Topics in
Macroeconomic
Policy,
Time Inconsistency
in Monetary and
Fiscal Policies

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Central Bank’s Object:
which yields


1
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u
Note that this solution deviates from
agents’ expectation,or equivalently,
central bank’s ex-ante commitment,This
is the time inconsistent problem found by
Kydland-Prescott (1977,JPE) and Barro-
Gordon (1983,JPE).
Time Inconsistency in
Monetary Policy
Agents’ Response,
Update Expectation
Find Rational Expectation
(Fixed Point Problem):
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The Consequent Equilibrium
Inflation becomes
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Note that this time consistent
solution is inferior to the solution
with zero inflation commitment.
First-Best Allocation,government is
allowed to manipulate agents’
expectation to achieve zero
unemployment and inflation.
Cheating (Time-Inconsistent)
Solution,government would like to
deviate from her precommitment,
because she always has incentive
to reduce unemployment by
unexpected inflation.
Precommitment Solution,
government is assumed to commit
to her ex-ante announced policy.
Time Consistent Solution,
government is unable to commit to
her policy.
Welfare Comparison:
1 > 2 > 3 > 4
Four Solutions
Time Inconsistency in
Fiscal Policy
The following model,developed
by Fischer (1981,JEDC),is a
benchmark to start with.
Household’s Behavior
glccU,,,m a x 21
Subject to
zlRkcykc lk 1121
This yields households’ response function
glgkgcgc lklklklk,,,,,,,,21
A benevolent government would like
to maximize households’ welfare,
subject to
zlkg lk
Four Solutions
First-Best Allocation,maximize
household’s welfare subject to
resource constraint.
Cheating (Time-Inconsistent)
Solution,government would like to
deviate from her previous policy at
the second period when k has been
accumulated,because she always
has incentive to reduce distortion by
only taxing k.
Precommitment Solution,Open-
Loop Solution in the Stackelburgh
Game.
Time Consistent Solution,Subgame
Perfect Equilibrium.
Welfare Comparison:
1 > 2 > 3 > 4
What Causes Time
Inconsistency
Insufficient Policy Instruments,One
can easily find that there is no time
inconsistency in the second model if
the government is allowed to levy
lump-sum tax (or proportional income
tax?),However,this explanation seems
not satisfactory since it can’t explain
why time inconsistency rises in the
first model.
Persson and Tabellini (1994) argue
that TI is due to the sequential nature
of policy-making,This,again,is only
partially correct,
Now most economists believe that TI
is induced by the heterogeneity of
interests,or equivalently,some
externality,This is first pointed out by
Chari,Kehoe and Prescott (1989).
Atomistic Assumption
Without loss of generality,we can
write down a,semi-indirect utility
function” for the ith household:
,,Kkv ii
Precommitment Solution (the first
equation is the households’ behavior and
the second is the government’s choice):
00




i
ii
i
i vK
K
v
k
v

Time Consistent Solution (the first
equation is the households’ behavior and
the second is the government’s choice):
00 i
i
i
i v
k
v
Clearly the discrepancy lies in atomistic
agents,or similarly,the externality of
aggregate capital K.
Application for Monetary
Policy
One can easily find that time
inconsistency in our first model
lies in the Phillips Curve.
However,the microfoundation of
the Phillips Curve again lies in
the heterogeneity of interests,
everyone hopes others to
underforecast inflation.
So,the Phillips Curve and
consequently the time
inconsistency problem no longer
exist in a,communist world”.
To conclude,time inconsistency
seems to be a general outcome
from the world with
heterogeneous interests.
A Formal Model to Study
Time Inconsistent Monetary
Policy
Product Market:
Monopolistic Competition,Price = W/λ
One cash good and one credit good,μ
part of goods are determined at the begin
of each period,i.e,the price,Psi,is sticky
during one period,The flexible price is
denoted by Pfi.
Aggregate price is Pi,dPi/ dPfi = 1-μ.
Households:

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Given prices,households choose M,B and n.
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Clearly the Precommitment Solution follows
Friedman Rule,i.e.,R = 1,For Time Consistent
Solution,Government chooses G ex-ante,or
equivalently Pf,subject to (1) and (2).
We claim that for some condition,TC
Solution is equivalent to Precommitment
Solution,
Method,Evaluate marginal returns to Pf,
denoted by L,at R = 1 with tight CIA,
Check if R = 1 can be sustained by Pf = Pe.
For the right derivative L+,i.e,Pf > Pe,we have
C1 = 1/P1 from CIA constraint,M has been
normalized to 1,
/
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Differentiating (2) yields
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Here we use the following equation,

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dP
PPd f
f
f
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22 1/
since we evaluate derivative at Pe = Pf = P and P1 =
P2 = P,
For the left derivative L-,i.e,Pf < Pe,we have
C1P1 = C2P2 from (1),Differentiation yields
Substitute (4) and (5) into (3) yields
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Here we use C1=C2 in the equilibrium,So L+ <
0 if μ <?.

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Together with (5),we have
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So L- > 0,To conclude,if μ <?,R = 1 is TC
solution,i.e.,Precommitment Solution is
equivalent to Time Consistent Solution if price
stickiness is sufficiently small,This is a simple
version of Proposition 3.3 in Albanesi,Chari
and Christiano (2003).
Clearly time inconsistency arises in the
case of μ >?,Intuitively,high price
stickiness induces incentive of inflation,
the underlying mechanism is the same as
Phillips Curve.
Specifically,“A benefit of expansionary monetary
policy is that it leads to an increase in demand for
goods whose prices are fixed,This increase in
demand tends to raise employment.”Note that
inefficient low employment is due to monopolistic
competition.,A principal cost of EMP is that it
tends to reduce employment in the cash good
sector”,Finally,“It is possible that the reduction in
employment in the cash good sector is so large
that overall employment and welfare fall.” Quoted
from Albanesi,Chari and Christiano (2003,pp,
136).
Note 1,This result is sensitive to the
modeling of money demand.
Note 2,We only study the local optimum
(global optimum has been checked
numerically).
Note 3,Here Time Consistent Solution is
(implicitly) contingent upon Pf,so ACC
call the solution,Markov Equilibrium”,
This equilibrium is unique under some
Assume CD production technology,
with capital output elasticity θ and fixed
labor (normalized to 1),Public
spending is financed by capital income
tax:
A Formal Model to Study
Time Inconsistent Fiscal
Policy
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Given g and balanced budget,households’
Behavior is the following:



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To obtain analytical solution,here we
assume 100% depreciation,Solving this
problem recursively,we have

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By deduction,one can obtain:
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It is clear that future taxation affects current
saving decision,Specifically,at time t,
government takes into consideration the
impact of τt+1 upon kt+1,However,at time t+1,
optimal taxation plan would change since kt+1
is fixed and the impact of τt+1 disappear.
Maximizing value function (static problem?)
yields TC solution,Precommitment (Ramsey)
Solution can’t be solved analytically (bellman
problem?).
Next we consider proportional income tax:
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By the same procedure,one can obtain:
Note that by proportional income tax,there is
no distortion in marginal returns of capital.
By deduction,one can obtain:

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It is clear that future taxation doesn’t affect
current saving decision,So one can expect that
Ramsey solution is time consistent
This is a classical model,but the above result
was recently emphasized by Klein,Krusell and
Rios-Rull (2002/3).
To conclude,time consistent taxation plan is
contingent upon current state variable,capital k,
That is why KKR call it,Markov equilibrium”
(TC solution is also called close-loop solution.),
In the above simple model with proportional
income tax,time consistent solution or Markov
equilibrium turns out to be irrelevant to k,So it
is equivalent to precommitment (open-loop)
solution.
Finally,the result is sensitive to preference
assumption,If utility is not logarithm,TC
solution is not equal to precommitment solution
since income effect can’t be offset by
substitution effect.
To Be Better Off,Some
Illustrations
Trigger Strategy,By the Folk
Theorem,the best outcome can be an
equilibrium if the discount rate is
sufficiently low in a infinite horizon
model (how about finite horizon?).
A possible strategy:
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One can check that if F and β are sufficiently
large,zero inflation will be a subgame
perfect equilibrium,
Note the concept of perfection,you not only
need to check deviations from the
equilibrium path,but also need to check zero
inflation is the best response of government
along off-equilibrium path,
Weakness,How to select an equilibrium
from so many possible candidates? This is a
typical problem in game theory.
Contract Approach
Government construct a contract to punish
central bank for inflation

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Time Consistent Solution is
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Clearly if government sets μ = -u,zero expected
inflation can be achieved,which is the first-best
outcome,This result was first pointed out by
Walsh (1995,AER),
Has the time inconsistency problem been killed
by this simple incentive contract? The answer is,
unfortunately,negative.
First note that the above result is crucially
dependent upon the linear Phillips Curve,Without
this ad-hoc assumption,it would be impossible to
derive any feasible incentive contract.
Second,contract implicitly removes time
inconsistency to another level,i.e,who insures the
government’s commitment to the contract?
Stories about Central Banker
A straightforward way to kill inflation is to
delegate central banker to a person with
infinite aversion to inflation,i.e,θ = ∞.
If there is no such a person,we can
delegate central banker to a person with
higher θ (than social level),One can
easily show that this can moderate time
inconsistency problem,This approach
was pointed out by Rogoff (1985,QJE).
Clearly public expectation towards
inflation is closely related to the
preference of central banker,But the
problem is how can public know the
banker’s preference,and consequently
adjust expectation of inflation?
This problem was first answered by
Backus and Driffill (1985,AER),They
assume there are two kinds of
policymaker,one with θT = ∞ (tough
banker) and one with θW < ∞ (weak
banker),Let pt denote the expected
probability of tough banker,The
For simplicity,we assume two period,If
π1 > 0,clearly p2 becomes 1,If π1 = 0
and q1 is the probability that weak banker
strategically choose π1 = 0 (q1 will be
determined later),p2 and hence the
expected inflation evolve according to the
above Bayes’ Rule.
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Note that once q1 < 1,p2 > p1 from (1),From
(2),one can see the benefit of weak banker
pretending to be tough,i.e,reducing the
expected inflation,Of course,the cost to play
the strategy is the loss caused by the deviation
from his best choice at the first period.
Result,If β is sufficiently large,pooling
equilibrium (q1 = 1),If β is sufficiently small,
separate equilibrium (q1 = 1),Otherwise,the
weak banker plays mixed strategy (0 < q1 < 1).
Bessatto (2002),
Multiple Equilibria
with Commitment