should gauge its actions in terms of monetary base growth
rates, rather than interest rates, and should provide stimulus by increasing the
growth rate of the monetary base. As can be seen in the bottom half of Figure
1, my base-growth-oriented policy rule would have called for about 11 percent
(per annum) growth rates over the period 1996–1998, rather than the values
of about half that magnitude that were actually recorded.
It is crucial to recognize, however, that just expanding the base growth rate
will not be effective, in the face of zero interest rates, unless nontraditional
assets are purchased. Normally, open market operations are conducted by
exchanging base money for short-term government bills. But when shortterm interest rates are near zero, such purchases will have virtually no effect.
One way to understand this point is to recall that both base money and bills
are nominally-denominated paper assets that are virtually free from default
risk. What then is the difference between them as assets; why do people and
firms hold money when bills normally provide the holder with a higher rate
of interest? The answer, from traditional monetary theory, is that money is a
generally accepted medium of exchange that provides transaction-facilitating
services to its holders—services not provided by bills.8 Rational economic
agents then adjust their holdings of these two assets so as to equalize their net
benefits at the margin. The sum of pecuniary interest earnings plus transactionfacilitating services is equated at the margin, for the two assets, with interest
earnings usually being lower and services higher for base money assets.
But when short-term interest rates fall to zero, then there is no difference
in the interest component of the net yield for the two assets, so their marginal
service yields will also be equal. That condition is brought about by holders
choosing to keep on hand a quantity of money large enough that its service
yield at the margin is driven down to zero. But then, at the margin, base money
and bills become perfect substitutes—the distinguishing characteristic of base
money is lost (at the margin, not overall)! Consequently, open-market operations that exchange bills for money in private portfolios have effects that are
like those of replacing a billion dollars’worth of $5 Federal Reserve Notes with
8 Or provided to a lesser extent by bills. For a review of traditional monetary analysis, see
McCallum and Goodfriend (1988).

Federal Reserve Bank of Richmond Economic Quarterly

a billion dollars’ worth of $10 Federal Reserve Notes. To an approximation, in
other words, there is no effect. Accordingly, an expansionary monetary policy
needs to be implemented in some nontraditional manner, e.g., by purchase of
nontraditional assets. Such a purchase would alter the composition, in private
portfolios, of these other assets relative to the sum of money plus government
bills, thereby stimulating some response on the part of private asset holders.9

SOME QUANTITATIVE RESULTS

Is there any empirical evidence supportive of the theoretical view just described? A very simple but straightforward way to approach that question is
to examine the relationship between base money growth and the growth rate
of nominal GDP. To that end, let us consider an updated and modified version of the simplest macroeconomic model of aggregate demand utilized in
McCallum (1993). It is a single-equation dynamic relationship of nominal income growth and its dependence on money base growth. Let xt and bt denote
logarithms of nominal GDP and the adjusted monetary base, respectively, so
that xt and bt are quarterly growth rates. The data series utilized extend
from 1970Q1 through 2001Q3 and are seasonally adjusted.10
Least-squares estimation over the period 1970Q3–2001Q3 yields the following relationship:
xt

= −0.0002 + 0.261 xt−1 + 0.344 xt−2 + 0.248 bt−1
(.0019) (.0873)
(.0840)
(.0887)

(3)

R 2 = 0.483 SE = 0.0116 DW = 2.15.
The numbers in parentheses are standard errors, so bt−1 evidently has a
highly significant effect on xt and its subsequent values. Thus if this relation
were structural, it would indicate that a money base rule could be devised to
keep nominal GDP growth reasonably close to a desired target path. A similar
relationship was utilized in that manner in McCallum (1993), where it provided
results quite comparable to those of small but somewhat more complex models
intended to be structural.11
The issue to be examined here, by contrast, is whether the relationship between base growth and nominal income has “broken down” in recent years—as
9 One reader has suggested that it would be stimulative for the central bank to simply print
base money and give it to private individuals. In fact such a process would create an imbalance
in private portfolios and thereby lead to some type of reaction. But such a scheme combines
both monetary and fiscal policy. It is equivalent to a fiscal transfer of government bills to private
agents plus an open-market purchase of bills.
10 These series were obtained from the web pages of the BOJ (monetary base) and the
Japanese government’s Economic and Social Research Institute (GDP).
11 For a discussion of the potential vulnerability of the relationship to the Lucas critique, see
McCallum (1993, 37–38).

B. T. McCallum: Japanese Monetary Policy

would arguably be the case with near-zero interest rates and traditional openmarket purchases of government bills. In fact, such an impression is supported
by visual inspection of a simple plot of those two variables against time. To
consider the matter more formally, however, I have reestimated relationship
(3) permitting crucial parameters to change in 1995Q1.12 Inclusion of a 0-1
dummy variable, that changes from 0 to 1 in 1995Q1, indicates a downward
shift in the equation’s constant term, with a highly significant t-statistic of
–3.05. If instead the slope coefficient on the base growth variable is permitted to change at that time, again a significant decrease is detected, with the
t-statistic being –3.31. Inclusion of both effects seems most appropriate (since
the two variables are highly collinear) and leads to the following estimates:
xt

= 0.0031 + 0.137xt−1 + 0.210xt−2 + 0.399
(.0022) (.091)
(.090)
(.103)

(4)

bt−1 − 0.318 D95 · bt−1 − 0.0045 D95
(.192)
(.0041)

R 2 = 0.531 SE = 0.0111 DW = 2.10.
Here we see that the estimate of the net effect of bt−1 on xt for the post1995 observations is 0.399−0.318 = 0.081, a very small value. Furthermore,
a Wald test of the hypothesis that the net effect equals zero gives a P-value
of 0.617, indicating that a zero-effect hypothesis could not be rejected at any
conventional significance level.13 For all practical purposes, then, the recent
effect on nominal GDP growth of additional money base growth has been zero,
according to these last estimates. This finding is consistent with the notion
that, in a situation with near-zero short-term interest rates, BOJ purchase of
treasury bills will be ineffective as means of stimulating aggregate demand.
Of course, the simple investigation just conducted falls well short of what
would be required for a convincing counterfactual policy exercise, which
would require a well-specified structural model. But the results here are not
being used in that manner, i.e., to assess the effects of an alternative policy
rule. Instead they are being used only to indicate that a substantial breakdown
in the money-GDP relationship has occurred. For that purpose, the foregoing
exercise should be adequate.
12 This break date, or one close to it, is suggested by the extensive recent empirical study
by Miao (2000).
13 Together the two shift terms are highly significant; a Wald test of the hypothesis that both
coefficients equal zero results in a P-value of 0.0022.

8
4.

Federal Reserve Bank of Richmond Economic Quarterly
POLICY PROPOSALS

Let us now turn to the crucial issue, namely, how monetary policy can be conducted in a situation with interest rates near zero. Several prominent monetary
economists have taken up this issue, including Marvin Goodfriend (1997,
2000), Paul Krugman (1998, 2000), Allan Meltzer (1998, 1999, 2000), Athanasios Orphanides and Volker Wieland (2000), Willem Buiter and Nikolaos Panigirtzoglu (2001), Lars Svensson (2001), and myself (McCallum 2000a, 2002).
Goodfriend (2000) and Buiter and Panigirtzoglu propose a tax on base money
that would keep it from being a perfect substitute for short-term securities and
thereby open the way for an effective monetary policy even when a zero-lowerbound situation is in effect. This scheme’s logic is evidently impeccable, but
the probable unpopularity of the explicit tax on money would seem to present
a formidable practical barrier (even though it would make possible a reduced
average level of the implicit tax on money). Accordingly we will focus on the
other proposals, which involve the central bank purchase with base money of
assets other than the traditional short-term yen securities. Meltzer (2000), for
example, suggests that the purchase of long-term Japanese government bonds
would be stimulative. McCallum and Svensson suggest instead the purchase
of foreign exchange (i.e., short-term securities that are claims to dollars or
other non-yen currencies). The general ideas behind these asset-purchase
proposals are basically similar, but there are important practical differences.
As explained above, the basic idea is that for increased growth of base
money to be stimulative, it is necessary that the assets bought from private
portfolios be ones that are not perfect substitutes for government bills (or
for money). Otherwise, the composition of private portfolios will not be
affected in an economically relevant manner so no response will be induced.
Obviously, longer term government bonds represent one leading possibility.
But according to the expectations theory of the term structure, which says
that long-term interest rates are appropriate averages of expected short-term
rates, long-term and short-term government securities are perfect substitutes.
Now, there is evidence strongly suggesting that this theory is not empirically
accurate, but there is no widely accepted alternative to rely upon. What is
needed is a theory of the term premium that relates variations in that premium
to asset positions. In the absence of any widely accepted theory of that type,
it is not obvious how to design an appropriate policy or even that purchases of
long-term bonds would have an effect on aggregate demand in the appropriate
direction.14
Consequently, I have suggested that the best course of action would be
for the BOJ—or any central bank in a similar situation—to purchase foreign
exchange (McCallum, 2000a, 2002). Lars Svensson (2001) has made a closely
14 A different and more optimistic position has recently been expressed by Goodfriend (2001).

B. T. McCallum: Japanese Monetary Policy

related proposal.15 The difference is that in this case there is a more well
understood transmission channel, working via the exchange rate. It is clear
that the purchase of foreign exchange tends to depreciate the value of the
yen. With prices in Japan initially rising less rapidly than the price of foreign
exchange,16 a real exchange rate depreciation would result, and this would
tend to stimulate exports and to increase Japanese income and production.
That is of course what is desired—to increase Japanese income and spending.
It is important to keep in mind, in this regard, that increases in income
have strong and reliable positive effects on imports. Indeed, the strength of
income effects on imports is probably strong enough that the overall effect
of the stimulative policy would be to increase Japan’s imports (in real terms)
from its trading partners. Under that assumption it is not the case that the
recommended policy would tend to depress aggregate demand in other nations.
Fear of that outcome is therefore not a sensible reason for avoiding stimulative
monetary policy.17 Indeed, it is not even clear that such a policy would induce
the real exchange rate to appreciate for more than a short period of time. These
issues will be quantitatively explored below, in Sections 5 and 6.
A few critics of the foreign-exchange strategy have contended that a central
bank cannot reliably influence its currency’s exchange rate. In that regard it
is widely believed by analysts that raising a currency’s real foreign-exchange
value by monetary policy is not possible, and that keeping its nominal value
high would require extreme measures that would not be tolerated for long in
a nation with democratic political processes. But to depreciate a fiat currency
in nominal terms is not difficult; the basic requirement is simply the creation
of an excess quantity of the currency.18 And a reduction in value is what is
needed in the case of Japan.19
Proceeding under the presumption that a central bank can exert adequate
control over its currency’s nominal exchange rate, McCallum (2000a, 2002)
has considered a policy rule for use in a zero-lower-bound situation of the
following form, with st representing the log of the home-country price of
15 It should be noted that a few economists, including myself, Marvin Goodfriend (1997),
Allan Meltzer (1998, 1999, 2000), and John Taylor (1997), have been urging a more expansionary
policy for the BOJ at least since 1995. Our first proposals did not, however, emphasize purchases
of foreign exchange per se.
16 Even in the event that Japanese domestic prices increased along with the price of foreign
exchange, there would be a benefit—this would raise nominal interest rates, leading to an escape
from the “liquidity trap” situation described above.
17 It is my impression that fear of this outcome did, in fact, keep U.S. and international
agencies from supporting policy proposals of the type expressed here, until recently. See Section
5 below.
18 For a more detailed argument, see McCallum (2000a, 2002).
19 Even a depreciation could not be effected if the currency were literally a perfect substitute
for foreign currencies, but such is not the case. Interesting new evidence of a market-microstructure
type has recently been developed by Evans and Lyons (2000, 2002).

Federal Reserve Bank of Richmond Economic Quarterly

foreign exchange:
st = µ0 + µ1 (2 − pt ) + µ2 (ȳt − yt ), µ1 , µ2 > 0.

(5)

Here the rate of depreciation of the exchange rate st is increased when
inflation and/or output are below their target values. Such a rule would be
implemented in a manner similar to that typically used with an interest-rate
instrument. Specifically, the central bank would observe the relevant asset
price almost continuously and make open-market purchases (sales) when a
depreciation (appreciation) is indicated.20 It is important to note that rule (5)
does not represent a fixed exchange rate. To the contrary, it represents a regime
that subordinates the exchange rate entirely to macroeconomic conditions.
Quite recently, in 2001 and 2002, the BOJ has taken actions that indicate an
intention to pursue a more stimulative policy than in the past. To date, however,
it has not given any official recognition to the possibility of purchasing foreign
exchange as a way of providing a more stimulative monetary policy.21 We
need to look into the reasons for this neglect, of which two are prominent.
One of these involves the BOJ’s legal charter and the other stems from beliefs
concerning the effects on other nations’ balance of payments magnitudes.
Since the latter topic is the more analytical in nature, it will be considered
first.

5. THE BALANCE OF PAYMENTS ISSUE
In this section we take up a major analytical issue concerning the policy position presented above. During the late 1990s, some leading officials of the
International Monetary Fund were opposed to monetary stimulus as a means
for combating Japan’s ongoing economic weakness.22 Their reason was a belief that monetary stimulus would lead to exchange rate depreciation, which
would be harmful to other nations seeking to expand (or, during the Asian
crisis, maintain) exports to Japan. This source of objection to a more stimulative monetary policy is, however, inappropriate. First, it is highly unlikely
that such a policy would lead to lessened imports by Japan, for an increase
20 As with current practice, market participants may to some extent move rates as desired
by the central bank, even without actual open-market operations, if the central bank’s intentions
are made clear.
21 In an interview with Bloomberg reported on July 19, 2001, Dr. Kunio Okina, Director of

the BOJ’s Institute for Monetary and Economic Studies, suggested that the BOJ should consider
purchase of foreign exchange as a tool of monetary policy, while leaving exchange rates to the
currency market. But on July 25, Mr. Sakuya Fujiwara, Deputy Governor of the BOJ, indicated (in
a question-and-answer session at the Tokyo Foreign Correspondents’ Club) that Okina’s suggestion
does not reflect BOJ policy.
22 This claim is based in part on personal conversations. For some evidence, see Fischer
(1998), which proposes fiscal expansion and banking reforms but does not mention monetary policy. In his very recent comment in the Brookings Papers, Fischer accepts the need for Japanese
monetary stimulus, but still labels this a “beggar-thy-neighbor” policy (2001, 165).

B. T. McCallum: Japanese Monetary Policy

in Japanese real income would tend to increase imports and most likely to
an extent greater than any decrease brought about by Japanese exchange rate
depreciation. Second, according to recent views of most academics and policymakers alike, monetary policy should be directed primarily toward keeping
inflation low (but non-negative!), with the avoidance of real cyclical fluctuations a possible secondary objective.23 From this perspective, fiscal and
structural policies are more appropriate tools to use in managing balance-ofpayments problems. Thus, if Japan is not going to share a common currency
with, e.g., the United States, then their bilateral exchange rate should be free
to float with each country managing its monetary affairs so as to keep a low
inflation rate.24 From this perspective, one could argue that the United States
should not try to use its political influence to prevent a depreciation of the yen.
More generally, it would seem undesirable for any country to attempt to induce
other nations to manage their monetary policy in ways that are domestically
harmful but temporarily helpful for the country in question.25 From a longterm perspective, the United States will benefit from having other important
nations conduct their monetary policies in a manner that yields low inflation
with domestic macroeconomic stability.
Not all analysts would agree, however, with the contention that monetary
stimulus of the type here suggested would not have a depressing effect on
other nations’ exports to Japan. Accordingly, this section and the next will be
devoted to a substantial consideration of that position. For such an issue it is
necessary to conduct analysis in a quantitatively specified structural model,
and the convincingness of the exercise will depend upon the qualifications of
the model utilized. The one that will be used was developed by McCallum
and Nelson (1999) and utilized subsequently by them (2000) in an exploration
of relationships between CPI inflation and exchange-rate depreciation. The
model is not econometrically estimated using Japanese time series data, but
is a “new open-economy macroeconomic model”—i.e., is based on dynamic
optimizing analysis assuming sticky-price adjustments and solved assuming
rational expectations—that has been calibrated to match certain characteristics
of the Japanese economy. It differs from other contributions in the area,
however, in the manner in which imported goods are treated. In particular, the
M-N model treats imports not as finished goods, as is common, but instead
as raw-material inputs to the home economy’s producers. This alternative
23 Real cyclical conditions should provide only a secondary objective for monetary policy because monetary effects on these conditions are temporary and poorly understood, whereas monetary
effects on prices (and thus on inflation rates) are long-lasting and well understood.
24 Moreover, decisions to share a common currency should be made on grounds of microeconomic efficiency, not in an attempt to solve macroeconomic stabilization difficulties.
25 Indeed, it may well have been U.S. pressure that led the BOJ to be somewhat too loose
(even on traditional standards that ignore asset price movements—see Figure 1, lower panel) during
1986–88, a stance that permitted Japan’s asset price boom of the late 1980s and set the stage for
a clampdown that began the past decade’s slump.

Federal Reserve Bank of Richmond Economic Quarterly

modelling strategy leads to a cleaner and simpler theoretical structure, relative
to the standard treatment, and is empirically attractive. Since the optimizing,
general equilibrium analysis has previously been presented in McCallum and
Nelson (1999), here I will take an informal expository approach designed to
facilitate understanding of the model’s basic structure.
It is well known that optimizing analysis leads, in a wide variety of infinitehorizon models that involve imperfect competition, to a consumption Euler
equation that can be expressed or approximated in the form
ct = Et ct+1 + b0 + b1 rt + vt ,

(6)

where ct is the log of a Dixit-Stiglitz consumption-bundle aggregate of the
many distinct goods that a typical household consumes in period t.26 In
(6), rt is the real interest rate on home-country one period bonds (private
or government) and vt is a stochastic shock term that pertains to household
preferences regarding present versus future consumption. In closed-economy
analysis, relation (6) is often combined with a log-linearized, per-household,
overall resource constraint to yield an “expectational I S function,” to use the
term of Kerr and King (1996). That step presumes that investment and capital
are treated as exogenous. The simplest version of that assumption is that the
capital stock is fixed; since that assumption is rather common in the literature,
it is adopted here.
For an open-economy extension, one might be tempted to write the resource constraint as
yt = ω1 ct + ω2 gt + ω3 xt − ω4 imt ,

(7)

where yt , gt , xt , and imt are logarithms of real output, government consumption, exports, and imports while ω1 , ω2 , ω3 , and ω4 are steady state shares
of output for consumption, government purchases, exports, and imports. But
if imports are exclusively material inputs to the production of home-country
goods, and Yt = ln −1 yt is interpreted as units of output, then the relevant
resource constraint is
yt = ω1 ct + ω2 gt + ω3 xt .

(7 )

It is desirable that import demand be modelled in an optimizing fashion.
Toward that end, assume that output of all consumer goods is effected by
producers that are constrained by production functions all of the same CES
form, with labor and material imports being the two variable inputs. Then the
cost-minimizing demand for imports is
imt = yt − σ qt + const.,

(8)

26 Thus c = ln C , with C = [ C (z)(θ−1)/θ dz]θ/(θ −1) , where θ > 1, z indexes
t
t
t
t

distinct
goods, and the integral is over (0, 1), while the corresponding price index is Pt =

[ Pt (z)1−θ dz]1/(1−θ) .

B. T. McCallum: Japanese Monetary Policy

where σ is the elasticity of substitution between materials and labor in production, and where “const.” denotes some constant.27 Also, qt is the log
price of imports in terms of produced consumption goods. We will refer to
Qt = ln−1 qt as the real exchange rate. Let Pt and St be the home-country
money price of goods and foreign exchange, with Pt∗ the foreign money price
of home-country imports. Then if pt , st , and pt∗ are logs of these variables,
we have
qt = st − pt + pt∗ .

(9)

Symmetrically, we assume that export demand is given as
xt = yt∗ + σ ∗ qt + const.,

(10)

where yt∗ denotes production abroad and σ ∗ is the price elasticity of demand
from abroad for home-country goods.
Now consider output determination in a flexible-price version of the model.
Taking a log-linear approximation to the home-country production function,
we have
yt = (1 − α)at + (1 − α)nt + α imt + const.,
where nt and at are logs of labor input and a labor augmenting technology
shock term, respectively. Suppose for simplicity that labor supply is inelastic, with 1.0 units supplied per period by each household. Thus with full
price flexibility we would have nt = 0 and the flexible-price, natural rate (or
“potential”) value of yt will be ȳt = (1 − α)at + α imt + const. so that
ȳt = (1 − α)at + α[ȳt − σ qt ] + const., or
ȳt = at − [σ α/(1 − α)]qt + const.

(11)

But while ȳt would be the economy’s output in period t if prices could adjust
promptly in response to any shock, we assume that prices adjust only sluggishly. And if the economy’s demand quantity as determined by the rest of
the system (yt ) differs from ȳt then the former quantity prevails—and workers depart from their (inelastic) supply schedules so as to provide whatever
quantity is needed to produce the demanded output, with imt given by (8).
In such a setting, the precise way in which prices adjust has a direct
impact on demand and consequently on production. There are various models
of gradual price adjustment utilized in the recent literature that are intended
to represent optimizing behavior. In the analysis that follows, I will use
pt = 0.5(Et pt+1 + pt−1 ) + φ 2 (yt − ȳt ) + ut ,

(12)

where ut is a behavioral disturbance. This form of equation has been fairly
prominent, primarily because it tends to impart a more realistic degree of
27 That is, the expression “const.” in different equations appearing through the remainder of
the article will typically refer to different constant magnitudes.

Federal Reserve Bank of Richmond Economic Quarterly

inflation persistence than does the Calvo-Rotemberg model (which is theoretically more attractive).28
A standard feature of most current open-economy models is a relation
implying uncovered interest parity (UIP). Despite its prominent empirical
weaknesses, accordingly, the basic M-N model includes one:
Rt − Rt∗ = Et st+1 + ξ t .

(13)

We include a time-varying “risk premium” term ξ t , however, that may have a
sizeable variance and may be autocorrelated.
It remains to describe how monetary policy is conducted. In most recent
research in monetary economics, it is presumed that the monetary authority
conducts policy by adjusting a one-period nominal interest rate in response
to prevailing (or forecasted future) values of inflation and the output gap,
ỹt = yt − ȳt :
Rt = (1 − µ3 )[µ0 + pt + µ1 (pt − π ∗ ) + µ2 ỹt ] + µ3 Rt−1 + et . (14)
Here µ3 > 0 reflects interest rate smoothing. Quantitative results reported
by M-N (1999, 2000) are based on estimated or calibrated versions of this rule,
in most cases with Et−1 applied to ỹt and pt .
To complete the model, we need only to include the Fisher identity,
(1 + rt ) = (1 + Rt )/(1 + Et pt+1 ), which we approximate in the familiar fashion:
rt = Rt − Et pt+1 .

(15)

Thus we have a simple log-linear system in which the ten structural relations
(6)–(15) determine values for the endogenous variables yt , ȳt , pt , rt , Rt , qt ,
st , ct , xt , and imt . Government spending gt and the foreign variables pt∗ , yt∗ ,
Rt∗ are taken to be exogenous—as are the shock processes for vt , at , et , and
ξt.
Of course, it would be possible to append a money demand function such
as
mt − pt = γ 0 + γ 1 yt + γ 2 Rt + ηt ,

(16)

and one of this general form—perhaps with ct replacing yt —would be consistent with optimizing behavior.29 But, as many writers have noted, that
equation would serve only to determine the values of mt that are needed to
implement the Rt policy rule.
With the structure given above, a useful measure of the balance on goods
and services account—i.e., net exports—is
nett = xt − (imt + qt ),
28 See Fuhrer and Moore (1995) and Clarida, Gali, and Gertler (1999).
29 See McCallum and Nelson (1999) or many other papers.

(17)

B. T. McCallum: Japanese Monetary Policy

where it is assumed that ω3 = ω4 . This measure is used in what follows. Also,
incidentally, it is possible to calculate the log of the GDP deflator as
ptDEF = [pt − ω3 (st + pt∗ )]/(1 − ω3 ).

(18)

Before moving on, it should be noted that an advantage of our strategy
of modelling imports as material inputs to the production process is that the
relevant price index for produced goods is the same as the consumer price index, which implies that the same gradual price adjustment behavior is relevant
for all domestic consumption. In addition, it avoids the unattractive assumption, implied by the tradeable versus nontradeable goods dichotomization, that
export and import goods are perfectly substitutable in production.
Theoretical advantages would not constitute a satisfactory justification,
of course, if in fact most imports were consumption goods. Such is not the
case, however, at least for the United States. Instead, an examination of the
data suggests that (under conservative assumptions) intermediate productive
inputs actually comprise a larger fraction of U.S. imports than do consumer
goods (including services).30
There is one way in which the model developed in McCallum and Nelson
(1999) differs significantly from the 10-equation formulation just presented.
Specifically, the M-N model includes a somewhat more complex form of
consumption versus saving behavior, one that features habit formation. Thus
in place of the time-separable utility function that leads to equation (6), we
assume that each period-t utility term includes ct /(ct−1 )h , with 0 ≤ h < 1,
rather than ct alone. That specification gives rise to the following replacement
for (6):
ct = h0 + h1 ct−1 + h2 Et ct+1 + h3 Et ct+2 + h4 (log λt ) + vt .

(6 )

Here λt is the Lagrange multiplier on the household’s budget constraint,
which obeys
log λt = const. + Et λt+1 + rt ,

(19)

and there are constraints relating the hj parameters to others in the system.
For details and additional discussion, see M-N (1999) and the recent study of
Fuhrer (2000).
Calibration of the model draws on M-N (1999) but differs in a few ways
that, in retrospect, seem appropriate. For the parameters governing spending
30 For the year 1998, imported consumer goods amounted to $453 billion while imports of
business inputs came to $624 billion, approximately. These figures are based on an examination
of categories reported in the August 1999 issue of the Survey of Current Business. For several
categories it is clear whether they are composed predominantly of consumer or business goods.
For others, judgmental assignments were required. Those assignments are as follows, with the reported figure being the fraction of the category classified as “business inputs”: automotive vehicles,
engines, and parts, 25 percent; travel, 25 percent; passenger fares, 25 percent; foods, feed, and
beverages, 50 percent; and other private services, 75 percent.

Federal Reserve Bank of Richmond Economic Quarterly

behavior, I retain here the h = 0.8 value taken from an early version of Fuhrer
(2000), but for the counterpart of b1 I now use −0.4, rather than −1/6, in
order to reflect the greater responsiveness of investment spending, which is
not included explicitly in the model.31 For σ , the elasticity of substitution in
production (and therefore the elasticity of import demand with respect to Qt ),
I again begin with 1/3—and for the elasticity of export demand with respect
to Qt the same value is used—but also consider larger values. In (11), the
labor-share parameter 1 − α equals 0.64. The steady state ratio of imports
(and exports) to domestic production is taken to be 0.10, a slightly lower value
than in M-N (1999) so as to reflect the Japanese degree of openness. For the
present application government consumption is included, with ω2 = 0.2.
In the price adjustment relation, the specification is that φ 2 = 0.03. The
latter value is based on my reading of a wide variety of studies, plus conversion into nonannualized fractional terms for a quarterly model. Policy
rule parameters should be thought of in relation to realistic values close to
µ1 = 0.5, µ2 = 0.4, and µ3 = 0.8, the latter reflecting considerable interest
rate smoothing. In the experiments reported in this paper, however, rule (14)
is replaced by the rule (5) that is designed for the zero-lower-bound situation.
In most cases, expectations based on t − 1 data are used for the pt and ỹt
variables appearing in the policy rule, in order to make the latter operational.
The stochastic processes driving the model’s shocks must also be calibrated, of course. For both foreign output and the technology shock, I have
specified AR(1) processes with AR parameters of 0.95, rather than the 1.0
values used in M-N (1999). The innovation standard deviations (SD) are 0.03
and (as before) 0.0035. The latter value might appear smaller than is usual, but
is appropriate to generate a realistic degree of variability in ȳt when the latter
is not exogenous but instead is dependent on qt . The UIP risk premium term
ξ t is generated by an AR(1) process with AR parameter 0.5 and innovation
0.04; these values are based on work reported in Taylor (1993b). Government
consumption (in logs) follows an AR(1) process, with AR parameter 0.99 and
innovation SD of 0.02. Finally, the vt , ut , and et shock processes are taken to
be white noise with SD values of 0.011, 0.002, and 0.0017, respectively.

SIMULATION RESULTS

In McCallum (2000a, 2002) I have conducted exercises with this model under
the assumption that the nominal interest rate is immobilized at zero, in order to
show that monetary policy conducted by means of a rule such as (5) would provide stabilizing influence despite the “liquidity trap” situation. Those policy
experiments were not designed, however, to reflect the transitional effects of
31 The parameter in question is the intertemporal elasticity of substitution in consumption
when h = 0.

B. T. McCallum: Japanese Monetary Policy

Figure 2 Responses to Policy Change, Initial Case

the adoption of such a rule; they were conducted as if the rule had been in
effect for a long period of time. In what follows, I will use a different strategy.
It will again be assumed that an exchange-rate rule has been in effect, but
the initial equilibrium is one that leaves the zero-interest situation intact. The
objective is to break out of that situation, so the “shock” to which the system
is subjected is an increase in the target rate of inflation. This is represented as
a permanent upward shock to π ∗ , the inflation target in the policy rule. Arbitrarily, the experiment assumes a 2.0 percent per-annum shock, e.g., from –1.0
percent inflation to +1.0 percent. In quarterly fractional units, that amounts
to an increase of 0.005 in π ∗ .32 The precise rule utilized is as follows, with
µ1 = 0.5 and µ2 = 0:
st = Et−1 pt + µ1 (π ∗ − Et−1 pt ) + µ2 (ȳt − yt ).

(20)

32 It should be said that I am not entirely comfortable with analysis of this type of “shock,”
which seems more like a regime change than the type of shock that RE policy analysis is best
designed to handle. Consequently, I would not take details of the dynamics too seriously, but
would limit attention to the general nature of the responses. (Many economists do, of course,
use rational expectations to analyze the effects of policy regime changes—i.e., to study transition
periods—but I have generally been skeptical of such studies.)

Federal Reserve Bank of Richmond Economic Quarterly

Figure 3 Responses to Policy Change, Preferred Case

The variable on whose response we shall focus is the home country’s—
i.e., Japan’s—net export balance in real terms. Since the model is formulated
to be linear in logarithms of most variables, the measure actually calculated
is the log of real exports minus the log of real imports. These have to be
expressed in common price units, so induced changes in the real exchange
rate have to be taken into account. The negative of that measure is taken to
reflect the increase in net real exports by Japan’s trading partners.
Results of the first experiment are shown in Figure 2. Responses over
40 quarter-years are shown for six variables: the log of real output (y), the
inflation rate (p), the nominal interest rate (R), the log of the real exchange
rate (q), the rate of depreciation of the nominal exchange rate (ds), and the
net export variable in (17) (net). In Figure 2 we see that the upward jump
in the target inflation rate (π), which occurs in period 1, does indeed induce
an exchange-rate depreciation rate that remains positive for over two years.
Inflation, not surprisingly, rises and stays above its initial value for over two
years, then oscillates and settles down at a new steady state rate of 0.005 (in
relation to its starting value). Quite surprisingly, p responds more strongly

B. T. McCallum: Japanese Monetary Policy

Figure 4 Responses to Policy Change, Alternative Case

than s so the real exchange rate appreciates.33 As expected, however, real
output rises strongly for two years. Most importantly, the real (Japanese)
export balance is so affected by the two-year increase in real output that it
turns negative and stays negative for almost two years, although it levels off
at a positive value. This pattern is only partly supportive of the argument
advanced above, but a single plausible change in the calibration alters it so as
to be almost entirely supportive.
The relevant point is that the parameter values used in Figure 2 include
figures of 1/3 for the import price elasticities (σ ) both at home and abroad.34
That figure, originally selected by McCallum and Nelson (1999) for reasons
that do not pertain in the present exercise, are quite small. Most specialists
contend that such magnitudes are substantially larger, at least large enough to
satisfy the venerable Marshall-Lerner condition (i.e., that their sum equals 1.0
or more). Accordingly, in Figure 3 the same experiment is repeated but with
33 It has been verified that steady state response value is zero, reflecting long-run monetary
neutrality. But it takes many years for q to return to its original vicinity.
34 In what follows, I will describe these elasticities as if they were both positive numbers.

Federal Reserve Bank of Richmond Economic Quarterly

Figure 5 Responses with Rt Kept at Zero

values of 1/2 for each of these import price elasticities. There the effect of
the real exchange rate appreciation is eliminated, and the net export balance
reflects only the movement of output. Thus Japan’s net exports remain negative
for about two years, briefly turn positive, and then finally stay slightly negative
for a long time (despite long-run neutrality). Setting each country’s import
price elasticity instead at 1.0, close to the conventional wisdom, gives an
entirely different picture, with net exports staying strongly negative for a very
long time—see Figure 4.
There are many parameter changes that could be considered, but the more
important order of business is to discuss the upward movement of the interest
rate Rt that occurs in Figures 2–4. It is clear that the long-run response
is a rise of 0.005, which must of course be the case if there is monetary
superneutrality and an upward jump in target inflation of that magnitude.
But how are the dynamics in Rt being modeled? One extreme possibility is
that uncovered interest parity is maintained throughout. But that would be
inconsistent with my basic position (and with huge quantities of empirical
research). Accordingly, my first attempts at this simulation exercise assumed
that the interest rate remains immobilized at its initial zero-lower-bound level.

B. T. McCallum: Japanese Monetary Policy

Figure 6 Responses with UIP Maintained Throughout

That specification leads, however, to the results shown in Figure 5. There the
responses are implausibly large, with inflation rising to almost 20,000 percent
per year. (Recall that the numbers shown are in fractional quarterly units.)
This might seem to reflect some kind of calculation error, but actually the
point is that if Rt were held unchanged, the increased inflation rate would
imply a reduction in the real interest rate of 2 percent per year, maintained
forever. In a forward-looking rational expectations model, such a change has
enormous effects. Furthermore, this way of treating the nominal interest rate
is inconsistent with superneutrality and inconsistent with what one believes
would happen in the face of a permanently increased inflation rate.
The other extreme treatment of Rt dynamics is to impose uncovered interest parity in all periods. In that case, which I have already described as
unrealistic, we have the results shown in Figure 6, where the responses are all
very small. Consequently, for the experiments reported in Figures 2–4, I have
imposed the following compromise formula:
uip

Rt = θ Rt

+ (1 − θ )Rt−1 , (0 ≤ θ ≤ 1),

(21)

Federal Reserve Bank of Richmond Economic Quarterly

Figure 7 Responses with Fast Rt Adjustment

uip

where Rt is the value that would prevail if uncovered interest parity held in
each period. The value used for θ in Figures 2–4 is 0.01, but the results for xt
are not much different qualitatively if one adopts a value of 0.1 or 0.001—see
Figures 7 and 8. That is, the net export variable follows a pattern of the same
shape; quantitatively the effects are larger the smaller is θ. In all of Figures
5–8, the import price elasticity was kept at 1/2.
In sum, the simulation results suggest strongly that the move to a more
expansionary monetary policy by the BOJ, implemented by policy rule (5),
would not have beggar-thy-neighbor effects on Japan’s trading partners but
instead would induce an increase in their net exports to Japan.

7. THE BANK OF JAPAN LAW, MONETARY POLICY, AND
EXCHANGE-RATE POLICY
Let us now turn to the second major objection that has been voiced to the adoption of a policy rule such as (5), namely, that foreign exchange purchases and
sales cannot legally be conducted by the BOJ according to its charter. Only
a few years ago, in 1998, did the BOJ gain monetary policy independence,

B. T. McCallum: Japanese Monetary Policy

Figure 8 Responses with Slow Rt Adjustment

i.e., the right and duty to conduct monetary policy as judged appropriate by
itself (rather than by the Ministry of Finance).35 The provisions of this independence are codified in a legal document that, in English, is termed “The
Bank of Japan Law.” The provisions of this law are of direct relevance because the BOJ evidently has seen the Law as an obstacle to a policy of the
type recommended above. Purchases of foreign exchange, it is contended, are
the province of the Ministry of Finance, not the BOJ. An unofficial English
translation of the Law, made by the BOJ, can be found on the BOJ’s web site
(http://www.boj.or.jp). The following comments and interpretation are based
on that version, as amended January 6, 2001.
The BOJ Law mentions foreign exchange purchases only in Articles 15,
40, 41, and 42. These references all simply presume that such purchases will
be made either for the purpose of “cooperating. . . with foreign central banks
and international institutions. . . ” or else “to stabilize the exchange rate of
the national currency.” Those activities, furthermore, are to be conducted
35 The law was promulgated on June 11, 1997, and put into effect on April 1, 1998. It has
been amended several times.

Federal Reserve Bank of Richmond Economic Quarterly

in a manner specified by the Ministry of Finance. So viewed alone these
passages do apparently suggest that the BOJ has no mandate to purchase
foreign exchange in the manner suggested above, i.e., for macroeconomic
demand management.
However, Articles 1 and 2 of the Law stipulate that a primary duty of the
BOJ is to “carry out currency and monetary control. . . ” in a manner “aimed at,
through the pursuit of price stability, contributing to the sound development
of the national economy.” Also, Article 3 states that “the BOJ’s autonomy
regarding currency and monetary control shall be respected.” Thus the Law
also gives support to the idea that foreign exchange purchases for the purpose
of monetary control would be consistent with the duties that are explicitly
assigned to the BOJ. Evidently, then, there is some internal inconsistency in
the Law.
Furthermore, Article 15 states that the Policy Board will decide on matters
inclusive of “determining or altering the guidelines for currency and monetary
control in other forms,” i.e., forms other than money-market operations. This
suggests that the Law could be interpreted as stating that the Policy Board has
the authority to adopt policies for exerting monetary control by the purchase or
sale of foreign exchange. In that regard it is important to emphasize again that
the purpose of the foreign exchange transactions in question is definitely not to
stabilize the exchange rate. Instead, the recommended policy makes the level
of the exchange rate subservient to monetary policy, with the latter directed
at maintaining price stability so as to promote the sound development of the
Japanese national economy. So Article 15 adds to the evident inconsistency
in the Law.
Finally, however, we need to consider Article 43, which states that the BOJ
“. . . may not conduct any business other than those prescribed by this Law
unless such business is necessary to achieve the Bank’s objectives prescribed
by this Law and the Bank obtains authorization from the Minister of Finance
and the Prime Minister.” It would appear that this article does not rule out the
suggested activities per se, because they are integral to the BOJ’s achievement
of its assigned objectives. Under recent conditions, moreover, they might
well be deemed “necessary.” Nevertheless, it would seem to be appropriate
for the BOJ to seek approval from the Minister of Finance and the Prime
Minister, since such a step would keep the proposed actions from conflicting
with Article 43. If the government were to favor more monetary stimulus, a
well-formulated proposal would presumably meet with approval (although it
is possible that political infighting could interfere).
That the BOJ Law does not recognize foreign exchange transactions as a
means for conducting monetary policy is illogical but not actually surprising,
partly because transactions involving government bills are satisfactory and desirable under normal conditions—i.e., with interest rates substantially above
zero. Furthermore, it must be noted that the Japanese arrangements are not

B. T. McCallum: Japanese Monetary Policy

out of line with those pertaining to central banks in other economies. In the
United States, for example, it is generally understood (despite unclear legislation) that foreign exchange policy is primarily the province of the Treasury.36
That assignment has not been troublesome for U.S. monetary policy in recent years, but arguably that is so because the Treasury has seen fit to let the
foreign exchange value of the dollar be determined by market forces without
substantial intervention. Even in the euro area, where monetary legislation
for the European Central Bank is expressly designed to protect central bank
independence and direct it toward price level stability, there is an anomalous
provision regarding exchange rates of the euro vis-a-vis the dollar, the yen,
and other currencies. This anomaly appears in Article 109 of the Maastricht
Treaty, which gives the E.U. Council of Ministers (i.e., the member nations’ finance ministers37 ) the power to make agreements on an exchange-rate system
for the euro (relative to non-EU currencies) or to adopt “general orientations”
for exchange-rate policy. These actions are supposed not to conflict with the
goal of price stability, but the provision could nevertheless create difficulties.
Despite the existence of these actual arrangements, I suggest that it is
a mistake to view monetary policy and exchange-rate policy as independent
entities, as they implicitly suggest. Indeed, although it would be a slight
exaggeration to claim that monetary and exchange-rate policies are merely
different aspects of one macroeconomic policy tool, that claim comes closer
to the truth than the view suggesting that there are two distinct tools. (In
making this statement, I am assuming that the nation under discussion does
not attempt to manage exchange rates by means of direct controls, which would
of course introduce serious microeconomic inefficiencies and inducements for
corruption.) Let us develop that argument in the remainder of this section.
One way to proceed is to recall the nature of monetary arrangements under
a gold standard (or any other metallic standard). Any such arrangement on
an international basis clearly dictates exchange rates among all nations that
adopt gold-standard regimes. But such regimes are simultaneously specifications of domestic monetary standards, ones that require monetary policy to
be governed by the overriding obligation of maintaining the domestic-money
price of gold—and consequently the value of money in terms of gold.
For fiat money systems the relevant analytical point is that, from a longrun perspective, money stock and exchange-rate paths cannot be independently
controlled or managed, basically as a consequence of the long-run neutrality
of money. Short-run non-neutralities are a fact of life, of course, so there is
some scope for temporary departures of exchange rates from the paths implied
by monetary policy. These departures can be effected by fiscal actions or
36 On this topic see Broaddus and Goodfriend (1996), which takes a position similar to that
of the present section, and Hetzel (1996). The quotes on page 21 of the latter are useful.
37 The Council members are finance or economics ministers when the business is finance or

economics, in which case the Council is known as Ecofin. For other issues, other ministers will
represent the member countries. When the Council is attended by the countries’ prime ministers,
the meeting becomes a “summit.”

Federal Reserve Bank of Richmond Economic Quarterly

possibly by sterilized—hence nonmonetary—exchange market interventions.
But since such departures will only be temporary, it is inappropriate (and
dangerous) to think of them as reflecting distinct macroeconomic policies.
A counterargument that some might raise would point out that real exchange rates can be affected permanently by fiscal stances. A higher steady
state ratio of government spending to income tends, for example, to generate
a higher real foreign-exchange value of a nation’s currency. But an increased
ratio of government consumption to income has a one-time effect on the real
exchange rate, not a continuing or ongoing effect. Thus a monetary policy that
generates an average inflation rate that is inconsistent with a fixed nominal
exchange rate—or more generally a specified nominal exchange-rate path featuring a nonzero rate of depreciation or appreciation—will eventually lead to a
breakdown. Fiscal policy cannot, that is, be used to overcome long-run inconsistencies between money stock, price level, and exchange-rate paths. Useful
papers elaborating on this point have been written by Bordo and Schwartz
(1996), Garber and Svensson (1995), and Obstfeld and Rogoff (1995).
Furthermore, it is important to keep in mind that a large fraction of fiscal policy actions involves switches between bond finance and tax finance
for given streams of government purchases. This reminder is relevant because many standard and widely-used macroeconomic models incorporate
the property of Ricardian equivalence, i.e., the property that switches between
bond and tax finance have no effect on macroeconomic variables of primary
importance, including real and nominal exchange rates (and net exports).38
Admittedly, it is quite unlikely that actual economies possess this Ricardian
property in full, but evidence suggests that deviations are fairly minor. Thus
for most fiscal policy actions, there will be at most minor or short-lived effects
on exchange rates.
The other possible way of exerting a policy effect on exchange rates is via
sterilized interventions, i.e., foreign exchange transactions that are offset so as
to result in no net change in the economy’s outstanding stock of base money.
It is widely agreed by students of the issue, however, that effects of sterilized
interventions are at most small and temporary.39 Thus they too cannot provide
a means for escaping the long-run links between money stock and exchange
rate magnitudes.
Yet another way to put the argument is as follows. Most economists
agree that central banks possess only one significant monetary policy tool.
Some would describe it as control over the monetary base whereas others
would emphasize the setting of short-term interest rates. But that distinction
is unimportant with regard to the issue at hand; what matters is that there is only
one significant tool. Consequently, if the central bank is required (externally or
38 An early statement of this result is provided by Stockman (1983, 151–2).
39 For a survey of the literature, see Edison (1993).

B. T. McCallum: Japanese Monetary Policy

by its own choice) to devote that policy tool to the achievement of some target
path for an exchange rate, then the tool is not available for achievement of a
domestic macroeconomic objective—be it expressed in terms of inflation alone
or, e.g., some combination of inflation and output deviations from their target
values. In short, legislation or arrangements that give exchange-rate control
to the finance ministry, or some other branch of government, are basically
inconsistent with central bank independence.

CONCLUSION

On the basis of the arguments developed above, plus those presented in previous papers, it would appear that an appropriate policy would be for the Bank
of Japan to temporarily maintain a growth rate of base money of 10–15 percent
per year, with most of the newly created base used to purchase foreign exchange (the remainder being used to purchase long-term government bonds).40
After a growth rate of nominal GDP of 4–5 percent has been achieved, policy could then revert to more normal arrangements, with a target of about 2
percent measured inflation or 4–5 percent nominal GDP growth.41
There have been two prominent objections to this type of proposal. One
is that the proposed policy would have undesirable “beggar-thy-neighbor”
effects on Japan’s trading partners. Simulations with a calibrated model of
the “new open-economy macroeconomics” type suggest, however, that the
policy’s expansionary effects on output would lead to an increase, not a decrease, in Japanese imports. Presentation of the model and the simulation
study constitutes a major undertaking of the paper.
The second main objection has been that purchase of foreign exchange
is inconsistent with the Bank of Japan Law. But the arguments developed
above indicate that purchase of foreign exchange for the purpose of monetary
control is basically consistent with those provisions of the Law that call for the
BOJ to exert monetary control so as to contribute to the sound development
of the Japanese economy. Therefore, since the Law does not mention this
reason for conducting foreign exchange transactions, the BOJ could overcome
the Law’s internal inconsistencies by requesting approval from the Minister
of Finance and the Prime Minister. It could also seek amendment of the
Law so as to recognize the close relationship between monetary policy and
exchange-rate policy, thereby strengthening Japan’s statutory basis for central
bank independence.

40 The range 10–15 percent is suggested, admittedly loosely, by the plots in the bottom panel
of Figure 1.
41 Studies including Shiratsuka (1999) suggest that measured overstates actual inflation in
Japan by about 1 percent per year.

Federal Reserve Bank of Richmond Economic Quarterly

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Broaddus, J. Alfred, Jr., and Marvin Goodfriend. 1996. “Foreign Exchange
Operations and the Federal Reserve.” Federal Reserve Bank of
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Buiter, Willem, and Nikolaos Panigirtzoglu. 2001. “Liquidity Traps: How to
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. “Comments.” 2001. Brooking Papers on Economic Activity
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, and George R. Moore. 1995. “Inflation Persistence.”
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Collapse of Fixed Exchange Rate Regimes.” Handbook of International
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. 2000. “Overcoming the Zero Bound on Interest Rate
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. 2001. “Financial Stability, Deflation, and Monetary
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. 2000. “Thinking About the Liquidity Trap.” Journal of the
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. 1993. “Specification and Analysis of a Monetary Policy
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. 2000. “Monetary Policy for an Open
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Efficient Public Investment
in a Model with Transition
Dynamics
Pierre-Daniel G. Sarte and Jorge Soares

long history of debate exists among macroeconomists regarding the
degree to which public capital contributes to overall economic activity.
Estimates of the output elasticity with respect to public capital range
anywhere from 0.06 in early work by Ratner (1983) to as high as 0.39 in a
widely cited study by Aschauer (1989). This elasticity refers to the percentage
change in GDP induced by a given percentage change in government capital.1
Glomm and Ravikumar (1997, 197) remark that economists have generally
been skeptical of Aschauer’s estimate, mainly because “the productivity of
public capital is simply not believed to be larger than the productivity of the
private capital stock (which is roughly 0.36).” That being said, few would
argue that public infrastructure plays no productive role, and estimates of the
public capital elasticity of output lying between 0.05 and 0.15 are often put
forward.2
In an economy where government investment in equipment and structures complements private investment, it is only natural to ask what factors
determine the optimal share of public investment in output. The trade-off involves public capital that is productive but that must also be financed through
distortional taxation.
Jorge Soares is assistant professor in the Department of Economics at George Washington
University, 2201 G Street NW, Washington, DC 20052.
We wish to thank Margarida Duarte, Marvin Goodfriend, and Alex Wolman for all of their
helpful suggestions. Any errors, however, are our own. The views expressed in this article are
the authors’ and do not necessarily represent those of the Federal Reserve Bank of Richmond
or the Federal Reserve System.
1 For any variables x and y, the elasticity of y with respect to x is defined as ε ≡

dy
dx

x .
y

Thus, if y = x θ , ε = θ .

2 See Glomm and Ravikumar (1997) for a survey of recent estimates.

Federal Reserve Bank of Richmond Economic Quarterly Volume 89/1 Winter 2003

Federal Reserve Bank of Richmond Economic Quarterly

The literature has shown that the optimal share of gross public investment
should equal the government capital elasticity of output. However, most influential papers in this literature utilize a very special class of growth models:
endogenous growth models without transition dynamics.3 In these models,
the elasticities of output with respect to private and public capital must add
up to one. This restriction is empirically implausible because the private capital elasticity of output is approximately 0.36 in the U.S. economy. Thus, if
this restriction were true, then the optimal share of public investment in GDP
would be 0.64.
In this article, we show that this implausible restriction is an implication
of the lack of transition dynamics. We revisit the optimal choice of public
investment in a more general and plausible model that allows for gradual transitions between steady states. Since endogenous growth is not essential to our
argument, we revert to the more conventional growth model with exogenous
technical progress.
Contrary to previous work, we show that (i) the optimal share of gross
public investment in output should be less than its elasticity along a balanced
growth path; (ii) this share depends in important ways on assumed preferences
and technology, including the underlying rate of productivity growth; (iii) the
optimal sequence of public investments is not time invariant, and furthermore,
a policy aimed at implementing it is not time consistent; and (iv) the government capital elasticity of output is likely to be relatively low at less than 0.1
if the observed U.S. ratios of gross public investment to output and of public
capital to private capital are approximately optimal.
This article is organized as follows: Section 1 gives an overview of public
investment in U.S. data. Section 2 sets the basic theoretical framework and
derives the optimal steady state share of public investment in output. In contrast, Section 3 describes the solution to the full optimal policy problem with
commitment. Section 4 offers concluding remarks.

PUBLIC INVESTMENT IN U.S. DATA

Panel A in Figure 1 depicts the behavior of U.S. gross public investment
relative to GDP during the postwar period. Observe first that the degree of
public investment in the United States is nonnegligible and roughly comparable as a percentage of GDP to that of its net exports. Public investment has
amounted to as much as 6 percent of GDP at its peak in the 1950s, but it has
also been relatively constant. The run-up in public investment apparent during
the 1950s and early 1960s captures a discrete increase in military spending
related to the Korean War, as well as increases in spending on schools and
3 See Barro (1990), Glomm and Ravikumar (1997), and Aschauer (2000), among many others.

P.-D. G. Sarte and J. Soares: Efficient Public Investment

Figure 1 Public Investment and Public Capital over the Postwar Period

highways. Outside of changes in military spending over time, the share of
public investment in output has stayed mostly flat at approximately 3 percent.
Panel B in Figure 1 shows the ratio of public to private capital from 1947
to today. Abstracting from public capital tied to national defense, which
includes aircrafts, ships, vehicles, electronic equipment, and missiles, this
ratio has never shown much variation, moving only between 0.20 and 0.25
during the entire period. We see a slight run-up in public capital during the
mid-1960s, which corresponds to the construction of the interstate highway
system. Interestingly, Fernald (1999) suggests that this construction provided
a significant onetime increase in productivity. Note in Panel C in Figure 1 that
private investment relative to output has always been much larger than public
investment, averaging around 15 percent since World War II.
The fact that some degree of public investment has consistently taken place
over the years suggests that the provision of infrastructure such as highways,
airports, and even public sector research and development indeed contributes to
economic activity. Thus, we will now study a simple economic environment
where public capital plays a productive role and examine the factors that
determine efficient public investment.

Federal Reserve Bank of Richmond Economic Quarterly

2. THEORETICAL FRAMEWORK
Consider a closed economy in which a large number of firms produce a single
final good according to the technology:
θ
Yt = Ktα (zt lt )1−α Kgt
,

(1)

where 0 < α < 1 and 0 < θ < 1 − α. The condition θ < 1 − α prohibits
the possibility of endogenous growth (see Barro and Sala-i-Martin [1995],
153). In equation (1), Kt and Kgt denote private and public capital at time t,
respectively, and zt lt represents the quantity of skill-weighted labor input. We
allow for exogenous technical progress in the use of labor input, lt , so that zt
grows at a constant rate over time:
zt = γ z zt−1 , γ z > 1, and z0 = 1.

(2)

This feature of the technology will make it possible for the economy to experience balanced growth in the long run at a rate related to γ z . We treat Kgt as
a pure public good that enhances each firm’s production.4
We assume that public investments are financed by a flat tax on income,
0 < τ t < 1, that can vary with time. Hence, we can express new outlays of
public capital, Kgt+1 , as
Kgt+1 =

τ t Yt

+ (1 − δ)Kgt ,

(3)

Public Investment

Kg0 > 0 given,
where 0 < δ < 1. Observe that τ t also represents the share of gross public
investment in GDP. Our main objective is to characterize the efficient ratio of
public investment to output.

Firms
Let rt and Wt denote, respectively, the rental price of private capital and the
wage at date t. Then, taking as given the sequence {rt , Wt }∞
t=0 , each firm
maximizes profits by solving
θ
max Ktα (zt lt )1−α Kgt
− rt Kt − Wt lt .
Kt , lt

(4)

The first order conditions associated with this problem imply that
θ
rt = αKtα−1 (zt lt )1−α Kgt

(5)

4 Thus, we abstract from congestion considerations and think of the aggregate stock of public
capital as being available to all firms. This assumption typically subjects the economic environment
to scale effects with respect to steady state allocations since public infrastructure is non-rival (see
Barro and Sala-i-Martin [1992]).

P.-D. G. Sarte and J. Soares: Efficient Public Investment

and
θ
.
Wt = (1 − α)Ktα zt1−α lt−α Kgt

(6)

Households
The economy is inhabited by a large number of identical households. Their
preferences are given by
U=

∞

t=0

βt

Ct1−σ
, with σ > 0,
1−σ

(7)

where 0 < β < 1 is a subjective discount rate. At each date, households
decide how much to consume or save and how much capital to rent to firms.
Each household is assumed to be endowed with one unit of time that they
supply inelastically. Their budget constraint is given by
Ct + It = (1 − τ t ) [Wt lt + rt Kt ] ,

(8)

Kt+1 = It + (1 − δ)Kt ,
K0 > 0 given.

(9)

where

In equation (8), Ct and It denote consumption and gross investment, respectively. Before proceeding with the household’s problem, we find it useful
to first derive the economy’s constant balanced growth rate in the steady state.
This will allow us to define a normalization of the economy’s variables that
will show explicitly why the optimal rate of public investment may depend
on the exogenous rate of technical progress, γ Z > 1. We denote the long-run
growth rate of a given variable x by γ x .
Equation (3) implies that if public investment ultimately grows at the constant rate γ Ig , then γ Ig = γ Kg . In addition, the household’s budget constraint
(8) implies that γ C = γ I = γ Ig = γ Y so that γ Kg = γ Y .5 Since equation
(9) also means that γ I = γ K , we further have that γ K = γ Y . Therefore, in
the end, all variables in the economy grow at the common growth rate γ Y . To
determine this growth rate, observe from the production technology described
in (1) that
Yt+1
=
Yt

Kt+1
Kt

γ 1−α
z

Kgt+1
Kgt

or
γ θKg .
γ Y = γ αK γ 1−α
z
5 Note that W l + r K = Y .
t t
t t
t

Federal Reserve Bank of Richmond Economic Quarterly

Since γ Kg = γ Y = γ K above, we can immediately obtain the economy’s
balanced growth rate, which we simply denote by γ :
1−α

γ = γ z1−α−θ .

(10)

Given this growth rate, a sensible normalization for our economy is one that
expresses our model’s variables in detrended form. Specifically, for any variable Xt that grows at rate γ along the balanced growth path, we express its
detrended counterpart in lowercase form:
xt =

Xt
.
γt

In the steady state, detrended variables will then be constant, while nondetrended variables will grow at the constant rate, γ .6
Taking the sequence of prices {rt ,wt }∞
t=0 , as well as the sequence of tax
rates {τ t }∞
,
as
given,
the
household’s
problem
may now be expressed as7
t=0
max ∞ U =

{ct , kt+1 }t=0

∞

t=0

1−σ
t ct
,
βγ (1−σ )
1 − σ

(11)

β∗

subject to
ct + γ kt+1 − (1 − δ)kt = (1 − τ t ) [wt lt + rt kt ] ,
k0 > 0 given.

(12)

The solution to the household’s dynamic optimization problem yields the familiar Euler equation,

−σ
γ ct−σ = β ∗ ct+1
(13)
(1 − τ t+1 )rt+1 + 1 − δ .
In the above expression, taxes distort private incentives to consume and
save but also finance public infrastructure that raises future returns to private
α−1 θ
investment; recall that rt+1 = αkt+1
kgt+1 . Given this trade-off, how should
society choose the share of public investment in output? An intuitive answer
to this question might be to select the ratio of public investment to output that
maximizes steady state welfare. This captures perhaps the simplest notion
of optimal policy. We show that such intuition indeed replicates conventional
wisdom obtained from endogenous growth frameworks. However, in both our
model and endogenous growth settings, ignoring transition dynamics turns out
to have crucial implications for policy.
6 Observe that if X grows at the rate γ , then X = γ t X . Therefore x = Xt = X .
t
t
t
0
0
γt
7 It should be noted that the modified discount rate, β ∗ , implicitly imposes a restriction on

the extent of technical progress, since γ (1−σ ) must be less than 1/β in order that the maximization
problem be well defined.

P.-D. G. Sarte and J. Soares: Efficient Public Investment

The Decentralized Steady State Equilibrium and a
Policy Golden Rule
We can describe the decentralized steady state equilibrium of the environment
laid out in this section most easily as a vector {c, i, k, kg } such that, given a
constant share of public investment in output, τ ,8

A) γ = β ∗ (1 − τ )αk α−1 kgθ + 1 − δ ,
B) c + i = (1 − τ )k α kgθ ,
C)

γ − (1 − δ) k = i,

γ − (1 − δ) kg = τ k α kgθ .

Conditions A) and B) are the households’ long-run consumption/savings
decision and budget constraint, respectively. Conditions C) and D) are the
normalized accumulation equations for private and public capital.
By combining conditions A) and D), and given the definition of β ∗ , it is
straightforward to show that the decentralized equilibrium ratio of public to
private capital satisfies
σ

γ
−
(1
−
δ)
τ
β
kg

.
(14)
=
k
α γ − (1 − δ) (1 − τ )
In particular, the higher the tax rate, the more public investment takes place
and the higher the steady state ratio of public to private capital. Furthermore,
one can show that the detrended level of private capital in the steady state is
given by
1−θ

θ
τ
(1
−
τ
)α
1−α−θ
k
= γσ
.
(15)
γ − (1 − δ)
− (1 − δ)
β
Observe in the equation above that k is a strictly concave function of τ .
In particular, setting τ = θ maximizes the steady state quantity of private
capital. Furthermore, we can make use of conditions A), B), and C) to solve
for consumption. Given the preferences in (7), it follows that steady state
welfare is
σ

(1−σ )
1 γ
−
(1
−
δ)
−
γ
−
(1
−
δ)
k
α
β
U=
,
(16)
1 − βγ (1−σ ) (1 − σ )
8 We use the expression “decentralized” to emphasize that steady state allocations obtain from
household and firm optimization conditional on policy.

Federal Reserve Bank of Richmond Economic Quarterly

so that U depends on τ only through private capital. Because U is a strictly
increasing function of k, while k is concave in τ , setting τ = θ also maximizes
steady state welfare.
To recapitulate, we have just shown in an exogenous growth context that
along the balanced growth path, the optimal share of gross public investment
in output, τ , must equal the public capital elasticity of output, θ . As it happens,
this is equivalent to the full optimizing solution in an endogenous growth model
without transition dynamics. Since endogenous growth requires α + θ = 1,
and since α is approximately 0.36 in the U.S. economy, endogenous growth
models imply that the optimal share of public investment in output should
be 0.64, an implausibly large number.9 Nothing in our exogenous growth
framework requires that α + θ = 1, and consequently, τ does not have to be
implausibly large at 1 − α.
There remains, however, another empirical puzzle to solve. Most estimates would place θ between 0.05 and 0.15. In contrast, we have seen that the
share of gross public investment in output (τ in our model) has consistently
remained only around 0.03 over the postwar period. To address this puzzle, we
now study the full welfare-maximizing policy problem that takes into account
both steady state considerations and transition dynamics.

EFFICIENT PUBLIC INVESTMENT WITH COMMITMENT:
THE RAMSEY PROBLEM

Unfortunately, there is no simple way to characterize the Ramsey problem for
our economy. Setting up and solving the full welfare maximization problem,
however, allows us to address explicitly two important notions associated with
efficient policy.
First, we show why the efficient sequence of public investment is not time
consistent. Second, we explain why Woodford’s (1999) “timeless perspective”
provides potentially one answer to this problem. In essence, the timeless
perspective advocates the implementation of the steady state solution to the
Ramsey problem at all dates. It thus considers optimality from the vantage
point of a date far in the past. Because the implied policy is one to which any
welfare-maximizing government would have wished to commit itself on that
date in the past, this solution concept avoids the problem of time consistency.
We will now address these issues in detail.
9 An implicit assumption here is that the share of private capital is measured adequately. An
alternative model that allowed for investment in human capital might allow θ to be significantly
less than 0.64. Indeed, with a broader concept of capital that allowed for not only physical but
also human capital, the share of capital in output might be closer to 0.75 (see Barro and Sala-iMartin [1992, 38]). Observe also that in an endogenous growth model that allowed for transition
dynamics (i.e., where the balanced growth path was reached asymptotically), the full welfareoptimizing solution would not necessarily prescribe τ = θ at all dates.

P.-D. G. Sarte and J. Soares: Efficient Public Investment

Consider a benevolent government that, at date zero, is concerned with
choosing a sequence of tax rates consistent with the development of public
infrastructure that maximizes household welfare. In choosing policy, this
government takes as given the decentralized behavior of firms and households
in the spirit of Chamley (1986). We further assume that at date zero, it can
credibly commit to any sequence of policy actions. The problem faced by this
benevolent government would then be to maximize (11) subject to equations
(12), (13), and (3) (in normalized form), and the corresponding Lagrangean
can be written as
∞

c1−σ
max ∞ L =
β ∗t t
(17)
{ct ,τ t ,kt+1 ,kgt+1 }t=0
1−σ
t=0
+

∞

−σ
β ∗t µ1t β ∗ ct+1
[(1 − τ t+1 )rt+1 + 1 − δ] − γ ct−σ

t=0

∞

β ∗t µ2t [τ t yt + (1 − δ)kgt − γ kgt+1 ]

t=0

∞

β ∗t µ3t [(1 − τ t )yt + (1 − δ)kt − γ kt+1 − ct ],

t=0

and rt = α(yt /kt ).
where yt =
The first constraint in (17) makes clear that our benevolent planner takes
households’ consumption/savings behavior as given. It can, however, influence the intertemporal allocations they choose by altering tax policy over time.
The optimal selection of τ t is governed by the following two equations:
∂L
: µ20 − µ30 = 0 for t = 0
(18)
∂τ t
and
αc−σ
∂L
: µ2t − µ3t − µ1t−1 t = 0 ∀t > 0.
(19)
∂τ t
kt
The fact that these first order conditions differ for t = 0 and t > 0 suggests
an incentive to take advantage of initial conditions in the first period with the
promise never to do so in the future. It is exactly in this sense that the optimal
policy is not time consistent, since once date zero has passed, a new planner
wishing to solve for the optimal policy at some date t > 0 would always choose
a tax rate on that date different from what had been prescribed at time zero. For
the purpose of this section, therefore, we imagine that the optimization takes
place only once, in period zero. Once our benevolent planner has decided on
a course of action, his hands are tied and he is precommitted to that course of
action.
θ
ktα lt1−α kgt

Federal Reserve Bank of Richmond Economic Quarterly

It should be noted that the choice of τ t introduces a lagged predetermined
variable, µ1,t−1 ≥ 0. At a purely mechanical level, the corresponding initial condition, µ1,−1 , serves as an artificial device that helps make stationary
the final system of difference equations that characterizes the optimal solution. However, unlike with fundamental state variables such as the private
or public capital stock, this initial condition is not arbitrary. Instead, since
the optimal choice of τ 0 should satisfy equation (18), it must be the case that
µ1,t−1 = 0 in (19) at t = 0. Alternatively, Dennis (2001, 6) points out that the
lagged Lagrange multiplier, µ1,t−1 , may be interpreted as the “current value
of promises not to exploit the initial state” and, in particular, to abide by past
commitments. However, since no history exists prior to period zero, there are
no past commitments on which to assign any value at that date. It is optimal,
therefore, to set µ1,−1 = 0.
The fact that the optimal policy is chosen once and for all in period zero
does not necessarily imply that it is not flexible. On the contrary, the solution
to the Ramsey problem provides a description of where to set τ t in every state
of the world. We shall see that it is also explicit about how the share of public
investment in output depends on the economic environment. Therefore, as
noted in Dennis (2001, 6), “if a change to one or more parameters takes place,
the policy rule automatically reflects this change; [and] there is no need for reoptimization to take place.” Accordingly, the remaining first order conditions
associated with problem (17) are
∂L −σ
: c0 + σ γ µ10 c0−σ −1 − µ30 = 0 for t = 0,
∂ct
and, ∀t > 0,
∂L −σ
: c − σ µ1t−1 ct−σ −1 [rt (1 − τ t ) + 1 − δ] + σ γ µ1t ct−σ −1 − µ3t = 0,
∂ct t
(20)
∂L
∂kt+1

−σ
: µ1t β ∗ ct+1
α(α − 1)

−µ3t γ + β ∗ µ3t+1

yt+1
(1 − τ t+1 ) + β ∗ µ2t+1 τ t+1 rt+1
2
kt+1

(1 − τ t+1 )rt+1 + 1 − δ
= 0,

(21)

and
∂L
−σ
: µ1t β ∗ ct+1
αθ
∂kgt+1

yt+1
(1 − τ t+1 ) − µ2t γ
kt+1 kgt+1

yt+1
+ 1 − δ + β ∗ µ3t+1 (1 − τ t+1 )θ
+β ∗ µ2t+1 τ t+1 θ
kgt+1
= 0.

yt+1
kgt+1
(22)

P.-D. G. Sarte and J. Soares: Efficient Public Investment

With these first order conditions in hand, we now turn to the long-run properties
of the efficient solution for public investment.

The Timeless Perspective and a Modified Policy
Golden Rule
As we have already seen, one of the features associated with the optimal policy
described in this section is that it requires the policymaker to commit once
and for all to his chosen rule at time zero. That is, optimal commitments are
generally not time consistent. However, in the context of monetary policy,
Woodford (1999) points out that “the optimal commitment fails to be time
consistent only if the central bank considers ‘optimality’ at each point in time
in a way that allows it to consider the advantages, from the vantage point of
that particular moment, of a policy change at that time that was not previously
anticipated.” One way around this problem, therefore, would be for the policymaker to adopt a pattern of behavior “to which it would have wished to
commit itself to at a date far in the past, contingent upon the random events that
have occurred in the meantime.” Woodford refers to this pattern of behavior
as the “timeless perspective.”
The attraction of Woodford’s timeless perspective is that it retains the solution to the Ramsey problem as the concept of optimal policy while getting
rid of the unique nature of date zero. Two potential problems with the notion
of timeless perspective are that it may be consistent with multiple policy outcomes (see Dennis 2001) and that welfare may be lower than that obtained
under alternative time consistent solutions (see Jensen and McCallum 2002).
Under the timeless perspective, the policy rule must ensure that the optimal
stationary equilibrium is eventually reached or, if already reached, that it
continues in that state. In our context, the optimal stationary equilibrium is
given by a vector {c, y, τ , k, kg , µ1 , µ2 , µ3 } that solves equations (19) through
(22), along with the Euler equation (13), the resource constraint (12), the
equation describing the evolution of public capital (3) (in normalized form),
θ
and the definition of output, yt = ktα lt1−α kgt
, all without time subscripts. The
optimal steady state equilibrium, therefore, can be characterized as a system of
eight equations in eight unknowns, which is shown in detail in the Appendix.
In the steady state, households’ optimal consumption/savings decisions
satisfy the familiar Euler equation,

y
γ − β ∗ (1 − τ )α
+ 1 − δ = 0.
(23)
k
Furthermore, the Appendix shows that the efficient share of public investment
in output must be such that

y
∗
γ −β θ
+ 1 − δ = 0.
(24)
kg

Federal Reserve Bank of Richmond Economic Quarterly

It follows from equations (23) and (24) that the Ramsey solution equates the
after-tax return to private investment, (1 − τ )α(y/k), with the marginal return
to public investment, θ (y/kg ). Consequently, we can immediately pin down
the ratio of public to private capital, given τ , as
kg
θ
=
.
(25)
k
α(1 − τ )
As expected, the higher the tax rate, the more public capital can be generated
relative to private capital.
The idea that the after-tax return to private investment must equal the
marginal return to public investment at the optimum also helps determine the
efficient share of public investment in output. Note that in our framework the
opportunity cost of one unit of resources invested in the public sector is the
after-tax return this unit would have otherwise earned in the private sector, or,
by equation (23),
y
γ
(1 − τ )α
= ∗ − (1 − δ).
(26)
β
k

return to private investment

The marginal benefit of investing one unit of resources in the public sector is
θ(y/kg ), and since public capital accumulates according to the law of motion
kg [γ − (1 − δ)] = τ y in the steady state, we have
γ − (1 − δ)
y
.
(27)
=θ
kg
τ
Hence, equating marginal cost and marginal benefit (i.e., the RHS of equations
26 and 27) directly yields the efficient solution for public investment as a
fraction of output:

γ − (1 − δ)
< θ.
(28)
τ = θ γσ
− (1 − δ)
β
θ

We can think of the equation above as a modified golden rule for policy.
Analogous to the modified golden rule for private capital in the one sector
growth model, the share of public investment in output given by (28) falls
short of the policy golden rule outlined in the previous section by an amount
that depends importantly on discounting. To see this, observe that when δ = 1
in equation (28), τ = βγ 1−σ θ = β ∗ θ < θ. Moreover, from (14) and (25),
it would then be the case that
kg
kg
θ
θ
= .
<
(29)
=
∗
∗
α(1 − β θ )
αβ (1 − θ )
k
k

Modified Policy Golden Rule

Policy Golden Rule

In other words, although the policy golden rule eventually leads to more public
infrastructure, and although public capital matters in production, the impatience reflected in the rate of time preference means that it is not optimal to

P.-D. G. Sarte and J. Soares: Efficient Public Investment

Figure 2 Efficient Public Investment

reduce current consumption through higher taxes to reach this higher ratio of
public to private capital.
Relative to most earlier work, the modified policy golden rule condition is
an important one in at least two respects. First, it implies that a high elasticity
of output with respect to public capital, θ , does not necessarily have to translate into a large share of public investment in output, τ . Therefore, observed
empirical estimates of θ lying between 0.05 and 0.15 do not have to be inconsistent with the 0.03 share of public investment in output. Second, the efficient
share of public investment in GDP now depends on a variety of preference
and technology considerations, including exogenous productivity growth, the
rate of depreciation, and the coefficient of intertemporal substitution.

Federal Reserve Bank of Richmond Economic Quarterly

Comparative Statics
Figure 2 depicts the effects of a rise in exogenous labor productivity growth on
the efficient ratio of public investment to output. We can see that an increase
in the rate of labor augmenting technical progress from γ to γ raises both
the marginal cost, βγ∗ − (1 − δ), and the marginal benefit, θ γ −(1−δ)
, of
τ
public investment. Intuitively, the returns to both types of capital increase,
and it is not clear whether investment in public capital should increase or fall.
Depending on the degree to which the marginal cost of public investment rises
in Figure 2, we can see that the optimal rate of public investment, τ ∗ , may
rise to τ or instead decrease to τ . In the case with 100 percent depreciation,
equation (28) reduces to τ ∗ = βγ (1−σ ) θ . Hence, it follows that
∂τ ∗
1
= (1 − σ )β −σ θ 0 ⇔ 1.
(30)
∂γ
σ
Put another way, when households are not particularly willing to substitute
consumption across time (i.e., 1/σ < 1), an increase in exogenous labor productivity growth leads them to want to increase present consumption relative to
output. Therefore, with fewer resources available for investment, the optimal
steady state share of public investment in output must fall.
To conclude this section, we calibrate the above example in order to determine what value of θ is implied by the theory. We shall think of our benchmark
as that of an economy resembling the United States. Other than for the public
capital elasticity of output, θ , the parameter values of the economy are selected
along the lines of conventional general equilibrium quantitative studies. Thus,
a time period represents one quarter, and we set the subjective discount rate,
β, to 0.99. The share of private capital in output, α, is set to 1/3, and σ is
chosen so as to make the coefficient of intertemporal substitution 1/2. Capital
is assumed to depreciate at a 10 percent annual rate. We let the rate of technical
progress vary so that the rate of growth in per capita output lies between zero
and 3.5 percent. In the United States, the rate of growth of per capita GDP
has averaged 2.5 percent since World War II.
Given these parameter values, equations (28) and (29) indicate that the
choice of θ simultaneously pins down two measures, namely, the share of
public investment in GDP and the ratio of public to private capital. There
is an obvious sense, then, in which our model can fail, since choosing θ to
match one measure for the United States would leave it free to miss its mark
on the other. We find that setting the public capital elasticity of output to
0.06 helps generate a share of public investment in output between 3.3 and 4
percent as in U.S. data. This is shown in the upper left-hand panel of Figure 3.
Interestingly, we find that this same public capital elasticity of output leads to
a ratio of public to private capital that matches well U.S. postwar experience
at approximately 20 percent (see the upper right-hand panel of Figure 3).
We conclude that if the share of public investment and the ratio of public to

P.-D. G. Sarte and J. Soares: Efficient Public Investment

Figure 3 Steady State Responses to Changes in Labor
Productivity Growth

private capital are approximately efficient in the United States, then the public
capital elasticity of output implied by the theory lies in the lower range of
most empirical estimates at around 0.06.
Figure 3 also shows that the efficient share of public investment in GDP
falls with increases in exogenous labor productivity growth. At the same time,
the consumption to output ratio rises in the lower right-hand panel of Figure 3.
In this calibrated example, as exogenous labor productivity growth increases,
households can afford to consume more relative to output and both private and
public investment as a fraction of GDP decreases.

48
4.

Federal Reserve Bank of Richmond Economic Quarterly
SUMMARY

In a setting where public infrastructure plays a productive role, we have characterized the main features of efficient public investment under commitment.
In contrast to most previous studies, we have shown that the optimal share
of public investment in output may be less than its elasticity in the long run
and, moreover, that this share may depend in important ways on assumed
preferences and technology. We have also stressed the crucial nature of the
commitment assumption, as the optimal sequence of public investments is
not time consistent. Finally, if the observed ratios of public investment to
GDP and of public to private capital are approximately optimal in the United
States, then our model suggests that the government capital elasticity of output
is likely to be relatively low at 0.06.

APPENDIX
As described in the text, the optimal stationary equilibrium in our framework
is a vector {c, y, τ , k, kg , µ1 , µ2 , µ3 } that solves eight equations. The first
four equations are
y − k α l 1−α kgθ = 0

(31)

y
+1−δ =0
γ − β ∗ (1 − τ )α
k

(32)

τ y + (1 − δ)kg − γ kg = 0

(33)

(1 − τ )y + (1 − δ)k − γ k − c = 0.

(34)

The next four equations are worked out in more detail.
Equation (21) reads as:
y
y
µ1 β ∗ c−σ α(α − 1) 2 (1 − τ ) + β ∗ µ2 τ α
k
k




y

∗

+
1
−
δ
−µ3 
(1
−
τ
)α
γ
−
β


k

= 0 by equation (32)

= 0.
Thus, we have
µ1 β ∗ c−σ α(α − 1)

y
k2

(1 − τ ) + β ∗ µ2 τ α

y
k

= 0,

P.-D. G. Sarte and J. Soares: Efficient Public Investment

or more simply
µ1 c−σ
(α − 1)(1 − τ ) + µ2 τ = 0.
(35)
k
In the steady state, equation (19) reads as
y
+ µ2 y − µ3 y = 0,
−µ1 c−σ α
k
so that
µ1 c−σ
α = µ2 − µ3 .
(36)
k
Equation (20) is

y
+ 1 − δ + σ γ µ1 c−σ −1 − µ3 = 0,
c−σ − σ µ1 c−σ −1 (1 − τ )α
k
or

y
c−σ − σ µ1 c−σ −1 (1 − τ )α
+ 1 − δ − γ − µ3 = 0,
k
so that by equation (32),

1
−σ
−σ −1
γ
− 1 − µ3 = 0.
(37)
c − σ µ1 c
β∗
Finally, equation (22) reads as

y
∗ −σ αθ y
∗
(1 − τ ) − µ2 γ + β µ2 τ θ
+1−δ
µ1 β c
kkg
kg
y
kg

+β ∗ µ3 (1 − τ )θ

= 0.

This simplifies to
β ∗θ

y
kg

(1 − τ )(µ2 − µ3 ) − µ2 γ + β ∗ µ2 τ θ
y
kg

+β ∗ µ3 (1 − τ )θ
by equation (36), which implies
∗

β θ

y
kg

(1 − τ ) − γ + β

τθ

+1−δ

y
kg

+ 1 − δ = 0,

y
+ 1 − δ = 0.
(38)
γ −β θ
kg
This last equation and equation (32) imply that the Ramsey solution equates
the marginal return to government capital with the after-tax return to private
capital.
∗

∗

y
kg

Federal Reserve Bank of Richmond Economic Quarterly

REFERENCES
Aschauer, David A. 1989. “Is Public Expenditure Productive?” Journal of
Monetary Economics 23 (March): 177–200.
. 2000. “Do States Optimize? Public Capital and Economic
Growth.” The Annals of Regional Science 34 (August): 343–63.
Barro, Robert J. 1990. “Government Spending in a Simple Model of
Endogenous Growth.” Journal of Political Economy 98 (October):
S103–25.
, and Xavier Sala-i-Martin. 1992. “Public Finance in Models
of Economic Growth.” Review of Economic Studies 59 (October):
649–61.
Chamley, C. 1986. “Optimal Taxation of Capital Income in General
Equilibrium with Infinite Lives.” Econometrica 54 (May): 607–22.
Dennis, R. 2001. “Pre-Commitment, the Timeless Perspective, and
Policymaking from behind the Veil of Uncertainty.” Working Paper
2001-19, Federal Reserve Bank of San Francisco (August).
Fernald, John G. 1999. “Roads to Prosperity? Assessing the Link between
Public Capital and Productivity.” American Economic Review 89 (June):
619–38.
Glomm, G., and B. Ravikumar. 1997. “Productive Government Expenditures
and Long-Run Growth.” Journal of Economic Dynamics and Control 21
(January): 183–204.
Jensen, C., and Bennett T. McCallum. 2002. “The Non-Optimality of
Proposed Monetary Policy Rules under Timeless-Perspective
Commitment.” Working Paper 8882, National Bureau of Economic
Research (April).
Ratner, J. B. 1983. “Government Capital and the Production Function of
U.S. Private Output.” Economic Letters 13 (2/3): 213–17.
Woodford, M. 1999. “Commentary on: How Should Monetary Policy Be
Conducted in an Era of Price Stability?” New Challenges for Monetary
Policy, a symposium sponsored by the Federal Reserve Bank of Kansas
City, 26–28 August.

Potential Consequences of
Linear Approximation in
Economics
Alexander L. Wolman and Elise A. Couper

he equilibrium of a dynamic macroeconomic model can usually be
represented by a system of nonlinear difference equations, and in contemporary models these systems can be large and difficult to solve.
Even for small models, it is common research practice to use approximations
that allow for analytical statements about the model’s behavior. The most
popular form of approximation is linearization around a steady state.1
To study linear approximations, economists have access to the methods
for solving dynamic linear models described in Sargent (1979) and Blanchard
and Kahn (1980). Blanchard and Kahn provide conditions under which there
exists a unique nonexplosive solution to a system of linear difference equations. In his analysis of first- and second-order linear difference equations,
Sargent explains how, in some cases, models based on optimizing behavior
justify the exclusion of explosive solutions as equilibria. Subsequently, attention has shifted toward explicitly nonlinear optimization-based models, but
the methods described by Sargent and Blanchard and Kahn have been widely
used to study the linear approximations to nonlinear models.
For as long as linear approximations have been used, economists have
been aware of certain limitations. In particular, linear approximations may be
quantitatively inaccurate unless one restricts attention to the model’s behavior near the point around which the linearization was taken. Recent work by
Benhabib, Schmitt-Grohé, and Uribe (2001) has highlighted an additional limitation of linearization that is potentially more severe: linearization may lead
This paper does not necessarily represent the views of the Federal Reserve System or the
Federal Reserve Bank of Richmond. The authors thank Marvin Goodfriend, Andreas Hornstein,
Tom Humphrey, and Ned Prescott for helpful comments.
1 A steady state is a point x̄ such that if x̄ is an equilibrium in any period t, then x̄ is also
an equilibrium in period t + 1.

Federal Reserve Bank of Richmond Economic Quarterly Volume

Federal Reserve Bank of Richmond Economic Quarterly

one to incorrect conclusions about the existence or uniqueness of equilibrium.
These scholars have argued, based on this reasoning, that a monetary policy
rule widely advocated for its stabilization properties may actually subject the
economy to multiple equilibria. Our purpose in this article is to provide a
simple exposition of the type of problems highlighted by Benhabib, SchmittGrohé, and Uribe. While we do not advocate that linearization be abandoned
entirely, it is important for users to be aware of the risks.
We use two simple models to illustrate the risks of linearization. In both
models, the dynamics boil down to one equation in one variable. This simplicity means that it is straightforward to compare the results based on linearization
to the model’s global properties.
In the first model, the single equation concerns the evolution of the stock of
government debt. We will show that analysis of this model based on linearization can lead one to an erroneous conclusion about whether an equilibrium
exists. A researcher might conclude, for example, that a particular tax policy
rule leads to the nonexistence of equilibrium when, in fact, an equilibrium
does exist for all but extreme initial values of debt. Or, linearization could
suggest that an equilibrium always exists when actually there is none outside
a narrow range of initial levels of debt.
In the second model, the single equation concerns the evolution of the
inflation rate. There, an equilibrium always exists, but naive analysis based
on linearization can lead one to erroneously conclude that there is only one
equilibrium when in fact there are many. Thus, a researcher using linearization
might advocate a particular policy rule based on its promise of delivering a
unique equilibrium when in fact that rule is susceptible to multiple equilibria.
This precise critique has been made by Benhabib, Schmitt-Grohé, and Uribe
against recent work advocating “active Taylor rules” for monetary policy.
In more complicated models, it may not be possible to determine whether
a linear approximation results in misleading conclusions about the uniqueness and existence of equilibrium. However, in closing we will offer some
suggestions for minimizing the risk of being misled.

MACROECONOMIC EQUILIBRIUM

A typical macroeconomic model consists of a set of maximization problems
and a set of market clearing conditions pertaining to a vector of variables.
Solving a model involves two steps: The first step is to derive the optimality
conditions that describe solutions to the maximization problems in isolation.
The second step is to collect these conditions with the market clearing conditions and manipulate them so that the variables whose values are not known
at the beginning of a period (for example, the price of a unit of capital) are
expressed as functions of the variables whose values are known at the beginning of a period (for example, the capital stock). We refer to the former set

A. L. Wolman and E. A. Couper: Potential Consequences

of variables as nonpredetermined, and the latter set as predetermined. If at
least one such vector-valued function exists, then an equilibrium of the model
exists. If there is exactly one such function, equilibrium is unique, whereas
multiple functions correspond to multiple equilibria. The second step can
be difficult, especially for models with many variables, and it often requires
some numerical approximations. The most popular approximation method is
linearization around a steady state.
We will describe two simple models, which will then be used in our analysis of linearization. In both models, there is assumed to be an infinitely lived
representative consumer who receives a constant endowment of consumption
goods each period. The consumer discounts future utility at rate β per period.
In the first model, there is a government that purchases a constant amount of
the consumption goods each period. The government issues debt and levies
lump- sum taxes in order to pay for its consumption. In the second model,
there is no government spending; however, the consumer derives utility from
real money balances as well as consumption. The government issues nominal
money by making lump-sum transfers to consumers.

A Model with One Dynamic Variable, Predetermined
The representative consumer has preferences for current and future consumption (ct ) given by the maximization problem

(1)
max
β t u(ct ),
subject to the budget constraint
ct + bt+1 + τ t ≤ y + rt−1 bt ,

(2)

where u () is an increasing and concave function; bt is the quantity of oneperiod, real government debt maturing in period t, paying a gross interest
rate of rt−1 ; τ t is the lump-sum tax levied in period t; and y is the constant
endowment received each period. Denoting by λt the Lagrange multiplier on
the budget constraint at time t, the first order condition for consumption is
u (ct ) = λt ;

(3)

the marginal value to the consumer of an additional unit of income is equal to
the marginal utility associated with using that income for consumption. The
first order condition for bond holding is
λt = βrt λt+1 ;

(4)

the marginal utility of income in the current period is equated to the present
discounted utility of converting current income into future income at the given
interest rate rt . The transversality condition is
lim β t λt bt+1 = 0;

t−→∞

(5)

Federal Reserve Bank of Richmond Economic Quarterly

this condition can be viewed as the first order condition for bond purchases
in the “final period.” It would be suboptimal for a consumer to accumulate
bonds so that the present utility value of consumption that could be realized
by selling those bonds in the distant future did not go to zero.
The government budget constraint is
bt+1 + τ t ≥ g + rt−1 bt ,

(6)

where g is the constant level of per-period government purchases of goods. The
left-hand side is the government’s sources of revenue, and the right-hand side
is the government’s uses of revenue. In equilibrium the goods market clears,
implying that government consumption plus private consumption equals the
total endowment of goods:
ct + g = y.

(7)

Because the endowment and government consumption are constant, private consumption must also be constant:
ct = c ≡ y − g

∀t.

(8)

Since equilibrium consumption is constant, the marginal utility of consumption is constant, and thus, from (3) and (4), the real interest rate is constant in
equilibrium:
rt = β −1 .

(9)

It remains to solve for the equilibrium quantity of government debt (bt+1 )
and the tax rate (τ t ). The two equations left to determine these variables
are the government budget constraint (6) and the transversality condition (5).
One might think that the consumer’s budget constraint (2) is an additional
equation. However, if we substitute the market clearing condition (7) into
the consumer’s budget constraint, the consumer’s budget constraint and the
government budget constraint become identical:
bt+1 = (g − τ t ) + β −1 bt .

(10)

This result is an implication of Walras’s law (see Varian [1992, 317]).
It is clear that the government budget constraint and the transversality
condition are not sufficient to determine a unique equilibrium path, or even a
finite number of equilibrium paths for bt+1 and τ t . In order to narrow the set
of equilibria, the standard research practice is to specify a policy rule for the
quantity of debt issued or, more commonly, for the tax rate. Given a rule that
sets the tax rate as a function of other variables, one can determine whether
equilibrium exists and is unique.
Note that if the rule makes the tax rate a function of no variables other than
bt+1 or bt , substituting the tax rule into (10) yields one equation that implicitly
determines the current period debt as a function of the predetermined debt from

A. L. Wolman and E. A. Couper: Potential Consequences

the previous period. Henceforth, we will assume that the tax rule sets the lumpsum tax as a function of only the predetermined level of debt, τ t = h (bt ) . In
this case, (10) explicitly describes the evolution of government debt:
bt+1 = g − h (bt ) + β −1 bt .

(11)

Below we will assume a particular form for h () , and thus we will be able to
determine whether equilibrium exists and is unique.

A Model with One Dynamic Variable,
Nonpredetermined
The second model we will consider is one in which we again end up with a
single dynamic equation, although in this case the equation will not contain
a predetermined variable. Here the representative consumer has preferences
for current and future real money balances (mt ) as well as consumption (ct ),
given by the maximization problem

max
β t [u(ct ) + v(mt )],
(12)
where u () and v () are increasing, concave functions.2 In this model, the government issues non-interest-bearing money by making lump-sum transfers to
consumers.3 The consumer maximizes utility subject to the budget constraint
Mt
Bt+1
Mt−1 Rt−1 Bt
T Rt
ct +
+
=y+
+
+
.
(13)
Pt
Pt
Pt
Pt
Pt
In (13), ct and y are as defined above. The new variables in (13) are the
nominal money supply (Mt = mt Pt ), the price level (Pt ), the quantity of oneperiod nominal bonds maturing in periods t + 1 and t (Bt+1 , Bt ), the nominal
interest rate on bonds maturing in the current period (Rt−1 ), and the quantity
of nominal transfers from the government to the household (T Rt ).4
Just as in the first model, the first order condition for consumption is given
by (3), with λt now the Lagrange multiplier on the budget constraint (13). The
first order condition for real money balances is
Pt
v (mt ) − λt + β
λt+1 = 0.
(14)
Pt+1
If the consumer increases real balances marginally in period t, he or she gains
current utility directly (v (mt ) > 0) but sacrifices current period consumption
2 We also assume lim

m→0 v (m) < u (y) and v (m̆) = 0, where m̆ < ∞. The former condition implies that, at a sufficiently high (finite) nominal interest rate, the economy will demonetize.
The latter condition implies that, at a nominal interest rate of zero, individuals become satiated
with a finite level of real balances.
3 See Brock (1975) and Obstfeld and Rogoff (1983) for further discussion of related models.
4 Although we have included nominal bonds in the consumer’s budget constraint, their quantity
will be zero in equilibrium (we assume that the government does not issue or purchase bonds,
and since households are identical, the quantity of bonds must be zero).

Federal Reserve Bank of Richmond Economic Quarterly

valued at λt . The same nominal balances are available in the next period as
a source of income to be used for consumption. However, the real value
in the next period of those nominal balances is deflated by the inflation rate
Pt+1 /Pt , and the marginal utility is discounted back to the current period by
the factor β. Condition (14) states that these effects are mutually offsetting:
optimal behavior implies that a marginal change in real balances leaves utility
unchanged.
The first order condition for holdings of nominal bonds is
λt = β

Pt
Rt λt+1 .
Pt+1

(15)

The interpretation of (15) is similar to that of (4). However, because here the
bonds pay off in dollars instead of goods, current real income is converted
t
into future real income at rate PPt+1
Rt . Finally, the transversality condition for
5
money is
lim β t λt mt = 0.

t−→∞

(16)

This has a similar interpretation to the bond transversality condition in the first
model.
The government budget constraint is
Mt /Pt + T Rt = Mt−1 /Pt ;

(17)

the left-hand side is the government’s sources of revenue, and the right-hand
side is the government’s uses of revenue. Because we assume that any changes
in the money supply are automatically accomplished by lump-sum transfers,
the government budget constraint does not play any role in the determination
of equilibrium.
In equilibrium the goods market clears, implying that private consumption
equals the total endowment of goods:
ct = y.

(18)

As before, constant consumption implies that the marginal utility of consumption is constant, and in this case, from (15), the equilibrium nominal interest
rate is equal to expected inflation divided by the discount factor
Pt+1
(19)
Pt
(this is a version of the Fisher equation relating nominal and real interest rates;
see Fisher [(1930) 1954]). Combining (15), (3), and (19), we see that there
is a simple relationship between the nominal interest rate and the marginal
Rt = β −1

5 There is also a transversality condition for bonds. However, since the quantity of bonds is
zero, this condition is automatically satisfied.

A. L. Wolman and E. A. Couper: Potential Consequences
utilities of consumption and real balances:

.
v (mt ) = u (c) 1 −
Rt

(20)

The marginal utility of consumption is known, so equation (20) then can be
used to express mt as a function of Rt ; it is a money demand function. In
turn, equation (19) determines Rt as a function of expected inflation. Without
a specification of monetary policy, however, we cannot determine the price
level or expected inflation. The standard research practice is to specify a policy
rule for the quantity of money or the nominal interest rate. Given a rule that
sets one of these variables as a function of other variables, one can determine
whether equilibrium exists and is unique.
Note that if the rule makes the nominal interest rate depend only on Pt or
Pt+1 , substituting the policy rule into (19) yields one forward-looking difference equation in the price level. Henceforth, we will assume that the monetary
policy rule sets the nominal interest rate as a function of the current inflation
rate, Rt = R (Pt /Pt−1 ) . In this case, the difference equation describes inflation, which we will denote by π (that is, π t = Pt /Pt−1 ):
β −1 π t+1 = R (π t ) .

(21)

Below we will assume a particular form for R () , and thus we will be able to
determine whether there is a unique nonexplosive equilibrium.
The reader may be struck by the fact that the difference equation in (21)
is independent of the preference specification in (12). It is a common feature
of simple monetary models that one can derive a difference equation in either
real balances or the price level (inflation is a transformation of the price level).
However, in general, one must bring in information from the “other” part
of the model in order to determine whether candidate paths are equilibria.
Anticipating the discussion below, here the linearization ignores information
from preferences, whereas the global analysis does not.

LINEARIZATION

The models we are working with contain just one dynamic variable and can
be written in the form
Et yt+1 = f (yt ),

(22)

where yt is the endogenous dynamic variable.
Linearization involves first computing a steady state of this equation and
then taking a first order Taylor-series approximation around that steady state. A
steady state of the difference equation system is a point ȳ such that ȳ = f (ȳ),
and the linear approximation around this steady state is
(Et yt+1 − ȳ) = f (ȳ)(yt − ȳ).

(23)

Federal Reserve Bank of Richmond Economic Quarterly

Figure 1 Dynamics of a Univariate Linear Difference Equation
1

This approximation is guaranteed to be valid only for small deviations from
the steady state. The univariate linear difference equation system (23) can be
written
Et ỹt+1 = Aỹt ,

(24)

where ỹt ≡ yt − ȳ.
Once we have the linearized equation, we can ask how many nonexplosive
solutions there are in the neighborhood of the steady state. In general, when yt
is predetermined, |A| must be less than one for a unique nonexplosive solution
to exist; when yt is nonpredetermined, |A| must be greater than one.
The logic behind these conditions is easy to see when A is positive.6
Figure 1 plots two possible cases for this univariate linear difference equation:
A > 1 and 0 < A < 1. The interpretation of these two cases depends on
whether the variable yt is predetermined.7
6 The case where A < 0 is similar. There, −1 < A < 0 produces dampened oscillations
rather than monotone convergence; A < −1 produces explosive oscillations.
The number of
equilibria can then be determined in the same manner as in the case where A > 0.
7 Using a plot of y
t+1 versus yt (such as Figure 1), it is simple to trace the time path for
yt starting from an initial point y0 . Start with y0 on the horizontal axis, and draw a vertical line
up to the function yt+1 . Then draw a horizontal line to the 45-degree line and a vertical line
back to the horizontal axis. This is y1 . Repeat to get y2 , etc. If yt is a predetermined variable,
then the initial condition is known, and the process just described reveals the path of yt (if there

A. L. Wolman and E. A. Couper: Potential Consequences

First, suppose yt is not predetermined, so that the initial condition y0 is
not known but instead needs to be determined in equilibrium. Then if A > 1,
any initial value y0 other than the steady state leads to yt exploding either
upward or downward: the steady state ȳ is the unique nonexplosive solution.
If A < 1, then yt will converge back to the steady state regardless of the
initial condition y0 : at any point in time there is a continuum of nonexplosive
solutions, one of which is the steady state.
Now consider the case where yt is predetermined, so that at any point in
time yt is known and yt+1 can be read off the graph. Then if A > 1, unless y0
happens to be equal to the steady state value, yt will explode over time: for
most initial conditions, a nonexplosive solution does not exist. If A < 1, then
yt will converge back to the steady state regardless of the initial condition y0 :
at any point in time there does exist a unique nonexplosive solution.
For models containing more than one variable, there are related conditions
involving the eigenvalues of a matrix A (see Blanchard and Kahn [1980]).
Sargent (1979, 177) describes the general principal as that of “solving stable
roots backward and unstable roots forward.”
Note that these conditions are concerned with the existence of nonexplosive solutions. It is common practice for researchers to restrict attention
to nonexplosive solutions. Sometimes equilibrium must be nonexplosive because of a transversality condition. In other cases, nonexplosiveness is not a
requirement of equilibrium, but researchers find other equilibria unappealing
on a priori grounds. Our tax model falls into the former category: explosive
behavior (at a rate greater than β −1 ) cannot occur in equilibrium because of
the transversality condition. In our monetary model, explosive behavior of the
price level cannot be ruled out as an equilibrium per se, but we will nonetheless
restrict attention to nonexplosive equilibria.8

3. THREE PITFALLS OF EXCESSIVE RELIANCE ON
LINEARIZATION
We have mentioned that linear approximations are less reliable far from the
steady state. This fact typically motivates researchers who use linearization
to study only examples in which there are small deviations around the steady
state. However, linearization may even give incorrect answers near the steady
state by suggesting an incorrect number of nonexplosive equilibria. This
possibility exists because a linear approximation treats the local properties of
the dynamic system as though they govern the model’s global behavior, and
is an equilibrium path). If yt is not predetermined, then the process reveals whether there is a
unique initial condition for which yt does not exhibit explosive behavior.
8 As is clear from Obstfeld and Rogoff (1983), in monetary models, ruling out candidate
equilibria based on simple explosiveness conditions is inappropriate. We use these conditions to
make our points more clearly.

Federal Reserve Bank of Richmond Economic Quarterly

Figure 2 Two Policy Rules for Lump-Sum Taxes

the global behavior can be crucial in determining the number of nonexplosive
equilibria. Locally the dynamics may imply that a variable explodes away from
the steady state, whereas the global dynamics exhibit sufficient nonlinearity
so that the explosiveness is shut off at some point. The opposite situation
can also occur. Using the models described earlier, we illustrate three ways
in which linear approximation can lead to an incorrect conclusion about the
number of nonexplosive equilibria in a model.

Spurious Nonexistence
It is possible that linearization suggests that there is no nonexplosive solution
when global analysis reveals that one in fact exists. In the tax model above,
the following tax policy gives such a result:
τ t = h(bt ) = τ̄ + τ 1 (bt − b̄)3 ,

(25)

where
b=

1
(g − τ̄ ).
1 − β −1

(26)

This policy rule is plotted in Figure 2 as the dashed line; it raises the lump-sum
tax when the level of debt is above b̄ and lowers the lump-sum-tax when the
level of debt is below b̄. The responsiveness of taxes to debt is nonlinear, rising
in magnitude the further the stock of debt is from b̄. This behavior appears

A. L. Wolman and E. A. Couper: Potential Consequences

reasonable, in that it might be expected to bring the level of debt back toward
a steady state from any initial condition.
Combined with the government budget constraint (10), the tax rule yields
an equation describing the evolution of the stock of government debt:
bt+1 = g − τ̄ − τ 1 (bt − b̄)3 + β −1 bt .

(27)

Linearizing (27) around b̄, which is a steady state, we get

bt+1 − b̄ = β −1 bt − b̄ .

(28)

Notice that in the linearized form of the tax policy, taxes do not respond to debt:
τ t = τ̄ . Given this nonresponsiveness, it is not surprising that an application
of the conditions discussed in Section 2 indicates that an equilibrium does not
exist unless the initial debt stock is equal to b̄. According to the linearized
model, for any initial debt level other than b̄, the quantity of debt will grow at
rate 1/β, violating the transversality condition.
The global analysis of (27) tells a very different story. A plot of bt+1
versus bt is given in Figure 3a.9 It turns out that there are three steady states:
b̄1 , b̄, and b̄2 . If the initial debt happens to be equal to one of those steady
state values, there is a unique equilibrium with constant debt. If the initial
debt is not equal to one of the steady state values, but it lies in one of the
intervals (b̄1 , b̄) or (b̄, b̄2 ), then there is a unique equilibrium in which the
debt converges to b̄1 or b̄2 , respectively. The debt levels bl and bu correspond
to a unique equilibrium in which the debt cycles between those two levels. If
the initial debt is between bl and the steady state b̄1 (or between b̄2 and bu ),
then there is a unique equilibrium in which the debt converges to one of the
steady states. Finally, if the initial debt is either below bl or above bu , then
there is no equilibrium, because the debt path implied by (27) violates the
transversality condition.
In this example, linearization leads us to conclude that an equilibrium
does not exist when in fact our analysis of the global dynamics shows that
there is an equilibrium for a wide range of initial conditions on the debt.
Since the nonexistence of equilibrium suggests that there is a fundamental
problem with a model, this possibility should lead to caution in interpreting
linearization when it results in a finding of nonexistence.

Spurious Existence
A second possibility is that there is a unique equilibrium to the linearized
model, but global analysis shows that there are no equilibria for a wide range
of initial conditions. Returning to the tax model, consider a tax policy given
9 In each panel of Figure 3, the dynamics of the linear approximation are indicated by a
dashed line.

Federal Reserve Bank of Richmond Economic Quarterly

Figure 3 Three Nonlinear Difference Equations

A. L. Wolman and E. A. Couper: Potential Consequences

by
τ t = h(bt ) = −(bt − a)(bt − b̄)(bt − c) + bt .

(29)

This function is plotted as the solid line in Figure 2, for carefully chosen values
of the parameters a, b̄, and c. For this rule, the tax rate rises with the debt stock
near the level b̄ but decreases with the debt stock far away from b̄. Substituting
(29) into the government budget constraint (10), the equation describing the
evolution of government debt is
bt+1 = g + (bt − a)(bt − b̄)(bt − c) − bt + β −1 bt .

(30)

Linearizing (30), we find
γ

bt+1 − b̄ = ((b̄ − a)(b̄ − c) − 1 + β −1 )(bt − b̄),

(31)

and we choose the parameters a, b̄, and c so that (b̄ − a)(b̄ − c) = 1 − β −1 ,
that is, γ = 0. For any starting value of bt , the linearized version implies that
government debt would converge immediately to the steady state b̄. This is
an example of the case where |A| < 1 and the one variable is predetermined.
Therefore, there appears to be a unique nonexplosive solution to the linearized
equations and thus a unique equilibrium.
The nonlinear difference equation (30) is graphed in Figure 3b. If the
initial debt is between b̄1 and b̄2 , then there is a unique equilibrium in which
the debt converges to b̄. However, if the initial debt is outside this range, no
equilibrium exists; the debt path implied by (30) violates the transversality
condition.
Here linearization leads us to conclude that there is a unique equilibrium,
whereas global analysis reveals that existence depends on the initial debt stock.
One could argue that if the initial debt stock is within a reasonable range, then
there is a unique equilibrium and the linear dynamics give a good approximation to that equilibrium. However, one could choose the parameters of
this example so that there is an arbitrarily small region in which the existence
results from the linear analysis hold.

Spurious Uniqueness
Finally, we can imagine a situation in which linearization suggests that there
is a unique nonexplosive equilibrium when in fact there are multiple nonexplosive equilibria. This is the case that has been highlighted in the recent work
by Benhabib, Schmitt-Grohe, and Uribe.10 Turning to the monetary model,
10 The research of Christiano and Rostagno (2001) is a related work that also uses global
analysis.

Federal Reserve Bank of Richmond Economic Quarterly

consider the following interest rate rule:
1
+ γ (π t − 1) ,
(32)
β
with γ > 1/β. This rule represents a well-defined, feasible policy for setting
the nominal interest rate, as long as the gross inflation rate is close to its targeted
steady state value of 1. Furthermore, this type of rule has been studied in both
empirical and theoretical contexts by authors such as Clarida, Gali, and Gertler
(2000). It is known as an active Taylor rule, because it is a Taylor-style rule
that raises the nominal interest rate more than one-for-one with the current
inflation rate.
Combining the policy rule with the Fisher equation relating inflation to
the nominal interest rate, we arrive at the following equation describing the
evolution of inflation:
Rt =

π t+1 = 1 + βγ (π t − 1) .

(33)

This difference equation has a unique steady state π̄ = 1 (the targeted steady
state). Furthermore, the equation is already linear, so we need merely note
that the coefficient on π t is greater than one to see that any path for inflation
other than the steady state will lead to inflation exploding upward (if π 0 > 0)
or downward (if π 0 < 0). Thus, there appears to be a unique nonexplosive
equilibrium.
The problem with the above reasoning is that along the explosive paths
on either side of the steady state, the policy rule (33) eventually implies an
infeasible choice of the nominal interest rate. First, consider an inflation path
in which the initial inflation rate is positive (π 0 > 1). Along such a path,
the inflation rate becomes arbitrarily high, and the path is hence ruled out as
explosive. But if the inflation rate becomes arbitrarily
high, at some point the

gross nominal interest rate exceeds R ∗ ≡ u (y) / u (y) − limm→0 v (m) .
At a gross nominal interest rate of R ∗ , the model economy demonetizes;
consumers will hold no money at interest rates equal to or greater than R ∗ ,
because the marginal benefit of real balances is bounded above by a number
less than the marginal interest cost of holding real balances. The economy thus
does not have a well-defined point-in-time equilibrium at a nominal interest
rate above R ∗ . Similarly, if the initial inflation rate is negative (π 0 < 1),
the dynamics in (33) indicate that the inflation rate will eventually become
arbitrarily large with a negative sign. But then at some point the policy rule
(32) requires that the gross nominal interest rate be less than unity. At a gross
nominal interest rate equal to unity, consumers are satiated with real balances.
The nominal interest rate cannot fall below unity because all agents would
choose to hold negative quantities of nominal bonds (money would have a
negative opportunity cost), and bonds are in zero net supply.
Because the interest rate rule given by (32) implies infeasible policy actions in certain situations, that rule is not a complete description of policy. A

A. L. Wolman and E. A. Couper: Potential Consequences

slightly modified rule that implies feasible policy actions in any situation is



1, if π t < 1 + γ1 1 − β1




1
1
1
1
1
∗
1
−
<
π
R
+
γ
−
1)
,
if
−
1
<
−
Rt =
(π
t
t
β
γ
β
γ
β



1
1
∗
∗

R , if γ R − β < π t − 1.
The difference equation describing equilibrium then becomes


1
1

β,
if
π
1
−
<
1
+

t

β


1
1
1 + βγ (π t − 1) , if γ 1 − β < π t − 1 < γ1 R ∗ − β1
π t+1 =





βR ∗ , if γ1 R ∗ − β1 < π t − 1.
This nonlinear difference equation is illustrated in Figure 3c. The consequences of modifying the policy rule so that it always delivers feasible policy
actions are dramatic. In the linear difference equation (33), paths beginning
from an initial inflation rate away from the steady state generated explosive
behavior of inflation (see the dashed line in Figure 3c). By contrast, the modified policy rule implies that there are two steady states (π = β and π̄ = βR ∗ )
in addition to the targeted steady state, and paths that begin away from the targeted steady state converge to one of these new steady states. Thus, instead of
there being a unique nonexplosive equilibrium, there is a continuum, indexed
by the initial inflation rate.

DISCUSSION

We have shown how approximating economic models by linearization around
a steady state may lead to incorrect conclusions about the existence or uniqueness of equilibrium. In each of our examples, the misleading results implied by
linearization were associated with a particular government policy rule. However, there is no reason to believe that it is only government policies that can
lead to these problems with linear approximations. One should not assume
that because a particular model has no role for government policy, a linear
approximation will necessarily give the right answers about the existence and
uniqueness of equilibrium.
In the simple models studied here, it was easy to see—and hence avoid—
the problems associated with linearization. Unfortunately, with larger models
it is harder to see the red flags signaling that linearization may be giving
incorrect answers. Furthermore, global analysis (i.e., analysis of the model
without any approximations) is infeasible with larger models.11 There are,
11 We should note that Benhabib, Schmitt-Grohé, and Uribe (2001) conduct global analysis

of a two-variable system in continuous time. Kuznetsov (1998) discusses global analysis of a
two-variable system in discrete time.

Federal Reserve Bank of Richmond Economic Quarterly

however, steps one can take to minimize the risk of falling victim to the
problems described above. Before linearizing it is important to determine
the number of steady states. If there is more than one steady state, it may
not be advisable to work with a linear approximation unless one has a strong
reason for believing that only one of the steady states is relevant. If there is
a unique steady state, then in some models a check on the results of linear
approximation can be provided by analyzing a simplified version of the model
in which it is feasible to compare the local linear and global dynamics. In the
case of a unique steady state, a promising approach currently receiving much
attention involves taking a local higher order approximation to the model’s
system of difference equations.12

REFERENCES
Benhabib, Jess, Stephanie Schmitt-Grohé, and Martin Uribe. 2001. “The
Perils of Taylor Rules.” Journal of Economic Theory 96 (January/
February): 40–69.
Blanchard, Olivier J., and Charles M. Kahn. 1980. “The Solution of Linear
Difference Models under Rational Expectations.” Econometrica 48
(July): 1305–12.
Brock, William A. 1975. “A Simple Perfect Foresight Monetary Model.”
Journal of Monetary Economics 1 (April): 133–50.
Christiano, Lawrence J., and Massimo Rostagno. 2001. “Money Growth
Monitoring and the Taylor Rule,” Working Paper 8539. Cambridge,
Mass.: National Bureau of Economic Research. (October).
Clarida, Richard, Jordi Gali, and Mark Gertler. 2000. “Monetary Policy
Rules and Macroeconomic Stability: Evidence and Some Theory.”
Quarterly Journal of Economics 115 (February): 147–80.
Fisher, Irving. 1954 [1930]. The Theory of Interest. New York: Kelley and
Millman, Inc.
Kuznetsov, Yuri A. 1998. Elements of Applied Bifurcation Theory. New York:
Springer Verlag.
Obstfeld, Maurice, and Kenneth Rogoff. 1983. “Speculative Hyperinflations
in Maximizing Models: Can We Rule Them Out?” Journal of Political
Economy 91 (August): 675–87.
12 See Sims (2000) and Schmitt-Grohé and Uribe (2002).

A. L. Wolman and E. A. Couper: Potential Consequences

Sargent, Thomas J. 1979. Macroeconomic Theory. New York: Academic
Press.
, and Neil Wallace. 1975. “ ‘Rational’ Expectations, the
Optimal Monetary Instrument, and the Optimal Money Supply Rule.”
The Journal of Political Economy 83 (April): 241–54.
Schmitt-Grohé, Stephanie, and Martin Uribe. 2002. “Solving Dynamic
General Equilibrium Models Using a Second-Order Approximation to
the Policy Function.” http://papers.ssrn.com/sol3/papers.cfm?
abstract id=277709 (January).
Sims, Christopher. 2000. “Second Order Accurate Solution of Discrete Time
Dynamic Equilibrium Models.” http://eco-072399b.princeton.edu/yftp/
gensys2/Algorithm.pdf (December).
Varian, Hal R. 1992. Microeconomic Analysis. New York: W.W. Norton and
Co.

The Cyclical Behavior of
Prices and Employee
Compensation
Roy H. Webb

re prices procyclical? For many economists, they clearly are. As
Lucas (1976, 104) put it, “The fact that nominal prices and wages
tend to rise more rapidly at the peak of the business cycle than they
do in the trough has been well recognized from the time when the cycle was
first perceived as a distinct phenomenon.” More recently, however, other
researchers have challenged the prevailing view. According to Kydland and
Prescott (1990, 17), “[T]he U.S. price level has clearly been countercyclical
in the post–Korean War period.”
The issue is of particular importance to macroeconomists who must choose
a model to work with. A monetary sector was an integral part of equilibrium
dynamic macro models that gained popularity in the 1970s, such as Lucas
(1972). Monetary misperceptions could then give rise to procyclical movements in prices. In contrast, the real business cycle models that later gained
popularity, such as Prescott (1986), did not have that property. If the behavior
of prices over the business cycle were a clearly established empirical regularity, that information would help choose the type of model to use for economic
analysis.
This paper attempts to better understand how respected economists can
hold such seemingly divergent views of the same data. By closely examining
the data on aggregate price measures, I will try to clarify why each view could
be correct under specific definitions of important terms. In doing so, I will
propose a way of viewing the data that may be useful in other circumstances.
The author gratefully acknowledges helpful comments from Marvin Goodfriend, Robert Hetzel, and Raymond Owens. Valuable research assistance was provided by Elliot Martin. The
views and opinions expressed in this article are solely those of the author and do not necessarily reflect those of the Federal Reserve Bank of Richmond or the Federal Reserve System.

Federal Reserve Bank of Richmond Economic Quarterly Volume 89/1 Winter 2003

Federal Reserve Bank of Richmond Economic Quarterly

In particular, the methodology that is employed to assess price cyclicality
can be easily used to study other variables of interest. The cyclical behavior
of wages has been a subject of controversy for over a half century and is
examined in the final section of the paper.

PRICES AND THE BUSINESS CYCLE

Much of our understanding of the complex phenomena that are unified under
the idea of the business cycle was developed by researchers associated with
the National Bureau of Economic Research (NBER) in the first half of the
twentieth century. Their initial approach was to describe the cycle, either
verbally or with voluminous statistics. Their conception of a typical business
cycle is now part of our common language, and many statistical regularities
that are usually thought to characterize cycles were first noted in their early
publications. An important early example of this line of research is Mitchell
(1913). Although his observations were based on an American economy much
different from our own, much of his account of the behavior of economic
aggregates anticipated later developments in economic activity. Prices played
a key role in his view of the cycle, as the following passages attest:
A revival of activity, then, starts with this legacy from depression: a level
of prices low in comparison with the prices of prosperity, [and]. . . drastic
reductions in the cost of doing business (150). While the price level is
often sagging slowly when a revival begins, the cumulative expansion
in the physical volume of trade presently stops the fall and starts a rise
(151). Like the increase in the physical volume of business, the rise in
prices spreads rapidly; for every advance of quotations puts pressure upon
someone to recoup himself by making a compensatory advance in the
prices of what he has to sell. . . . Retail prices lag behind wholesale. . . and
the prices of finished products [lag] behind the prices of their raw materials
(152). [O]ptimism and rising prices both support each other and stimulate
the growth of trade (153). Among the threatening stresses that gradually
accumulate within the system of business during seasons of high prosperity
is the slow but sure increase in the costs of doing business (29). The
price of labor rises. . . . The prices of raw materials continue to rise faster
on the average than the selling prices of products (154). [T]he advance
of selling prices cannot be continued indefinitely. . . [because] the advance
in the price level would ultimately be checked by the inadequacy of the
quantity of money (54). [Once a downturn begins] with the contraction
in trade goes a fall in prices (160). [T]he trend of fluctuations [in prices]
continues downward for a considerable period. . . . [T]he lowest level of
commodity prices is reached, not during the crisis, but toward the close
of the subsequent depression, or even early in the final revival of business
activity. The chief cause of this fall is the shrinkage in the demand for
consumers’ goods, raw materials, producers’ supplies, and construction

R. H. Webb: Cyclical Behavior

work (134). [E]very reduction in price facilitates, if it does not force,
reductions in other prices (160). Once these various forces have set trade
to expanding again, the increase proves cumulative, though for a time the
pace of growth is kept slow by the continued sagging of prices (162).

Note that this account was based on economic activity under the gold
standard at a time when no trend would be expected in the price level. Evidence during that time generally supported the behavior Mitchell described.
Zarnowitz (1992, ch. 4), for example, found strong evidence of procyclical
prices in the first 150 years of U.S. history. In contrast, under our current fiat
money system, the CPI has risen in each of the past 47 years, with an average
annual increase of 4.1 percent. This change in monetary regime leads to an
immediate modification of Mitchell’s analysis that preserves its spirit while
conforming to recent evidence. Inflation can be substituted for the level of
prices in the writing above, and the logic is preserved; a recession1 is thus associated with falling inflation and consequently the inflation rate is relatively low
at the beginning of a cyclical expansion. Then, as the expansion progresses,
the rate of inflation rises, led by relatively large increases in commodity prices.
The evidence presented below is consistent with that analysis.
The controversy, though, concerns the cyclical behavior of the price level
in the last half century. In order to understand the challenge to the conventional
wisdom that the price level is procyclical, we need to investigate the exact
meaning of cyclical price movements when prices are continually rising. The
following section thus examines filtering, that is, removing some measure of
a long-run trend from a series in order to study shorter-run movements.

FILTERING ECONOMIC TIME SERIES

Consider a series of data generated as
Xt = (1 + g)Xt−1 (1 + ε t ),

(1)

where X is a data series, the subscript t indexes time, g is a fixed positive
number, and ε is a random variable with zero mean. The series would grow,
on average, at rate g, and a graph of X versus time would eventually appear
nearly vertical. A common first step in studying the series would be to take
logarithms, which would change the time-series plot to a series fluctuating
around a straight line with slope 1 + g. In this case, an obvious filter for
removing the long-run trend would be to divide each observation Xt by (1+g)t .
In the typical case where g is not known, one can estimate the coefficients in
1 Mitchell used the word depression where we would use recession today.

Federal Reserve Bank of Richmond Economic Quarterly

Figure 1 GDP Price Index and Trend

the following regression
ln Xt = α + βTt + υ t ,

(2)

where T is a trend variable, taking a value of 1 in the first period, 2 in the
second, and so forth; β̂ is the estimated growth rate of the series; and υ is
assumed to be white noise. In this case, the antilog of the estimated residual,
eυ t , would be the detrended value of the observation Xt . This method is widely
referred to as linear detrending. In some cases, a linear trend can fit the data
well over a lengthy interval; for example, in Webb (1993) it is shown that real
per capita GDP in the United States has fluctuated around a stable linear trend
for over 100 years.
This method of detrending is not always appropriate. Suppose that g
varied substantially over time in equation (2). Then imposing a linear trend
could lead to long swings above or below trend, and the detrended data would
be difficult to analyze. Price data, in particular, are not always and everywhere
consistent with a fixed, linear trend; monetary regimes have varied, and within
regimes the monetary authority may not have had a constant inflation target.
Thus several methods of estimating a flexible, or time-varying, trend have
been proposed that could be applied to prices. A conceptually simple method
is to estimate the trend by a centered moving average. Thus letting the trend
value of X be denoted X∗ , then
Xt∗ =

1
Xt−k ,
2k + 1 t=−k

(3)

R. H. Webb: Cyclical Behavior

Figure 2 Annualized Percentage Change in GDP Price Index

and the detrended value can be either the difference between actual and trend,
or the ratio of actual to trend.
Many macroeconomists use a flexible trend that is produced by a method
known as the Hodrick-Prescott (HP) filter (1980). They calculate the trend
terms xt∗ to minimize
N
N−1

∗
∗
(xt − xt∗ )2 + λ
[(xt+1
− xt∗ ) − (xt∗ − xt−1
)]2 ,
t=1

(4)

t=2

where the small x and x ∗ terms are logarithms of their counterparts using
capital letters, N is the number of observations, and λ is a fixed number. For
analyzing quarterly macroeconomic data, Hodrick and Prescott recommend
a value of 1600 for λ, which will be used below. Intuitively, minimizing the
expression (4) trades off deviations from trend, given by the first term, against
changes in the trend value, given by the second term.
A final method of removing the trend is to simply take a difference in logs
or, similarly, look at percentage changes in a variable. A disadvantage of this
method is that the changes over a short period can be dominated by erratic
factors.
These methods of removing the trend can be seen in Figures 1 through 4.
In Figure 1 the logarithm of the GDP price index is first graphed, with shaded
areas denoting cyclical recessions as defined by the NBER. Also included is
the trend, estimated with the HP filter. In Figure 2, the quarterly percentage
change is graphed, which effectively removes the trend. In Figure 3, the
trend of the price index is estimated by the HP filter and a nineteen-quarter

Federal Reserve Bank of Richmond Economic Quarterly

Figure 3 GDP Price Index Trends

moving average filter. Both trends appear similar, and indeed, the correlation
coefficient between the two is 0.999. Finally, the detrended values are plotted
in Figure 4, and again both methods give somewhat similar estimates; in
this case, the correlation coefficient is 0.95. Thus, when thinking about the
meaning of filtered data, the intuitive moving average filter can be substituted
for the less intuitive HP filter, if desired. All three methods indicate that
inflation has been highly variable in the post–World War II period, and thus
some form of a flexible trend is necessary in order to study price data.

3. THE ASSERTION OF COUNTERCYCLICAL PRICES
Kydland and Prescott (1990) studied the cyclicality of prices by examining the
correlation of real GNP with the CPI and of real GNP with the GNP implicit
price deflator. They found a sizable negative correlation between GNP and
each price index and interpreted that negative correlation as demonstrating
that the price level is countercyclical. In their words,
This myth [that the price level is procyclical] originated from the fact that,
during the period between the world wars, the price level was procyclical.
But. . . no one bothered to ascertain the cyclical behavior of the price level
since World War II. Instead, economists just carried on, trying to develop
business cycle theories in which the price level plays a central role and
behaves procyclically. The fact is, however, that whether measured by

R. H. Webb: Cyclical Behavior

Table 1 Series with the Segmented Cyclical Trend Removed
Series

Phase 2

Phase 3

−1.0

0.6

1.1

GDP Price Index

0.4

−0.4

Personal Consumption
Expenditure Price Index

0.5

−0.4

Consumer Price Index

0.7

−0.6

Producer Price Index

1.1

−0.9

−1.6

Average Hourly Compensation
Real Average Hourly
Compensation

Real GDP

Journal of Commerce Index

Phase 1

Phase 4

Phase 5

−1.2

−3.3

0.4

0.9

−0.1

0.6

1.0

−0.1

0.9

1.3

1.1

1.4

0.8

1.3

0.1

−5.0

0.9

−0.8

0.2

0.8

0.3

−0.3

0.3

0.2

−0.2

the implicit GNP deflator or by the consumer price index, the U.S. price
level clearly has been countercyclical in the post–Korean War period (17).

Cooley and Ohanian (1991) provided even more evidence of a negative
correlation. They examined data over a longer time span and used a variety
of methods to remove the trend in prices. An important part of their analysis
was to apply the same filter to both prices and output data and then to examine
the correlations. For 1948 Q2 to 1987 Q2, using a simple linear trend resulted
in a correlation of –0.67; using log-differenced data resulted in a correlation
of –0.06; and using HP-filtered data resulted in a correlation of –0.57. They
interpreted these results as contradicting the view that prices are procyclical.
A common feature of these articles is that they discussed the cyclicality
of prices by either redefining or ignoring the traditional business cycle. The
traditional definition of business cycles was given by NBER researchers Burns
and Mitchell (1946, 3):
Business cycles are a type of fluctuation found in the aggregate economic
activity of nations that organize their work mainly in business enterprises:
a cycle consists of expansions occurring at about the same time in many
economic activities, followed by similarly general recessions, contractions,
and revivals which merge into the expansion phase of the next cycle; this
sequence of changes is recurrent but not periodic; in duration business
cycles vary from more than one year to ten or twelve years; they are

Federal Reserve Bank of Richmond Economic Quarterly

Figure 4 Detrended GDP Price Index

not divisible into shorter cycles of similar character with amplitudes
approximating their own.

There are many valid reasons to study detrended macroeconomic variables, but they do not necessarily reveal much about business cycles as defined
by the NBER. For example, by definition a detrended series will be symmetric,
with positive observations balanced by negative observations. However, the
business cycle has been notably asymmetric in the post–World War II United
States. Most obviously, expansions last much longer than recessions. The
length of the average recession has averaged 10.5 months, whereas expansions have averaged over five times as long, 56.9 months. In fact, the one
expansion from 1991 to 2001 lasted 120 months, while all ten recessions from
1948 to date have totaled 105 months.2
Another property of focusing on detrended data is that results may be
crucially dependent on the particular method used to detrend the data. As
Canova (1998, 475) puts it, based on a study of data on real economic activity,
“[Stylized facts] of U.S. business cycles vary widely across detrending methods, and . . . alternative detrending filters extract different types of information
from the data.” This effect can be seen in Figures 2 and 4 for the GDP price
2 For this calculation it is assumed that the recession that began in March 2001 ended in
December 2001.

R. H. Webb: Cyclical Behavior

Figure 5 GDP Price Index, Segmented Trend Removed

index. In Figure 2, differencing the data produces a series that tends to rise in
cyclical expansions and fall in recessions. Conversely, applying the HP filter
or a moving average filter to the same data series, as shown in Figure 4, yields
a series that tends to fall in cyclical expansions and rise in recessions. Thus,
whenever an assertion is based on detrended data, one should ask if the assertion is sensitive to the detrending method. Notice that the detrended prices in
Figure 4 tend to be negative in the middle of cyclical expansions. That could
be due to falling prices, but it could also be due to the rising trend, as both
methods illustrated tend to have increasing trends in cyclical expansions. The
next section thus takes a different approach to the question of price cyclicality.

NEW EVIDENCE ON PRICE CYCLICALITY

Assertions of procyclical prices have relied on purely statistical methods that
ignored the traditional business cycle. Does that make a difference? This section looks at evidence based on a statistical method that is based on traditional
business cycle dates. The method will be to take a simple trend, as shown in
equation (2), that is defined only for a specific business cycle. In this paper,
business cycles will be defined from trough to trough, where the date of the
trough has been determined by the NBER. For the recession that began in
March 2001, the NBER has not yet determined the date of the trough; in this
paper, December 2001 will be used in place of an official date of the recession’s trough. This method will be referred to below as the segmented cyclical
trend, or SCT, method. It is illustrated in Figure 5. Most data series extend
back to 1947, which allows nine complete busines cycles to be examined.

Federal Reserve Bank of Richmond Economic Quarterly

In order to assess the cyclicality of price movements, it is useful to divide each cycle into five separate phases to allow distinctive behavior to be
observed. All calendar quarters will be classified as being in an expansion
or a recession. An expansion begins in the quarter after the one that contains
a trough, ends in the quarter containing the peak, and is divided into three
phases.3 The first phase, referred to here as early expansion, contains the first
fourth of the number of quarters in the cyclical expansion. The second phase,
or middle expansion, covers the next half of the number of expansion quarters.
The final phase comprises the remaining one-fourth of the number of expansion periods. Recessions begin in the quarter following the peak and end in the
quarter containing the trough. Since recessions are on average much shorter
than expansions, they can be divided into a first half and a second half.4 In the
author’s experience, this has been a useful classification for post–World War
II business cycles, but many others can be imagined. In particular, Burns and
Mitchell (1946) divided business cycles into nine phases for their analysis.
This cyclical classification is applied in Table 1. Several measures of
prices are examined, including the price index for GDP and the price index for
personal consumption expenditure from the Bureau of Economic Analysis; the
consumer price index and the producer price index for finished goods from the
Bureau of Labor Statistics; and the Journal of Commerce Index of commodity
prices. The final two lines are discussed in the section below. All data series
are seasonally adjusted. Each entry in the table is an average over a cyclical
phase for the nine business cycles of items with the segmented linear trend
removed.
The first series in the table is real GDP, which is often taken as the prototypical cyclical variable. Its high point is reached in Phase 3, which contains
the cyclical peak. Similarly, its low point is reached in Phase 5, which contains
the cyclical trough. Thus real GDP behaves as would be expected and is a
useful benchmark for the series of prices.
The next four series are broad measures of prices of finished goods. Their
behavior is quite different from real GDP. The GDP price index is typical,
with its low point in Phase 2 and its high point in Phase 5. This different
behavior of output and prices would seem to be consistent with the finding
of countercyclical prices. This behavior can also be examined with other
methods of detrending. Table 2 presents series of percentage changes, and
Table 3 presents data detrended with the HP filter.
3 This classification was motivated by the casual observation that growth was often very rapid
near the beginning of expansions and was often subpar near the end of expansions.
4 What if the length of expansion is not evenly divisible by four? For purposes of this
section, if there is a nonzero remainder after dividing the number of quarters in an expansion
by four, then the number of quarters in the remainder is added to the middle expansion phase.
Similarly, if the number of quarters in a recession is odd, that first phase will be one quarter
longer than the second.

R. H. Webb: Cyclical Behavior

Table 2 Series Expressed as Percentage Changes
Series

Phase 1

Phase 2

2.6

0.8

−0.4

−6.2

−3.9

GDP Price Index

−0.5

−0.2

0.6

0.7

0.2

Personal Consumption
Expenditure Price Index

−0.7

−0.3

0.9

1.0

Consumer Price Index

−1.0

−0.4

1.5

1.2

−0.4

Producer Price Index

−1.4

−0.4

1.9

1.5

−0.5

4.1

−0.2

2.8

−9.0

−8.6

−0.4

−0.3

1.1

0.2

−0.6

0.2

−0.8

−0.6

Real GDP

Journal of Commerce Index
Average Hourly Compensation
Real Average Hourly
Compensation

0.3

Phase 3

Phase 4

Phase 5

The entries in Table 2 illustrate the importance of the detrending method.
For real GDP, the highest value now occurs in Phase 1, rather than in Phase
3. This means that the real growth rate tends to be highest in the early phase
of an expansion, even though from Table 1 we know that the level of GDP
tends to be highest above trend in the late expansion phase. With prices, it
is harder to discuss the detrended level of each series intuitively. Note in
Figure 1 how the price level has risen consistently over the past half century.
Any cyclical tendencies are small relative to the dramatic increase over time.
Moreover, the rate of increase is significantly more rapid from the mid-1960s
to the early 1980s than at other periods. For many purposes these broad trends
may be more important than the cyclical movements. That said, in Table 2, the
movements in prices over the business cycle are somewhat different than real
GDP, which is again consistent with the assertion of countercyclical prices.
Note that in this table inflation is highest when real growth is lowest, in Phase
4. Similarly, real growth is highest when inflation is lowest, in Phase 1. But
also note that the entries for finished goods prices tend to increase during
expansions, hit their highs in the early recession phase, and decline in the
late recession phase, hitting their low point in the early expansion. Thus this
general conformity with the business cycle could be viewed as a procyclical
movement, but with a one-phase lag.
Finally, HP-filtered data are presented in Table 3. These data resemble
those in Table 1. Real GDP is highest in the late expansion phase and lowest
at the beginnning of expansions. Prices of final goods are below trend when
GDP is above trend, and vice versa.

Federal Reserve Bank of Richmond Economic Quarterly

Table 3 Series with the HP Trend Removed
Series

Phase 2

Phase 3

−1.3

0.5

1.5

−0.1

−0.3

GDP Price Index

0.2

−0.3

−0.1

0.6

0.8

Personal Consumption
Expenditure Price Index

0.2

−0.4

−0.1

0.9

1.1

Consumer Price Index

0.2

−0.5

−0.1

1.3

1.5

Producer Price Index

0.3

−0.7

1.6

1.7

−2.5

0.1

3.2

1.4

−4.9

Average Hourly Compensation

0.3

−0.4

0.8

0.7

Real Average Hourly
Compensation

0.1

−0.1

−0.3

Real GDP

Journal of Commerce Index

Phase 1

0.1

Phase 4

Phase 5

So far, then, the evidence seems, on balance, to support the assertion of
countercyclical prices. Another interpretation is also possible. Until now the
language of leading or lagging indicators has not been used, although it has a
long tradition in discussions of cyclical behavior. Looking at the price indexes
for finished goods in Table 1, one can see that these series reach their peak two
phases after real GDP reaches its peak. This could be due to price stickiness,
which is an integral feature of many macroeconomic models, for example,
Goodfriend and King (1997). Thus, if changes in aggregate demand affect
output before affecting prices of finished goods, that relationship could make
a price index a lagging indicator as in Table 1.
Further evidence can be found by looking at commodity prices. Since
commodity prices are often determined in spot markets, they can be
immediately affected by supply or demand shifts. In contrast, finished goods
prices are often set by explicit or implicit contracts and thus do not immediately display the total impact of supply or demand changes. That consideration
suggests that commodity prices should be more of a coincident indicator. And
in Tables 1 and 3, notice that the Journal of Commerce Index of commodity
prices, like real GDP, hits its peak in Phase 3 and hits its low point in Phase
5. This behavior supports the view that commodity prices are a coincident
indicator while finished goods prices are a lagging indicator. Thus, these data
are consistent with many models that incorporate fluctuations of aggregate
demand.

R. H. Webb: Cyclical Behavior

Figure 6 Real Average Hourly Compensation

EVIDENCE ON EMPLOYEE COMPENSATION

The cyclical behavior of wages has a long history of controversy, which began
when Keynes (1936) asserted that real wages were countercyclical. Many
articles have been written on the subject, and it is possible to find respected
authors arguing for a procyclical pattern of real wages, a countercyclical pattern, or no meaningful pattern. For example, see Abraham and Haltiwanger
(1995) for selected quotes and a discussion of recent evidence.
Unfortunately, consistent series on wages are not as plentiful as series on
prices. This paper examines one particular series, employee compensation,
which is available in quarterly form beginning in 1947. It includes wages,
salaries, and fringe benefits. The nominal series is deflated with the PCE price
index to obtain the real series and is graphed in Figure 6. Here the fluctuations
around a trend are quite small, especially before 1973. For many analysts, the
main issue is the significant growth in real wages over a half century, with a
noticeable slowing between the early 1970s and the mid-1990s.
Both detrended nominal and real compensation are included in the tables.
In Tables 1 and 3, the nominal series behaves somewhat like detrended prices
of final goods. Both prices and compensation have low points in Phase 2 of
the business cycle. Compensation is notably above trend in Phases 4, 5, and
1. In Table 2, the average growth rate of nominal wages is procyclical, hitting
its high point in Phase 3 and its low point in Phase 5. Since nominal wage
stickiness is often taken as a stylized fact, it may not be surprising that the
nominal wage level behaves as a lagging indicator, too, as indicated in Tables
1 and 3.

Federal Reserve Bank of Richmond Economic Quarterly

The controversy has dealt with real wages, however, and the evidence is
mixed. The growth rate of real wages seems procyclical in Table 2. That
growth rate rises in expansions and declines in recessions. There is also
evidence of procyclical real wage behavior in Table 3. But in Table 1 the real
wage is above trend in Phases 1, 3, and 4, but below trend in Phases 2 and
5. Here again the choice of filter is important. This illustrates the limits of
letting the data speak for themselves; in this case, some theory is needed just to
choose a filter to remove the long-run growth trend of real compensation. And
it is not surprising that authors have differed on the cyclicality of real wages.
None of the evidence, though, supports Keynes’s assertion of countercyclical
real wages.

CONCLUSION

Data averaged over phases of post–World War II business cycles were examined for evidence of price cyclicality. The behavior of the level of final goods
prices is consistent with the view that prices are countercyclical. Another interpretation, however, is that final goods prices are a lagging indicator, possibly
due to price stickiness. Evidence of a procyclical level of commodity prices
supports the latter interpretation. Phase-averaged data are also examined for
employee compensation. Nominal compensation behaves much like finished
goods prices, which would not surprise an analyst who believed that both
wages and final goods prices are sticky. Real wage behavior is more difficult
to characterize; however, it is difficult to reconcile the evidence presented with
Keynes’s original assertion of countercyclical wages.

REFERENCES
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the Business Cycle.” Journal of Economic Literature 33: 1215–64.
Burns, Arthur F., and Wesley C. Mitchell. 1946. Measuring Business Cycles.
New York: National Bureau of Economic Research.
Canova, Fabio. 1998. “Detrending and Business Cycle Facts.” Journal of
Monetary Economics 41: 475–512.
Cooley, Thomas F., and Lee E. Ohanian. 1991. “The Cyclical Behavior of
Prices.” Journal of Monetary Economics 28: 25–60.
Goodfriend, Marvin, and Robert King. 1997. “The New Neoclassical
Synthesis and the Role of Monetary Policy.” NBER Macroeconomics

R. H. Webb: Cyclical Behavior

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Keynes, John Maynard. [1936] 1964. The General Theory of Employment,
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Mitchell, Wesley Clair. [1913] 1941. Business Cycles and Their Causes.
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Prescott, Edward C. 1986. “Theory Ahead of Business Cycle Measurement.”
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Activity.” Federal Reserve Bank of Richmond Econoimic Quarterly
(Spring): 68–94.
Zarnowitz, Victor. 1992. Business Cycles: Theory, History, Indicators, and
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Full text of Economic Quarterly (Federal Reserve Bank of Richmond) : Winter 2003, Vol. 89, No. 1

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