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Working Paper Series

Should Optimal Discretionary Monetary
Policy Look at Money?

WP 02-04

Michael Dotsey
Federal Reserve Bank of Philadelphia
Andreas Hornstein
Federal Reserve Bank of Richmond

This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/

Should Optimal Discretionary Monetary Policy Look
at Money?¤
Michael Dotseyyand Andreas Hornsteinz
Federal Reserve Bank of Richmond Working Paper No. 02-04
November 2002

Abstract
This paper examines whether monetary indicators are useful in implementing optimal discretionary monetary policy when the policy maker has incomplete information
about the environment. We …nd that money does not contain useful information for
the policy maker, if we calibrate the model to the U.S. economy. If money demand
were to be appreciably less variable, observations on money could be useful in response
to productivity shocks but would be harmful in response to money demand shocks. We
provide an incomplete information example where equilibrium welfare declines when
the money demand volatility decreases.
JEL Nos: C61, E52, E58
Keywords: monetary policy, sticky prices, optimal time-consistent policy, asymmetric
incomplete information

¤

We would like to thank Per Krusell, Alex Wolman, Robert King and seminar participants at the Federal
Reserve Bank of Richmond for helpful comments. Errors are our own. The views expressed in this paper
are those of the authors and do not necessarily represent those of the Federal Reserve Banks of Philadelphia
and Richmond, or the Federal Reserve System.
y
Federal Reserve Bank of Philadelphia (michael.dotsey@phil.frb.org)
z
Federal Reserve Bank of Richmond (andreas.hornstein@rich.frb.org)

1. Introduction
Over the years we have participated in many discussions concerning appropriate monetary
policy actions. While most central banks use an interest rate instrument in the pursuit of
monetary policy, participants in these discussions frequently suggest that a key consideration
for the setting of this instrument should be the behavior of money. One of the most prominent
advocates of this approach, Friedman (1969), suggests that if money demand is well behaved,
then monetary policy should respond to deviations of money growth from a preset target.
Although there have been periods when the behavior of money has in‡uenced the setting
of short term interest rates, Friedman’s prescription has not been followed in general. This
is re‡ected in empirically estimated policy rules, such as Taylor (1993), which suggest that
monetary authorities adjust their interest rate instrument in response to the behavior of
in‡ation and some measure of real economic activity, but not in response to the behavior of
money.
Is the current neglect of money in the pursuit of monetary policy justi…ed? We reconsider
this issue within the context of optimal monetary policy in an explicitly speci…ed general
equilibrium environment. In particular we study optimal time-consistent monetary policy
in an economy where prices are sticky. If the policymaker has complete information about
the state of the economy, optimal policy does respond to the state of the economy, but
that state does not include the nominal money stock. Although monetary policy does not
respond to the behavior of money, it may appear to do so to an outside observer, if that
observer does not have complete information on the state of the economy, and the behavior
of money re‡ects the behavior of the state. If the policymaker has incomplete information
about the state of the economy, then optimal policy may respond to the behavior of money,
if that behavior contains useful information for the policymaker about the underlying state
of the economy.1 For this to hold, money demand needs to be more stable than we observe
for the U.S. economy. With incomplete information, however, it is not necessarily true that
more information improves welfare. In particular, we provide an example where with more
stable money demand the policymaker responds more aggressively to movements in money
and thereby reduces economic welfare.
The plan of our paper is as follows. In section 2 we give a brief overview of monetary
policy in the postwar United States and describe the extent to which the Federal Reserve
System has used money to guide policy. We also investigate whether adding money to a
Taylor-type rule would help explain Federal Reserve behavior. Only for the early to mid
1

Recent work on optimal policy under incomplete information include Aoki (2000), Orphanides (1998),
Smets(1998), Svensson and Woodford (2000, 2001), and Tetlow (2000).

2

1970s and the early 1980s do we …nd some evidence that the Federal Reserve raised interest
rates in response to high money growth.
In section 3 we describe an economy with sticky prices due to staggered price setting
in the spirit of Taylor (1980). In the model, real balances enter the representative agent’s
utility function, and we can derive a money demand function where velocity shocks represent
preference shocks. Besides velocity shocks, we also consider productivity shocks. We parameterize the model based on the U.S. economy, and calculate a linear approximation to the
optimal time-consistent monetary policy under full information. We characterize optimal
monetary policy through impulse response functions and the behavior of estimated Taylor
rules. We …nd that the optimal monetary policy responds to both shocks, but that the e¤ects
of productivity shocks dominate. In response to a positive productivity shock, optimal policy
lowers interest rates. Taylor rules estimated in the model capture some but not all features
of empirically estimated Taylor rules: while higher in‡ation is associated with higher interest
rates, above average output is associated with lower interest rates. The last feature re‡ects
the response to productivity shocks which are the major source of ‡uctuations in the model
economy. Estimated policy rules also seem to indicate a desire for interest rate smoothing,
although the underlying true policy rule has no role for this behavior. Finally, for some of
the estimated policy rules, interest rates are negatively associated with money growth.
In section 4 we analyze our model when the private sector continues to have full information, but the monetary authority has incomplete information. In particular, we assume
that the policymaker does not observe the state of the economy, but receives a noisy contemporaneous signal of the money stock and lagged noisy information about output. Our
basic …nding is that for a parameterization of the signal noise in money and the volatility
of velocity, which is consistent with that observed in the United States, observing money in
addition to output does not change the dynamics of the economy substantially. However, for
appreciably less money demand volatility, information on money does improve the reponse
to productivity shocks. This improved reponse to productivity shocks comes at a cost, in
the sense that money demand shocks are now misconstrued as negative productivity shocks,
and the policymaker’s response to the signal can cause a recession. Overall, the economy
with a more stable money demand is actually worse o¤ in terms of unconditional expected
utility. We conclude with a brief summary and some thoughts for future work. In the appendix we characterize time-consistent optimal policy for linear-quadratic control problems
with incomplete and asymmetric information.

3

2. Evidence of the use of money in monetary policy
In this section we provide some evidence on the use of money in monetary policy. Our
discussion concentrates on post-World War II Federal Reserve policy, but it is clear from work
such as Bernanke and Mishkin (1992), Clarida and Gertler (1997), and Rich (1997) that other
central banks occasionally do pay attention to the behavior of some monetary aggregate. We
investigate the use of money in two ways. First, we ask if the federal funds rate was adjusted
in response to deviations of actual money growth from an explicitly stated money growth
target. Second, we ask if the inclusion of past money growth enters signi…cantly into Taylorrule type estimates of monetary policy.
2.1. Descriptive evidence
There is evidence that U.S. monetary policy responded to the behavior of M1 during the
…rst half of the seventies and the …rst half of the eighties. For the …rst period, Hetzel (1981)
describes how beginning in September 1972 an M1 target was speci…ed in terms of a twoquarter growth rate. Target is somewhat of a misnomer because the Fed did not conduct
monetary policy with the sole intention of hitting some …xed growth rate. However, the
behavior of M1 did in‡uence the setting of the funds rate at FOMC meetings and served
as a device for indicating how the open market desk should vary the funds rate between
meetings. For example, during the joint intervals October 1972 to August 1974, April 1975
to October 1975, and February 1977 to September 1979, the projected growth of M1 at the
prevailing funds rate was above the midpoint of its tolerance range at 44 meetings and below
the midpoint only 4 times. As a result, the FOMC raised the funds rate at 37 of these
meetings and lowered the rate only seven times. Conversely, over the intervals September
1974 to March 1975 and November 1975 to January 1977, projected M1 growth was below
target 14 times and above it on only three occasions. Over this period, the FOMC lowered
the funds rate 13 times and raised it only four times. In particular, from the spring of 1973
to the fall of 1974, one could argue that the behavior of M1 constrained monetary policy in
the sense that the Fed successfully hit its money growth targets over that period.
Although M1 targets were de-emphasized in 1982, there is still evidence that M1’s behavior in‡uenced policy over the period 1983-85 (see Dotsey [1996])2 . During this period the
ordering of variables in the FOMC’s directive to the open market desk continually changed.
The February, March, and May 1983 directives emphasized the behavior of money. As the
year progressed, business activity and in‡ation became increasingly important, but in 1984
2

There also exists a debate whether M1’s behavior a¤ected policy during the 1980-81 period. For di¤ering
views see Broaddus and Goodfriend (1984) and Bernanke and Mishkin (1992).

4

the growth rate of money was emphasized again. As documented in Goodfriend (1993), the
Fed was faced with an in‡ation scare and as a result policy tightened. The Fed’s focus on
strong growth in both M1 and M2 no doubt provided useful political cover for tighter policy.
But with the containment of in‡ation, emphasis on money waned and by 1985 there is no
mention of it in the directive. Unfortunately, by mid-1985 the ordering of variables in the
directive no longer changes and thus provides no information concerning the importance of
money’s behavior for policy. Further, M1 targets were formally abandoned in February of
1987.
2.2. Statistical evidence
To further investigate whether the behavior of money played any role in policy, we look
at a Taylor-type reaction function augmented by the growth rate of M1, Taylor (1993) .
Speci…cally we run the following regression using quarterly data,
Rt = a0 + a1 gapt + a2 (log Pt ¡ log Pt¡4 ) + a3 Rt¡1 + a4 (log M 1t ¡ log M1t¡4 ) + et ;
where R is the federal funds rate, gap is the output gap de…ned as the deviation of the log
output from a quadratic trend where the estimated trend only uses information available
during the relevant period. Thus, the gap is reestimated at each date. Pt is the consumer
price index, and M1 is the money stock. The regression uses 40-quarter rolling windows and
the estimated values of the coe¢cients together with their two-standard-deviation-con…dence
bands are depicted in Figure 1.3 We use rolling windows because of the well-known time
varying behavior of monetary policy. The coe¢cient on M1 growth is positive and takes on
its largest value from the early to mid 1970’s and in the mid 1980’s, which is consistent with
the descriptive evidence presented in the previous section. However, the coe¢cient is only
statistically signi…cant at the 5 percent level for the earlier sub-period, but is signi…cant at
the 10 percent level in the latter sub-period. Furthermore, starting in the late 1980s the
con…dence bands not only include zero, but the absolute value of the coe¢cient on money
growth declines. As noted, this was when the Federal Reserve formally dropped M1 targets.

3. Optimal monetary policy with full information
We have seen that monetary policy occasionally responds to the behavior of money, but not
always. We now ask whether monetary policy should respond to the behavior of money and, if
3

The main message of the results are not appreciably di¤erent if we replace the output gap with growth
in the output gap or output growth, or if we extend our rolling windows to 60 quarters.

5

yes, under what circumstances. To answer this question we study optimal monetary policy
in an explicit dynamic general equilibrium framework. This approach has the advantage
that the objective of the policymaker is well-de…ned, namely the welfare of the agents in the
economy. We choose to look at time-consistent or discretionary monetary policy because we
feel it may be a better representation of actual policy than full commitment, although both
are extreme cases. At least in the United States, policy is decided as a sequence of individual
policy actions rather than adherence to a well-de…ned rule. Indeed much academic policy
advice suggests that a rule-like behavior be adopted. It is, however, true that current policy
actions are somewhat constrained by long-run in‡ation objectives, but concerns about future
in‡ation also enter into the discretionary policymaker’s decision through his value function.
He, however, takes policy decisions with respect to future in‡ation as outside his control.
Therefore, policy issues that concern reputation and credibility are outside the scope of our
modeling strategy. Given the experience with in‡ation scares and the variability of monetary
policy over the last 30 years, neither the time-consistent nor the full commitment approach
seems adequate. We feel there is something to be learned from both approaches.
3.1. The model
We wish to develop a model where the behavior of money is potentially important in the
design of optimal policy. We do this by incorporating two important channels through which
money may in‡uence policy. First, changes in the demand for money arise from changes in
preference parameters. In this setting, a monetary authority concerned with maximizing
welfare may wish to react to changes in the demand for money. Second, in a world of
incomplete information, money may convey useful information about state variables in the
monetary authority’s reaction function. In this section we abstract from the second channel
and study the case of full information.
Our basic model includes an in…nitely lived representative household with preferences
over consumption, leisure, and real balances. The consumption good is produced with a
large number of di¤erentiated intermediate goods. Each intermediate good is produced by a
monopolistically competitive …rm with labor as the single input. Each intermediate goods
…rm sets a nominal price for its product, and this price is …xed for a …nite number of
periods. In particular, an equal number of …rms can change their price each period. This
type of staggered time-dependent pricing behavior, referred to as a Taylor contract, is a
common methodology for introducing price stickiness into an otherwise neoclassical model.

6

3.1.1. The household
The representative household’s utility is a function of consumption ct , real money balances
mt , and the fraction of time spent working nt ,
"1
(
)#
1¡Á
X
(1
¡
n
)
¡
1
t
1¡½ ½ 1=½
U = E0
¯ t log[µc½t + (1 ¡ µ) zmt
mt ] + Â
;
(3.1)
1
¡
Á
t=0

where Â; Á ¸ 0, ½ · 1, 0 < ¯ < 1, and zmt is a preference shock. The notation Et is used to

denote the expectations conditional on the information available to the household at time t.
The household’s period budget constraint is
Pt ct + Bt+1 + Mt+1 · Wt nt + Rt¡1 Bt + Mt + ¦t ;

(3.2)

where Pt (Wt ) is the money price of consumption (labor), Bt+1 (Mt+1 ) are the end-of-period
holdings of nominal bonds (money), and Rt¡1 is the gross nominal interest rate on bonds.4
The agent owns all …rms in the economy, and ¦t is pro…t income from …rms. In the following
we will use the term real to denote nominal variables de‡ated by the price of consumption
goods, and we use lower case letters to denote real variables. In particular, real balances are
de‡ated end-of-period nominal balances mt = Mt+1 =Pt .
The …rst order conditions of the representative household’s problem can be written as
¸t wt = Â (1 ¡ nt )¡Á ;
·
¸
Pt
¸t = ¯Rt Et ¸t+1
;
Pt+1
µ
¶1=(1¡½)
1 ¡ µ Rt
mt = zmt ct
;
µ Rt ¡ 1
¸t = ¤ (Rt ; zmt ) =ct ;
)
(
µ
¶1=(1¡½) µ
¶½=(1¡½)
1¡µ
Rt
¤ (Rt ; zmt ) ´ 1= 1 +
zmt .
µ
Rt ¡ 1

(3.3)
(3.4)
(3.5)
(3.6)

Equation (3.3) states that the marginal utility of leisure equals the real wage weighted by the
marginal utility of consumption. Everything else unchanged, the consumer will work more
with higher wages. Equation (3.4) describes the optimal savings behavior of individuals. If
the return to saving rises, then households will consume less today, save more, and consume
more in the future. Equation (3.5) is a money demand relationship, where real money
demand depends on consumption and the opportunity cost of holding money. Equation
(3.6) de…nes the Lagrange multiplier on the resource constraint, that is the marginal value
of consumption.
4

In an equilibrium, bonds are in zero net supply.

7

3.1.2. Firms
The consumption good is the …nal output of a constant returns to scale technology, which
uses a continuum of di¤erentiated intermediate goods as inputs, indexed j 2 [0; 1]. Total
i"=("¡1)
hR
1
("¡1)="
, where
dj
output as a function of intermediate goods y (j) used is c = 0 y(j)
" > 1. Producers of the …nal good behave competitively in their markets, and given prices
P (j) for the intermediate goods, the nominal unit cost and price of the …nal good is
·Z 1
¸1=(1¡")
1¡"
P =
P (j) dj
:
(3.7)
0

For a given level of production, the cost-minimizing demand for an intermediate good j is
y(j) = [P (j)=P ]¡" c:

(3.8)

Each intermediate good is produced by a single …rm, and j indexes both the …rm and
the good. Output of …rm j is a function of labor n (j) only,
yt (j) = zyt nt (j),

(3.9)

where zy is an aggregate technology shock. Each …rm behaves competitively in the labor
market, and takes wages as given. Real marginal cost in terms of …nal goods is then à t =
wt =zyt . Alternatively, the average mark-up in the economy is 1=Ã t . Since each intermediate
good is unique, intermediate goods producers have some monopoly power, and they face
downward sloping demand curves (3.8).
A …rm is allowed to adjust its nominal price once every J periods, and it chooses a price
that will maximize the expected value of the discounted stream of pro…ts over that period.
Since all intermediate goods producers are identical except for the time when they can adjust
their price, we consider only symmetric equilibria where producers di¤er only according to
¤
how much time has elapsed since they last changed their price. Let p¿ ;t = Pt¡¿
=Pt denote
¤
the time t relative price of a …rm which has set its price ¿ periods ago, Pt¡¿
. These relative
prices change with the price level
p¿+1;t+1 =

p¿ ;t
for ¿ = 0; : : : ; J ¡ 2.
Pt+1 =Pt

(3.10)

Given the sequence of relative prices the expected present value of intermediate goods producers can be de…ned recursively as5
v0;t = max f¼t (p0;t ) + Et [¢t;t+1 v1;t+1 (p1;t+1 )]g
p0;t

v¿ ;t (p¿ ;t ) = ¼t (p¿ ;t ) + Et [¢t;t+1 v¿ +1;t+1 (p¿ +1;t+1 )] for ¿ = 1; : : : ; J ¡ 2

5

vJ¡1 (pJ¡1;t ) = ¼t (pJ¡1;t ) + Et [¢t;t+1 v0;t+1 ] for ¿ = J ¡ 1,

We have suppressed the dependence of the …rm’s value on variables other than it’s own relative prices.

8

¿
where real pro…ts are ¼ t (p¿ ;t ) = (p¿ ;t ¡ Ã t ) p¡"
t;¿ ct , and ¢t;t+¿ = ¯ ¸t+¿ =¸t is the discount
factor according to which the representative household evaluates future consumption relative

to current consumption. Assuming that the value functions are di¤erentiable, we get a
recursive de…nition of the marginal values of intermediate goods producers
£
¤
0
0 = ¼ 0t (p0;t ) + Et ¢t;t+1 v1;t+1
(p1;t+1 ) (Pt =Pt+1 )
(3.11)
£
¤
v¿0 ;t (p¿ ;t ) = ¼ 0t (p¿;t ) + Et ¢t;t+1 v¿0 +1;t+1 (p¿ +1;t+1 ) (Pt =Pt+1 ) for ¿ = 1; : : : ; J ¡ 2

0
vJ¡1;t
(pJ¡1;t ) = ¼ 0t (pJ¡1;t ) :

Repeated substitution for the marginal value of a …rm with preset prices yields the
following representation of the pro…t-maximizing relative price choice
¤
PJ¡1 £
"
"
Pt¤
¿ =0 Et ¢t;t+¿ Ã t+¿ (Pt+¿ =Pt ) yt+¿
=
:
PJ¡1
"¡1 y
Pt
"¡1
t+¿ ]
¿ =0 Et [¢t;t+¿ (Pt+¿ =Pt )

(3.12)

If there is zero in‡ation and marginal cost is constant, the optimal relative price is a constant
markup over marginal cost, ¹ = "= (" ¡ 1). In general, however, a …rm’s pricing decision
depends on future marginal costs, the future aggregate price level, future aggregate demand,
and future discount rates. For example, if a …rm expects marginal costs to rise in the future,

or if it expects higher rates of in‡ation, it will choose a relatively higher current price for its
product.
We obtain an “aggregate” production function from the demand function (3.8) for intermediate goods, the production function of intermediate goods (3.9), and labor market
clearing

"

J¡1
1 X ¡"
p
yt = zyt at nt with at =
J ¿ =0 ¿ ;t

#¡1

(3.13)

Production e¢ciency requires equal use of intermediate inputs, given that the …nal goods
production function is symmetric and concave with respect to intermediate inputs. Yet, in
an economy with sticky prices and in‡ation, di¤erent intermediate goods producers charge
di¤erent prices and therefore sell di¤erent quantities. Production in the economy with sticky
prices is then in general ine¢cient, and the allocational e¢ciency coe¢cient at · 1 re‡ects
the distortion introduced by unequal relative prices. Finally, these relative prices have to
satisfy the following adding-up constraint, derived from the aggregate price index (3.7),
J¡1
1 X 1¡"
p :
1=
J ¿ =0 ¿ ;t

(3.14)

To complete the model, we assume that aggregate productivity and the money demand
9

preference shifter follow stationary AR(1) processes
ln zyt = ° y ln zy;t¡1 + uyt

(3.15)

ln zmt = ° m ln zm;t¡1 + umt
with j° i j < 1, E [uit ] = 0, E [u2it ] = ¾ 2i , for i = y; m.
3.2. Optimal time-consistent monetary policy
The policymaker follows a policy which maximizes the expected present value of the representative agent’s lifetime utility subject to the restriction that the allocation can be supported
as a competitive equilibrium. The agent’s current period utility u (x; y; R) is a function of
the state of the economy x, other non-predetermined variables y, and the policy instrument,
which we take to be the nominal interest rate R. We assume that the policymaker cannot
commit to future policy actions, and for this reason we study Markov-perfect equilibria.
In a Markov-perfect equilibrium we can view policy as being determined by a sequence of
independent policymakers, and today’s policymaker assumes that future policymakers will
select the policy instrument as a given function of the state, Rs = Fs (xs ) for s > t. Also,
given the decision rules of future policymakers, next period’s non-predetermined variables
and the lifetime utility of the representative agent from period t + 1 on will be given by the
functions Gt+1 (xt+1 ) and Vt+1 (xt+1 ). We represent the competitive equilibrium restrictions
through a system of equations that represent the law of motion for state variables Cx , and
restrictions on non-predetermined variables derived from market-clearing and optimizing behavior Cy . Given these restrictions, the policymaker chooses the nominal interest rate and
non-predetermined variables optimally6
Vt (xt ) = maxRt ;yt ;xt+1 u(xt ; yt ; Rt ) + ¯Et [Vt+1 (xt+1 )]
s.t. xt+1 = Cx (xt ; yt ; Rt ; ux;t+1 )
Et [Cy1 (xt+1 ; yt+1 ; Rt+1 )] = Cy0 (xt ; yt ; Rt )
yt+1 = Gt+1 (xt+1 ) and Rt+1 = Ft+1 (xt+1 ):

(3.16)

The utility maximizing choice implies policy functions for today’s instrument and nonpredetermined variables, Rt = Ft (xt ) and yt = Gt (xt ), and a value function Vt (xt ) which
re‡ects maximal lifetime utility of the representative agent from today on. A stationary
Markov-perfect equilibrium for this problem is characterized by the triple Z = (F; G; V )
such that (3.16) maps Z into itself.
6

When the policy maker has full information, the distinction between policy instruments and nonpredetermined variables has no substantive content.

10

We study a linear-quadratic approximation of the model, and a full description of our
methodology can be found in the technical appendix of Dotsey and Hornstein (2001). Brie‡y,
conditional on an initial guess of the steady state nominal interest rate, we can solve for the
steady state of the competitive equilibrium. We then construct a linear-quadratic approximation of the objective function and a linear approximation of the competitive equilibrium
constraints around this steady state. We solve for the Markov-perfect equilibrium of the
linear-quadratic approximation following Svensson and Woodford (2000) and obtain an optimal policy Rt = F xt and yt = Gxt . We use the optimal policy together with the linearized
law of motion for the state variables to determine the steady state of the approximation,
which includes the steady state of the nominal interest rate. We adjust the initial guess of
the steady state nominal interest rate until the two rates are the same.
3.2.1. The representation of our speci…c problem
In a standard rational expectations equilibrium, when the policy rule speci…es the choice
of policy instrument Rt as some given function of state and ‡ow variables, we treat the
lagged relative prices p¿ ;t¡1 for ¿ = 0; : : : ; J ¡ 2, which have been set by …rms in the past

J ¡ 1 periods as state variables. From the point of view of the planning problem, however,
nominal levels are of no concern. The equations that characterize the competitive equilibrium
in any given period involve real variables, relative prices, the nominal interest rate, and the
in‡ation rate, but not the current-period price level. Given that the price level is arbitrary,
past nominal prices impose restrictions on the current allocation only through their relative
prices. To clarify this point de…ne the normalized lagged prices as q¿ ;t = p¿ ¡1;t¡1 =pJ¡2;t¡1 ,

for ¿ = 1; : : : ; J ¡ 2. Using the transition equation for relative prices (3.10), we can rewrite
the constraint on relative prices (3.14) as
"
à J¡1
!µ
¶1¡" #
X
Pt¡1
1 1¡"
1=
p0;t +
q¿ ;t 1¡"
pJ¡2;t¡1
J
Pt
¿ =1

with qJ¡1;t ´ 1. Since the policymaker is free to choose the current in‡ation rate, the level of

lagged relative prices, that is pJ¡2;t¡1 does not represent a restriction on the policymaker’s
choices; it is not pay-o¤ relevant. Only normalized lagged prices constrain the policymaker’s
choices and should therefore be included as state variables in a Markov-perfect equilibrium.
Finally, the normalized lagged prices evolve according to
q¿ ;t+1 =

p¿ ¡1;t
for ¿ = 1; : : : ; J ¡ 2.
pJ¡2;t

(3.17)

We use the equations that de…ne the competitive equilibrium, (3.3), (3.4), (3.5), (3.6),
(3.11), (3.13), (3.14), (3.15), and (3.17) to de…ne the dynamic constraints of the planning
11

problem. The state variables are the normalized lagged prices and the exogenous shocks,
xt = [q1t ; : : : ; qJ¡2;t ; zyt ; zmt ]. A convenient choice of the ‡ow variables includes consumption,
the relative price of the current price adjusting …rm, and the marginal value of …rms that
0
0
have changed their prices in previous periods, yt = [ct ; p0t ; v1t
; : : : ; vJ¡2;t
]. Solving for the
behavior of these variables allows us to recover the behavior of all the other variables in the
model.
In order to perform our numerical analysis we have to parameterize the model economy.
Table 1 lists the parameter values and implied steady state values. We choose the time
preference parameter ¯ such that the annual real rate of interest is 4 percent. We select the
leisure parameters  and Á such that agents work approximately 25 percent of total hours
and the implied labor supply elasticity is slightly greater than one, which is consistent with
estimates in Mulligan (1998). We choose ½ so that the interest elasticity of money demand
is ¡0:1. This elasticity is within the bounds of most empirical estimates for M1. Likewise
we choose µ to make the velocity of money equal to 1:12, which is the average value of M1
velocity over the period 1959-99. Finally, consistent with the work of Basu and Fernald
(1997), ² is calibrated to yield a markup of 10 percent. The autocorrelation coe¢cients
and variances for the technology shock are roughly consistent with the values used in the
literature on quantitative dynamic general equilibrium models. The process for the money
demand shock is derived from an M1 demand function estimated for the United States from
1970 to 1999. Thus our parameterization is broadly consistent with both the literature and
U.S. data.
3.3. The optimal time-consistent policy function
We now characterize our approximation of the Markov-perfect equilibrium. We have outlined
our solution procedure in section 3.2. We …rst describe the steady state nominal interest
rate, and then interpret the behavior of the nominal interest rate o¤ the steady state. The
two parts are interrelated since in Markov-perfect equilibria the steady state depends on
o¤-steady state policy decisions.
Conditional on the parametrization of our economy we …nd that the steady state nominal
interest rate is 3:52 percent on a quarterly basis. Given the 1 percent real interest rate, this
implies an annual in‡ation rate of about 10 percent. This number is large relative to current
in‡ation rates in most OECD countries, although many countries have experienced in‡ation
rates of that magnitude over the last 30 years. The steady state in‡ation rate is also high
relative to what would occur under a policy of full commitment.
The high optimal in‡ation in this model occurs for the following reasons. In our economy,

12

the policymaker faces several distortions. There is the standard feature that real balances are
too low unless the nominal net-interest rate equals zero. There are also two other distortions
that tend to become more important in an economy where the policymaker cannot commit
to future policy choices. First, monopolisticaly competitive …rms set their price as a markup
over marginal cost, and production is ine¢ciently low. This distortion creates a desire to
in‡ate, which lowers the average markup because not all …rms can adjust their price. Second,
with in‡ation and staggered price setting, …rms di¤er according to the prices they charge,
and the quantities they produce and sell. Because the di¤erent …rms’ intermediate goods
enter …nal goods production symmetrically, this dispersion in the production allocation is
ine¢cient. The optimal response to this distortion is to lower the in‡ation rate, which
reduces the price and output dispersion across intermediate goods-producing …rms.7
For a policymaker who cannot commit to future actions, the relative importance of the
markup and relative price distortion changes directly with the number of periods for which
prices are …xed. Loosely speaking, if prices are preset for a longer duration, then the impact
of contemporaneous in‡ation on the current average markup increases, and the impact on
the current relative price distortion decreases. This happens because there are relatively
more …rms with preset prices, and the fraction of …rms whose relative price can be a¤ected
declines. Since the policymaker cannot commit to future actions, he tends to focus on the
contemporaneous impact of his actions and discounts the impact on future average markups
and relative price distortions.
In experiments, not reported here, we have found that the steady state in‡ation rate is
reduced by one-half if J = 2, and that the steady state in‡ation rate is close to the Friedman
rule if there is full commitment. Thus our solution leads to two observations. One is that
monetary authorities have done better than the discretionary equilibrium, perhaps because
they have access to some form of commitment technology. The other is that time-dependent
pricing may not be an accurate depiction of the way …rms behave. We conjecture that a
state-dependent pricing mechanism would result in greater price ‡exibility and less incentive
for the monetary authority to in‡ate.
Optimal policy lowers the interest rate in response to higher productivity and increases
the interest rate in response to higher money demand. The response of the interest rate to
percentage deviations of the state variables from their steady state values is8
^ t = 0:22^
R
q1t + 0:019^
q2t ¡ 0:13^
zyt + 0:0058^
zmt :
We have found Figure 2 useful for interpreting how policy responds to deviations of
7
8

For a detailed discussion see Khan, King, and Wolman (2000).
Hats denote percentage deviations from steady state values.

13

state variables from their steady state. Figure 2 displays the inital relative price p0t and
marginal cost à t , conditional on the contemporaneous policy and state variables Rt , q1t ,. . . ,
0
qJ¡2;t , zyt , zmt , and next period’s variables ¸t+1 , p1;t+1 , v1;t+1
. The initial relative price and
marginal cost re‡ect the two distortions the policymaker tries to a¤ect: a higher marginal

cost is equivalent to a lower average markup 1=Ã t , and a higher initial relative price implies a
lower allocational e¤ciency at .9 The curve labeled LL is based on the household’s optimality
conditions (3.3), (3.4), and (3.6), and market clearing conditions (3.13), (3.14). The curve
labeled PP is based on the optimal price-setting equation (3.11). For our discussion we
assume that the utility function is logarithmic in leisure, Á = 1, and the two curves are given
by
¤ (Rt ; zmt )
1
;
Rt
¯¸t+1 p1;t+1
= Â= f¤ (Rt ; zmt ) (zyt =ct ¡ 1=at )g ;

ct = p0t

Ãt
£
¤
1¡"
0
p0t
(" ¡ 1) ¡ "Ã t p¡1
= ¯¸t+1 v1;t+1
p1;t+1 =¤ (Rt ; zmt ) :
0t

(LL)
(PP)

The intersection of the two curves represents a “temporary” equilibrium in which both the
…rm’s and the agent’s …rst order conditions are satis…ed. It is a “temporary” equilibrium,
since it is conditional on given values of future variables that may depend on current choices.
Both curves are upward sloping, but numerical analysis shows that at the steady state
the LL curve is steeper than the PP curve. Regarding the LL curve, a higher p0 implies
a higher in‡ation rate Pt+1 =Pt = p0t =p1;t+1 , that is a lower real rate, and therefore higher
consumption today. Labor demand increases because production increases, and due to the
higher p0 production is less e¢cient (a declines). Overall the real wage and therefore marginal
cost has to increase along LL. The left-hand side of equation (PP) is proportional to the
negative of marginal pro…t of a price-adjusting …rm, and because the steady state marginal
pro…t of such a …rm is negative and period pro…ts are concave in p0 , the direct e¤ect of an
increase in p0 is positive. For given values of future variables, marginal cost has to increase
to maintain the equilibrium.
The policymaker faces a trade-o¤ between the markup and the relative price distortion.
A higher nominal interest rate increases the real interest rate, consumption declines, and the
LL curve shifts down. A higher nominal interest rate also shifts down the PP curve, since ¤
9

From equations (3.13) and (3.14), the elasticity of the relative price distortion with respect to the relative
price is
@a¡1 p0
p1¡²
p0 a¡1 ¡ 1
0 =J
=
²
> 0:
1¡²
¡1
@p0 a
1 ¡ p0 =J p0 a¡1

Notice that the magnitude of the derivative declines with the number of periods J over which prices cannot
be adjusted.

14

is decreasing in the nominal interest rate. For our parametrization, the e¤ect of a nominal
interest rate change on the PP curve is small relative to the e¤ect on the LL curve, and we
will ignore the e¤ect on the PP curve. Given the positive slope of the PP curve, marginal
cost and the initial relative price increase.
We interpret the response of optimal policy to deviations of state variables from their
steady state as an attempt of the policymaker to return the economy to a desired combination
of markup and relative price distortion. Consider …rst an increase in the normalized lagged
price qj for j = 1; 2. Near the steady state the relative price distortion is increasing in qj , that
is allocational e¢ciency a declines. The decline in a reduces the denominator of the term on
the right hand side of (LL) and marginal cost must rise for any given value of p0 : Therefore,
the LL curve shifts up. This shift reduces the marginal cost à and the initial relative price
p0 , that is it increases the average markup and and increases allocational e¢ciency. In order
to o¤set the impact on the distortions, the policymaker increases the nominal interest rate
and reverses the shift of the LL curve.
Notice that as the number of periods increases for which a …rm’s price is preset, the
policymaker’s impact on the distribution of relative prices declines, because much of the
distribution is inherited, cf footnote 99. In terms of Figure 2 this means that the impact
of the initial relative price p0 on the relative price distortion becomes smaller, and the (LL)
curve becomes less steep. Therefore, a nominal rate increase has a bigger e¤ect on the
average markup compared to the relative price distortion, and the monetary authority on
average chooses a higher nominal interest rate and in‡ation rate.
Now consider the interest rate response to a productivity increase. Higher productivity
shifts the LL curve down, which in equilibrium lowers the mark-up distortion and increases
the relative price distortion. In order to return the economy towards the original combination
of distortions, the interest rate is lowered and the LL curve shifts back towards its original
position. In response to a money demand increase, the PP curve shifts down very slightly
implying a higher markup and lower p0 : In order to move back in the direction of the steady
state trade-o¤ between the two distortions, the LL curve must also shift down. This shift
is accomplished by an increase in the nominal interest rate. In each case the interest rate
response is to return the economy toward its steady state equilibrium.
3.4. The behavior of the economy with full information
The preceding discussion gives some intuition on how a policymaker will respond to various
shocks. The discussion, however, holds …xed the values of future variables, and it does not
consider the dynamic interaction of private sector decisions and policy decisions. We now

15

describe the qualitative and quantiative dynamic features using impulse response functions.
We also discuss whether our model generates reduced form policy rules which are similar to
Taylor-rules estimated for the U.S. economy.
3.4.1. Impulse response functions
We display the impulse reponse functions for a productivity shock (panel A) and a money
demand shock (panel B) in Figure 3. Following a productivity shock the policymaker lowers
the nominal interest rate, and the economy’s behavior is remarkably close to that of an
economy without frictions. In an economy with ‡exible prices the income and substitution
e¤ects of higher real wages cancel, and employment does not move. The output response
then just mirrors the time path for productivity. In an economy with sticky prices, optimal
monetary policy with commitment almost perfectly replicates the outcome for the economy
without frictions, Khan, King, and Wollman (2000). As we can see here, optimal policy
without commitment appears to be a bit more opportunistic in the sense that it generates
a small but noticable increase of employment, and therfore a bigger output response. In
lowering the nominal interest rate, the policymaker is able to reduce the markup, as is
evident from the small increase in marginal costs, and simultaneously achieves an increase
in e¢ciency, as indicated by the decline of the initial relative price. The increase in output,
as well as the decline in the nominal interest rate, increases real balances. One period after
the shock, adjusting …rms also lower their price as do the remaining …rms when their turn
for price adjustment occurs.
The e¤ects of a money demand shock are quantitatively small, relative to the e¤ects of
a productivity shock. The only exception are real balances.10 For output movements, a
comparison of panels A and B in Figure 3 shows that productivity shocks are much more
important than are money demand shocks. Thus without observations on the underlying
shocks, one is more likely to attribute a given increase in output to a productivity shock.
This feature will be important when we study optimal policy under incomplete information
in the next section.
3.4.2. Reduced form feedback rules
In this section we wish to investigate whether monetary policy derived from optimizing behavior gives rise to estimated behavior that resembles a Taylor rule. Surprisingly, estimation
10

Our result are therefore little di¤erent from what occurs in a model where money enters separably. This
is consistent with McCallum (2000), who shows that for reasonably calibrated money demand behavior,
separability is not a bad approximation.

16

of policy rules with model-generated data yields statistical relationships that are somewhat
similar to policy rules estimated for the U.S. economy, in the sense that monetary policy apparently increases interest rates in response to in‡ation, and that policy apparently smooths
the behavior of interest rates. We present two examples, …rst the policy rule (T) based on
Taylor (1993) and second the rule (CGG) based on Clarida, Gali, and Gertler (2000):
¶
µ
¶
µ
Mt
Pt
(T )
¡ 0:47 log yt + 0:16 log Rt¡1 ¡ 0:001 log
log Rt
= 0:13 + 0:20 log
(0:03)
(0:04)
(0:001)
(0:01)
(0:12)
Pt¡4
Mt¡1
¶
µ
¶
µ
Mt
Pt+1
(CGG)
¡ 0:39 Et¡1 log yt + 0:19 log Rt¡1 ¡ 0:014 log
:
log Rt
= 0:09 + 1:74 Et¡1 log
(0:04)
(0:10)
(0:004)
(0:01)
(0:60)
Pt
Mt¡1
The regression coe¢cients represent averages of 200 regressions run on 100 quarters worth
of data, with standard deviations in parentheses. Our model does not contain a trend, we
therefore do not correct output by a potential output measure to obtain an output gap
measure as is usual in the literature on policy equations11 . For speci…cation (CGG) we
estimate a bivariate VAR with output and prices for each sample, and use the VAR to
generate one-quarter-ahead forecasts of in‡ation and output, with standard deviations in
parentheses.
The qualitative properties of the two speci…cations are very similar. In both speci…cations
output appears with the “wrong” sign, the coe¢cient on in‡ation has the “correct” sign, and
the coe¢cient on output tends to be more signi…cant than the coe¢cient on the in‡ation rate.
The positive coe¢cient on in‡ation re‡ects the fact that under optmal policy the interest rate
and in‡ation behave similarly to technology and money demand disturbances. The negative
coe¢cient on output re‡ects the fact that optimal policy lowers the nominal interest rate
relatively aggressively when there is a positive shock to technology. Hence, output increases
are inversely related to the interest rate. Both speci…cations appear to indicate a desire
for interest rate smoothing, which re‡ects the high degree of autocorrelation in the nominal
interest rate. This autocorrelation is a result of the autocorrelation in the shock processes
and not of any fundamental desire of the monetary authority to smooth rates12 . For the
(CGG) speci…cation, the regressions seems to indicate that the behavior of nominal money
is important for monetary policy: nominal money growth enters with a signi…cant negative
coe¢cient.
11

For each speci…cation quarterly output, the nominal interest rate, and growth rates are annualized. For
speci…cation (T) in‡ation is the average in‡ation over the last four quarters.
12
Model-generated data indicate that output, in‡ation, and the nominal interest rate are all highly autocorrelated. Their …rst-order autocorrelation coe¢cients are 0.87, 0.95, and 0.90, respectively. The high
autocorrelation is due to the highly autocorrelated shocks.

17

4. Optimal monetary policy with incomplete information
We have seen that optimal monetary policy is unresponsive to the behavior of money when
the monetary authority has full information about the state of the economy. It is unlikely,
however, that the monetary authority ever sees technology or money demand shocks, nor
does it observe the true price distribution. In an environment where the policymaker does
not have complete information on the state of the economy, optimal policy may respond to
money because the behavior of money contains information about the unobserved state.
We model the notion of money as a signal for an information-constrained policymaker
who never observes the true state of the economy as follows. In each period the policymaker
receives two signals: a signal s0t on money at the beginning of the period, contemporaneously with the decision to be made, and a signal s1t on output at the end of the period,
after a decision has been made. Given our linear approximation of the economy, the policymaker’s information about the state xt is summarized by his conditional expectations.
The policymaker starts out with an expectation about the current state x¹tjt¡1 conditional on
information from past periods. Given the signal s0t the policymaker updates the expectation
to x¹tjt . After the equilibrium has been determined the policymaker receives the signal s1t and
he updates his conditional expectation to x¹tjt+ . Using the law of motion for state variables
he also forms expectations about next period’s state to x¹t+1jt . We continue to assume that
private agents have full information. We do so largely to make the problem more tractable.13
We show that certainty equivalence holds in the sense that the policymaker’s decision
depends only on his expected value of the state variables, Rt = Ft x¹tjt .14 We can therefore
use the results on optimal full-information policy rules from the previous section, and simply
replace actual values of state variables with the policymaker’s conditional expectations. The
equilibrium values of all nonpredetermined variables will depend on the actual state and
the policymaker’s conditional expectation of the state yt = G1;t xt + G2;t x¹tjt . We solve the
inference problem for the policymaker using a methodology based on the Kalman-…lter. The
inference problem is fairly complicated and the equilibrium outcome depends on the policymaker’s policy rule. Thus, the separation of inference from optimization which Svensson
and Woodford (2000) have shown to hold under common incomplete information is no longer
13

As pointed out in Aoki (2000), private agents may face measurement problems less serious than those
that face the central bank. Firms may have a better idea of the state of technology and consumers may not
need to know the price of all goods when optimizing. Thus modeling private agents as having information
superior to that possessed by the monetary authority seems a reasonable strategy. Even so, our information
assumptions are severe.
14
A detailed explanation of our procedure is contained in the appendix. Similar derivations can be found
in Svensson and Woodford (2001).

18

present.
In most developed countries data on real GDP, nominal GDP, and the money stock are
readily available for a policymaker. Furthermore, reliable information on money is available
earlier than is information on GDP. We represent this information structure in our model as
follows.15 At the beginning of the period, before the policy instrument is set, the policymaker
receives a signal on real balances ln(Mt =Pt )o + ´ R Rt = yt + zmt + À mt , where À mt is the
measurement error. In reality policymakers observe separate signals on the nominal money
stock and the price level. To keep the problem tractable, we have combined both signals
into the real balance signal, which conveys more information than would the nominal money
stock alone. At the end of period, after the policy instrument has already been determined,
the policymaker receives a signal on real output, yto = yt + À yt , where À yt is the measurement
error on real output. The standard deviation of the measurement error of the nominal
disturbance is 0:003, and the one for the output signal is 0:005. These values are consistent
with the variance of revisions for real GDP and nominal M1.16 Using these two signals, the
monetary authority forms expectations of the economy’s current state and conducts policy
accordingly.
4.1. The behavior of the economy under incomplete information
We …rst study how the information structure a¤ects the response of monetary policy to
productivity shocks. We …nd that when the policymaker has only lagged information on
output, a productivity shock results in an initial decline of output with subsequent humpshaped convergence towards the full-information path. If the policymakers receives additional
contemporaneous information on money, the policymaker can respond to the productivity
shock and the initial negative e¤ect on output is weakened. The more stable money demand
is, the better can the policymaker respond to a productivity shock. We then show that with a
stable money demand the policymaker is more likely to confuse money demand disturbances
for productivity shocks. Since the policymaker responds strongly to perceived productivity
shocks, this introduces undesirable volatility into the economy. For our example, having more
information can reduce welfare. Finally, we estimate Taylor rules for our model economy and
…nd that money growth enters signi…cantly with a negative coe¢cient in these equations. For
15

We assume that these measures are never updated, and that the authority never observes the actual
state variables. For an alternative information setting in which the monetary authority observes the true
state with a one period lag, see Aoki (2000) and Swensson and Woodford (2001).
16
Because we parameterize the “nominal” noise using only the standard deviation of the money stock
noise, we underestimate the magnitude of the “nominal” noise, implicitely giving money more signal value
than it actually has.

19

the analysis of the model economy under incomplete information, we maintain the calibration
of the previous section.
4.1.1. The e¤ects of a productivity shock
Figure 4 displays the economy’s response to a productivity shock: the actual values and
the policymaker’s expectations of state variables in panel A, and the actual values of nonpredetermined variables in panel B. Because the policymaker has no contemporaneous information, there is no immediate response to the shock. From the standpoint of the full
information case this inaction means that the nominal rate is roughly 50 basis points too
high. With a one-period delay, the policymaker becomes aware of the productivity increase
and understands that his inaction has resulted in a decline in q1 : In response to this new
information, the policymaker now drastically lowers the nominal interest rate, but brings
it back up to the full information case within three periods. We can see how incomplete
information magni…es the policymaker’s response to a shock: relative to full information
the nominal interest rate movements are about four times as big. In turn, the policymaker’s
delayed and magni…ed response also introduces additional output and employment volatility.
Firms who adjust their price in the period that the shock occurs know that it will take time
for output to rise. Hence the path of wages and marginal cost will be lower than under full
information and as a result …rms slash prices. This price reduction reduces the anticipated
in‡ation rate, and since the nominal interest rate is unchanged, it increases the real rate of
interest. Although, future consumption is expected to increase, after the policymaker recti…es the mistake, the real rate increase is su¢cient to induce a reduction in current-period
consumption and output.
Could the excess volatility that occurs under incomplete information relative to what
occurs under full information be reduced if the policymaker were to obtain contemporaneous
information on money in addition to lagged information on output? It turns out that for our
parameterization of money demand uncertainty, the nominal money stock does not contain
much useful information about output. The impulse response functions with contemporaneous information are essentially the same as the ones in Figure 4. Basically, money demand
is too volatile for observations on money to be of much value. Since observed movements in
money can be largely attributed to the money demand shock, the inference of the technology shock and of relative prices is unchanged, and observing money does not improve the
policymaker’s knowledge of the state of the economy.
The property that money does not contain any useful information about output and
hence technology disturbances is an empirical matter. If money demand is more stable, for

20

example if we reduce the variance of the money demand shock by a factor of ten, the outcome
is quite di¤erent (see Figure 5). In this situation contemporaneous money is sending a much
sharper signal of contemporaneous output, and following a productivity shock the policymaker attributes much of the movement in money to that shock. The policymaker, however,
also attributes some of the increased output to relatively low normalized lagged prices and
a greater degree of inherited e¢ciency. The combination of beliefs leads to a decline in the
nominal interest rate of 50 basis points, almost the identical decline as under full information. Yet output does not respond in nearly the same way as it does under full information.
The key to the di¤erence is subtle and involves the fact that the policymaker only gradually
learns the true nature of the shock, but that the private sector already anticipates further
nominal rate reductions induced by this learning process. As the policymaker learns more
about the productivity shock, he implements further substantial nominal rate reductions
that increase output further and reduce prices. The anticipation of big price reductions in
the future induces current producers to reduce their prices substantially, and the expected
in‡ation rate declines and the real rate increases. The real rate increase is then consistent
with the hump-shaped response of output.
4.1.2. The e¤ects of a money demand shock
Money demand shocks do not play an important role under full information. We would then
expect that in an incomplete information environment a policymaker might ignore money
demand shocks alltogether. This is actually the case when the policymaker has only lagged
information on output.17 We have also seen that the policymaker’s response to a productivity
shock is improved when he has contemporaneous information on money and money demand
volatility is low. However, more information is not necessarily welfare improving in our
environment. When the policymaker believes that the contemporaneous signal is a more
reliable indicator of productivity shocks, the consequences are more severe when he mistakes
a money demand shock for a productivity shock.
Figure 6 shows the e¤ects of a money demand shock when the central bank observes
money and money demand is stable; that is the variance of the money demand shock is
one-tenth of our baseline speci…cation. With this speci…cation, the policymaker believes not
only that a money demand shock has occurred, but also believes that the economy has been
subject to a negative technology disturbance. This latter result occurs, because the inference
problem and the policy response are no longer separate. Note that the contemporaneous
17

We do not report the results here, but in this case the policy maker never updates his estimate of the
money demand shock, and he responds only to perceived productivity and normalized lagged price shocks.

21

signal contains the sum of the money demand shock and the output response (ct + zmt +
À mt < 0). In our case the planner sees a negative signal, which he interprets as a negative
productivity shock and a positive money demand shock. The policymaker ignores the money
demand shock, but responds to the productivity shock and increases the nominal rate; the
higher interest rate, in turn, reduces consumption, which leads to a negative signal. Thus
the response and the inference are internally consistent. Unlike the case when the shock
is actually a productivity shock, more information is not preferable and less variability in
money demand actually inhibits accurate inference of the true disturbance.
Simulations of the model economies show that the economy with contemporaneous observation on money and low money demand volatility is much more volatile than the economy
with information on lagged output only. This indicates that more information is not necessarily welfare improving in our environment.18 We can verify this for our linear-quadratic
approximations of the utility functions. In particular, we calculate the consumption equivalent welfare loss relative to full information (1) with lagged output only as 0:0072 percent
of steady state consumption; (2) with contemporaneous money and high money demand
volatility as 0:0058 percent; and (3) with contemporaneous money and low money demand
volatility as 0:0713 percent.
4.1.3. Reduced form feedback rules
Policy rules estimated from data generated by a model where the policymaker has incomplete
information show a policymaker who is more responsive to money stock changes. When we
estimate the two equations (T) and (CGG) from section 3.4.2 for the baseline parameterization of the incomplete information model with a contemporaneous money signal and a
lagged output signal, we obtain
µ
¶
µ
¶
Pt
Mt
(T )
¡ 0:76 log yt + 0:33 log Rt¡1 ¡ 0:11 log
log Rt
= 0:14 + 0:10 log
(0:17)
(0:10)
(0:02)
(0:02)
(0:086)
Pt¡4
Mt¡1
µ
¶
µ
¶
Pt+1
Mt
(CGG)
log Rt
= 0:12 + 1:75 Et¡1 log
¡ 0:65 Et¡1 log yt ¡ 0:15 log Rt¡1 ¡ 0:06 log
:
(0:02)
(0:20)
(0:16)
(0:09)
(0:02)
Pt
Mt¡1
The Taylor speci…cation (T) now displays a weaker response to annual in‡ation, a stronger
negative response to the output gap, more interest rate smoothing, and a signi…cant negative
response to money growth. The Clarida-Gali-Gertler speci…cation displays an unchanged
positive response to predicted in‡ation, a stronger negative response to the predicted output
18

This is not an entirely new observation. Pearlman (1992) has pointed out that more information need
not be welfare improving in optimal control problems with partial information, even when the inference and
the control problem are separable.

22

gap, no signi…cant interest rate e¤ect, and also a statistically negative dependence on money
growth.
Incomplete information makes the dynamics of the model su¢cienctly complicated such
that the simple policy rules no longer capture the dynamics of the interest rate so well.19
The increased importance of the model’s internal propagation relative to the exogenous
productivity shock is also re‡ected in the reduced persistence of variables. The …rst-order
serial correlation coe¢cients on output, in‡ation, and the nominal interest rate are now
.63, .66, and .51. Thus incomplete information and its subsequent e¤ect on policy has a
signi…cant e¤ect on how an econometrician would view this economy. This is true even
though the private sector’s behavior is unchanged.

5. Conclusion
In this paper we have attempted to evaluate how useful money is for the pursuit of monetary
policy. We have done so for an environment where we have explicitly speci…ed why money
has real e¤ects (sticky prices due to staggered price setting), and what are the policymaker’s
objective (maximize the expected utility of the representative agent) and constraints (choose
a time-consistent policy rule). We have found that even though money communicates information on aggregate output, it is of limited use for a policymaker. We should emphasize
that money’s usefulness as a signal is an empirical matter, and that if the money demand
was more stable than it appears to be, the value of money as a signal could dramatically
increase. In particular money would be a useful signal in an environment driven by productivity shocks, but using it as a signal would have adverse consequences in the presence
of money-demand disturbances. This …nding suggests that time variation in the behavior
of money-demand disturbances and in the types of shocks primarily a¤ecting the economy
could imply time variation in a policymaker’s responsiveness to money.
Importantly, however, the dynamic behavior of an economy can be very di¤erent depending on what information is available to the policymaker. The dynamic behavior di¤ers
because policy responds very di¤erently under full information from the way it responds
under incomplete information. As stressed in Dotsey (1999), the form of the policy rule can
be crucial in the type of model studied here. Thus, it may pay to model the economy’s information structure accurately if one is to explain economic behavior. Information and signal
extraction problems were a centerpiece of the early literature on rational expectations, these
same items may again prove to be important for the analysis of optimal monetary policy.
19

The average adjusted R2 for the two speci…cations are 0:73 (0:05) for (T) and 0:85 (0:04) for (CGG).
Under complete information they were 1:03(0:01) and :91(:05); respectively.

23

Finally, the joint speci…cation of private pricing decisions and discretionary policy tends
to produce somewhat high steady state rates of in‡ation. It would be interesting to relax
both of these features of the model by allowing for state-dependent pricing decisions or some
form of partial commitment. We hope to investigate these avenues in future work.

24

References
[1] Aoki, K. 2000. Optimal commitment policy under noisy information. Kobe University
working paper.
[2] Basu, S., Fernald, J., 1997. Returns to scale in U.S. production: estimates and implications. Journal of Political Economy 105, 249-83.
[3] Bernanke, B., Mishkin, F. 1992. Central bank behavior and the strategy of monetary
policy: observations from six industrialized countries. In Blanchard, O., Fischer, S. (Ed.)
NBER Macroeconomics Annual 1992, MIT Press, Cambridge, 183-227.
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the Federal Reserve Board. Carnegie-Rochester Conference Series on Public Policy 47,
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[5] Broaddus, A., Goodfriend, M.,1984. Base drift and the longer run growth of M1: experience from a decade of monetary targeting. Federal Reserve Bank of Richmond Economic
Review 70, 3-14.
[6] Calvo, G., 1983. Staggered prices in a utility maximizing framework. Journal of Monetary Economics 12, 383-398.
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[11] Dotsey, M.. 1999. The importance of systematic monetary policy for economic activity.
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25

[13] Hetzel, R. 1981. The Federal Reserve System and control of the money supply in the
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26

Technical Appendix
A1. Introduction
This appendix describes the algorithms we use to …nd the steady state of an economy
with discretionary optimal policy and incomplete information. We …rst describe a Markovperfect equilibrium for the optimal control problem when the policymaker cannot commit
to future policy choices. We describe a simple algorithm to …nd a linear approximation to
the equilibrium. For this case we assume that the policymaker has complete information.
We then consider the case where the policymaker has incomplete information, in particular
he has less information than the private sector. This analysis takes as a starting point the
linear approximation of the environment. The description of the linear-quadratic optimal
control and estimation under incomplete information essentially follows Svensson and Woodford (2000). The section on incomplete information extends SW to the case of asymmetric
information between the private sector and the policymaker.

A2. The model with full information
In this section we describe the optimal control problem of a policymaker under full
information. The policymaker chooses allocations which satisfy the constraints that support
the outcome of a market equilibrium given the actions of the policymaker. The policymaker
maximizes an intertemporal objective function. For our applications this objective function
is the expected present value of the representative agent’s utility. We assume that the
policymaker cannot commit to future policy choices, therefore we study a Markov-perfect
equilibrium.
The constraints of the policymaker, are the …rst order conditions and market clearing
conditions of the competitive equilibrium. It is useful to divide these constraints into two
blocks: one that contains the evolution of the predetermined state variables, x; and denoted
Cx , and the other that involves the non-predetermined ‡ow variables, y; and denoted Cy :
Formally, the constraints can be represented by
xt+1 = Cx (xt ; yt ; Rt ; ux;t+1 )
Et Cy1 (xt+1 ; yt+1 ; Rt+1 ) = Cy0 (xt ; yt ; Rt )

(5.1)
(5.2)

where R denotes the policy instruments and u is an iid random variable with mean zero.
There are nx state variables, ny ‡ow variables, and nR instruments. De…ne Zt ´ [x0t ; yt0 ; Rt0 ]0 .

Equation (5.1) de…nes the law of motion for the state variables, and equation (5.2) re‡ects
the fact that the chosen allocation has to satisfy the private sector’s optimality conditions
27

in a market economy. The function Cx is vector-valued of dimension nx, Cy0 and Cy1 are
vector-valued of dimension ny , and Et denotes the private sector’s expectations conditional
on the information set It . In this section we assume that the policymaker and the private
sector have complete information at time t, It = fZ¿ : ¿ · tg. Note that (5.1) and (5.2)
de…ne an incomplete dynamic system, since the policymaker’s decision rule with respect to
the policy variables has not been speci…ed. If we were to specify the policy variables as a

function of the state and/or ‡ow variables, such as in a Taylor-rule, we could solve (5.1) and
(5.2) for the implied rational expectations equilibrium.
The objective function of the policymaker is
E0

1
X

¯ t U (Zt )

(5.3)

t=0

where 0 < ¯ < 1. In our example U is the utility function of the representative agent, which
generates the competitive equilibrium. We are looking for a time-consistent policy. For this
purpose we solve for the set of Markov-perfect equilibria where the policymaker’s decision
depends only on pay-o¤ relevant state variables, that is
Rt = Ft (xt )

(5.4)

and the equilibrium outcome is such that the ‡ow variables depend only on the current state
yt = Gt (xt ) :

(5.5)

For the Markov-perfect equilibrium, the policymaker takes next period’s outcome functions
Ft+1 and Gt+1 and the continuation value function Vt+1 (xt+1 ) as given, and the optimization
problem is
Vt (xt ) =

max Et [U (xt ; yt ; Rt ) + ¯Vt+1 (xt+1 )]

xt+1 ;yt ;Rt

(5.6)

s.t. xt+1 = Cx (xt ; yt ; Rt ; ux;t+1 )

(5.7)

Et [Cy1 (xt+1 ; Gt+1 (xt+1 ) ; Ft+1 (xt+1 ))] = Cy0 (xt ; yt ; Rt ) :

(5.8)

A Markov-perfect equilibrium is characterized by the triple (F; G; V ) such that (5.6), (5.7),
and (5.8) maps (Ft+1 ; Gt+1 ; Vt+1 ) = (F; G; V ) to (Ft ; Gt ; Vt ) = (F; G; V ).
For the solution we proceed in several steps. We …rst construct a linear-quadratic approximation of the problem around a steady state indexed by the policy instrument R¤ . We then
¤
solve the LQ approximation, and derive the steady state of the approximation RLQ
. This
de…nes a mapping from R¤ to R¤LQ . We solve for a steady state choice of policy instruments
¤
such that R¤ = RLQ
.
28

Solving for an approximate steady state
Step 1. Suppose the steady state values of the policy instruments are given by R¤ .
Conditional on R¤ we can solve (5.1) and (5.2) for the steady state values of the state and
‡ow variables (x¤ ; y ¤ ). Now derive a linear approximation of the constraints (5.1) and (5.2)
xt+1 = Cx;Z Zt + ux;t+1
Et [Cy1;Z Zt+1 ] = Cy0;Z Zt

(5.9)
(5.10)

and a quadratic approximation of the period utility function20
2
32
3
Uxx Uxy Uxz
xt
6
76
7
Zt0 UZt = [x0t ; yt0 ; Rt0 ] 4 Uyx Uyy Uyz 5 4 yt 5 :
Uzx Uzy Uzz
Rt

Step 2. Given the linear-quadratic structure, we guess that next period’s non-predetermined
variables are linear functions of next period’s state variable and that the continuation value
from next period on is a quadratic function of next period’s state variable. Speci…cally,
Rt+1 = Ft+1 xt+1

(5.11)

yt+1 = Gt+1 xt+1

(5.12)

and x0t+1 Vt+1 xt+1 is the continuation value:
We now determine the equilibrium outcome for the current period and next period,
conditional on the current state and policy choice. Assume that the matrix
"
#
Ix
0xy
C1;t =
Cy1;x + Cy1;R Ft+1 Cy1;y
is invertible, then we can rewrite the constraints as
"
#
"
#
"
#"
# "
#
"
#
xt+1
C
A
A
x
B
u
x
xx
xy
t
x
x;t+1
¡1
= C1;t
Zt =
+
Rt +
: (5.13)
Et yt+1
Cy0;Z
Ayx;t Ayy;t
yt
By;t
0
We now proceed as in Svensson and Woodford (2000). Substitute the transition equations
for the state variables (5.13) in our guess for next period’s ‡ow variables (5.12) and take
conditional expectations
Et [yt+1 ] = Gt+1 (Axxxt + Axy yt + Bx Rt ) :
20

In the following we will interpret the variables in terms of deviations from the steady state. Furthermore,
the …rst derivative of the utility function is implicitly included by adding a constant term to the vector of
0
state variables [1; x0 ] . We also have normalized the covariance matrix of ux such that the derivative of
Cx;ux = I.

29

Now substitute these expectation for next period’s ‡ow variables in the optimality conditions
from the competitive equilibrium (5.13) and get
Gt+1 (Axx xt + Axy yt + Bx Rt ) = Ayx;t xt + Ayy;t yt + By;t Rt :
Assuming that Ayy;t ¡ Gt+1 Axy is invertible, we can solve for this period’s ‡ow variables
~t Rt with
yt = A~t xt + B
A~t ´ (Ayy;t ¡ Gt+1 Axy )¡1 (Gt+1 Axx ¡ Ayx;t )

(5.14)

~t ´ (Ayy;t ¡ Gt+1 Axy )¡1 (Gt+1 Bx ¡ By;t ) :
B

Substituting (5.14) for this period’s ‡ow variables in the transition equation (5.9) yields
xt+1 = A¤t xt + Bt¤ Rt + ux;t+1 with

(5.15)

A¤t ´ Axx + Axy A~t

Bt¤ ´ Bx + Axy;t Bt :
After substituting for yt from (5.14) and for xt+1 from (5.15) in the period t optimization
problem
we get

©
£
¤ª
max Zt0 UZt + ¯Et x0t+1 Vt+1 xt+1
s.t. (5.14) and (5.15).
h
i
0
0
max xt ; Rt

Qxx;t Qxz;t

#"

xt

#

Qzx;t Qzz;t
Rt
£ ¤
¤
+¯E (At xt + Bt¤ Rt + ux;t+1 )0 Vt+1 (A¤t xt + Bt¤ Rt + ux;t+1 )
Rt

with

"

Qxx;t
Qxz;t
Qzz;t

i
h
= Ix ; A~0t

"

Uxx Uxy

#"

Ix
A~t

#

Uyx Uyy
("
# "
#"
#)
h
i
U
U
U
0
xz
xx
xy
= Ix ; A~0t
+
~t
Uyz
Uyx Uyy
B
"
#"
#
"
#
h
i U
h
i U
U
0
xx
xy
xz
~t
~t0
= Uzz + 0; B
+ 0; B
~
Uyx Uyy
Bt
Uyz
"
#
0
+ [Uzx ; Uzy ]
:
~t
B

The …rst order condition for the optimal choice of the policy instrument is
0 = (Qzz;t + ¯Bt¤0 Vt+1 Bt¤ ) Rt + (Qzx;t + ¯Bt¤0 Vt+1 A¤0
t ) xt
30

(5.16)

(5.17)

Assuming that Qzz;t + ¯Bt¤0 Vt+1 Bt¤ is invertible we can solve for the policy instrument and
de…ne this period’s policy function Rt = Ft xt and ‡ow variable equation yt = Gt xt with
¡1

Ft = ¡ (Qzz;t + ¯Bt¤0 Vt+1 Bt¤ )
~t Ft :
Gt = A~t + B

(Qzx;t + ¯Bt¤0 Vt+1 A¤0
t )

(5.18)
(5.19)

Substituting for the policy function in (5.16) yields the current period value as a quadratic
function of the current period state de…ned by the matrix
(
"
#
)"
#
¤0
A
I
t
Vt = [I; Ft0 ] Qt + ¯
Vt+1 [A¤t ; Bt¤ ]
¤0
Bt
Ft

(5.20)

A stationary Markov-perfect equilibrium to the policymaker’s decision problem is a triple of
matrices (F; G; V ) which is a …xed point of the mapping de…ned by equations (5.13) through
(5.20).
Step 3. Calculate the steady state of the Markov-perfect equilibrium. Substituting the
policy rule F and the equilibrium function G into the transition equation for the state
variables we get
x¤LQ = Axx x¤LQ + Axy Gx¤LQ + Bx F x¤LQ :
Recall that constant terms are implicit in this equation through the de…nition of the state
variable which contains a constant term. We can solve this expression for the steady state
¤
value of the linear approximation x¤LQ and RLQ
= F x¤LQ . Since we started with the assumption that the steady state of the Markov perfect equilibrium is R¤ we adjust R¤ until
¤
RLQ
= R¤ .

Implementation of the algorithm
The implementation of the algorithm is straightforward, with two minor exceptions. The
derivation of the Markov-perfect equilibrium suggests that we use a simple iteration scheme:
given an assumption on the triple (Ft+1 ; Gt+1 ; Vt+1 ) use equations (5.13) through (5.20) to
obtain values for the triple (Ft ; Gt ; Vt ), and iterate until convergence. There are two problems
with simple iterations, which relate to the fact that we have not shown that such a process
will indeed converge.
First, we have found that frequently we obtain convergence on the linear terms of the
functions F , G, and V , but we cannot obtain convergence once we include the constant terms
in these functions. This problem is easily dealt with. Given the linear quadratic structure
we know that the linear terms are independent of the constant terms, and in a …rst run we
ignore the constant terms and interpret the model in terms of deviations from the steady
state. Essentially this means ignoring the linear terms of the utility function. Usually we
31

obtain convergence on the linear terms for this simpli…ed model. After we have obtained
the linear terms, the constant terms can be obtained as a solution to a linear system of
equations. We need to know the constant terms since they will determine the steady state
of the approximation economy.
The second problem is that occasionally the simple iteration scheme does not converge
for the linear terms. In this case we solve the equations (5.13) through (5.20) as one big
system of non-linear equations. This procedure tends to be slower than simple iterations
and good starting values are required.

A3. The model with incomplete and asymmetric information
We now consider a special case of incomplete information: we assume that the private
sector of the economy has complete information, whereas the policymaker has incomplete
information. In particular, we assume that the policymaker receives two types of signals
about the economy. The …rst signal s0t is received at the beginning of the period, and
its value is determined simultaneously with the values of the ‡ow variables and the policy
instrument. The second signal s1t is received at the end of the period, after the ‡ow variables
and the policy instrument have been determined. We call the information contained in the
…rst and second signal contemporaneous respectively lagged. The signals are related to the
true state of the economy according to
0

sit = Si [x0t ; yt0 ] + eit

(5.21)

where eit is iid with mean zero and covariance matrix §ei , for i = 0; 1. The policymaker’s information set at the beginning of period t is I¹t = f(Rt ; s0t ) and (R¿ ; s0¿ ; s1¿ ) for ¿ < tg and
at the end of the period it is I¹t+ = I¹t [fs1t g. We denote the policymaker’s expectations about
£
¤
the current state of the economy conditional on the available information as x¹tjt = E xt jI¹t ,
£
¤
respectively x¹t+1jt = E xt+1 jI¹t+ . The private sector has complete information, that is the

beginning and end-of-period information sets are It = f(xt ; yt ; zt ; s0t ) and (x¿ ; y¿ ; z¿ ; s0¿ ; s1¿ )
for ¿ < tg and It+ = It [ fs1t g, respectively, and the conditional expectations are denoted
xtjt and xt+1jt .

The optimal control problem.
We guess that next period’s policymaker’s decision is
Rt+1 = Ft+1 x¹t+1jt+1

(5.22)

and that next period’s ‡ow variables depend on the actual values of the state variables and
32

the policymaker’s perceptions of the state variables
yt+1 = G1;t+1 xt+1 + G2;t+1 x¹t+1jt+1 :

(5.23)

Also assume that the continuation value of next period’s policymaker is given by
£
¤
E x0t+1 Vt+1 xt+1 jI¹t+1 :

When the policymaker forecasts next period’s ‡ow variables, the distinction between actual
and forecasted variables is irrelevant
y¹t+1jt = Gt+1 x¹t+1jt and Gt+1 ´ G1;t+1 + G2;t+1 :
Proceeding as before we can obtain the policymaker’s expectation about the current ‡ow
variables
~t Rt
y¹tjt = A~t x¹tjt + B
and his forecast of next period’s state variables
x¹t+1jt = A¤t x¹tjt + Bt¤ Rt
~t , A¤t , and Bt¤ are de…ned as in (5.15) and (5.14).
where A~t , B
The policymaker’s value function is now given by

where
lt ´ E

£
¤
0
E Zt0 UZt + ¯x0t+1 Vt+1 xt+1 j I¹t = Z¹tjt
UZ¹tjt + ¯ x¹0t+1jt Vt+1 x¹t+1jt + lt
h¡

i
¡
¢0 ¡
¢
¢0
¡
¢
Zt ¡ Z¹tjt U Zt ¡ Z¹tjt + ¯ xt+1 ¡ x¹t+1jt Vt+1 xt+1 ¡ x¹t+1jt j I¹t :

(5.24)

(5.25)

Notice that if the policymaker’s inference and forecast errors are independent of current
policy, then the term lt is independent of current policy. For this “certainty equivalence”
case, optimization proceeds with respect to the policymaker’s forecasts of the state variables,
and we can use the results from the full information case where we replace actual values with
the policymaker’s best forecasts.

Inference: the Kalman …lter
In this section we solve the optimal signal extraction problem while assuming that the
policy is as in the full information case. From the solution to the optimal control problem
we have Rt = Ft x¹tjt and y¹tjt = Gt x¹tjt . In order to determine the realized values of the ‡ow
variables we also need the components of Gt as de…ned in (5.23). Below we will construct G1t
and G2t , but for now we take them as given. In the following we construct the Kalman-…lter
33

recursively. The equilibrium we study in the paper is a …xed point of the mapping implied
by this procedure.
We …rst use the policy rule (5.22) and the equilibrium ‡ow variables equation (5.23) to
eliminate Rt and yt from the transition equation (5.13) and the signal equation (5.21):
xt+1 = Jt x¹tjt + Ht xt + ux;t+1 with

(5.26)

Ht ´ Axx + Axy G1;t
Jt ´ Axy G2;t + Bx Ft
and
sit = Mit x¹tjt + Lit xt + ei;t with

(5.27)

Lit ´ Six + Siy G1;t
Mit ´ Siy G2;t
These equations essentially de…ne the Kalman-…lter problem: equation (5.26) is the transition equation for the unobserved state, and equation (5.27) is the measurement equation
which relates the signals to the state. The only non-standard feature is that the policymaker’s updated estimate of the state appears on the right-hand side of each equation. We
now introduce the following notation for the derivation of the Kalman …lter. First, de…ne
the forecast errors for the state and signals as
x~t ´ xt ¡ x¹tjt¡1 and s~0t ´ s0t ¡ s¹0;tjt¡1 at the beginning of the period, and

x~+
´ xt ¡ x¹tjt
t

and s~1t ´ s1t ¡ s¹0;tjt

at the end of the period.

The forecast update is then x^t = x¹tjt ¡ x¹tjt¡1 . Second, in the standard Kalman-…lter the
update of the state variable is a linear function of the signal forecast error. Here we follow
Svensson and Woodford (2000) and guess that at the beginning of the period the policymaker’s update of the state is a linear function of a transformation of the signal, s~0t ¡M0t x¹tjt ,
x^t = K0t (L0t x~t + e0t )

(5.28)

This guess will be veri…ed below. We construct the Kalman-…lter in two steps, for each signal
separately.
Beginning of the period. From equation (5.27) and the law of iterated expectations,
the policymaker’s forecast of this period’s contemporaneous signal, based on the information
available at the end of last period, is
s¹0;tjt¡1 = (M0t + L0t ) x¹tjt¡1
34

We subtract this expression for the prior expectation of the contemporaneous signal from its
actual value (5.27), and obtain the forecast error for the signal
¡
¢
¡
¢
s~0t = L0t xt ¡ x¹tjt¡1 + M0t x¹tjt ¡ x¹tjt¡1 + e0;t = L0t x~t + M0t x^t + e0t :

Substituting our guess (5.28) for x^t , the expression for the contemporaneous forecast error
simpli…es to
(5.29)

s~0t = N0t x~t + º t
with N0t ´ (I + M0t K0t ) L0t

º t ´ (I + M0t K0t ) e0;t and

§º;t ´ Et [º t º 0t ] = (I + M0t K0t ) §e0 (I + M0t K0t )0 :
From equation (5.29) we get the update of the state variable x^t as a linear projection of x~t
onto s~0t
£
¤ £
¤
+
+ ¡1
^ 0t s~0t
s~0t = K
(5.30)
x^t = E x~t s~00t jI¹t¡1
E s~0t s~00t jI¹t¡1
¡
¢
¡1
0
^ 0t ´ Ptjt¡1 N0t0 N0t Ptjt¡1 N0t
+ §º;t
and
with K
£ 0 + ¤
Ptjt¡1 ´ E x~t x~t jI¹t¡1 .

Based on the posterior of the state variable we get an expression for the state variable forecast
error
^ 0t s~t = x~t ¡ K
^ 0t (N0t x~t + º t )
x~+
¹tjt = xt ¡ x¹tjt¡1 ¡ K
(5.31)
t = xt ¡ x
which we can use to get an estimate of the variance of the update error
Ptjt = E

h¡

xt ¡ x¹tjt

¢¡

´
¢0 i ³
^ 0t N0t Ptjt¡1 :
xt ¡ x¹tjt jI¹t = I ¡ K

(5.32)

We now have two expressions for how to use the contemporaneous signal to update the
estimate of the state variables: our initial guess from equation (5.28) and the implied Kalman
…lter from equation (5.30). For our initial guess to be correct we need that
^ 0t (I + M0t K0t ) :
K0t = K

(5.33)

^ 0t from (5.30) and for N0t and §v;t from equation (5.29)
Substituting for the de…nition of K
and simplifying we get
¡
¢¡1
K0t = Ptjt¡1 L00t L0t Ptjt¡1 L00t + §e0
:

(5.34)

Finally note that (5.33) together with the de…nition of N0t from (5.29) also implies that
^ 0t N0t :
K0t L0t = K
35

(5.35)

End of the period. We now turn to the second stage when the policymaker receives
signal s1t . From equation (5.27) and the law of iterated expectations, the policymaker’s
forecast of this period’s lagged signal based on the information available at the beginning of
the period is
s¹1;tjt = (M1t + L1t ) x¹tjt .
We subtract this expression for the prior expectation of the lagged signal from its actual
value (5.27), and obtain the forecast error for the signal
¡
¢
s~1t = L1t xt ¡ x¹tjt + e1;t = L1t x~+
t + e1t :

(5.36)

From equation (5.36) we get the update of the state variable x^+
~+
t as a linear projection of x
t
onto s~1t
£
¤ £
¤¡1
^ 1t s~1t with
x^+
= E x~+
~01t jI¹t E s~1t s~01t jI¹t
s~1t = K
t
t s
¡
¢
^ 1t ´ Ptjt L01t L1t Ptjt L01t + §e1 ¡1 and
K
£
¤
Ptjt ´ E x~t x~0t jI¹t .

(5.37)

The updated conditional expectation of this period’s and next period’s state variables then
are
©
ª
^
¹tjt
¹+
x¹+
tjt + K1t s1t ¡ (M1t + L1t ) x
tjt = x

x¹t+1jt = Jt x¹tjt + Ht x¹+
tjt :

From this expression and the law of motion for state variables (5.26) we obtain the forecast
error for the state variables at the beginning of next period as a function of this period’s
beginning-of-period forecast error
x~t+1 = xt+1 ¡ x¹t+1jt
³
´
= Ht xt ¡ x¹+
tjt + ux;t+1
³
´
^ 1t L1t x~t + ux;t+1 ¡ Ht K
^ 1t e1t ;
= Ht I ¡ K

(5.38)

and we can use this expression to update the
period’s beginning-of-period
h¡ estimate of ¢next
i
¡
¢0
forecast error covariance matrix Pt+1jt = E xt+1 ¡ x¹t+1jt xt+1 ¡ x¹t+1jt jI¹t+
³
´
³
´0
0
^ 1t L1t Ptjt I ¡ K
^ 1t L1t Ht0 + Ht K
^ 1t §e1 K
^ 1t
Pt+1jt = Ht I ¡ K
Ht0 + §u :

It remains to show how the equilibrium function (5.23) can be de…ned.

36

(5.39)

The equilibrium relation G
Suppose the equilibrium rule is as de…ned in equation (5.23). Then the policymaker’s
and the private sector’s forecast of next period’s ‡ow variables, and the implied di¤erence
in their forecasts is
y¹t+1jt = (G1;t+1 + G2;t+1 ) x¹t+1jt

£
¤
yt+1jt = G1;t+1 xt+1jt + G2;t+1 E x¹t+1jt+1 jIt
¡
¢
¡ £
¤
¢
yt+1jt ¡ y¹t+1jt = G1;t+1 xt+1jt ¡ x¹t+1jt + G2;t+1 E x¹t+1jt+1 jIt ¡ x¹t+1jt :

(5.40)

We can write the policymaker’s conditional expectation at the beginning of next period as
this period’s beginning-of-period conditional expectations plus the sum of this period’s three
updates
³
´ ³
´ ¡
¢
+
+
x¹t+1jt+1 = x¹tjt + x¹tjt ¡ x¹tjt + x¹t+1jt ¡ x¹tjt + x¹t+1jt+1 ¡ x¹t+1jt :
The public’s expectations of these three updates are in turn
h
i
^ 1;t L1;t x~+
E x¹+
¡
x
¹
jI
= K
tjt t
t
tjt
i
h
^ 1;t L1;t x~+
= x¹t+1jt ¡ x¹tjt + (Ht ¡ I) K
E x¹t+1jt ¡ x¹+
t
tjt jIt
³
´
£
¤
^ 0;t+1 N0;t+1 Ht I ¡ K
^ 1;t L1;t x~+
E x¹t+1jt+1 ¡ x¹t+1jt jIt = K
t :

Substituting these expressions yields
h
³
´i
£
¤
^
^
^
E x¹t+1jt+1 jIt ¡ x¹t+1jt = ªt x~+
with
ª
=
H
K
L
+
K
N
H
I
¡
K
L
:
t
t 1;t 1;t
0;t+1 0;t+1 t
1;t 1;t
t
(5.41)
From the transition equation for the state variables (5.13), we get that the di¤erence between
the private sector’s forecast and the policymaker’s forecast of the state variables is
¡
¢
¡
¢
xt+1jt ¡ x¹t+1jt = Axx xt ¡ x¹tjt + Axy yt ¡ y¹tjt :

(5.42)

Using (5.42) and (5.41) in (5.40) yields

¡
¢
¡
¢
yt+1jt ¡ y¹t+1jt = (G1;t+1 + G2;t+1 ªt ) xt+1jt ¡ x¹t+1jt + G1;t+1 Axy yt ¡ y¹tjt

On the other hand, the optimality constraints on allocations (5.13), imply that the di¤erence
in forecasts is given by
¡
¢
¡
¢
yt+1jt ¡ y¹t+1jt = Ayx;t xt ¡ x¹tjt + Ayy;t yt ¡ y¹tjt :

Equating the last two expressions and solving for the current value of the ‡ow variables
yields
¡
¢
yt = y¹tjt + (Ayy;t ¡ G1;t+1 Axy )¡1 (G1;t+1 Axx + G2;t+1 ªt ¡ Ayx;t ) xt ¡ x¹tjt
37

or
(5.43)

yt = G1;t xt + G2;t x¹tjt
where G1;t ´ (Ayy;t ¡ G1;t+1 Axy )¡1 (G1;t+1 Axx + G2;t+1 ªt ¡ Ayx;t )
G2;t ´ Gt ¡ G1;t .

Note that the matrix G1;t is also implicitly de…ned as
Ayy;t G1;t + Ayx;t = G1;t+1 Ht + G2;t+1 ªt+1 :

(5.44)

The interaction of optimal control and inference
We have solved the optimal control problem under the assumption that certainty equivalence holds, that is the policymaker’s decision is a function of the policymaker’s expectation
of the state of the economy, conditional on the available information at the time the decision
is made. For this to be true it remains to be shown that the term lt in the policymaker’s
value function (5.24) is indeed independent of the policy choice. From equation (5.25) it
follows that lt is a function of the update error xt ¡ x¹tjt ;
xt ¡ x¹tjt = x~t ¡ x^t
¡
¢
yt ¡ y¹tjt = G1;t xt ¡ x¹tjt
¡
¢
¡
¢
xt+1 ¡ x¹t+1jt = Axx xt ¡ x¹tjt + Axy yt ¡ y¹tjt + ut+1 ;

and it is enough to show that the update error for the state variable xt ¡ x¹tjt is independent
of policy. The update error has two components, the forecast error from the previous period
x~t , and the update of the state variable estimate x^t . As of time t the forecast error x~t is
independent of the current policy choice. This follows from equation (5.38).
Notice, that the only way the policy rule a¤ects the signal extraction problem is through
the determination of Gt in equation (5.19). Thus if we can show that the update of the state
variable estimate does not depend on Gt , then the update error is independent of current
policy. From equation (5.28) the time t update of the state variable depends on the matrices
K0t and L0t , and the forecast error x~t . From equation (5.34), K0t depends on Ptjt¡1 and L0t ,
and L0t in turn depends on G1;t . From equation (5.39), Ptjt¡1 depends on past outcomes
such as Ht¡1 , but not current choices, so it is independent of current policy. Thus we have
to show that G1t is independent of current policy. From the de…nition of G1t in equations
^ 1;t L1t in (5.37), L1t in (5.27),
(5.41) and (5.43), together with the de…nition of Ht in (5.26), K
and Ptjt in (5.32) it follows that G1t is a function of G1;t+1 and Gt+1 , but not of Gt . Therefore
G1t is independent of current policy Ft .
38

We have just shown that certainty equivalence holds, that is conditional on past and
future decision rules, the inference problem of the current policymaker is not a¤ected by
the policymaker’s own choices. This does not mean that the signal extraction problem is
independent of the optimal control problem. Since the Markov-perfect equilibrium we study
is a …xed point of the mapping de…ned by the inference problem, and since G is a function
of the policy rule F , the inference problem cannot be separated from the control problem.

39

¯ = 0:99,

Table 1. Parameters and Steady State
Parameters
µ = 0:73,
½ = ¡17:52, Â = 1:30,
Á = 3:0,
" = 10,
° y = 0:9,

y = 0:265, c = 0:265,
q0 = 1:042,

° m = 0:9,

¾ y = 0:01,
Steady State

n = 0:265,
q1 = 1:016,

y0 = 0:175, y1 = 0:225,

R = 1:036,
q2 = 0:991,

¾ m = 0:026
P^ = 1:025,
q3 = 0:966;

y2 = 0:289, y3 = 0:370

40

w = 0:897;

J = 4,

Figure 1. Estimated Taylor-Rule for the U.S. Economy

41

LL
ψ

PP

p0

Figure 2. Temporary Equilibrium

42

Panel A. The Response to a Productivity Shock
Output, Employment, and Real Balances

Nominal Interest Rate, Inflation Rate, and Real Rate
1.5

1.5

0.0

0.0

Pt+1/Pt
1.0

ct
0.5

rt

1.0

mt

nt
2

3

4

5

-1.0

-1.0

0.0

0.0
1

0

6

Relative Prices

1

2

3

4

5

p3t

0.05

0.10

0.020

0.05

0.015

0.00

0.010

-0.05

0.005

-0.10

0.000

0.020

0.00

p1t
p0t
0

1

2

3

4

5

6

0.015

at

p2t

-0.10

6

Allocational Efficiency and Marginal Cost

0.10

-0.05

-0.5

0.5

zyt

0

Rt

-0.5

0.010

ψt

0.005
0.000
0

1

2

3

4

5

6

Panel B. The Response to a Money Demand Shock
Output, Employment, and Real Balances

Nominal Interest Rate, Inflation Rate, and Real Rate

1.5

1.5
1.0

0.00

-0.5
1

2

3

4

5

-0.05

6

-0.05
0

Relative Prices

1

2

3

4

0.05

0.000

6

0.00

-0.005

-0.05

-0.010

0.000

ψt

p0t

0.00

5

Allocational Efficiency and Marginal Cost

0.05

p1t

0.00

rt

0.0

-0.5
0

0.05

Rt

0.5

nt

ct

Pt+1/Pt

0.05

mt

0.0

0.10

1.0

zmt

0.5

0.10

at

p2t

-0.005

p3t

-0.05
0

1

2

3

4

5

6

-0.010
0

1

2

3

Figure 3. The Full-Information Case

43

4

5

6

Panel A. Conditional Expectations of State Variables
Lagged Relative Price

1.0

0.0

1.0

1.0

0.0

0.0

Lagged Relative Price

1.0
0.0

_

-1.0

-1.0

-1.0

-2.0

-2.0

-3.0

-3.0

q2,t|t-1

-1.0

_

q1,t|t-1

-2.0

q1,t
-3.0
0

1

2

3

4

5

Productivity Shock

1.0

1.0

zy,t

-3.0
0

6

-2.0

q2,t
1

2

3

4

5

6

Money Demand Shock

0.1

0.1

zm,t
0.0
0.5

0.0

0.5

_

zm,t|t-1

-0.1

_

-0.1

zy,t|t-1

0.0

0.0
0

1

2

3

4

5

-0.2

6

-0.2
1

2

3

4

5

6

7

Panel B. Realizations of Flow Variables
Output, Employment, and Real Balances
2.0
1.0

mt

0.0

ct
nt

-1.0
-2.0
0

1

2

3

4

5

Nominal Interest Rate, Inflation Rate, and Real Rate
2.0

4.0

1.0

2.0

0.0

0.0

-1.0

-2.0

-2.0

-4.0

6

p3t

0.0

p1t

-1.0

p0t

-2.0
-3.0
0

1

2

3

0.0

Rt

-2.0

Pt+1/Pt

-4.0

1

2

3

4

5

6

Allocational Efficiency and Marginal Cost

2.0

p2t

2.0

rt

0

Relative Prices

1.0

4.0

4

5

2.0

1.0

1.0

0.0

0.0

-1.0

-1.0

-2.0

-2.0

-2.0

-3.0

-3.0

-3.0

-4.0

6

1.0

at

0.0
-1.0

ψt

-4.0
0

1

2

3

4

5

6

Figure 4. The Response to a Productivity Shock with Information on Lagged Output

44

Panel A. Conditional Expectations of State Variables
Lagged Relative Price

Lagged Relative Price

0.5
0.0
-0.5

_

q1,t|t-1

-1.0

_

q1,t|t

-1.5

q1,t

-2.0
0

1

2

3

4

5

0.5

0.0

0.0

-0.5

-0.5

-1.0

-1.0

-1.5

-1.5

-2.0

-2.0

1.0

zy,t

0.5
0.0

-2.0
1

0.0
5

5

6

0.1

0.0

zm,t|t-1

-0.1

_

zm,t|t

zy,t|t-1
4

4

_

-0.1

3

3

zm,t

_

2

2

Money Demand Shock

0.0

1

-1.0
-1.5

0.5

0

-0.5

q2,t

0.1

zy,t|t

0.0

q2,t|t-1

q2,t|t

_

0.5

_

_

0

6

Productivity Shock

1.0

0.5

-0.2

6

-0.2
0

1

2

3

4

5

6

Panel B. Realizations of Flow Variables
Nominal Interest Rate, Inflation Rate, and Real Rate

Output, Employment, and Real Balances
1.5

1.5

mt

1.0

1.0

1.0

0.5

0.0

0.0

0.5

ct

Rt

-1.0

0.0

0.0

-0.5
-1.0

2

3

4

5

1.0

-3.0
0

1

2

3

4

5

6

Allocational Efficiency and Marginal Cost
2.0

1.0

1.0

at

1.0

p3t

0.0

p2t

0.0

-3.0

6

Relative Prices

2.0

-2.0

Pt+1 /Pt

-1.0
1

-1.0

-2.0

-0.5

nt
0

1.0

rt

0.0

0.0

ψt

p1t

-1.0

-1.0

-1.0

-1.0

p0t
-2.0

-2.0
0

1

2

3

4

5

-2.0

6

-2.0
0

1

2

3

4

5

6

Figure 5. The Response to a Productivity Shock with Information on
Contemporaneous Money and Lagged Output

45

Panel A. Conditional Expectations of State Variables
Lagged Relative Price

Lagged Relative Price
0.5

0.5

0.5

_

q1,t|t

0.0

0.0

0.0

-0.5

-0.5

-0.5

-1.0

-1.0

_

q1,t|t-1

-1.0
-1.5

q1,t

-2.0
0

1

2

3

4

5

-1.5

-2.0

-2.0

0.5

1.0

q2,t|t

0.0

-0.5

q2,t|t-1

-1.0

q2,t

-1.5
-2.0

1

2

3

4

5

6

Money Demand Shock

1.0

zm,t
_

_

zm,t|t

zy,t|t-1

zy,t

0.0
_

0

6

Productivity Shock

0.5

-1.5

0.5

_

0.0

0.5

0.5

_

zm,t|t-1

_

zy,t|t

-0.5

-0.5
0

1

2

3

4

5

0.0

6

0.0
0

1

2

3

4

5

6

Panel B. Realizations of Flow Variables
Nominal Interest Rate, Inflation Rate, and Real Rate

Output, Employment, and Real Balances
2.0

2.0

1.0

1.0

3.0

3.0

2.0

mt
0.0

0.0

yt

1.0

0.0

-2.0
-2.0
1

2

3

4

5

-3.0
0

Relative Prices
1.0

4

5

6

1.0

at

0.0
-1.0

ψt

0.0

p1t
-1.0

-1.0

p0t
-2.0

-2.0
2

3

-1.0

p2t

1

2

0.0

1.0

p3t

0

1

Allocational Efficiency and Marginal Cost
2.0

1.0

-2.0

Pt+1/Pt

-3.0

6

2.0

0.0

-1.0

-1.0

-2.0
0

0.0

Rt

-1.0

nt

-1.0

2.0

rt

1.0

3

4

5

-2.0

-2.0

-3.0

-3.0

-4.0

6

-4.0
0

1

2

3

4

5

6

Figure 6. The Response to a Money Demand Shock with Information on
Contemporaneous Money and Lagged Output

46