GPEFM Macro II Part 1 Ramon Marimon 1
Credible policies and learning(mostly from Sargent’s
“Conquest of Amerincan Inflation”)
GPEFM, UPF Macro II, Part 1 Ramon Marimon Winter 2006
GPEFM Macro II Part 1 Ramon Marimon 2
A simple credibility problem
• A simple Phillips curve example
• Government’s preferences
)(* eUU ππθ −−=
( )2221),( παππ +−= Ur e
GPEFM Macro II Part 1 Ramon Marimon 3
Credibility problem
• Rational expectations equilibrium (REE)
• Gov’t best response
ππππ =eeU satisfying ),,(
ea?
a?*a?a?e p UB 2
2
2 11 )(
+++=π
GPEFM Macro II Part 1 Ramon Marimon 4
Credibility problem
• Nash equilibrium
• The Ramsey outcome solves
ππππ
ππ
== ee
e
B ii) and )( i)
:satisfying ),(
REE) (i.e., subject to ),( max πππππ =eer
GPEFM Macro II Part 1 Ramon Marimon 5
Credibility problem
• The –optimal-- Ramsey outcome is based on commitment and rational expectations
• The Nash equilibrium imposes rational expectations but does not require commitment
UU *RRe === 0;ππ
U; UUpp *NNe === *αθ
GPEFM Macro II Part 1 Ramon Marimon 6
Credibility problem & delegation
• Let the central banker choose• The ‘right’ mandate (McCallum)
• The ‘right’ central banker (Rogoff’s‘strategic delegation’)
00 ==⇒= RN ππθ
00 ==⇒= RN ππα
GPEFM Macro II Part 1 Ramon Marimon 7
New lessons for strategic delegation
• With uncertainty, inflation variability matters ⇒ tradeoff between price and output stability
• Clarida, Galí & Gertler (1999), Herrendorf & Lockwood (1997) and Svensson (1997): inflationary bias vanishes if
a ? 0 and 0ifnot but ,0 >==α
GPEFM Macro II Part 1 Ramon Marimon 8
Perceived and Actual Law of Motion
• Preferences
• Perceived inflation
• Actual inflation
πηππ ˆ sets CB ,ˆ +=
βπ =E
ηβπ += )(B
∑∞
=−
0),()1(
t
ett
t rE ππδδ
GPEFM Macro II Part 1 Ramon Marimon 9
Perceived and Actual Law of Motion
• Perceived Law of Motion → Actual Law of Motion
• PLM →ALM characterizes possible outcomes
• REE is a fixed point of this map
)(ββ B→
GPEFM Macro II Part 1 Ramon Marimon 10
Rationalizability
Agents understand B( ) (rationalizability)
• therefore their only consistent belief (rationalizable) is the Nash equilibrium
,)( ,
and )( , e
e
eeN
eeN
B
B
ππππ
ππππ
<>∀
><∀
GPEFM Macro II Part 1 Ramon Marimon 11
Basic learning
Agents use a learning (forecasting) rule & actual motion given by B( ).
i. Cournot’s forecasting
ii. Mean forecasting (Bray, 1982)
Nt
et
ett
e B ππηπππ →+== −− then ,)( 11
∑ −
==
1
01
t
n ntte ππ
GPEFM Macro II Part 1 Ramon Marimon 12
Basic learning
1)log(t)( ,)(
))((1
1
))((1
1noise, stationary variance)(finite and large for
))()(/1(
))(/1(
1
11
111
111
+=−≈
+−+
≈
+−+
=−
+−+=
−+=
+
++
−−−
−−−
tB
BEt
Bt
t
Bt
t
ee
tte
te
t
tte
te
te
te
tte
te
te
te
te
tte
te
d
edτππ
ηππ
ηππππ
ηππππ
ππππ
τττ
τπ
GPEFM Macro II Part 1 Ramon Marimon 13
E-stability
• Stochastic Approximation theory provides conditions guaranteeing that the stochastic difference equation (of the learning process) asymptotically convergence to a fixed point of B( ) if and only if the following ODE is locally stable at this point:
ττ ππτ
τπ eeBd
ed−= )(
GPEFM Macro II Part 1 Ramon Marimon 14
E-stability
• It follows that the mean learning process converges to the Nash equilibrium REE, since
01
11)( 2 <
+−=−′
αθπ NB
GPEFM Macro II Part 1 Ramon Marimon 15
Adaptive expectations
iii. Adaptive expectations (Friedman and Cagan)
parameter tracking'' theis )1,0( where
))((
)(
111
111
∈+−+=
−+=
−−−
−−−
ληππλππ
ππλππ
tte
te
te
te
te
tte
te
B
GPEFM Macro II Part 1 Ramon Marimon 16
Adaptive expectations
• Stochastic approximation theory provides conditions guaranteeing that there is a stationary distribution with positive mass in the fixed point of B( ).
N tomass the
all assigns where0, as
then on,distributi ergodic theis if and
π
λµ
µ
ππλ
λ
NN II ↓→
GPEFM Macro II Part 1 Ramon Marimon 17
Learning with Central Bank foresight
The policy maker understands agents follow a learning or forecasting rule.
• For example, Cournot forecasting• Now the CB problem becomes
intertemporal
),()1(0 1
1∑∞
= ++−
t ttt r ππδδ
GPEFM Macro II Part 1 Ramon Marimon 18
Learning with CB foresight• Recall, when it takes only current
expectations as given,
• Now, instead
*ea?
a?*a?a?e a?Up UB =+=
++N
11 and , )( 2
2
2 ππ
*
ta?a?a?
ta?a?a?*
a?a?
a?t
Ua?
p U
)1(
and
ˆ
C
11111
)1(22
2
22
2
22
δπ
ππδ
δδδ
δ
−=
++= +++−++++−
GPEFM Macro II Part 1 Ramon Marimon 19
Learning with CB foresight
• Similarly, with Adaptive Expectations, the CB problem takes the recursive form
• which results in a unique stationary REE solution
))1((),()1()( te
tte
te VrV πλλπδππδπ −++−=
)1/()1( A δλδδπ +−−= *a?U
GPEFM Macro II Part 1 Ramon Marimon 20
Induction and foresight
• With adaptive expectations,
• If the CB maintains an inflation target, then agents eventually form expectations consistent with the target (Induction Hypothesis, Cho and Matsui).
• A patient CB will behave almost as a Ramsey planner.
∑ −
= −−−=
1
11)1(
t
n ntn
te πλλπ
1 as ,0 A →→ δπ
GPEFM Macro II Part 1 Ramon Marimon 21
General beliefs
• Beliefs, however, can be fairly arbitrary
• Agent i forecast according to
),,...,,(),...,( 110010 −−− ≡= tttt UUsss ππ
)( ti
et sf=π
GPEFM Macro II Part 1 Ramon Marimon 22
Strategic forecasting and reputation
• Discontinuous beliefs
**
*
**
~ & 0 e.g.,
,1,...,0 if ~1,...,0 ,if
)(
U
ptn
tn psf
NR
n
nti
αθππππ
ππ
ππ
====
≠−=∃
−===
GPEFM Macro II Part 1 Ramon Marimon 23
Strategic forecasting and reputation
• Maintaining reputation through incentive constraints
)~()),(()1(
)(),()1()(**
****
πδππδ
πδππδπ
VBr
VrV
+−≥
+−=
GPEFM Macro II Part 1 Ramon Marimon 24
Supporting Ramsey through reputation
31for satisfied IC then the1 if .,.
)1()1)(1( 1
)1(5.)),((
)1(5.),()(
5.),()(
212
122*
22*
2*
≥=
+++−≤
+−=
+−==
−==
−
−
δαθ
αθδαθδ
αθαππ
αθαπππ
απππ
ge
UBr
UrV
UrV
RR
NNN
RRR
GPEFM Macro II Part 1 Ramon Marimon 25
Beliefs and reputation
• The same argument can be applied to many inflation rates, even if the deviation is to zero inflation
• Why should agents coordinate beliefs this way?• Learning is an inductive process through common
experience.
** ~ and 0 e.g., UNN αθππππ ==<<
GPEFM Macro II Part 1 Ramon Marimon 26
Equilibria selection
• In a Cash-in-Advance Monetary Model (or in a OLG model) with seignorageand equilibrium indeterminacy:– Learning selects the ‘classical equilibrium’
as the ‘learnable’ one.– A fiscal constraint may help to stabilize
prices when agents learn, but not when REE is postulated.
– Experimental evidence supports the learning theory prescriptions.
GPEFM Macro II Part 1 Ramon Marimon 27
The ‘seignorage model’
• In a simple Cash-in-Advance Monetary Model (or an OLG model) with seignorageand equilibrium indeterminacy,
otherwise 0 ,0 if
)(
1
11
≥−
−===
+
++
et
et
et
dt
t
dt
b
bmmp
M
π
ππ
GPEFM Macro II Part 1 Ramon Marimon 28
The ‘seignorage model’
tet
d
et
d
t
ttdt
dt
st
dttt
st
st
dmm
dmm
MMpdMM
−=
+=⇒
=+=
+
−
−
)()(
,
1
1
1
ππ
π
π
GPEFM Macro II Part 1 Ramon Marimon 29
The ‘seignorage model’
• If d is sustainable there are – Two Stationary Rational Expectations
Equilibria
– and a continuum of non stationary REE, with long run inflation close to
1 , and ≥> LHHL ππππ
Hπ
GPEFM Macro II Part 1 Ramon Marimon 30
Some lessons from experimental data
• Long-run behavior consistent with adaptive learning predictions
• Short-run behavior not consistent with simple AL algorithms (e.g., OLS)
• ‘Sunspots’ do not arise from spontaneous coordination of beliefs, perhaps through ‘common experience with fluctuations’
• ‘Rational agents’ may opt for tracking observed behavior; a source of instability
GPEFM Macro II Part 1 Ramon Marimon 31
Misspecified policies and self-confirming equilibria
• Two canonical models in one
• For b ∈(0,1) it is an ‘asset price model’.• Alternatively, for b < 0 is a demand-supply
‘Cobweb model’
tett bpap υ++= +1
GPEFM Macro II Part 1 Ramon Marimon 32
The ‘cobweb model’
tett
ttett
ds
tet
s
ttd
bpap
eipp
pQ
pQ
υ
ηηγβδαβ
ηγδ
ηβα
++=
−+−−=
⇒=
++=
+−=
+
+−−
+
1
211
11
21
1
.,.)()(
GPEFM Macro II Part 1 Ramon Marimon 33
Rational Expectations
ttt
tt
tttt
ttet
ba
ba
bap
ba
pE
pbEapE
pEp
υυ +−
=+
−+=
−=
+=
=
−
+−−
++
11
1
i.e., ,
1
111
11
GPEFM Macro II Part 1 Ramon Marimon 34
From PLM to ALM revisited• PLM
• ALM
• E-stability
β=+etp 1
tt bap υβ ++=
1for stable-E)1()(
)(
<−+=−
+=
bbaT
baTβββ
ββ
GPEFM Macro II Part 1 Ramon Marimon 35
Misspecified beliefs• Adaptive expectations with ‘tracking’
• PLM
• The ‘correct’ perception (linear least square forecast) if the price follows the process
i.e., ),( 111 −−− −+= te
tte
te pppp λ
1)1(1 −−−
= tte p
Lp
λλ
11 )1( −− −−+= tttt pp ελε
te
t pp −=t where ε
GPEFM Macro II Part 1 Ramon Marimon 36
Misspecified beliefs• ALM
• Agent i misspecified belief istt
tt
LfpLb
Lb
ap
υλα
υλλ
λλ
);()1(1
)1(11
+=−−−
−−+
−=
it
it
it
i
t
Lgp
LL
p
ελ
ελ
);(
)1()1(1
=
−−−
=
GPEFM Macro II Part 1 Ramon Marimon 37
Misspecified beliefs equilibrium
• Agent i chooses the tracking that minimizes expected (squared)errors
• Equilibrium with misspecified (constant tracking) beliefs:
2
);();(
argmin)( i
+= iLg
LfEB
λλα
λλ
)( λλ B=
GPEFM Macro II Part 1 Ramon Marimon 38
Misspecified beliefs equilibrium
• The misspecification introduces a ‘unit root’ (i.e., a permanent effect of a shock).
• Given a shock, an equilibrium with misspecified beliefs has less short term effects and more long term effects than a REE equilibrium.
GPEFM Macro II Part 1 Ramon Marimon 39
Misspecification in the Phillips Curve problem with foresight
• Recall,
• Being a quadratic problem,))1((),()1()( t
ett
et
e VrV πλλπδππδπ −++−=
tt
tte
t
te
te
t
Lba
ba
baB
ηπλ
λ
ηππ
πππ
+−−
+=
++=
+==
−1)1(1
)(ˆ
GPEFM Macro II Part 1 Ramon Marimon 40
Misspecification in the PC model
• PLM
• ALM
gt
gt
gt
g
t
Lg
LL
ελπ
ελ
π
);(
)1()1(1
=
−−−
=
tt
gtt
LhLbLL
Lb
a
ηλαπ
ηλ
λπ
);()1(1
)1(11
+=−−−
−−+
−=
GPEFM Macro II Part 1 Ramon Marimon 41
Misspecification in the PC model
• Again, a constant term in ALM must be estimated ‘as a unit root’ in PLM
• An equilibrium with misspecified beliefs is a fixed point of
2
);();(
argmin)( i
+= iLg
LfEB
λλα
λλ
GPEFM Macro II Part 1 Ramon Marimon 42
Sargent’s conclusions
• In both models the misspecification results in long-run effects, in discrepancies with the true process at low frequencies (e.g., a non-zero long-run inflation response).
⇒It may take too long to detect the ‘misspecification’
• As expected, in the PC model the resulting inflation → 0 as δ → 1
GPEFM Macro II Part 1 Ramon Marimon 43
Self-confirming equilibria in the Phillips Curve model
• Recall the Phillips Curve model
• An old macro debate was whether to fit unemployment on inflation (Classical fit) or, as most CBs did, inflation on unemployment (Keynesian fit).
• Recall that, in general, agents beliefs are given by
ttt
ettt tUU
2
*
ˆ)( 1
υππυππθ
+=+−−=
$ ( )x f ztt=
GPEFM Macro II Part 1 Ramon Marimon 44
Classical fit of the PC• Government, given beliefs γ, solves
• Phillips Curve is satisfied• Agents have Rational Expectations• Orthogonality conditions underlying Gov’t
beliefs are also satisfied– Resulting in a policy
tt
tt
tt
tt
zfUts
UE
εγπγγ
πδ
+++=
+∑∞
=
)( ..
)(min
210
2
0
2
x h ztt= ( , )γ
GPEFM Macro II Part 1 Ramon Marimon 45
Self-confirming equilibria• Candidates for Self-Confirming equilibria
– Beliefs with non-degenerate parameters:
– Sargent’s special misspecified model:
– In this special case,
– i.e., there is no direct effect of today’s chosen inflation on future inflation
γ γ1 20 0≠ ≠,
γ γ2 10 0= ≠,
x h z htt= =( , ) ( )γ γ
GPEFM Macro II Part 1 Ramon Marimon 46
Self-confirming equilibria
• Self-confirming equilibria– Given the true model
– and the perceived one
– together with
– then
U U y f zt tt
t= − − +* ( ( ))θ υ1
U y f zt tt
t= + + +γ γ γ ε0 1 2 ( )
f z xtt( ) $=
γ γγ γ θ
2 1
2 1
0 00
= ≠⇒ = = −
,,
GPEFM Macro II Part 1 Ramon Marimon 47
Classical self-confirming eq.
• Estimating the (classical) Phillips curve
θπ
πγ
θσπ
σπ
σσθ
επγγ
−==
−=
=
+=
++=
)var(),cov(
),cov(
)var(
)var(
1
22
22
21
22
2
10
t
tt
tt
t
t
ttt
U
U
U
U
GPEFM Macro II Part 1 Ramon Marimon 48
Classical self-confirming eq.
• The CB solution is given by
*21010
*
21
10
)1(
then,1
ˆ
UU C
Ct
γγπγγ
πγγγ
π
+=⇒+=
≡+
−=
GPEFM Macro II Part 1 Ramon Marimon 49
Classical self-confirming eq.1. Given perceived parameters for the PC
(PLM), γ, the CB solves its max. problem.2. Agents forecast taken the policy of the CB,
π = h(γ) as given (Rational Expectations) 3. The (perhaps aumented) PC determines the
actual tradeoff and, therefore, theunemployment level
4. CB’s beliefs satisfy orthogonallity conditions
−+
==′θθ
γγγ
)1( )),((
2ChT
GPEFM Macro II Part 1 Ramon Marimon 50
Keynesian self-confirming eq.
• Estimating the (classical) Phillips curve
*
1
102
22
21
1
21
22
2
22
1
10
1 ,1
)var(),cov(
U
UU
U
k
kkk
t
tt
ttt
γγ
γσθ
σθγ
σσθθσπ
β
ηββπ
+−=
+−=
+−
==
++=
GPEFM Macro II Part 1 Ramon Marimon 51
Cho & Sargent Escape routes
GPEFM Macro II Part 1 Ramon Marimon 52
Cho & Sargent Escape routes
GPEFM Macro II Part 1 Ramon Marimon 53
Different perceptions
• But even in the US recent perceptions differ
US 1995-1999
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
3.5 4 4.5 5 5.5
Unemployment
Infla
tion
GPEFM Macro II Part 1 Ramon Marimon 54
Different perceptionsUS 1989-1993
0
0.5
1
1.5
2
2.5
3
4 4.5 5 5.5 6 6.5 7 7.5 8 8.5
Unemployment
Infl
atio
n
GPEFM Macro II Part 1 Ramon Marimon 55
More US perceptions (1951.1-2005.1)
USA 1951-2005 Jan
-2
0
2
4
6
8
10
12
14
16
0,0 2,0 4,0 6,0 8,0 10,0 12,0
Unemployment
Infl
atio
n
GPEFM Macro II Part 1 Ramon Marimon 56
US 1989-1993
USA 1989-1993
0
1
2
3
4
5
6
7
4,5 5,0 5,5 6,0 6,5 7,0 7,5 8,0
Unemployment
Infl
atio
n
USA 1995-1999
0
0,5
1
1,5
2
2,5
3
3,5
3,5 4,0 4,5 5,0 5,5 6,0
Unemployment
Infl
atio
n
USA 2000-2005 Jan
0
0,5
1
1,5
2
2,5
3
3,5
4
3,0 3,5 4,0 4,5 5,0 5,5 6,0 6,5
Unemployment
Infl
atio
n
US 1995-1999
US 2000-2005
GPEFM Macro II Part 1 Ramon Marimon 57
and EMU perceptions(1993-2004)
EMU 1993-2004
0
0,5
1
1,5
2
2,5
3
3,5
4
7 7,5 8 8,5 9 9,5 10 10,5 11 11,5
Unemployment
Infl
atio
n
GPEFM Macro II Part 1 Ramon Marimon 58
EMU 1995-1999
0
0,5
1
1,5
2
2,5
3
8 8,5 9 9,5 10 10,5 11 11,5
Unemployment
Infl
atio
n
EMU 1995-1999
EMU 2000-2004
1
1,5
2
2,5
3
3,5
7,8 8 8,2 8,4 8,6 8,8 9 9,2
Unemployment
Infl
atio
n
EMU 2000-2004
GPEFM Macro II Part 1 Ramon Marimon 59
in a heterogeneous EUGermany 2000-2004
0
0,5
1
1,5
2
2,5
3
6 6,5 7 7,5 8 8,5 9 9,5 10 10,5 11
Unemployment
Infl
atio
n
France 2000-2004
0
0,5
1
1,5
2
2,5
3
7 7,5 8 8,5 9 9,5 10 10,5
Unemployment
Infl
atio
n
Germany 2000-2004
France 2000-2004
GPEFM Macro II Part 1 Ramon Marimon 60
Different perceptions
France 1995-1999
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
11.2 11.4 11.6 11.8 12 12.2
Unemployment
Infla
tion
Germany 1995-1999
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
8 8.5 9 9.5 10
Unemployment
Infla
tion
GPEFM Macro II Part 1 Ramon Marimon 61
in a heterogeneous EU
Spain 2000-2004
Spain 2000-2004
1,5
2
2,5
3
3,5
4
4,5
10 10,5 11 11,5 12 12,5
Unemployment
Infl
atio
n
Italy 2000-2004
1,5
1,7
1,9
2,1
2,3
2,5
2,7
2,9
3,1
3,3
6 7 8 9 10 11 12
Unemployment
Infl
atio
n
Italy 2000-2004
GPEFM Macro II Part 1 Ramon Marimon 62
with different time-perceptions
Spain 1995-1999
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
1,8
12 13 14 15 16 17 18 19Unemployment
Infla
tion
Italy 1995-1999
0
0,5
1
1,5
2
2,5
11,7 11,8 11,9 12 12,1 12,2 12,3 12,4
Unemployment
Infla
tion
GPEFM Macro II Part 1 Ramon Marimon 63
Self-fulfilling equilibria
• In any case, Self-Fulfilling Equilibria (with their transitional dynamics) provide a new framework to analyze these shifting curves with their interplay between data and policy
GPEFM Macro II Part 1 Ramon Marimon 64
the economy as a recursive adaptive system
• From Arrow-Debreu equilibria to• Prescott et al. ‘Recursive Competitive
Equilibria’
•From Prescott et al. ‘Recursive Competitive Equilibria’ to •Sargent et al. ‘Recursive Adaptive Competitive Equilibria’
GPEFM Macro II Part 1 Ramon Marimon 65
policies vs. plans
• Sharper characterizations• by capturing basic
structures• Stationarity even in
transitions• Rat. Exp. on policies!• Learnable and
computable
• Fine to get GE results• and to define
separate time series• Stationarity is missing
in transitions• Rat. Exp. on plans!• Difficult to learn and
compute
GPEFM Macro II Part 1 Ramon Marimon 66
from Perceived Law of Motionto Actual Law of Motion
• PLM• ALM
• REE
• PLM• ALM
• REE
GPEFM Macro II Part 1 Ramon Marimon 67
With learning theory we learn
• To start modeling the PLM to ALM as a recurrent process
• For policy makers and for agents• Reinforcement learning opens the possibility to
encompass fundamentals (preferences & technologies) into this process
• Whether this exercise is worth it depends on...
The ability of the Macro Learning Theory to better explain data and help policy modeling
GPEFM Macro II Part 1 Ramon Marimon 68
end of Macro II Part 1!
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