(mostly from Sargent’s “Conquest of Amerincan Inflation...

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


A simple credibility problem

• A simple Phillips curve example

• Government’s preferences

)(* eUU ππθ −−=

( )2221),( παππ +−= Ur e


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


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


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


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


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


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


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→


Rationalizability

Agents understand B( ) (rationalizability)

• therefore their only consistent belief (rationalizable) is the Nash equilibrium

,)( ,

and )( , e

e

eeN

eeN

B

B

ππππ

ππππ

<>∀

><∀


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


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


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−= )(


E-stability

• It follows that the mean learning process converges to the Nash equilibrium REE, since

01

11)( 2 <

+−=−′

αθπ NB


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


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 ↓→


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


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

−=

++= +++−++++−


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


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


General beliefs

• Beliefs, however, can be fairly arbitrary

• Agent i forecast according to

),,...,,(),...,( 110010 −−− ≡= tttt UUsss ππ

)( ti

et sf=π


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

====

≠−=∃

−===


Strategic forecasting and reputation

• Maintaining reputation through incentive constraints

)~()),(()1(

)(),()1()(**

****

πδππδ

πδππδπ

VBr

VrV

+−≥

+−=


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


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 ==<<


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.


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



tet

d

et

d

t

ttdt

dt

st

dttt

st

st

dmm

dmm

MMpdMM

−=

+=⇒

=+=

+

−

−

)()(

,

1

1

1

ππ

π

π



• 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


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


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


The ‘cobweb model’

tett

ttett

ds

tet

s

ttd

bpap

eipp

QQ

pQ

pQ

υ

ηηγβδαβ

ηγδ

ηβα

++=

−+−−=

⇒=

++=

+−=

+

+−−

+

1

211

11

21

1

.,.)()(


Rational Expectations

ttt

tt

tttt

ttet

ba

ba

bap

ba

pE

pbEapE

pEp

υυ +−

=+

−+=

−=

+=

=

−

+−−

++

11

1

i.e., ,

1

111

11


From PLM to ALM revisited• PLM

• ALM

• E-stability

β=+etp 1

tt bap υβ ++=

1for stable-E)1()(

)(

<−+=−

+=

bbaT

baTβββ

ββ


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


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

=

−−−

=


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=


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.


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


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

+=−−−

−−+

−=


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


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


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=


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


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


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

= ≠⇒ = = −

,,


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


Classical self-confirming eq.

• The CB solution is given by

*21010

*

21

10

)1(

then,1

ˆ

UU C

Ct

γγπγγ

πγγγ

π

+=⇒+=

≡+

−=


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


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

+−=

+−=

+−

==

++=


Cho & Sargent Escape routes


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


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


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


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


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


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


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


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


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


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


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


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’


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


from Perceived Law of Motionto Actual Law of Motion

• PLM• ALM

• REE

• PLM• ALM

• REE


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


end of Macro II Part 1!

(mostly from Sargent’s “Conquest of Amerincan Inflation...

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