Topics in
Macroeconomic
Policy:
Optimal Monetary
and Fiscal Policies
A Standard Framework
To Study Monetary Policy
Monopolistic Competition:
attatttt aanay 1
Production Technology:
vapwmc tttt
Government Spending [assume GS is a
fraction τ of output and hence g = -ln(1-τ)]:
gttgtttt gggcy 1
Household Behavior:
tttttttt
t
t
t
t
DCPArNWA
tsNC
11
0
1
1
..
1
lnm a x
By some approximations,we also have
ln11 tttttt cEErc
The supply of labor satisfies:
tttt ncpw
From the labor supply equation,we have
The intertemporal equation yields
Finally,assume money demand as
)1(1 vgaymc tttt
tttt rypm
)3(ttt gay
)2(111 tgtttttt gyEEry
Assume flexible price,from (1) and (2)
one can see that money policy is neutral:
Note,(3) implies first best allocation
can be achieved by setting v = μ,
Moreover,(4) indicates optimal policy
rule of real interest rate (mechanism),
Finally,(3) and (4) show this is a model
with prominent Keynesian properties.
)4(11 tgtat gar
1/1,/v? where
Staggering Price
Assume each firm resets price in any
period only with probability 1-θ,So
the evolution of aggregate price can
be approximated by the following
)5(*11 ttt ppp
Note that this is different from simple
sticky price assumption,Firm solves the
following programming to set pt*
ts sss ICIR?m a x
Subject to (5) and pt = pt*,With the static
price rule p = mc + μ,we obtain
)6(1* 0 n sttsst mcEp
Here mcn denotes nominal marginal cost,
From (1) and (3),we know
ttttt yyxxmc 1
Then from (5) and (6),one can obtain a
New Phillips curve
)7(1 tttt xE
κ is a positive parameter (?),From (2)
and (4),one can further obtain
)8(11 ttttttt xErErx?
(7) and (8) characterize the dynamics of
the model,
Note that there are 4 distortions in the model,
First,money-holding cost (Friedman Rule),We
ignore this because of the ad hoc money
demand assumption,Second,static distortion
from imperfect competition (solved by the
subsidy v),The Third is firms’ inability to
adjust prices and the Fourth is the relative
price distortion (due to the lack of
synchronization in pricing),which induces
allocation inefficiency,Intuitively,the optimal
monetary policy requires xt = πt = 0 to eliminate
the last two distortions (mechanisms).
The Taylor Rule
Taylor (1993) finds U.S,monetary policy
follows a simple rule (why not M?):
ttt xr 5.05.1c o n
Generally the Taylor Rule is not optimal,
compared with (4),So why we need it? The
answer is positive since xt = πt = 0 is not
implementable by the optimal rule (4),To see
this,substituting (4) into (8),together with
(7),one can obtain:
1
111
tt
tt
t
t
E
xEx
Clearly xt = πt = 0 is a solution,Uniqueness
requires both of the eigenvalues of the
coefficient matrix lie in the unit cycle,to
eliminate bubble solutions (see Yuan and
Song,pp,67-70 for a non-technical
explanation),However,one can verify that
only the smaller one lies in [0,1],Hence,the
optimal rule (4) can’t implement xt = πt = 0.
Use a specific Taylor Rule can avoid the
indeterminacy
)9(txttt xrr
Then one can verify (?) that uniqueness
requires the following condition
)10(011 x
However,many economists argue that (9)
itself is not practicable since central bank
doesn’t have sufficient knowledge about
equilibrium real interest rate (why?),One
can instead assume a simple Taylor Rule,
)11(txtt xr
It is straightforward to see (10) ensures
uniqueness,But there are many other
possible rules to avoid indeterminacy,So
again we have to answer why we need
Taylor Rule? Gali (2003) compares
Taylor Rule with two other candidates,
Money Growth Peg and Interest Rate Peg
(with implementation rule).
Gali finds that Taylor Rule is superior to
Money Growth Peg and Interest Rate Peg,
Specifically,under some reasonable
parameters,the volatility of inflation and
output gap is close to zero (0.21 and 0.16)
and the welfare loss is impressively close
to zero (0.002%),substantially lower than
MPG (2.30,0.98 and 0.22%) and IRP (2.31,
0.99 and 0.22%),
1) Taylor Rule overwhelming MGP and
IRP is not a robust result,One can see if the
dynamics is characterized by more than
two variables,i.e,xt and πt,it is possible
that Taylor Rule performs worse than some
simpler policy (e.g,capital - CKM?).
2) There is no inflation-output trade-off in
this model (since v = μ),Consequently
there is no time inconsistency problem
(why?),which seems unpleasant,So one
can assume (why?)
ttttt uxE 1
Some Related Researches
Benhabib,Schmitt-Grohe and Uribe (2001,AER,
2002,JET) find that Taylor Rules are possible to
generate multiple equilibria (indeterminacy and
bifurcation) and hence lead to high inflation,The
mechanism lies in the active response of interest
rate to inflation,i.e,фπ > 1 in (9),
SU (2003,JET) assume distortion from
monopolistic competition can’t be removed by
subsidy (with sticky price we have inflation
incentive to increase employment) and
government has no access to lump-sum tax
(hence inflation plays as a non-distortionary tax
to finance government spending),They also
assume sticky price and CIA (welfare loss of
inflation to firms and agents),Then the optimal
(with commitment) volatility of inflation is close
to zero (compared with Gali,2003).
ACC (2002,pp.30) find that lack of commitment,
together with CIA and sticky price,can generate
multiple equilibria (non-monotone cost function
of inflation – first decrease and then increase –
compared with ACC,2003,which assumes
different money demand and technology).
Some Indirect Evidences
Jordi Gali and his coauthors hope that the
above model could be a standard
framework to study monetary policy
(from both of the normative and positive
aspects),They provide some supportive
evidences to convince their colleagues,
First,small variance of money supply is
able to generate large volatility of output,
This wins support from many monetary
economists.
Second,it is possible to generate liquidity
trap (low 1/σ),This wins support from
many Keynesian economists.
Finally,it is consistent recent empirical
finding,positive technological shock
leads to decrease in employment,There
are many explanation to this interesting
phenomenon (example),Next we discuss
the explanation from Gali (1999,AER).
We only discuss the basic mechanism of
Gali (2003),Suppose a simple money
demand equation:
ttt ypm
Equilibrium requires:
)12(tttt napm
Given flexible price and fixed money
supply,at increases 1%,pt decreases 1%
and employment is irrelevant to
technology shock (check this is a special
case for logarithm utility and the relation
is positive for lower and more realistic
intertemporal substitution),
However,if assuming staggering price,
one can see that pt can’t decreases 1%
w.r.t,1% increase in at,So (12) implies nt
must decrease (if 1/σ > 1?).
Intuition,price becomes relatively higher
and hence the demand becomes relatively
lower,which suppress employment.
Optimal Fiscal Policy
Lucas Second Critique (2003,AER),
welfare loss from different monetary
policies is generally very small,
often less than 1% (w.r.t,
consumption),However,the impact
of different fiscal policies to the
welfare is generally much larger,
often more than 10% (w.r.t,
consumption).
Precommitment Solution of Optimal
Taxation,Need some techniques
(see Lucas,1990,Chari,Kehoe and
Christiano,1995,Chari and Kehoe,
1999),Early researches
(Chamberlin,1986,Judd,1985,
Jones,Manuelli and Ross,1993)
simply prove a very straightforward
result,Recent researches
incorporate business cycle and
some other factors.
Macroeconomic
Policy:
Optimal Monetary
and Fiscal Policies
A Standard Framework
To Study Monetary Policy
Monopolistic Competition:
attatttt aanay 1
Production Technology:
vapwmc tttt
Government Spending [assume GS is a
fraction τ of output and hence g = -ln(1-τ)]:
gttgtttt gggcy 1
Household Behavior:
tttttttt
t
t
t
t
DCPArNWA
tsNC
11
0
1
1
..
1
lnm a x
By some approximations,we also have
ln11 tttttt cEErc
The supply of labor satisfies:
tttt ncpw
From the labor supply equation,we have
The intertemporal equation yields
Finally,assume money demand as
)1(1 vgaymc tttt
tttt rypm
)3(ttt gay
)2(111 tgtttttt gyEEry
Assume flexible price,from (1) and (2)
one can see that money policy is neutral:
Note,(3) implies first best allocation
can be achieved by setting v = μ,
Moreover,(4) indicates optimal policy
rule of real interest rate (mechanism),
Finally,(3) and (4) show this is a model
with prominent Keynesian properties.
)4(11 tgtat gar
1/1,/v? where
Staggering Price
Assume each firm resets price in any
period only with probability 1-θ,So
the evolution of aggregate price can
be approximated by the following
)5(*11 ttt ppp
Note that this is different from simple
sticky price assumption,Firm solves the
following programming to set pt*
ts sss ICIR?m a x
Subject to (5) and pt = pt*,With the static
price rule p = mc + μ,we obtain
)6(1* 0 n sttsst mcEp
Here mcn denotes nominal marginal cost,
From (1) and (3),we know
ttttt yyxxmc 1
Then from (5) and (6),one can obtain a
New Phillips curve
)7(1 tttt xE
κ is a positive parameter (?),From (2)
and (4),one can further obtain
)8(11 ttttttt xErErx?
(7) and (8) characterize the dynamics of
the model,
Note that there are 4 distortions in the model,
First,money-holding cost (Friedman Rule),We
ignore this because of the ad hoc money
demand assumption,Second,static distortion
from imperfect competition (solved by the
subsidy v),The Third is firms’ inability to
adjust prices and the Fourth is the relative
price distortion (due to the lack of
synchronization in pricing),which induces
allocation inefficiency,Intuitively,the optimal
monetary policy requires xt = πt = 0 to eliminate
the last two distortions (mechanisms).
The Taylor Rule
Taylor (1993) finds U.S,monetary policy
follows a simple rule (why not M?):
ttt xr 5.05.1c o n
Generally the Taylor Rule is not optimal,
compared with (4),So why we need it? The
answer is positive since xt = πt = 0 is not
implementable by the optimal rule (4),To see
this,substituting (4) into (8),together with
(7),one can obtain:
1
111
tt
tt
t
t
E
xEx
Clearly xt = πt = 0 is a solution,Uniqueness
requires both of the eigenvalues of the
coefficient matrix lie in the unit cycle,to
eliminate bubble solutions (see Yuan and
Song,pp,67-70 for a non-technical
explanation),However,one can verify that
only the smaller one lies in [0,1],Hence,the
optimal rule (4) can’t implement xt = πt = 0.
Use a specific Taylor Rule can avoid the
indeterminacy
)9(txttt xrr
Then one can verify (?) that uniqueness
requires the following condition
)10(011 x
However,many economists argue that (9)
itself is not practicable since central bank
doesn’t have sufficient knowledge about
equilibrium real interest rate (why?),One
can instead assume a simple Taylor Rule,
)11(txtt xr
It is straightforward to see (10) ensures
uniqueness,But there are many other
possible rules to avoid indeterminacy,So
again we have to answer why we need
Taylor Rule? Gali (2003) compares
Taylor Rule with two other candidates,
Money Growth Peg and Interest Rate Peg
(with implementation rule).
Gali finds that Taylor Rule is superior to
Money Growth Peg and Interest Rate Peg,
Specifically,under some reasonable
parameters,the volatility of inflation and
output gap is close to zero (0.21 and 0.16)
and the welfare loss is impressively close
to zero (0.002%),substantially lower than
MPG (2.30,0.98 and 0.22%) and IRP (2.31,
0.99 and 0.22%),
1) Taylor Rule overwhelming MGP and
IRP is not a robust result,One can see if the
dynamics is characterized by more than
two variables,i.e,xt and πt,it is possible
that Taylor Rule performs worse than some
simpler policy (e.g,capital - CKM?).
2) There is no inflation-output trade-off in
this model (since v = μ),Consequently
there is no time inconsistency problem
(why?),which seems unpleasant,So one
can assume (why?)
ttttt uxE 1
Some Related Researches
Benhabib,Schmitt-Grohe and Uribe (2001,AER,
2002,JET) find that Taylor Rules are possible to
generate multiple equilibria (indeterminacy and
bifurcation) and hence lead to high inflation,The
mechanism lies in the active response of interest
rate to inflation,i.e,фπ > 1 in (9),
SU (2003,JET) assume distortion from
monopolistic competition can’t be removed by
subsidy (with sticky price we have inflation
incentive to increase employment) and
government has no access to lump-sum tax
(hence inflation plays as a non-distortionary tax
to finance government spending),They also
assume sticky price and CIA (welfare loss of
inflation to firms and agents),Then the optimal
(with commitment) volatility of inflation is close
to zero (compared with Gali,2003).
ACC (2002,pp.30) find that lack of commitment,
together with CIA and sticky price,can generate
multiple equilibria (non-monotone cost function
of inflation – first decrease and then increase –
compared with ACC,2003,which assumes
different money demand and technology).
Some Indirect Evidences
Jordi Gali and his coauthors hope that the
above model could be a standard
framework to study monetary policy
(from both of the normative and positive
aspects),They provide some supportive
evidences to convince their colleagues,
First,small variance of money supply is
able to generate large volatility of output,
This wins support from many monetary
economists.
Second,it is possible to generate liquidity
trap (low 1/σ),This wins support from
many Keynesian economists.
Finally,it is consistent recent empirical
finding,positive technological shock
leads to decrease in employment,There
are many explanation to this interesting
phenomenon (example),Next we discuss
the explanation from Gali (1999,AER).
We only discuss the basic mechanism of
Gali (2003),Suppose a simple money
demand equation:
ttt ypm
Equilibrium requires:
)12(tttt napm
Given flexible price and fixed money
supply,at increases 1%,pt decreases 1%
and employment is irrelevant to
technology shock (check this is a special
case for logarithm utility and the relation
is positive for lower and more realistic
intertemporal substitution),
However,if assuming staggering price,
one can see that pt can’t decreases 1%
w.r.t,1% increase in at,So (12) implies nt
must decrease (if 1/σ > 1?).
Intuition,price becomes relatively higher
and hence the demand becomes relatively
lower,which suppress employment.
Optimal Fiscal Policy
Lucas Second Critique (2003,AER),
welfare loss from different monetary
policies is generally very small,
often less than 1% (w.r.t,
consumption),However,the impact
of different fiscal policies to the
welfare is generally much larger,
often more than 10% (w.r.t,
consumption).
Precommitment Solution of Optimal
Taxation,Need some techniques
(see Lucas,1990,Chari,Kehoe and
Christiano,1995,Chari and Kehoe,
1999),Early researches
(Chamberlin,1986,Judd,1985,
Jones,Manuelli and Ross,1993)
simply prove a very straightforward
result,Recent researches
incorporate business cycle and
some other factors.
