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Working Paper Series
Taylor Rules in a Limited Participation
Model
Lawrence J. Christiano and
Christopher J. Gust
Working Papers Series
Research Department
Federal Reserve Bank of Chicago
March 1999 (W P-99-3)
FEDERAL RESERVE BANK
O F CHICAGO
T a y lo r R u le s in a L im ite d P a r t i c i p a t i o n M o d e l*
Lawrence J. C hristiano and Christopher J. G ust
March 1, 1999
Abstract
We use the limited participation model of money as a laboratory for
studying the operating characteristics of Taylor rules for setting the rate
of interest. Rules are evaluated according to their ability to protect the
economy from bad outcomes such as the burst of inflation observed in the
1970s. Based on our analysis, we argue for a rule which: (i) raises the
nominal interest rate more than one-for-one with a rise in inflation; and (ii)
does not change the interest rate in response to a change in output relative
to trend.
JEL numbers: El, E4, E52, E58
^ P o rtio n s o f th is d o c u m e n t are rep rin ted , w ith th e p e r m issio n o f th e U n iv e r sity o f C h ic a g o
P r e ss, from a c o m m e n t b y u s th a t is fo r th c o m in g in T aylor (1 9 9 9 a ). T h e first a u th o r is g r a te fu l
to th e N a tio n a l S c ie n c e F o u n d a tio n for a g ra n t to th e N a tio n a l B u rea u o f E c o n o m ic R e sea rch .
T h e c o n te n ts o f th is p a p er d o n o t reflect th e v ie w s o f t h e F ed era l R e serv e B o a r d or o f th e F ed era l
R e serv e B a n k o f C h ica g o .
1. I n t r o d u c t i o n a n d O v e r v i e w
Much research in monetary economics is stimulated by the burst of inflation ex
perienced by a number of countries in the 1970s. This research addresses two
questions: ‘why did this costly failure of monetary policy occur?’, and ‘what can
be done to prevent it from happening again?’
This introduction begins by briefly reviewing the evolution of thinking on these
questions, from the focus on institutional reform in the 1980s, to the focus on the
design of monetary policy rules more recently. We go on to discuss Taylor rules
specifically, and why it is of interest to consider their operating characteristics in
a limited participation model of money. We then summarize the results obtained
when we do this. An implication of one of our results is that further progress on
the analysis of monetary policy rules would benefit from addressing some of the
issues of credibility considered in the earlier literature on institutional reform.
1.1. Identifying G o o d Institutions
The initial body of research addressing the two questions in the opening paragraph
was stimulated by the seminal papers of Kydland and Prescott (1977) and Barro
and Gordon (1983). This work suggested that there was an inflation bias inherent
in monetary institutions and that some sort of institutional reform was required
to prevent a recurrence of 1970s-style inflation. Examples of such institutional re
form include legislative changes that focus a central bank’s mission more sharply
on inflation and that grant central banks more independence from the rest of the
government. Barro and Gordon’s analysis led to the prediction that, absent such
reform, inflation would move up and down as the incentives to inflate moved up
and down. To operationalize the theory, they made the assumption that the cen
tral bank’s incentive to inflate is measured by the natural rate of unemployment.
However, the Barro and Gordon theory lost some of its appeal in the two decades
since they wrote their paper, when the incoming evidence appeared to contradict
it.1 In the United States, a major, persistent drop in the rate of inflation occurred
starting in 1980, about three years before the unemployment rate started to come
down. In Europe and other countries, the incentive to inflate stood at a post-war
high in the 1980s and 1990s because the unemployment rate was so high, and yet
E v i d e n c e th a t does su p p o r t th e K y d la n d a n d P r e s c o tt (1 9 7 7 ) - B arro a n d G o rd o n (1 9 8 3 ) id e a
c o n c e r n s th e r e la tio n sh ip b e tw e e n in fla tio n a n d c e n tr a l b a n k in d e p e n d e n c e . S ee, for e x a m p le ,
t h e su r v e y in B la n ch a rd (1 9 9 7 , p. 55).
2
inflation was very low.2 Both sets of observations are puzzling from the Barro and
Gordon perspective, particularly because they were not preceded by significant,
formal institutional reform.3
1.2. Identifying G o o d Policy Rules
Alternative approaches to the two questions driving this literature were developed.
These place less emphasis on issues of commitment and on the notion that there is
an inflation bias in modern monetary institutions. To explain this, the concept of
a monetary policy ‘rule’ is useful. This specifies how the monetary authority varies
the instruments at its command as a function of the state of the economy. The
recent research focuses on identifying simple monetary policy rules that will reduce
the likelihood of a recurrence of a 1970s style inflation outbreak. The underlying
vision is that the poor economic outcomes of the 1970s were a consequence of the
poor monetary policy rule in place at that time. The notion that improvements
in our understanding of the economy that have occurred since then, arising both
from conceptual advances and from increased data, put us in a position to design
a better rule now.4
2S e e C h r istia n o a n d F itz g e r a ld (1 9 9 9 ) an d F r ie d m a n a n d K u ttn e r (1 9 9 6 ) for a n e la b o r a t io n
o n th e s e o b se r v a tio n s.
3V a rio u s m o d ific a tio n s o f th e B arro an d G o r d o n a p p r o a c h c a n p o te n tia lly r e c o n c ile t h e o b
s e r v a tio n s o n in fla tio n a n d u n e m p lo y m e n t w ith th e th eo ry . For e x a m p le , o n e c a n p o s it t h a t
th e r e is v a r ia tio n over tim e in p o lic y m a k e r p refe r e n c e s (s e e B a ll (1 9 9 5 ), C u k ie r m a n a n d M e ltz e r
(1 9 8 6 ), or R o g o ff (1 9 8 5 )). A lte r n a tiv e ly , b y a d o p tin g a v e r sio n o f th eir th e o r y in w h ic h th e e q u i
lib r iu m v a r ia b le s are a fu n c tio n o f th e h isto r y o f p a s t g o v e r n m e n t a c tio n s, it is p o s s ib le to h a v e
e q u ilib r ia in w h ic h cen tr a l b a n k s are ‘p u sh e d ’ in to s u p p ly in g m o re or le ss in fla tio n in r e s p o n s e
to m o v e m e n ts in v a ria b les o th e r th a n th e n a tu r a l r a te o f u n e m p lo y m e n t (s e e C h a r i, C h r is tia n o
a n d E ic h e n b a u m (1 9 9 8 ).) T h is c a n p o te n tia lly a c c o u n t for th e p u z z lin g o b se r v a tio n s j u s t c it e d .
W e c o n sid e r th is b elo w .
4For a s o m e w h a t p e ss im istic a ss e ssm e n t o f th e o u tlo o k for th is a p p ro a ch , se e S a r g e n t (1 9 9 9 ).
H e c o n s tr u c ts a v a r ia n t o f th e K y d la n d -P r e s c o tt/B a r r o -G o r d o n m o d e l in w h ic h th e p o lic y m a k e r
m o d ifie s its v ie w s a b o u t th e str u c tu r e o f th e e c o n o m y as n e w d a ta c o m e in. A s th e s e v ie w s e v o lv e ,
th e p o lic y m a k e r a d ju sts its m o n e ta r y p o lic y ru le. In S a r g e n t’s e x a m p le , th is p r o c e ss d o e s n o t
co n v e r g e . It sim p ly lea d s to a n e n d le ss r e p e titio n o f in fla tio n ta k e -o ff’s like th a t o b s e r v e d in
th e 1 9 7 0 s, fo llo w ed b y in fla tio n c o lla p se s. S a r g e n t’s e x a m p le is im p o r ta n t b e c a u s e it a r tic u la te s
c le a r ly a p o te n t ia l p itfa ll a ss o c ia te d w ith th e d e sig n o f m o n e ta r y p o lic y ru les. S till, th e d e ta ils
o f h is m o d e l are reje c te d in th e se n s e th a t it is n o t a b le to a c c o u n t for d u r a tio n o f th e h ig h
in fla tio n in th e 197 0 s. T h e r e a so n is th a t th e p o lic y m a k er in S a r g e n t’s m o d e l, w h e n c o n fr o n te d
w it h th e s im u lta n e o u s rise in in fla tio n an d u n e m p lo y m e n t o b se r v e d in th e e a r ly 1 9 7 0 s, w o u ld
h a v e in ferred th a t h ig h in fla tio n is not a p r o d u c tiv e w a y to r e d u c e u n e m p lo y m e n t. A c c o r d in g to
3
In the quest for good monetary policy rules, rules for setting the interest rate
have taken a particularly prominent role. Such rules are called ‘Taylor rules’ after
John Taylor, who has played an important role in popularizing this research.
The work has attracted so much attention in part because the interest rate is
what central bankers view themselves as controlhng. As a result, the research
on interest rate rules has substantial potential practical relevance. Although this
research is still fairly new, a consensus has already begun to emerge. To explain
this, consider the following typical Taylor rule
r t
=
c + prt_i + m r£ +
p y t ,
(1.1)
where 7rt is the annualized rate of inflation, r t is the annualized Federal Funds
rate and y t is the log deviation of output from trend. The emerging consensus
is that a Taylor rule characterized by an aggressive response of the interest rate
to high inflation and high output is likely to yield good results.* For example,
5
Taylor (1999) urges the implementation of a rule with p = 0, 0 = 1 and a — 1.5.
1.3. The Limited Participation Model as a Laboratory
The strategy of the existing literature evaluates monetary policy rules by studying
their operating characteristics in quantitative, economic models. For the most
part, the models used in this literature are sticky price, rational expectations
versions of the IS-LM model.6 The question naturally arises: are the existing
results robust to alternative, plausible models? We investigate this in the context
of one such model. In particular, we investigate the performance of Taylor rules in
a simple limited participation model recently studied by Christiano, Eichenbaum
and Evans (1998) (CCE).7 The mechanisms in this model differ from those in
the existing literature. In particular, the friction which generates monetary non
neutrality is a credit market friction, not stickiness in price setting. In addition,
the channel from expected inflation to output in this model differs from what it
S a r g e n t’s m o d e l, th e p o lic y m a k e r ’s r e a c tio n to th is d isc o v e r y w o u ld h a v e b e e n to keep in fla tio n
low . S ee S a r g e n t’s c h a p te r 9 for a fu rth er d isc u ssio n .
5S ee th e p a p ers in T ay lo r (1 9 9 9 a ). S ee a lso C la rid a , G a li an d G ertler (1 9 9 7 ) and K in g an d
K err (1 9 9 6 ).
6W h e n resea rch ers a d o p t m o d e ls n o t in th is p a ra d ig m , th e y o fte n g e t d ifferen t r e su lts. S ee,
for e x a m p le , B e n h a b ib , S c h m itt-G r o h e an d U rib e (1 9 9 8 ).
7For a c o m p a r iso n o f th e em p ir ic a l p erfo rm a n ce o f s tic k y p rice v e r su s lim ite d p a r tic ip a tio n
m o d e ls, s e e C h r istia n o , E ich e n b a u m an d E v a n s (1 9 9 7 ).
4
is in the sticky price, rational expectations version of the IS-LM model. Since the
source of monetary frictions and the channels from expected inflation to output
are not yet well understood, we view our analysis as providing a useful robustness
check on the existing literature.
In evaluating a particular parameterization of the Taylor rule, we focus pri
marily on its ability to rule out bad outcomes.8 In particular, we want to ensure
that the monetary policy rule is not itself a source of welfare-reducing instability
for the economy.9 This can happen for at least two reasons: (i) the rule may en
able expectations of inflation to become self fulfilling, a situation that can occur
when the steady state equilibrium of the nonstochastic version of the economy is
‘indeterminate’ and (ii) the rule may cause the economy to react explosively to
shocks.
1.4. Our Results
Three results are reported below that we wish to emphasize here. First, ag
gressiveness in a Taylor rule is a good idea, but only in response to inflation.
Aggressiveness in the response to deviations in output from trend is a bad idea in
our model, and can produce welfare-reducing volatility of the kind cited in (i) and
(ii) in the previous paragraph. For example, we find that Taylor’s recommended
values for a , p,(3 places too much weight on output, and result in explosiveness.10
Second, when we incorporate the monetary policy rule estimated by Clarida, Gali
and Gertler (1997) to have been followed by the US Federal Reserve in the 1970s
into our model, we find that the model exhibits equilibrium indeterminacy. As a
result, our model is able to articulate the view that the burst of high inflation in
8W e d o n o t seek to id e n tify p o lic y ru le p a ra m eter v a lu e s th a t o p tim iz e u tility in ou r m o d e l,
a n d w e m a k e n o a tte m p t to co m p a re th e p erfo rm a n ce o f T ay lo r ru les w ith th e u n c o n str a in e d
o p tim a l m o n e ta r y p o licy . In our e x p e r ie n c e , first-ord er w elfa re g a in s are to b e h a d b y a v o id in g
th e ‘b ad o u tc o m e s ’ liste d n e x t in th e te x t. O n ce th e se o u tc o m e s h ave b e e n a v o id e d , th e r e is
r e la tiv e ly le ss to b e g a in e d from m o v in g to th e g lo b a lly o p tim a l sp e c ific a tio n . T h is is c o n s is te n t
w ith fin d in g s re p o r te d in R o te m b e r g an d W o o d fo rd (1 9 9 9 ), w h o d isp la y a m o d e l in w h ic h th e
w elfa re fu n c tio n is r e la tiv e ly in s e n sitiv e to a lte r n a tiv e sp e c ific a tio n s o f in te r e st r a te ru les, a s lo n g
a s o n ly p a r a m e te r v a lu es in th e reg io n o f eq u ilib riu m d e te r m in a c y are co n sid e r e d .
9O th e r resea rch th a t a d o p ts th is p e r sp e c tiv e on th e d e sig n o f m o n e ta r y p o lic y ru les in c lu d e s
C a r lstr o m a n d F u erst (1 9 9 8 , 1 999) an d B en h a b ib , S c h m itt-G r o h e an d U r ib e (1 9 9 8 ).
10For a n o th e r m o d e l w ith th is prop erty, see Isard, L a x to n an d E lia sso n (1 9 9 9 ).
5
the 1970s was due to higher expectations of inflation.11 According to the model,
these expectations were translated into higher actual inflation because the policy
rule implemented in the 1970s was insufficiently aggressive with respect to infla
tion. In this respect, our result is similar to the one reported for the sticky price,
rational expectations version of the IS-LM model considered by Clarida, Gali and
Gertler (1997). Still, our result does differ from theirs in one potentially impor
tant respect. In our model, a rise in inflation expectations that is self-fulfilling
acts to weaken the economy. In a model like that of Clarida, Gali and Gertler
(1997), such a rise in inflation expectations drives output up. This distinction
between these two classes of models may provide a way to discriminate between
them, since the 1970s are thought to be a period when output was low relative to
trend.
The basic intuition underlying these different implications of our model and
versions of the standard IS-LM model is simple. The latter emphasize that higher
anticipated inflation leads to a reduction in the real rate of interest, which in
turn results in a rise in output and actual inflation by stimulating the investment
component of aggregate demand.12 If the central bank adopts a tight money policy
every time output and/or inflation is high, this chain of causation from expected
inflation to actual inflation is cut. Thus, a high a and/or a high ft eliminates
equilibria in these models in which high inflation is self-fulfilling.
Now consider our model. Here, higher anticipated inflation induces households
to substitute out of cash deposits in the financial sector and towards the purchase
of goods. The resulting shortfall of cash in the financial sector puts upward
pressure on the nominal rate of interest. If a in the central bank’s policy rule is
small, it has to inject liquidity into financial markets in order to prevent a large
rise in the rate of interest. This expansion of liquidity would produce the increase
in inflation that people anticipated. This is the intuition underlying our finding
that a small value of a increases the likelihood that expectations of inflation can
11 T h is is a v ie w t h a t is also a r tic u la te d in C h a ri, C h r istia n o a n d E ich en b a u m (1 9 9 8 ) an d
C la rid a , G a li a n d G ertler (1 9 9 7 ).
12T h e b a sic lo g ic c a n b e illu str a te d u sin g a te x t b o o k A g g r e g a te S u p p ly -A g g r e g a te D e m a n d
d ia g r a m , w ith p r ic e o n th e v e rtica l a x is an d o u tp u t o n th e h o riz o n ta l. In th e u su a l w ay, a fa ll
in e x p e c te d in fla tio n sh ifts A g g r e g a te D e m a n d to th e rig h t. P r ic e s rise a s th e e c o n o m y m o v e s
u p a lo n g th e A g g r e g a te S u p p ly cu rve. T h e r e su ltin g rise in p rice co rr e sp o n d s an a c tu a l rise in
in fla tio n . T h is c h a in lin k in g e x p e c te d in fla tio n to a c tu a l in fla tio n is b rok en if th e a u th o r itie s
sh ift th e A g g r e g a te D e m a n d C u rve to th e left w h e n e v e r th e y s e e o u tp u t or in fla tio n risin g. H ig h
v a lu e s o f a a n d (3 d o ju s t th a t.
6
be self-fulfilling. Similarly, a large value of a reduces the likelihood that this type
of equilibrium could exist.
The previous intuition also shows why a large value of 0 can actually in c r e a s e
the likelihood that inflation expectations are self-fulfilling in our model. T hat
is because the rise in the interest rate that occurs with a rise in inflation under
the Fed’s policy rule also produces a reduction in output. With a large 0 , th at
fall in output operates to offset the Fed’s policy of raising the interest rate when
a > 0. In effect, raising 0 cancels out the indeterminacy-fighting properties of a
high value of a .
Our third and final result that deserves emphasis is the following. Our analysis
suggests that the literature on monetary policy rules may have been too quick to
abandon the issues of commitment raised by the analysis of Kydland and Prescott
(1977) and Barro and Gordon (1983). Our results suggest that a Taylor rule th at
is sufficiently aggressive to inoculate the economy against a 1970s style inflation
outburst may lack credibility because there is a strong - perhaps irresistible incentive to deviate from it. We computed an example in which a benevolent
central bank has an incentive to deviate from such a rule when there is a supply
shock which drives prices up and output down simultaneously. In the example,
the increased welfare gains from deviating to a k % rule at that time are the
equivalent of about 0.3% of consumption, forever. To get a sense of the magnitude
of this, it corresponds roughly to the amount the federal government spends on
the administration of justice, or on general science, space, and technology.13 This
is a substantial amount, and may be difficult to resist for a central bank. A more
complete analysis of the concerns raised in this example requires spelling out more
clearly the details of the environment. This is beyond the scope of our analysis.14
13T h e p r e lim in a r y e s tim a te for 1 9 9 7 o f c o n s u m p tio n o f n o n d u r a b le g o o d s a n d s e r v ic e s in th e
1 9 9 8 E c o n o m ic R e p o r t o f th e P r e sid e n t is $ 4 .8 tr illio n , s o t h a t 0.3% o f th is is $ 1 6 b illio n . T h e
fed er a l e x p e n d itu r e s in fisca l year 1 9 9 7 o n g e n e r a l s c ie n c e , sp a c e , a n d te c h n o lo g y w a s $ 1 7 b illio n ,
a n o n th e a d m in istr a tio n o f ju s tic e it w a s $20 b illio n .
14R o te m b e r g an d W o o d fo rd h a v e p o in te d o u t to us in p r iv a te c o n v e r sa tio n th a t a s t ic k y p r ic e
m o d e l m a y n o t suffer from th e so rt o f c r e d ib ility p r o b le m e m p h a s iz e d h ere. In a s t ic k y p r ic e
m o d e l, th e r e is a te n d e n c y for o u tp u t to fa ll b y le ss th a n th e efficien t a m o u n t, a fte r a b a d
te c h n o lo g y sh o c k . A c c o r d in g to th is m o d e l, im p le m e n tin g a tig h t m o n e ta r y p o lic y a t s u c h a
tim e m ig h t a c tu a lly im p ro v e th e w elfa re o f p r iv a te a g e n ts .
7
1.5. Rules and Credibility
These results on credibility highlight a different possible answer to the two ques
tions posed in the first paragraph. It may be that the problem in the 1970s was
not lack of knowledge that a higher value of a might have prevented the infla
tion take off. Instead, reasoning as in Chari, Christiano and Eichenbaum (1998),
that episode may have reflected a weakness in monetary policy institutions, which
simply could not resist accommodating higher inflation expectations in a faltering
economy.
That these concerns may be of more than academic interest is suggested by
the statements on inflation by Arthur Burns, who was chairman of the Federal
Reserve in the 1970s. These suggest that his failure to raise interest rates in line
with the dictates of a more aggressive Taylor rule did not reflect ignorance about
the connection between money and inflation. He claimed that, instead, it was
his fear of the social consequences of such an action that prevented him from
implementing a high interest rate policy.15 Thus, both history and theory suggest
that credibility issues should also be considered when designing monetary policy
rules.
The next section briefly describes our model. Results are presented in the
following section. We close with a brief conclusion.
2. M o d e l
In this section, we describe the model used in our analysis and we present some
empirical evidence in its favor.
15A n e x c e r p t from a sp e e c h b y A rth u r B u rn s in 1 9 7 7 su m m a r iz e s v ie w s th a t h e r e p e a te d o fte n
d u rin g h is te n u r e a s c h a irm a n o f th e F ed eral R eserve: ‘W e w e ll k n o w -a s d o m a n y o t h e r s - t h a t
if th e F ed era l R e se r v e sto p p e d c r e a tin g n ew m o n ey , or if th is a c tiv ity w ere slo w ed d r a stic a lly ,
in fla tio n w o u ld s o o n e ith e r c o m e to an en d or b e s u b s ta n tia lly ch eck ed . U n fo r tu n a te ly , k n o w
in g th a t tr u th is n o t as h elp fu l as o n e m ig h t su p p o se . T h e c a tc h is th a t n o w a d a y s th e r e are
tr e m e n d o u s n o n m o n e ta r y p ressu res in our e c o n o m y th a t are te n d in g to d riv e c o sts an d p r ices
h ig h e r ....I f th e F ed era l R e serv e th e n so u g h t to c r e a te a m o n e ta r y e n v ir o n m e n t th a t se r io u sly
fell sh o r t o f a c c o m m o d a tin g th e n o n m o n e ta r y p ressu r e s th a t h a v e b e c o m e c h a r a c te r istic o f our
tim e s , se v e r e str e sse s co u ld b e q u ick ly p ro d u ced in ou r eco n o m y . T h e in fla tio n ra te w o u ld p rob
a b ly fa ll in th e p r o c e ss b u t so , to o , w o u ld p r o d u c tio n , jo b s , a n d p ro fits. T h e ta c tic s a n d str a te g y
o f th e F ed era l R ese r v e S y s t e m -a s o f an y c en tra l b a n k -m u s t b e a ttu n e d to th e s e r e a litie s .’ For
a d d itio n a l d is c u ssio n o f B u r n s’ (1 9 7 8 ) sp eech es, s e e C h a ri, C h r istia n o an d E ic h e n b a u m (1 9 9 8 ).
8
We examine the operating characteristics in our model of the following three
variants on (1.1):
r t — c + p r t_ i
a E t 7rt+ i + f i y t ,
+
(Clarida-Gali-Gertler)
p r t_i
+
a ir t
r t = c + p r t_ i
+
otTTt-\ + P y t - 1 ,
rt = c
+
+
Pyt,
(Generalized Taylor)
(Lagged Taylor)
As before, r t is the (annualized) nominal rate of interest that extends from the
beginning of quarter t to the end of quarter t. Also, 7rt = l o g ( P t) —o g ( P t _ i ) ,
l
7 = l o g ( P t ) — l o g ( P t _ A), and y t = lo g ( Y t ), after a trend has been removed. We
ft
refer to the above as the Clarida, Gali, and Gertler (1997) (CGG), the Generalized
Taylor (GT) and Lagged Taylor (LT) policy rules, respectively.
We study the performance of these three rules in the CEE model. A detailed
discussion of the model appears in CEE, and so we describe it only very briefly
here. Apart from two modifications, it is basically a standard limited participation
model. One modification is that, in addition to having a technology shock, it also
has a money demand shock. Traditionally, an important rationale for adopting an
interest rate targeting rule was to eliminate the effects of money demand shocks
from the real economy (see, for example, Poole (1970).) So, if anything, including
them in the analysis should bias the results in favor of the interest rate targeting
rule. A second difference is that, although there is still a monetary authority on
the sidelines transferring cash into and out of the financial system in our model
economy, those transfers are endogenous when the monetary authority conducts
its operations with the objective of supporting an interest rate targeting rule.
The representative household begins period t with the economy’s stock of
money, M t , and then proceeds to divide it between Q t dollars allocated to the
purchase of goods, and M t — Qt dollars allocated to the financial intermediary. It
faces the following cash constraint in the goods market:
Qt
+
W t L t > P t (C t
+ I t) ,
where I t denotes investment, C t denotes consumption, L t denotes hours worked,
and W t and P t denote the wage rate and price level. The household owns the
stock of capital, and it has the standard capital accumulation technology:
K t+ i
=
It +
(1 - 0.02) K t .
9
The household’s assets accumulate according to the following expression:
M t+ 1
=
Qt
+
W tL t — Pt (Ct
+
It)
+
R t(M t
—Q t +
X t)
+
+
Dt
VktKt ,
where X t is a date t monetary injection by the central bank and R t denotes the
gross quarterly rate of return on household deposits with the financial intermediary.16
Also, D t denotes household profits, treated as lump sum transfers, and r^t is the
rental rate on capital. An implication of this setup is that the household’s date
t earnings of rent on capital cannot be spent until the following period, while
its date t wage earnings can be spent in the same period. As a result, inflation
acts like a tax on investment. The household’s date t decision about Q t must be
made before the date t realization of the shocks, while all other decisions are made
afterward. This assumption is what guarantees that when a surprise monetaxy
injection occurs, the equilibrium rate of interest falls, and output and employ
ment rise. To assure that these effects are persistent, we introduce an adjustment
cost in changing Q t , H t = H
, where H t is in units of time, and H is an
increasing function.17 The household’s problem at time 0 is to choose contingency
plans for C t , It, Q t, M t+ \, L t , K t+ 1 , t = 0,..., oo to maximize
o°
£
E oE ^ - 03-'25) U ( C t , L t , H t ) ,
O
(L + H ) (l+V
U ( C , L , H ) = log
C -ip o
1+
t0
=
$
subject to the information, cash, asset accumulation and other constraints. Here,
ip = 1/2.5 and tpQ is selected so that L t = 1 in nonstochastic steady state.
Firms must finance J t of the wage bill by borrowing cash in advance from the
financial intermediary, and 1 —J t can be financed out of current receipts. The
random variable, J t , is our money demand shock, and it is assumed to have the
following distribution:
l o g ( J t ) = 0 .9 5 lo g ( J t _ i )
+
e Jtt,
16W e h a v e rt = 4 (Rt- 1).
17T o a ssu r e th a t th e in te r e st ra te e ^ e c t is p e r siste n t, w e in tr o d u c e a c o s t o f a d ju s tin g
H
w h ere
x
Qt
Q t-i
= 4
ex p
Qt
-b ex p
Q t -1
d e n o te s th e a v era g e r a te o f m o n ey g ro w th . W e s e t
10
—
c
Q
—
Q t- 1
d = c=
1
2 and
—
X
Q t:
)H
x=
0 .0 1 .
where e j j has mean zero and standard deviation 0 .0 1 . All of the rental payments
on capital can be financed out of current receipts. This leads to the following first
order conditions for labor and capital:
W t [R tJt
+ 1 — Jt]
_ fl,,t
Pt
V
Tkt _ f{(,t
’
Pt ~
V
’
where p = 1.4 is the markup of price over marginal cost, reflecting the existence
of market power. Also, f i% represents the marginal product of factor i, i = L , K ,
t
and
f ( K t , L t , v t ) = e x p ( v t ) K ™ 6L°t 64,
where
vt =
0.95ut_i +
e Vyt,
and £Vit has mean zero and standard deviation 0 .0 1 .
Finally, we specify monetary policy in four ways. In the first, money growth
is purely exogenous, and has the following second order moving average form:
Xt — x
+ 0.08£j + 0.26et_i + 0.1 l^t—,
2
where e t is a mean zero, serially uncorrelated shock to monetary policy and
x = 0.01. This representation is Christiano, Eichenbaum and Evans (1998)’s
estimate of the dynamic response of M l growth to a monetary policy shock, after
abstracting from the effects of all other shocks on monetary policy. Other repre
sentations of monetary policy analyzed here include the CGG, the GT and the LT
rules presented above. In these cases, the response of x t to nonmonetary shocks
is endogenous, although we preserve the assumption throughout that E x t — x .
Figure 1 presents the dynamic response of the model’s variables to an e t shock
in period 2. The percent deviation of the stock of money from its unshocked
growth path is displayed in panel c. The magnitude of the shock was chosen so
that the money stock is eventually up by 1 percent. Panels a, b and f indicate
that the impact effect on output of the monetary policy shock is so great that the
price response is nil. Afterward, the price level rises slowly, and does not reach
its steady state position until around one year later. The reasons for this sluggish
response in the price level are discussed in detail in Christiano, Eichenbaum and
Evans (1997).18 Next, note the hump-shaped responses of employment, output,
18T h e b a sic id e a is a s fo llo w s. A p o s itiv e m o n e ta r y in je c tio n h a s tw o effects: (i) it s t im u la t e s
11
consumption and investment. Finally, there is a persistent fall in the interest
rate. As emphasized in Christiano, Eichenbaum and Evans (1998), these patterns
are all qualitatively consistent with the data. They support the notion that our
model represents a useful laboratory for evaluating the operating characteristics
of alternative monetary policy rules.
3. R e s u l t s
This section presents our quantitative results. We first display the regions of
the policy parameter space in which indeterminacy, determinacy and explosive
ness occur. Loosely, determinacy corresponds to the case where equilibrium is
(locally) unique, so that self-fulfilling inflation episodes are not possible. Indeter
minacy corresponds to the case where such equilibria are possible. Explosiveness
corresponds to the case in which a shock causes the economy to diverge perma
nently from its initial position.19 In the subsequent two subsections we report
some calculations to illustrate the economic meaning of the indeterminacy and
explosiveness findings. In addition, we discuss the credibility difficulties that may
exist in implementing an interest rate rule in practice.
3.1. Indeterminacy, Determinacy and Explosiveness
Figures 2, 3 and 4 report regions of a , ft where equilibrium is determinate (white),
indeterminate (grey) and explosive (black), for p = 0.0, 0.5, 1.5. The results are
for the CGG, GT and LT rules, respectively.
We begin with a discussion of the results for the CGG rule, displayed in Figure
2 . Consider the case, p = 0, first. We find that when (3 = 0, then determinacy
requires a > 7 , where 7 is a number just below unity .20 This is analogous to
findings reported in Kerr and King (1996) for the IS-LM model (see also CGG). In
that model, the value of 7 where the economy switches between determinacy and
d e m a n d b y p u tt in g m o re c a sh in th e h an d s o f h o u se h o ld s an d (ii) it stim u la te s su p p ly b y r e d u c in g
th e r a te o f in te r e st. T h e effect o f (i) alo n e is to in c r e a se th e p rice lev el. T h e e ffect o f (ii) is
to d e c r e a se th e p rice lev el. If th e s e su p p ly a n d d e m a n d e ffe c ts trig g ered b y a m o n e ta r y sh o c k
r o u g h ly c a n c e l, th e r e is o n ly a sm a ll effect o n th e p rice level.
19T e c h n ic a lly , d e te r m in a c y , in d e te r m in a c y a n d e x p lo s iv e n e ss c o r r e sp o n d to th e n u m b er o f
e x p lo s iv e e ig e n v a lu e s in th e m o d e l’s red u ced form , a s in th e a n a ly s is o f B la n c h a r d a n d K a h n
(1 9 8 0 ).
20N o te from F ig u re 2 a th a t d e te r m in a c y a lso req u ires th a t
12
a
not be
to large.
o
indeterminacy is 7 = 1 . Our results resemble those of Kerr and King (1996) and
CGG in supporting the notion that an aggressive response to expected inflation
reduces the likelihood of indeterminacy. In contrast to CGG, however, we find th at
the likelihood of indeterminacy and explosiveness increase with 0 . The intuition
for the former result was discussed in the introduction.
Now consider the case p — 0.5. When (3 = 0, then determinacy requires a > 7 ,
where 7 is a number just below 0.5. This result, and others not reported, are
consistent with the notion that the condition for determinacy is similar to what
it was in the case of p = 0, as long as it is placed on a / ( 1 —p ), and not a . T hat is,
in several quantitative experiments we found that with [3 — 0 and for 0 < P < 1,
determinacy requires a / ( l — p) > 7 , where 7 is slightly below unity. Interestingly,
a / ( l —p) corresponds to the long run cumulative impact on the interest rate of a
one-time increase in expected inflation.21 This suggests that what is important,
in guaranteeing equilibrium determinacy, is that the cumulative effect over time
of an increase in expected inflation be greater than unity. The precise timing of
the response of the interest rise to an increase in inflation matters less. Note also
that, like in the p = 0 case, raising (3 increases the likelihood of indeterminacy or
explosiveness.
Finally, consider the case p = 1.5. As is to be expected from the p = 0.5 result,
the range of a 's which generate determinacy is larger here. As in the other cases,
increasing 0 raises the likelihood of indeterminacy or explosiveness.
Now consider the results reported in Figure 3 for the GT rule. Taylor (1999)
suggests that a good parameterization for (1.1) is p = 0, a = 1.5 and 0 = 1 .
Interestingly, Figure 3 indicates that, for our model, this parameterization lies
in the explosiveness region. Thus, our model indicates that the economy would
perform very poorly with this parameterization of the policy rule. According to
the results in Rotemberg and Woodford (1999), when p = 0, a > 0, then increasing
0 raises the likelihood of equilibrium determinacy. In our model, this is not the
case. Either we enter the explosiveness region for large 0 , or we enter the region
of indeterminacy. Interestingly, as p increases, the region of determinacy expands.
The results in Figure 4 for the LT policy rule resemble those in Figure 3. The
preferred parameterization of Rotemberg and Woodford (1999), a = 1.27, 0 =
0.08 and p = 1.13 lies in the determinacy region for our model, if we extrapolate
21T h u s, s u p p o se th e r e is a o n e -tim e p u lse o f m a g n itu d e u n ity in E tr+i. T h e im p a c t e ffe c t o n
it
r t is a. T h e la g o n e e ffect is ap, an d th e la g i effect is apl for 1 — 1 . 2 . 3 , . . . . T h e s u m o f t h e s e
,
e ffe c ts, a s lo n g a s
\\ <
p
1, is
a/(1
—p.
)
13
between the p = 0.5 and p = 1.5 graphs in Figure 4. A notable feature of the LT
policy rule is that with p large, the determinacy region is reasonably large and
resembles the determinacy region for the GT rule.
To summarize, an aggressive response to inflation (or, expected inflation) in
creases the likelihood of determinacy. However, a more aggressive response to
output has the opposite effect in our model. In addition, our results support the
notion that choosing a high value of p increases the likelihood of determinacy.
Finally, the CGG rule appears to have the smallest region of determinacy.
3.2. Illustrating Indeterminacy
We report some calculations to illustrate what can happen when there is indeter
minacy. To this end, we worked with two versions of the CGG rule. The first is
useful for establishing a benchmark, and uses a version of the CGG rule for which
there is a locally unique equilibrium, (p = 0.66, 0 = 0.16, a — 0.61). The second
uses a version, (p — 0.66, 0 = 0.16, a — 0.32), of the CGG rule for which there is
equilibrium indeterminacy. We refer to the first rule as the stable CGG rule and
to the second as the unstable CGG rule. We consider the dynamic response of
the variables in our model economy to a one standard deviation innovation in J t
in period 2 .
Figure 5 displays the results for economy operating under a k% money growth
rule (dotted line) and under the stable CGG rule. Note that under the k % rule,
the results are what one might expect from a positive shock to money demand:
interest rates rise for a while and inflation, output, employment, consumption and
investment drop. Now consider the economy’s response to the money demand
shock under the stable CGG rule. As one might expect, this monetary policy
fully insulates the economy from the effects of the money demand shock. Figure
5c indicates that this result is brought about by increasing the money stock. Not
surprisingly, the present discounted utility of agents in the economy operating
under the stable CGG rule, 74.092, is higher than it is in the economy operating
under the k % rule, 74.036. These present discounted values are computed under
the assumption that the money demand shock takes on its mean value in the
initial period, and the capital stock is at its nonstochastic steady-state level.
Now consider the results in Figure 6, which displays the response of the model
variables to a money demand shock in two equilibria associated with the unstable
CGG policy rule. In equilibrium # 2 (see the dotted line), the economy responds
14
in essentially the same way that it does under the stable CGG rule. Now consider
equilibrium # 1 (the solid line). The money demand shock triggers an expec
tation of higher inflation. Seeing the inflation coming, the central bank raises
interest rates immediately by only partially accommodating the increased money
demand.22 In the following period households, anticipating higher inflation, shift
funds out of the financial sector and towards consumption (Figure 6b shows th at
Q t rises, relative to its steady state path, in period 3). The central bank responds
by only partially making up for this shortfall of funds available to the financial
sector. This leads to a further rise in the interest rate and in the money supply.
In this way, the money stock grows, and actual inflation occurs. Employment and
output are reduced because of the high rate of interest. Investment falls a lot
because the higher anticipated inflation acts as a tax on the return to investment.
In addition, the rental rate on capital drops with the fall in employment.
The utility level associated with equilibrium # 1 is 73.825 and the utility level
in equilibrium # 2 is 74.110. The utility numbers convey an interesting message.
On the one hand, if the stable CGG rule is implemented, then agents enjoy higher
utility than under the k % rule. On the other hand, if the unstable CGG policy
rule is used, then it is possible that utility might be less than what it would be
under the k % rule. In this sense, if there were any uncertainty over v/hether a
given interest rate rule might produce indeterminacy, it might be viewed as less
risky to simply adopt the k % rule. In a way, this is a dramatic finding, since the
assumption that money demand shocks are the only disturbances impacting on
the economy would normally guarantee the desirability of an interest rate rule like
(1.1).
3.3. Illustrating Explosiveness and Implementation Problems
We now consider a version of our model driven only by technology shocks. We
consider two versions of the LT policy rule. One adopts the preferred parameter
ization of Rotemberg and Woodford (1999): a = 1.27, f3 = 0.08, p = 1.13. The
other adopts a version of this parameterization that is very close to the explosive
region in which (3 is assigned a value of unity. Figure 7 reports the response of
the economy to a one standard deviation negative shock to technology under two
22T h is is d iffic u lt to se e in F ig u re 6 c b e c a u se o f sc a le . M o n e y g r o w th in p e r io d 2 is n e a r ly 6
p e r c e n t, a t a n a n n u a l r a te , in eq u ilib riu m 2. A c c o r d in g to F ig u re 6 g , th is is e n o u g h to p r e v e n t
a rise in th e in te r e st r a te in th a t eq u ilib riu m . M o n e y g r o w th in p e r io d 2 o f e q u ilib r iu m # 1 is
le s s, n a m e ly 5 .5 p e r c e n t, a t an a n n u a l ra te.
15
specifications of monetary policy. In one, monetary policy is governed by a k %
rule (see the dotted line), and in the other it is governed by the LT rule just
described (see the solid line).
Consider first the k % rule. The technology shock drives up the price level,
which remains high for a long period of time. Employment, investment, con
sumption and output drop. There is essentially no impact on the rate of interest.
The present discoimted value of utility in this equilibrium is 74.095. Consider by
contrast the LT rule. The rise in inflation in the first period leads the central
bank to cut back the money supply in the following period (recall, this policy
rule looks back one period). This triggers a substantial rise in the interest rate,
which in turn leads to an even greater fall in employment, output, consumption
and investment than occurs under the k % rule. The present discoimted value of
utility in this equilibrium is 74.036. It is not surprising that in this case, the k %
rule dominates the monetary policy rule in welfare terms, and in terms of the
variability of output and inflation.
Now consider the operation of the nearly explosive policy rule, in Figure 8.
With this rule, responses are much more persistent than under the previous rule.
The response looks very much like a regime switch, with money growth and the
interest rate shifting to a higher level for a long period of time. Given all the
volatility in this equilibrium, it is not surprising that welfare is lower at 73.549.
These examples illustrate the practical difficulties that can arise in imple
menting an interest smoothing rule like (1.1). In a recession, when output and
employment are already low, the rule may require tightening even further. The
social cost of doing that may be such that the pressures to deviate may be ir
resistible. Numerical results to support this proposition were summarized in the
introduction.23
4. C o n c l u s i o n
One interpretation of the high inflation experience of the 1970s is that it was
the outcome of the Federal Reserve implementing a policy rule which permitted
23C la r id a , G a li a n d G ertler (1 9 9 7 a ) argu e for a sp e c ific a tio n in w h ic h y t is th e d e v ia tio n fro m
p o te n tia l o u tp u t , ra th er th a n from tren d , as w e d o h ere. W e s u s p e c t th a t if w e rep la c e yt in
th e T a y lo r ru le w ith th e d e v ia tio n from p o te n tia l, th e c r e d ib ility p r o b le m w ith ou r p o lic y ru le
w o u ld b e w o rse, for ( > 0. T o se e w hy, n o te th a t w ith 0(yt— zt , w h e r e zt is p o te n tia l o u tp u t ,
3
)
a fa ll in p o te n t ia l a fter a te c h n o lo g y sh o ck w o u ld a c t to r a ise th e r a te o f in te r e st e v e n m o re.
16
inflation expectations to be self-fulfilling. An important objective of monetaryanalysis is to design rules which will not allow bad outcomes like this to happen
again. This paper studied the operating characteristics of Taylor rules in the
context of a limited participation model of money. In this model, monetary nonneutrality arises from a particular friction in the household’s portfolio decision.
Equilibria in which expectations about inflation are self-fulfilling are eliminated
when the Taylor rule responds aggressively to inflation and very little to output.
A strong response to output risks destabilizing the economy. In this respect,
the model’s implications differ from those of standard sticky price models, which
suggest that the possibility of self-fulfilling inflation expectations are ruled out
when the Taylor rule responds aggressively both to inflation a n d output.
So, which model should be taken more seriously for purposes of designing
monetary policy? We have pointed out that under a sticky price model, equilibria
in which inflation expectations are self-fulfilling tend, other things the same, to
be associated with high output and investment. The limited participation model
has the opposite property. This suggests that the latter may have an easier time
explaining the 1970s than the former, since this was a period when output and
investment were generally low. If a more formal analysis turns out to support this
possibility, then the policy implications of the limited participation model would
need to be taken seriously.
But, suppose it is not so easy to determine which model, the sticky price
model or the limited participation model, is closer to the truth? Robustness
considerations suggest picking a rule which works well in either model. And, each
model has the implication that bad outcomes are avoided by Taylor rules which
respond aggressively to inflation and not to output. So, we conclude that if a
Taylor rule is to be adopted, then it should be of this type.
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17
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18
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M o n e t a r y P o l ic y R u les,
19
University of Chicago Press,
Figure 1
Response of Model to an Exogenous Monetary Policy Shock
a: Price Level - % dev from SS
e: Inflation Rate - A PR
6.0000
5 00
.50
5 00
.00
450
00
4 00
.00
3 00
.50
3 00
.00
0 t 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9
f: Output - % dev from SS
d: Consumption - % dev from SS
% dev from SS: deviation from unshocked nonstochastic steady state growth path expressed in percent terms
APR: annualized percentage rate
F ig u r e 2
R e g io n s o f U n iq u e n e s s , E x p lo s iv e n e s s a n d In d e te r m in a c y
C la r id a - G a li- G e r t le r R u le
]
U n iq u e n e s s
E x p lo s iv e n e s s
in d e te r m in a c y
F ig u r e 3
R e g io n s o f U n iq u e n e s s , E x p lo s iv e n e s s a n d In d e te rm in a c y
G e n e ra liz e d T a y lo r R u le
U n iq u e n e s s
E x p lo s iv e n e s s
I n d e te r m in a c y
F ig u r e 4
R e g io n s o f U n iq u e n e s s , E x p lo s iv e n e s s a n d In d e te r m in a c y
L a g g e d T a y lo r R u le
]
U n iq u e n e s s
E x p lo s iv e n e s s
I n d e te r m in a c y
R
e s p o n s e
t o
a
M
F ig u r e 5
o n e y D e m a n d
S h o c k
U n d e r T w o
P o lic y
g: Interest Rate - APR
0 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8
d: Consumption - % dev from SS
Stable CGG Rule
K% Rule ............
See F g r 1f rNotes
iue o
R u
R
e s p o n s e
t o
a
M
F ig u r e 6
o n e y D e m a n d
S h o c k
U n d e r U n s ta b le
h: Investment - % dev from S S
Equilibrium 1
Equilibrium 2
See F g r 1f rNotes
iue o
C G G
R
e s p o n s e
t o
a
F ig u r e 7
N e g a t iv e T e c h n o lo g y
S h o c k
U n d e r T w o
h: Investment - % dev from SS
RW Lagged Response Rule
K% Rule ...............
See F g r 1f rNotes
iue o
P o lic y
R
e s p o n s e
t o
Perturbed RW Lagged Response Rule
K% R u l e ...............
See F g r 1f rNotes
iue o
a
F ig u r e 8
N e g a t iv e T e c h n o lo g y
S h o c k
U n d e r T w o
P o lic y
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