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Working Paper Series
Optimal Disinflationary Paths
WP 95-01
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Peter N. Ireland
Federal Reserve Bank of Richmond
WORKING
PAPER95-1
OPTIMALDISINFLATIONARY
PATHS
Peter N. Ireland*
Research Department
Federal Reserve Bank of Richmond
January 1995
*I would like to thank John Cochrane, Thomas Cooley, Jeffrey Lacker, Kevin
Reffett, Stacey Schreft, Alan Stockman, John Weinberg, and seminar
participants at Florida State, Iowa, and the Minneapolis Fed for helpful
comments and suggestions. The opinions expressed herein are my own and do not
necessarily represent those of the Federal Reserve Bank of Richmond or the
Federal Reserve System.
Abstract
This
paper
general
characterizes
equilibrium
Starting
setting.
disinflation
fully
optimizing
from a steady
state
of disinflation
using
other
in the context
agents
and staggered
with
policy
when the monetary
to be welfare-improving,
be replaced
policy
positive
when announcements
Disinflationary
and employment
fails
monetary
model with
is desirable
credible.
the benefits
optimal
yields
authority
still
exceed
however,
distortionary
of
credibility;
the costs.
when lost
taxes.
a rapid
monetary
substantial
losses
policy
are
in output
nevertheless,
Disinflation
seignorage
a
price
inflation,
future
lacks
of
often
revenues
must
1.
Introduction
The inflation
peak of
over
stabilized
should
at an average
annual
the average
inflation
rate
1983,
of price
monetary
model to suggest
in fact,
still
stabil1ty.l
reaching
inflation
further,
n
after
however,
This
how the final
paper
stage
a
has
The Federal
of 4 percent.’
rate
goal
States
Reserve
if
uses
it
is
to
a general
of disinflation
proceed.
simple
work with
answer
like.
agents’
to the question
and Hansen
variables
general
to the extent
decisions
as they
balances.
Optimal
the Friedman
the nominal
interest
wages and prices
is
either
short-run
costs
Friedman
rule
monetary
that
inflation
acts
of
zero.
Moreover,
the
these
Hence,
be adopted
the models
an immediate
completely
offset
the gains
and Taylor
(1983)
analyze
is,
(1987)
real
distorting
on their
cash
tax by
to make
assume that
that
nominal
the Phillips
they
indicate
identify
that
no
the
immediately.
or wages are not perfectly
of
imply
that
or positively-sloped;
affects
inflation
models
a
path
the money supply
so they
flexible,
policy
as a tax,
contracting
provides
and Huffman
to economize
eliminates
disinflation.
should
costs
in efforts
rule,
are perfectly
When prices
be short-run
rate
of Greenwood
instance,
policy
models
disinflationary
for
(1969)
vertical
models
engage
monetary
monetary
of what the optimal
(19891,
only
following
equilibrium
In the cash-in-advance
and Cooley
curve
in the United
Since
long-run
Previous
looks
sharply
in 1980.
its
equilibrium
fell
13 percent
must reduce
achieve
rate
flexible,
disinflation
from removing
the problem
the
that
inflation
of disinflation
1
however,
partially
tax.
with
there
may
or
Phelps
models
(1979)
featuring
overlapping
labor
contracts
Both Phelps
periods.
disinflationary
employment
the rate
path
along
of money growth
price
setting
can actually
money supply
is
the
chosen
and Taylor’s,
on output
suggest,
however,
wages are
output,
of
path
the striking
of
(19941
result
that
provided
that
and
a reduction
uses
a model of
the path
called
output
in
a quick
in Ball’s
disinflation
several
a
output
Ball
decreases
for
involves
Nevertheless,
This
paper
flexible
current
adds to the
goods
that
economizing
Here,
that
is
for
for
the
model as in
by the
and employment
in
by Danzlger
prices.
setting
none of
on their
the presence
than on welfare.
differs
from
Danzlger
(1988)
to be welfare-improving
similar
of
research
or
the optimal
shows that
an
in a model of
initiated
of past
are
2
used
provides
in the face
a cash-in-advance
inflation
between
and those
models
cash balances
and
the consequences
difference
here
by Phelps
by considering
the effects
the earlier
results
to Ball’s.
and Ball
developed
Their
path when prices
significantly
Indeed,
The major
of
of disinflationary
disinflationary
line
in a model where
price
is
may fall
the effects
rather
case.
setting
and continued
disinflation
consider
the optimal
not perfectly
price
tax.
this
gradual.
models
and Ball
that
staggered
for
only
and employment
disinflation
however,
initial
the instantaneous
immediate
staggered
levels
increase
in the flexible-price
Taylor
the
show that
is
in advance
the existence
appropriately.
Taylor,
policies
into
that
wages
run.
Phelps,
of
both
cash-in-advance
short
path
which
nominal
demonstrate
to derive
disinflation
flexible-price
fix
and Taylor
are maintained;
staggered
Phelps
that
the model of
in previous
agents
of
constraint
built
with
a positive
implies
work,
a motive
inflation
that
the
inflation
tax has the
traditional
The model can therefore
be used
disinflation
removing
due to nominal
the
inflation
from conventional
permits
of
policy
with
section
Section
3 then defines
Section
4 derives
alternatives:
(1986)
prescribed
taxes.
with
from
here
the
for
it
implications
the optimal
called
differs
to which
Thus,
disinflation
section
selgnorage
staggered
switch
of price
policy
to zero
stability,
money growth,
and the
monetary
by the
emphasize
the effects
and Hansen
(1991)
an immediate
revenues
setting.
only
must be replaced
their
of
result
Finally,
section
3
the
to the
of
model.
the monetary
disinflationary
of disinflation
when
to an announced
change
cash-in-advance
to be welfare-improving
using
carries
with
models.
use a flexible-price
falls
switch
to the basic
gradually
disinflation
to two
consistent
the role
5 investigates
equilibrium.
it
cash-in-advance
the outcome
respond
setting.
immediate
two modifications
(19931
price
and compares
in determining
6 asks whether
price
staggered
the model economy's
by flexible-price
expectations
Cooley
Section
monetary
and Goodfriend
to show that
when lost
of
benefits
in the extent
by comparing
the model of
on to consider
credibility
agents'
in policy.
model
goes
Thus,
private
only
and characterizes
goal
The paper
policies.
outlines
an immediate
rule
authority's
long-run
dlsturbances.3
immediate
the optimal
Reserve's
Sargent
the
costs
the model presented
models
isolated
on money demand.
the short-run
the
In addition,
are
effects
models.
The next
Friedman
with
to monetary
setting
here
flexible-price
Federal
rigidity
tax.
to adjust
price
found
to compare
cash-in-advance
prices
staggered
dlstortlonary
other
over
dlstortlonary
to the environment
7 concludes.
2.
A Model of Staggered Price Setting
As indicated
monetary
model
above,
that
this
borrows
most of
cash-in-advance
models
Hansen
The model departs
( 1989 I.
introducing
(19831,
is
an element
a staggered
not derived
as a tractable
continuously
their
provide
nominal
goods
(see,
Cooley
nominal
appear
consider
rigidities
do not all
a list
price
price
as in Blanchard
rigidity.
Here,
structure
of
and second,
is
simply
that
firms
do so simultaneously.
Ohanlan
that
document
by
and
serves
cannot
when firms
studies
imposed
The framework
that
empirical
however,
do change
and Stockman
rigidity
in
prices.
most of
market
find
rather
that
the nature
as well
(19941,
than
nominal
the focus
in the
nominal
models
While
for
of
optimal
as price
1990).
research.
4
of
left
thought
real
hand,
job
correlations
it
would
in economies
is
of
do a better
that
policy
are widely
behavior
models
rigidity
emphasis
On the other
a variety
this
on nominal
models
work indicates
rigidities,
the contributions
This
the cyclical
wage setting
monetary
is
market.
wage contracting
at matching
their
here
labor
Manklw 1987 and King
that
including
literature,
implications
setting
in the data.
the recent
and Ball
instance,
(1992)
prices
and
ones,
first,
their
and Cooley
from more conventional
two ideas:
counterfactual
for
(1987)
way of
the fact
to have
and Huffman
setting
capturing
equilibrium
from the flexible-price
frictions.
(19881
in the goods
structure
underlying
Following
by Danzlger
price
a general
from more basic,
they
(1994)
of nominal
develops
its
Greenwood
nominal
adjust
prices,
reflects
of
section
Cho and
than
that
be useful
with
as a task
wages
wage
for
future
to
2.1.
The Economic Environment
The economy consists
government
(or monetary
by le{l,2,3,4);
there
different
produce
there
types
are four
of
infinitely-lived
are N identical
types
different
of goods
by firms
of
Each consists
of
members:
three
At the beginning
sequence
each
of
{Ht)rzo
household's
supply
a worker,
bond trader
t=O,
at each
according
four
indexed
each
Firms of
consumption
also
types,
type.
Hence,
goods.
indexed
by 1, where good
are all
a shopper,
the monetary
transfers
date
identical.
and a bond trader.
authority
that
it
announces
plans
t=O,l,Z,....
M;+Hr'
where MS
t+l denotes
the per-household
and the beginning
of
the government's
are of
the
to make to
The nominal
money
to
Mt+l=
now,
and a
by t=0,1,2,....
of period
period
households,
The N households
i.
lump-sum monetary
then evolves
of
perishable
type
and indexed
firms
in the economy,
1 is produced
Time ls'dlscrete
The firms
authority).
firms,
t+l.
activity
t=0,1,2,...,
money supply
By choice
is
limited
of
at the end of
nominal
units,
to making these
period
Mi=l.
t
For
monetary
transfers.
Firms
of
its
monetary
households
behavior
2.2.
and households
act
of
policy
as if
each,
perceive
{M:+r}to
they
given
the government's
as fully
have perfect
the announced
time
credible.
foresight
policy
Hence,
from
{M~+l)~zo,
t=O announcement
firms
and
t=O forward.
is described
The
next.
Firm Behavior
Each firm
technology
for
of
type
producing
1 has access
good
1.
If
to a constant
returns
the representative
to scale
type
1 firm
hires
n it units
of
then
labor,
Firms are
Specifically,
output
is fixed
interval,
the firm
setting
is
setting
new prices
to sell
at prices
dates
its
while
all
set
new prices
each
date
this
at which
t.
its
four-period
one type
types
are
set
Price
are
constrained
new prices
at
at dates
at dates
firm
for
price.
teT3={3,7,11,...},
teT4={4,8,12,...).
conditions;
the first
of
Type 1 firms
new prices
price
fixed
three
new prices
at dates
its
at time
setting
a nominal
firms
the other
set
output
During
all
period.
2 firms
3 firms
until
to set
on demand at
of
of
and price
interval.
output
firms
as initial
price
required
in a previous
1, type
taken
production
in each period,
1, type
set
is
units
ylt=nit
a four-period
so that
set
4 firms
initial
each
must supply
staggered
teT2={2,6,10,...
at t=O are
in their
over
teT’={l,5,9,...
and type
can produce
constrained
decisions.
that
it
is
it
Goods prices
constrained
is
to sell
permitted
to set
at
a
new price.
An individual
type
other
firms
as given.
price
p and all
individual
je{O,
prices
firm
1,2,3),
other
1 firm
type
sets
If
and time
other
sets
choose
type
to set
units
time teT’
i firm
price
of
p,,
output
taking
chooses
then
the
to set
the
at each
date
t+j,
where
is
t+j.4
firms’
at
individual
c I,+,(p,p,)
C ,,+,(P,P,)
and c it+,(p)
the
1 firms
must produce
a new price
=
the representative
That is,
price,
the same price
it
if
the
if
P<P,
cit+,(pI
0
lf
P’P,
if
P’P,
household’s
individual
must satisfy
as the other
NCit+,(Pl
demand for
firm
sets
the demand of
firms,
6
it
(1)
all
must satisfy
its
good
price
1 at price
below
N households.
p
the
If
it
the same demand as
the other
If
firms.
attracts
it
the gross
individual
its
its
price
above
the other
firms
price,
it
no demand.
let
For t=0,1,2,...,
denote
sets
type
price
is
nominal
interest
1 firm's
fixed
wt denote
the nominal
rate
discounted
between
profits
at p and the other
type
wage at time
periods
over
the
1 firms
t and Rt
The
t and t+l.
interval
price
during
is
fixed
which
at p,
are
X(P.P,l
(p-wt)Clt(P’Pt)+ (P-Wt+1 )Clt+&P’Ptmt
=
(21
+ (p-w
since
the
t+2
1c it+2(P,pt)/(R
R
t t+1
individual
firm
p to maximize
interest
setting
rates
p,
This
are
0
all
type
=
i firms
and all
competed
zero-profit
to determine
type
of
labor
the representative
type
demands ~~~+,(p,p~),
1 firms'
price
p,,
to meet
1 firm
wages w
t+J'
and the price
as given.
in equilibrium
profits
teT',
taking
Rt+J, the other
rules
Since
x(p,pt),
units
c It+,(p,pt)
must hire
At each date
demand at time t+j.
chooses
1 + (P-w~*,)c~~+~(P,P~)/IR
R R
1,
t t+1 t+2
are
will
earn
profits
away in equilibrium,
condition,
along
the equilibrium
with
price
of
will
choose
the same price
Positive
rt=n(pt,pt).
so that
equations
rrt=O for
(1)
p, as the solution
+ (Pt-wt+1 )c&p,VR
(Pt-wt)cJPt)
all
identical,
all
and (21,
t=0,1,2,....
is
sufficient
to
t
(3)
+ (Pt-W,+2)Clt+2(Pt)/(RtRt+l)
With p, given
C It+,(p1
below,
Thus,
by the solution
consistent
the
with
individual
the nominal
price
+ (P,-~~+~)c~~+~(P,)/(R~R~+~R~+~).
to
the optimizing
type
p,,
1 firm
of
good
(3)
and the demand functions
behavior
maximizes
of
r(p,pt)
1 at time t is
7
households
by choosing
given
by
described
p=p,.
P 1t
Pt
=
for
teT'
for
MT'
(4)
p,,-,
t
for
and by initial
t=l,2,3,...
Although
the course
firm
competition
of
each
rt=n(p,,p,)
eliminates
interval
may make a nonzero
conditions
during
profit
must be zero
for
is nonzero
firms,
nit
covering
t,
then
this
price
any single
it
If
is
profits
fixed,
period.
over
an individual
That is,
may be that
nit,
while
given
by
(Plt-wt)cJPJ'
type
t=0,1,2,....
its
discounted
and 1~{1,2,3,4).
to the representative
period
in period
2.3.
any t=0,1,2,...
so the representative
profits
each
for
which
t=0,1,2,...,
=
It
it
t=O.
each firm's
during
all
for
1 firm
The households
delivers
household's
nltCO,
so that
the representative
its
current-period
bond trader
the type
own the
at the end of
1 firm
incurs
bond trader
is
responsible
preferences
are described
a loss
for
loss.
Household Behavior
The representative
utility
household's
by the
function
(5)
where
time
work,
c It is
t.
the household's
During
shop,
each
and trade
period
bonds
The representative
supply
to firms
of
each
consumption
1 and xt is
t=0,1,2,...,
the three
in spatially
distinct
household's
type
of good
worker
1~{1,2,3,4)
8
members of
leisure
at
the household
markets.
decides
during
its
each
how much labor
period
nit
t=0,1,2,...
to
subject
to the constraints
4
1
E:
xt +
t=0,1,2,....
1 nit,
(61
I=1
The time endowment in equation
The representative
consumption
Since
he shops
worker
his
goods
is
shopper,
of each
in different
meanwhile,
from those
required
to unity.
type during
markets
he is
beginning-of-period
advance
is normalized
household's
from firms
employed,
(6)
to pay for
money holdings
each
period
in which
all
of
That is,
Mt.
purchases
t=0,1,2,....
his
his
household's
purchases
he faces
out of
the cash-ln-
constraints
4
Mt
=
t=0,1,2,....
CPAs
(7)
i=l
At the end of period
trader
receives
the shopper,
from
the wage earnings
and current-period
the government.
accumulate
profits
the cash Mt+i that
traders
from other
bonds.
Bonds paying
dollars
at the end of
Bt+Mt+H
his
4
4
n
i=l
It.
+
will
c
any unspent
carry
the bond trader
of
bond
from
transfer
funds
into
to
the following
from or lends
or purchasing
time
cash
and the
sources
borrows
at the end of
Thus,
t c
these
also
by issuing
time t.
+w
uses
household
Bt+I dollars
household's
from the firms,
bond trader
households
t
from the worker,
The bond trader
The representative
period.
the representative
t=O,l,Z,...,
to
one-period
discount
trade
Bt+I/Rt
t+l
faces
for
the constraints
4
a
2
It.
c
PC
+M
+B
m9
it 1t
t+i
t+1
t
(8)
IS1
I=1
t=0,1,2,....
The representative
M
t+1'
and Bt+l to maximize
household
the utility
chooses
sequences
function
9
for
(51 subject
tit,
nit,
to the
xt,
constraints
(6)-(81,
taking
the
initial
conditions
3.
Equilibrium Defined and Characterized
such
discounted
profits
(ill)
households
labor
markets
is defined
that:
(1)
bonds
over
each
behave
clear.
optimally;
=
and labor
markets
=c
It
conditions
for
household's
it'
n
and writ and
for
prices
each
firm
earns
its
price
is
the money,
bond,
goods,
which
M
it'
problem.
clear
in its
t+1’
These
supply,
and
zero
fixed;
and
when
)....
the bond market
clears
when
t=0,1,2,....
0,
=
(ii)
t=0,1,2
net
when
t=0,1,2,...,
household's
linear,
c
during
Mt+l'
It’
The representative
are
p,,,
sequences
optimally;
and (iv)
in zero
t+1
constraints
of
The money market clears
are available
n
behave
interval
B
The goods
Ht, wt. Rt,
as a collection
firms
M:+l
Since
for
MO=1 and Bo=O as given:
An equilibrium
quantities
the sequences
objective
choice
function
variables.
and B
first
lo{l.2,3.4).
t+1
order
is
Thus,
characterize
conditions
concave,
the first
the solution
imply
that
and its
order
to the
in
equilibrium,
n
It
=c
=
1t
W
M;/(4pJ,
t =
orM;+1/(4PL
t=0,1,2,..., 1~{1,2,3,4),
t=0,1,2,...,
(9)
(101
and
Rt
Equations
(3)
=
M,S+2/(PM;+1I,
and (4)
summarize
t=0,1,2,....
the implications
(11)
of
firm
optimization
and the zero-profit
In light
condition.
function
in equation
(91-(11)
imply
(3)
that
is given
equation
(3)
of
equation
(91,
by c it+J(p1=MI+J/(4p).
can be rewritten
the demand
Thus,
equations
as
MI + @M;+l + 8G;+2 + s":+,
P,
(12)
#
=
+ @MS /MS
t+2 + s":+2m:+3
t+1
+ 133Mi+3/M:+4
I
t=1,2,3,....
Equations
in terms of
supply
(4)
the
initial
sequence
its
they
prices
and quantities
and the money
show how equilibrium
monetary
equilibrium
money growth
equilibrium
M:=l and {p,,,p2,,p3,,p4,~
Thus,
varies
on the economy's
the moment that
express
conditions
{M:+l)Fzo.
as the government
policy
and (9)-(12)
policy.
Two key effects
are best
;r,=M:+r/MI is
outcomes
illustrated
constant,
of
change
monetary
by assuming
with
r,=r
for
for
all
t=0,1,2,....
Equation
the nominal
Greenwood
positive
nominal
teT1.
(10)
wage wt also
During
type
under
also
the constant
constant,
and Cooley
rate
provides
on their
cash
The government
the Friedman
representative
is
(1987)
interest
economize
Equation
that
with
money growth
Rt=R=~/p
model as in the flexible-price
and Huffman
consumption.
following
rate
In this
nominal
inefficiently
less
indicates
interest
t=o, 1,2 ) . . . .
of
(11)
(1969)
grows
three
sets
its
periods,
price
with
a strictly
to
more leisure
and
inefficiency
by
of
7=/3.
rate
above
money growth
r.
When r>l,
the nominal
the wage rises
11
models
an incentive
source
the constant
at the constant
1 firm
the next
under
with
(19891,
by enjoying
can remove this
rule,
shows that
households
r,
all
cash-in-advance
and Hansen
balances
for
rate
while
rate
r,
the
the
wage at date
the price
remains
zero
fixed;
eventually
discounted
profits
profits
early
firm
sets
fall
during
relative
that
its
price
below
the next
r=l,
efficiency
This
periods
condition
model,
only
therefore,
contracting
the money supply,
the optimal
monetary
utility-maximizing
4.
that
smaller
cost
sees
however,
a tension
policy
The next
is
is
the
tension
period
consistent
inflation
tax by
efficiency
which
is
with
the monetary
productive
section,
manner
constant.
between
can remove
the wage
requires
wt in every
monetary
the
than the fixed
efficiency,
shows how this
then
moves in a saw-tooth
but can guarantee
policy,
teT1,
when the money supply
fixed.
when r<l,
only
characterizes
resolved
in a
way.
Optimal Disinflation
Let
inflation:
initial
goods
prices
to continue
At t=O, however,
the monetary
follow
believe
come from an equilibrium
p4,=gp30=g2p2,=g3pIo,
the money supply
will
price
The government
the money supply
is
marginal
features
objectives.
by holding
its
implies
authority's
it
Productive
equal
(10)
until
earns
by making positive
Conversely,
wage at date
The firm
price.
interval
later.
nominal
wage.
p,,
Equation
t=0,1,2,....
losses
the firm's
price
the fixed
the four-period
the nominal
three
to the nominal
the firm's
this
over
on and incurring
Unless
price.
the wage exceeds
this
an alternative
announcement
where g>l.
expanding
at the
authority
path
Prior
{M~+l}~=o instead.
and act
from t=O forward
12
to t=O,
inflation
announces
with
that
positive
agents
rate
g>l
expect
forever.
the money supply
Firms and households
with
perfect
foresight,
taking
prices
perfectly
set
optimal
utility
disinflationary
To compute
set
quarter
equal
fixed
preference
wages,
neither
any of
monetary
Rt-1,
rate
of
policy.
by Ball
by equations
(4)
the representative
the economy's
the preference
in the model
(19941,
(9)-(12)
the equilibrium
levels
these
variables
Thus,
a is
goods
indicate
levels
of
equal
xtrO and nit20
do not bind
prices
that
the
a affects
rate
to 10, which
one
consumption,
are determined,
nor the growth
set
parameter
represents
individual
Equations
only
constraints
the other
of
money
guarantees
without
results.
It
on the the
and output.
is
constitutes
path,
period
Once their
inflation.
policy
each
intervals.
1 shows the optimal
annual
maximizes
disinflationary
CLaffects
the nonnegativity
percent
constraints
and prices.
the optimal
Figure
to these
so that
one-year
the growth
affecting
that
as suggested
parameter
leisure,
'M~+ll~zo
the optimal
to 0.99,
for
are determined
announcement
path.
Thus,
year.
remain
that
subject
the policy
as given;
and quantities
The sequence
household's
under
Prices
credible.
and (9)-(121.
#3 is
at t=O and earlier
disinflationary
traces
out
inflation
Aggregate
output
path
the effects
rate,
the net
is measured
starting
of
from a 4
the optimal
nominal
interest
rate
as
4
yt = c%t
I=1
and is normalized
so that
and labor
implies
that
Inflation
is
by dividing
calculated
nominal
Yo=l.
Yt also
The market clearing
serves
as an index
as the growth
expenditure
rate
by real
13
of
output:
condition
of
total
the price
for
goods
employment.
level
Pt,
obtained
4
=
pt
Y;' 1 P,&,.
I=1
1, money and prices
In figure
which
period,
rate
corresponds
and aggregate
steady
rate
state
of
prices
prior
the slower
maintaining
requires
a similar
determined
initial
their
increase
per year,
are
when the economy
constant
The optimal
at t=O, which
policy
that
and employment
money holdings
calls
does
later.
in his
model of
Since
not
percent
interest
in its
initial
on real
Phelps
in the
goods
(1979)
finds
disinflation
households
translate
per
an increase
the effect
MOwhen the new policy
in money growth
is
for
occurs
jump in money growth.
0.985
and the nominal
counteracts
money growth
output
of
to 4 percent
to t=O.
money growth
of
that
output
grow at the rate
have already
is
into
announced,
the
a distortlonary
tax.
Real
response
lower
output
to the unexpected
inflation
addition,
following
starts
balances
tax,
reduces
t=O,
between
t=O and t=l,
at t=O,
and increases
household's
of
consumption
demand for
slower
immediately,
even
though
partly
but also
real
money growth
at t=l.
during
money growth
in
because
balances
and money demand increasing,
In
increases
the
the
does
a
inflation
not begin
to
t=l.
money growth
r=g and r=l.
converges
Thus,
the benefits
of
the Friedman
and the benefits
of
zero
efficiency.
between
of money growth
leisure
With output
to fall
until
After
lies
burst
t=O in anticipation
period.
decelerate
that
tax
by 2 percent
the representative
at the end of
rate
increases
Disinflation
is
quickly
in the
to a new steady
long
run,
the optimal
rule,
which
eliminates
money growth,
which
ensures
largely
complete
14
after
state
the
value
policy
inflation
productive
one year.
Since
the
inflation
tax
exactly
is permanently
as in flexible-price
Figure
growth
2 displays
policy,
percent
annual
initial
burst
falls
the effects
As before,
slightly
inflation
with
Table
take
but
eliminating
percentage
of
percent
annual
welfare
gains
adopting
smaller.
policy
consumption.
rule
partially
1 also
setting,
These
monetary
Here,
come close
results
policy
with
4
without
the
policy.
as the effects
2 nor
off
of
Output
reducing
that
the
the
percent
those
under
while
of
the
obtained
benefits;
are optimal
announcements
that
it
is desirable
are perfectly
15
that
inflation
the gain
path
is
from
0.0223
eliminating
a 10
These
from immediately
adjustment
flexible-price
costs
of
hence,
the zero
switch
goods
continuing
(19891
neither
measured
four
consumption.
the net
under
gain
is
money growth
to the Friedman
to the optimum in welfare
suggest
all
the optimal
and Hansen's
immediate
models
of
from optimally
long-run
that
of disinflation,
It shows that
using
the short-run
reveals
by flexible-price
both
0.0874
in Cooley
offset
shown in figure
prescribed
state
sharply
in consumption
inflation
equals
model.
Table
money
stability
the optimal
consequences
The gain
are much smaller
the Friedman
disinflation
steady
under
a zero
price
falls
policies.
annual
inflation
cash-in-advance
rate
as well
the disinflationary
total
in its
later
increase
household
a 4 percent
percent
increases
adopting
Reserve's
occurs
on the welfare
makes the representative
is under
that
higher,
over.
1 reports
as the permanent
as it
the Federal
money growth
permanently
models.
immediately
The inflation
at first,
tax
of
the economy begins
inflation.
of
remains
cash-in-advance
consistent
objective.
output
reduced,
staggered
rule
price
terms.
to disinflate
credible.
Sargent
quickly
(1986)
when
and
Goodfriend
(19931,
credibility
practice.
respond
however,
can be a major
stumbling
Under the optimal
policy
to promises
initially
of
different
they
instead
observe
the next
changes
if
of
section
agents
this
even
this
to the faster
that
how the optimal
announcements
of
policy
1, households
of
money growth
authority's
as fully
perfect
Fischer
in
and firms
as money growth
policy
might
money growth
they
be
that
are promised.
disinflationary
path
are no longer
perceived
credible.
money growth
policy.
As before,
money growth
policy
rate
'M;+l';zo
and households
(19861,
foresight
future
on their
and their
respond
g>l.
that
only
it
plans
agents'
incorporated
the monetary
into
expectations
to an announced
in a steady
to implement
each
of
that
t forward.
is
state
of
change
under
authority
in
a constant
announces
at the beginning
of
a new
t=O.
Firms
as follows.
of
knowledge
that
gradually
As before,
then react
have observed
credibility
the economy begins
expectation
g-t i from time
partial
model by assuming
At the beginning
they
consequence
investigates
lack
disinflationary
in the future
respond
the slower
for
shown in figure
inflation
agents
the money authority's
Optimal Disinflation With Partial Credibility
Following
based
that
block
The welfare
when the monetary
by private
5.
lower
accelerates.
considerably
Thus,
emphasize
period
teT',
the actual
the magnitude
of
i firms
set
beginning-of-period
the money supply
Households
type
will
make their
grow at
money supply
the constant
time t decisions
the time t transfer
16
new prices
Ht.
Thus,
M:
rate
after
they
make their
decisions
based
money supply
MI+i and their
the constant
rate
=
rate
of
g,-,
If
constant
7,
also
then
(13)
converge
from those
prices
their
(131
the expected
by the fraction
(p of
of money growth
during
more rapidly
to observed
can be characterized
is defined
in section
that
of
eventually
rate
as more
converges
of
to some
money growth
as a collection
and households
of firms
3, however.
time
teT'
on their
will
of
behave
gt
expand
type
sequences
optimally
and households
Since
based
the money supply
replaced
=
Thus,
rates
the expected
decisions
is
P,
that
The optimal
described
(12)
money growth
that.firms
at the beginning
equation
rate
to
t=0,1,2,...,
g.
adjust
policy
such
expectation
rate
so that
an equilibrium
clear.
grow at
to 7.
and quantities
markets
expectations
implies
will
according
(pe(O,l),
and expected
of
end-of-period
the money supply
downward in each period
the actual
the actual
As before,
prices
inflation
between
the actual
gt evolves
by the initial
adjusts
of
forward.
)-gtml 1 s
in money growth,
credible.
that
of money growth
I As (p increases,
the period.
changes
expectation
+ (P[ (M;+/$
money growth
the difference
will
rate
is given
where g-,
knowledge
gt from time t+l
The expected
gt
on their
and
now differ
i firms
knowledge
forever
for
set
new
of M: and
at rate
gt 1,
by
wztml (l+Pg t-1+P'g: - 1+P3gf-1 MSt
t=1,2,3,....
,
(14)
413( 1+P+P2+B3I
Note that
which
lack
for
p, now depends
may differ
credibility.
t=1,2,3,...
on firms'
from the actual
Individual
and by initial
expected
rate
goods
rate
of money growth
of money growth
prices
conditions
17
when announcements
are determined
for
t=O.
gtei,
by equation
(4)
Households
MI+l
and their
rate
gt.
make their
expectation
Thus,
continue
decisions
while
that
t based
the money supply
equilibrium
to be described
at time
labor
will
supplies,
by equations
(9)
on their
expand
knowledge
forever
consumptions,
and (IO),
nominal
of
at
and wages
interest
rates
are now
=
Rt
Since
the current
future
which
interest
money growth,
may differ
(15)
are not
fully
initial
path
when the
sequence
is defined
prices
of
during
by gt,
time t.
equilibrium
prices
M:=l and
{Mr+l}F=o when policy
As in the case
household's
initial
express
conditions
expectations
Rt is determined
of money growth
credible.
the representative
that
and (131-(15)
the
(15)
on households'
indicates
rate
(IO),
disinflationary
constraints
depends
and the money supply
announcements
maximizes
(91,
in terms of
{p,,.p,o,p,,,p,o~
the optimal
equation
(41,
and quantities
rate
from the actual
Equations
t=0,1,2,....
g/is,
of
full
as the sequence
utility
subject
come from a steady
credibility,
{M:+llT-o
that
to these
state
with
positive
inflation.
Figure
Fischer
occurs
(19861,
calls
it
is
unanticipated
of
credible,
a sharp
expectations
back,
but
negative.
monetary
is
Output
falls
contraction
burst
the optimal
begin
still
path when,
the initial
contraction
can be read directly
bounces
remains
disinflationary
Instead
fully
for
inflationary
expectations
growth
(p=O.125.
when policy
credibility
Thus,
3 shows the optimal
off
of
slower
the
policy
at once;
interest
with
since
rate
than expected,
by almost
and remains
18
of money growth
below
its
that
at t=O.
Rt=gt/P,
series.
at t=l;
1.4 percent
by
partial
the money supply
to ease
of
as suggested
these
Money
thereafter,
in response
initial
level
to the
for
Eventually,
two full
years.
inflation
tax
that
between
lies
While
the Friedman
a comparison
a big
difference
for
table
2 indicates
those
from the full
percent
consumption
these
percent
gain
alternative
inflation
using
figures,
under
full
rule
Thus,
of
continue
yields
authority
lacks
that
long-run
benefits
run costs.
practice,
Cooley
in magnitude
revenues.
Cooley
cash-in-advance
of
to disinflationary
other
other
money growth
however;
19
when the
here
exceed
reason
taxes
their
show
the short-
why,
in
taxes
to replace
analysis
the next
section
presents
in the model with
setting.
terms.
to be welfare-improving:
distortionary
distortionary
or the
(19931,
still
and Hansen conduct
model,
the
the results
another
might fail
policy
to the 0.0223
and employment
disinflation
of
when q=O.750.
and Goodfriend
suggest
percent
Moreover,
to zero
Nevertheless,
(19911
seignorage
1.
a 4
0.0184
consumption
in output
resemble
when policy
to the optimum in welfare
a quick
to increase
challenge
similar
credibility.
policies
and employment,
easier
equals
switching
may have
the presence
policy
immediately
of
is
of
losses
makes
credibility
percent
(1986)
state
credibility
from eliminating
shown in table
government
whether
the gain
credibility
and Hansen
disinflationary
flexible-price
instance,
significant
that
partial
Disinflation
are
as the
to a new steady
on output
under
the optimal
levels
money growth.
case.
by Sargent
monetary
the
gains
to come close
as suggested
converges
of disinflation
however,
higher
1 and 3 reveals
the welfare
for
reaches
and zero
figures
credibility
policies
disinflation
rule
the effects
that
again
when (p=O.125 and 0.0214
Both of
Friedman
of
so that,
annual
output
Money growth
recedes.
is more credible
however,
lost
in a
considers
a serious
staggered
the
price
6.
Disinflation With Other Distortionary Taxes
Suppose
government
now that,
levies
the proceeds
each
period
in addition
a flat-rate
to households
to controlling
tax tt on workers'
in the form of
labor
income,
household's
the
returning
a lump-sum transfer
The representative
t=0,1,2,....
the money supply,
at
budget
the end of
constraints
become
4
4
Bt + Mt + Tt + (l-tt)wt~nIt
+ 1 nit
I=1
4
L cP&t+Mt+l
+Bt+l/Rtp
l=l
(16)
1=1
t=0,1,2
In equation
revenues
(161,
as well
{T Ia0 is
t t=o
taken
in equilibrium
the total
lump-sum transfer
as the newly
as given
created
Tt includes
money Ht; that
by the representative
by the government's
budget
is,
income
while
household,
it
)....
tax
the sequence
is determined
constraints
4
=
Tt
t=0,1,2,....
Ht + ttwt c nit,
(17)
i=l
Cooley
to establish
for
all
and Hansen
two results.
t=0,1,2
exceed
(1991)
those
First,
, - * * , they
when z=O.
Thus,
cost
the welfare
transfers
Tt/Pt
improving
when ~~ must adjust
constant,
They conclude
that
holding
show that
increases
(17).
use a flexible-price
of
they
the tax
the benefits
the presence
inflation.
show that
to satisfy
disinflation
cash-in-advance
of
rate
of
20
holding
the real
ceases
the government
tax
value
to be welfare-
budget
desirable
tt=~
when 00
distortionary
disinflation
is no longer
with
disinflation
another
Second,
constant,
model
constraints
when the
of
.
government
must use other
distortionary
taxes
to replace
lost
seignorage
revenues.
Following
the
income
Cooley
tax rate
economy begins
policy
is
announcements
money growth
When t0=0.23,
policy
benefits
of
to zero
0.448
of
government
beginning
of
hold
transfers
money growth
both
to zero
t=O, but
policy
of
table
both
rule
and monetary
When t"=O,
are
that
The
from immediately
there
is no
adopting
the
the same as in table
policies
increase
annual
inflation,
a welfare
gain
the Friedman
rule
switching
of
0.231
yields
percent
a gain
of
equal
the economy begins
money growth
increases
Tt/Pt
3,
to yield
annual
inflations.
disinflationary
income
The table
continues
inflation
or the Friedman
the labor
fixed.
in a steady
rule
g>l.
when t'=O,
welfare
gain
The Friedman
rule,
starting
from a 4 percent
policies
yield
sizable
The
at the
tax tt as necessary
shows that
a small
rate
state
to
the zero
starting
from
however,
inflation.
losses
in welfare
when t0=0.23.
Thus,
findings
1.
and Hansen to match the US data,
to and a constant
to be welfare-improving
Moreover,
3 assumes
t=0,1,2,....
inflation,
gains
yields
adopting
all
table
credible.
disinflationary
tax rate
4 and 10 percent
ceases
both
part
labor
switches
real
fully
for
of
consumption.
In the second
a constant
positive
by Cooley
money growth
Immediately
with
with
from a 4 percent
consumption.
percent
rt=~'
again
chosen
part
with
and the Friedman
Starting
immediately
the first
tax and the welfare
the value
dramatically.
state
are once
labor
the welfare
constant,
in a steady
distortionary
zero
and Hansen,
the results
by showing
that
displayed
in table
the welfare
3 confirm
consequences
21
Cooley
and Hansen's
of disinflation
depend
to
critically
on how the government
revenues.
On the one hand,
the government
On the other
cuts
hand,
spending
be replaced
through
7.
so that
higher
the income
ceases
so that
income
loss
becomes
often
fixed,
the
of
its
seignorage
much more attractive
tax does
not have
when
to rise.
to be welfare-improving
all
of
the
lost
when
seignorage
must
taxes.
Conclusion
Inflation
the past
has averaged
decade.
principal
the best
flexible-price
Cooley
Since
long-run
What is
that
(1989)
Danziger
(19881,
rate
there
run?
equal
rate
price
for
of
answer
should
to zero,
are nominal
(1994)
lists
of Greenwood
a simple
Previous
and Ball
may then differ
models
in the United
question
of money growth
interest
in the short
Reserve
the average
provide
the rate
annually
a central
way to reduce
But what if
costly
the Federal
cash-in-advance
make the nominal
(1969).
4 percent
objectives,
and Hansen
indicating
path
spending
is held
with
disinflation
disinflation
government
copes
considerably
monetary
that
policy
The
and Huffman
(1987)
question
as called
and
by
to
by Friedman
make disinflation
(19791,
Taylor
the optimal
from an immediate
is:
immediately
for
that
during
among its
inflation?
be reduced
work by Phelps
suggests
stability
to this
rigidities
States
(1983).
disinflationary
switch
to the Friedman
rule.
None of
with
these
an incentive
positive
inflation
earlier
models
to economize
tax,
however.
with
on their
nominal
cash
balances
The model of
22
rigidities
provides
in the face
staggered
price
agents
of
setting
a
developed
here
shows that
inflation
on money demand is
disinflation.
optimal
disinflationary
in output
zero
path
models,
yields
following
the optimal
in output
announcements
determinant
Sargent's
are not fully
the effects
(1986)
studies
analysis
of
disinflation
long-run
benefits
consequences
of
reducing
tax
of
the results
increases
as that
price
obtained
from
Thatcher
exceed
money growth
as a chief
corroborate
hyperinflations
1979.
policy
credibility
the results
as well
still
in significant
authority's
By isolating
the European
support
Cooley
distortionary
taxes
inflation.
of
the
as Goodfriend's
1920's
and
(1993)
In the model,
the short-run
however,
costs.
and the Friedman
by increasing
that
reducing
and Hansen's
the
Even
rule
has major
(1991)
effects
The model shows that
to be welfare-improving
indicate
to
continue
policies.
other
that
credible.
zero
of
results
of
no
by flexible-price
results
when the monetary
of disinflation
revenues
as large
disinflation
in the US since
the presence
seignorage
that
under
to be welfare-improving
fails
is almost
with
switching
goal
for
involve
the
one year
Reserve's
as called
of disinflation,
credibility,
Finally,
rule,
within
of
does
inflation,
immediately
the Federal
that
and employment
experience
often
gain
of
partial
with
indicates
the British
with
In addition,
policy
annual
inflation
effect
path.
The model also
losses
from a 4 percent
the Friedman
a welfare
distortionary
the optimal
for,
eliminates
consistent
or adopting
traditional
accounted
and employment.
money growth,
stability,
this
Starting
a rapid
losses
once
when the government
the tax on labor
the government's
are not necessary
is
23
crucial
that
on the welfare
disinflation
must replace
income.
revenue
finding
Thus,
the
requirements
to the success
of
lost
any
so
disinflationary
policy.
Aiyagari
price
stability
Aiyagari's
leather
other
hand,
that
tax
requires
increases
discussed
income
variability
with
the US currency
consideration
of
losses
authority
required
the model does
by Aiyagari,
the
level
supply
these
of
lacks
issues
addresses
inflation,
of
future
24
seignorage
of
the other
of
overseas.
provided
lost
greater
research.
On the
The model
the costs,
and the fact
to be held
constraint.
several
lack
some of
and employment,
to make up for
the
Reserve's
the shoe
credibility.
not capture
awaits
the Federal
reduces
exceed
association
appears
here
in output
including
the possible
of
cash-in-advance
of disinflation
are not
tax,
critique
disinflation
a binding
short-run
the benefits
However,
revenues.
capital
with
when the monetary
indicates
elements
On the one hand,
associated
it
a detailed
The model developed
objective.
costs
income
provides
concerns.
particularly
of
(19901
indexation
that
in the US
inflation
that
a large
A full
fraction
Notes
'These
for
figures
all
refer
urban
consumers,
President (1994,
2Black
(1990)
Reserve's
Lucas
good
focus
i.
list
and the prices
will
of
of
the consumer
price
index
in the Economic Report of the
is
closest
(19921,
be accounted
stability
in spirit
as the Federal
of
the other
three
price
however,
conditions.
25
developed
and Stockman
demand for
suppresses
i firms'
for,
to those
and Ohanian
household's
used here
type
price
objective.
the representative
on the effects
dependence
long-run
The notation
rate
as reported
(1991)
Cho and Cooley
income
growth
B-59).
the model
respect,
41n general,
of
CPI-U,
and Hoskins
(19881,
on its
Table
principal
31n this
to the annual
goods
good
(1994).
i will
as well
by
depend
as the price
this
dependence
in order
setting
decisions.
The
in deriving
equilibrium
to
References
Aiyagari,
Reserve
Ball,
1990,
S.R.,
Bank of Minneapolis
L.,
American
Black,
1994,
Credible
Economic
1990,
R.P.,
Blanchard,
O.J.,
R. Dornbusch
Press,
1983,
84,
and T.F.
Manuscript
(Queens
76.
of
Price
with
14,
inflation,
Federal
2-11.
staggered
price
stability,
asynchronization
price-setting,
Federal
Reserve
Bank of
eds.,
and price
Inflation,
debt,
level
inertia,
and indexation
in:
(MIT
MA).
Cooley,
1992,
University,
Hansen,
model,
American
Economic
T.F.
and G.D.
Hansen,
Journal
1988,
of Money,
Danziger,
L.,
Costs
inflation
and disinflation,
The business
Kingston,
and G.D.
inflations,
Review
zero
3-6.
T.F.
Cooley,
for
282-289.
and M.H. Simonsen,
J.O.
cycle
Quarterly
In support
Review
Cambridge,
Cooley,
the case
disinflation
Review
Richmond Economic
Cho,
Deflating
1989,
1991,
79,
tax
costs
and Banking
adjustment
American
Economic
26
nominal
contracts,
in a real
business
733-748.
The welfare
Credit,
with
Ontario).
The inflation
Review
of price
cycle
23,
of
moderate
483-503.
and the welfare
Review
78,
economics
633-646.
of
Economic
Report
Office,
of
Washington,
Fischer,
inflation,
Contracts,
and economic
Friedman,
M.,
1969,
1979-1992,
Federal
Greenwood,
J.
inflation
Interest
Reserve
Reserve
1991,
1990,
Rochester,
Lucas,
1988,
D.,
contracts,
Management,
rate
Printing
money,
in:
and the
A dynamic
inflation:
Quarterly
Review
cycles,
The optimum quantity
inflation
Quarterly
Economics
15,
for
IL).
scare
equilibrium
all
Indexing,
MA).
Company, Chicago,
of Monetary
zero
in:
79,
problem:
l-24.
model of
19,
naught,
203-228.
Federal
16-20.
Manuscript
(University
of
NY).
Price
Working
policy
1987,
Money and business
Rochester,
of
Publishing
Journal
Defending
Bank of Minneapolis
R.G.,
Government
Cambridge,
Bank of Richmond Economic
and G.W. Huffman,
W.L.,
States
and disinflation,
(MIT Press,
(Aldine
and unemployment,
Hoskins,
King,
policy
essays
1993,
M.,
credibility,
The optimum quantity
of money and other
Goodfriend,
(United
DC).
1986,
S.,
1994,
the President,
and interest
Paper
Department
rate
6 (Northwestern
of Finance,
dynamics
University,
Evanston,
27
IL).
induced
by multiperiod
Kellogg
School
of
1987,
Mankiw, N.G.,
annual
1987
Ohanian,
prices
(MIT Press,
L.E.
are
Comment, in:
and A.C.
sticky,
Cambridge,
Stockman,
Federal
Reserve
S. Fischer,
ed.,
NBER macroeconomics
MA).
1994,
Short-run
effects
of
money when some
Bank of Richmond Economic
Quarterly,
forthcoming.
Phelps,
monetary
York,
1979,
E.S.,
policy,
in:
Disinflation
Studies
without
recession:
in macroeconomic
adaptive
theory
(Academic
guideposts
Press,
and
New
NY).
Sargent,
T.J.,
New York,
Taylor,
Economic
1986,
Rational
expectations
and inflation
(Harper
and Row,
NY).
J.B.,
1983,
Review
73,
Union wage settlements
981-993.
28
during
a disinflation,
American
Table
1 .--Welfare
Gain from Monetary
Policy
Initial
Annual
Inflation
Rate
Monetary
Policy
Optimal Path
Zero Money Growth
Friedman Rule
4
Percent
10
Percent
0.0223
0.0196
0.0874
0.0831
0.0174
0.0803
Figures
refer
to the permanent percentage
increase
in
consumption
that makes the representative
household
as well off
under continuing
inflation
at the initial
rate as it is under
the optimal policy,
zero money growth, or the Friedman rule.
Notes:
Table
2. --Welfare
Gain from Monetary
Credibility
Policy
With Partial
Initial
Annual
Inflation
Rate
Monetary
Policy
4
Percent
10
Percent
0.0184
0.0171
0.0136
0.0749
0.0739
0.0694
0.0202
0.0185
0.0159
0.0809
0.0793
0.0759
0.0211
0.0192
0.0170
0.0839
0.0820
0.0791
0.0214
0.0195
0.0173
0.0849
0.0828
0.0800
Q=O.l25
Optimal Path
Zero Money Growth
Friedman Rule
Q=O.250
Optimal Path
Zero Money Growth
Friedman Rule
Q=O.500
Optimal Path
Zero Money Growth
Friedman Rule
Q=O.750
Optimal Path
Zero Money Growth
Friedman Rule
Notes:
See notes
to table
1.
Table
3. --Welfare
Gain from Monetary Policy
Labor Taxation
With Distortionary
Labor Tax Rate Constant
Initial
Annual
Inflation
Rate
Monetary
Policy
4
Percent
10
Percent
0.0196
0.0174
0.0831
0.0803
0.231
0.448
0.595
0.812
TOSO.
00
Zero Money Growth
Friedman Rule
t0=0.23
Zero Money Growth
Friedman Rule
Government
Spending
Constant
Initial
Annual
Inflation
Rate
Monetary
4
Percent
Policy
10
Percent
2O=o. 00
Zero Money Growth
Friedman Rule
0.00520
-0.00212
0.0307
0.0220
r0=0.23
Zero Money Growth
Friedman Rule
Notes:
See notes
-0.112
-0.243
to table
1.
-0.298
-0.456
(Money Growth 1
0’11”1”‘11”1’111111”“111”1’11’1’~11’
’
-4
0
4
8
12
16
20
24
28
32
36
20
24
28
32
36
t
j Inflation 1
1.025
1.5
1.02
1
E8
G
9.
1.015
0.5
1.01
0
1.005
-0.5
1
0.995
-1
-4
0
4
8
12
16
20
24
28
32
36
L
-4
0
4
8
12
16
t
t
Fig. 1. Optimal Monetary Policy
-4
0
4
8
12
16
20
24
28
32
36
O.B"~~~~'~~~.~~.,~~"~."'~~~,.~'~"~'~~'~'~
-4
0
4
8
12 16 20 24 28 32
t
t
1Inflation 1
'
36
I
) output
1
1.5
1
2
8
G
a
0.5
0
-0.5
-1
-4
0
4
8
12
16
20
24
28
32
36
-4
0
4
8
12
16
t
t
Fig. 2. Zero Money Growth Policy
20
24
28
32
36
1Interest Rate 1
-1.5'
-4
0
4
8
12
16
20
24
28
32
36
32
36
~~~~~~~~~~~~~~~~~~'~~~~~~~~'~~'~~~'~~~~~~
-4
0
4
8
12
16 20 24 28 32
36
-4
0
4
8
12
16
20
24
28
t
t
1.02
’
-4
0
4
8
12
16
t
20
24
28
32
36
0.98
t
Fig. 3. Optimal Monetary Policy With Partial Credibility, phi=0.125
@FedFRASER