On Credible Monetary Policies with Model Uncertainty Anna Orlik and Ignacio Presno New York University
February 22, 2011
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Central Banking as the Management of Expectations
State of private inflation expectations as crucial for good monetary policy.
Svensson (2004): ”monetary policy is to a large extent the management of expectations”.
Woodford (2005): ”not only do expectations about policy matter, but, very little else matters”.
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
2 / 42
How Can a Central Banker Manage Private Expectations? Common unique prior (Chari, Christiano, and Eichenbaum (1998), Albanesi, Chari and Christiano (2002)) it = E [it |It ] → decision rule = system of private expectations
Unique but different priors (Cukierman and Melzer (1986), Cogley, Matthes, Sbordone (2010), Morris and Shin (2005)) it = E [it |It ] → decision rule = system of private expectations
This paper e [it |It ] it = E → decision rule = system of private expectations chosen by the planner
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
3 / 42
Bernanke on Knightian Uncertainty
”Most economic researchers continue to work within the classical paradigm that assumes rational behavior and the maximization of ”expected utility” (...). An important assumption of that framework is that, in making decisions under uncertainty, economic agents can assign meaningful probabilities to alternative outcomes. However, during the worst phase of the financial crisis, many economic actors–including investors, employers, and consumers– metaphorically threw up their hands and admitted that (...) they did not know what they did not know. The idea that, at certain times, decisionmakers simply cannot assign meaningful probabilities to alternative outcomes is known as Knightian uncertainty”
Bernanke (2010), Implications of the Financial Crisis for Economics
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
4 / 42
This Paper
Model: stochastic version of Calvo (1978) and Chang (1998) and with model uncertainty. We extend the existing concepts of a Ramsey plan and sustainable plans (Chari, Kehoe (1990)) to account for model uncertainty. Policy questions in the presence of model uncertainty I I I
what are the sustainable values of welfare? is the long-run inflation affected? is the optimal deflation more/less gradual?
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Outline of the Talk
Model.
Solution methodology.
Computation.
Policy results.
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Model: Outline
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
7 / 42
Model: Outline
Time: t = 0, 1, 2, ....
Players: planner (central banker) and a representative household.
Uncertainty: st ∈ S. s t ≡ (s0 , s1 , ..., st ) ∈ St+1
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Model: Central Banker Policy action: ht (s t ) = Mt−1 s t−1 /Mt (s t ).
Budget constraint xt (s t ) = qt (s t ) Mt−1 s t−1 − Mt (s t )
Taxes and transfers are distortionary yt (s t ) ≡ f (xt (s t ), st )
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
8 / 42
f(x,1)
x
0
f(x,2)
Figure: f (x, s) for s = 1, 2
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Model: Structure of Uncertainty Planner’s approximating model π(s t ). Assume that st follows a Markov process with π (st+1 |st ).
Households’ set of alternative models, each of which denoted by π e (s t ) and t t t t t t+1 e (s ) = 0, ∀s ∈ S . such that π e (s ) π (s ) i.e. π (s ) = 0 ⇒ π
By Radon-Nikodym theorem, Dt (s t ) =
π e(s t ) π(s t )
Conditional likelihood ratio dt+1 (st+1 |s t ) ≡
Anna Orlik (NYU Stern)
≥ 0.
Dt+1 (s t+1 ) Dt (s t )
Credible Monetary Policies with Model Uncertainty
=
π e(st+1 |s t ) π(st+1 |st ) .
February 22, 2011
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Model: Preferences
VH =
max
min
t t ∞ {ct (s t ),mt (s t )}∞ t=0 {Dt (s ),dt+1 (st+1 |s )}t=0
∞ X
βt
t=0
+θβ
X
n π(s t )Dt (s t ) u ct s t + v (mt s t )
st
X
o π(st+1 |st )dt+1 (st+1 |s t ) log dt+1 (st+1 |s t )
st+1
where
qt (s t ) Mt
(s t )
= mt
(s t ),
θ ∈ (θ, +∞] and
(i) regularity conditions for u and v (ii) Inada conditions (iii) ∃m < +∞ such that v 0 (m) = 0
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Model: Constraints
qt s t Mt s t ≤ yt s t − xt s t − ct s t + qt s t Mt−1 (s t−1 ) qt s t Mt s t ≤ m Dt+1 (s t+1 ) = dt+1 (st+1 |s t )Dt (s t ) X π(st+1 |st )dt+1 (st+1 |s t ) = 1 st+1
D0 = 1, M−1 and s0 given
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Model: Timing Protocol
Both Ramsey plan and sustainable plans. Within-period timing: shock realizes, government chooses the rate of money growth, households choose real money holdings, taxes are collected, consumption takes place.
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Competitive Equilibrium with Model Uncertainty
Definition Given s0 , M−1 and D0 , competitive equilibrium with model uncertainty is given by a sequence of allocations (m, x, h) = {mt , xt , ht }∞ t=0 , prices ∞ , and values of households utility , belief distortions d = {d } q = {qt }∞ t+1 t=0 t=0 t H such that for all t and all s }∞ VH = {Vt+1 t=0 (i) given q and the belief distortions d, taxes x and money growth rates h, m, VH solve households’ maximization problem; (ii) given q and m, x, h, VH , d solves the alter ego’s minimization problem; (iii) government’s budget constraint holds; (iv) markets clear, i.e. ct (s t ) = yt (s t ) and mt (s t ) = qt (s t )Mt (s t ).
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
14 / 42
Competitive Equilibrium with Model Uncertainty Proposition Acompetitive equilibrium is completely characterized by sequences m, x, h, d, VH such that all s t mt (s t ) ∈ M, xt (s t ) ∈ X, for all t and H t t+1 ∈ D, and Vt+1 (s t ) ∈ V and ht (s ) ∈ Π, dt+1 s
(1) Euler equation for a representative households holds;
(2) optimality condition for alter ego holds;
(3) lifetime utility value to the household is defined given (1) and (2);
(4) planner’s budget constraint is satisfied.
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Competitive Equilibrium with Model Uncertainty (1)
Euler equation of a representative household X qt+1 s t+1 u 0 (ct s t ) = v 0 (mt s t ) + β π(st+1 |st )dt+1 (st+1 |s t )u 0 (ct+1 s t+1 ) qt (s t ) st+1 ≤ 0 if mt s t = m
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
(1)
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Competitive Equilibrium with Model Uncertainty (1)
Euler equation of a representative household
u 0 (ct s t ) − v 0 (mt s t ) st+1 |s t u 0 (ct+1 s t+1 )mt+1 s t+1 ht+1 s t+1 {z } |
mt s t =β
X
π(st+1 |st )dt+1
st+1
(1)
µt+1 (s t+1 )
≤ 0 if mt s
Anna Orlik (NYU Stern)
t
=m
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Competitive Equilibrium with Model Uncertainty (2) Optimality condition for alter ego exp dt+1 (st+1 |s t ) = P
st+1
Anna Orlik (NYU Stern)
V H (s t+1 ) − t+1 θ
(2)
V H (s t+1 ) π(st+1 |st ) exp − t+1 θ
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Competitive Equilibrium with Model Uncertainty (2) Optimality condition for alter ego exp dt+1 (st+1 |s t ) = P
st+1
V H (s t+1 ) − t+1 θ
(2)
V H (s t+1 ) π(st+1 |st ) exp − t+1 θ
Example: Assume S = {1, 2}. Then dt+1 (st+1 = 1|s t ) =
Anna Orlik (NYU Stern)
1 H V H (s t+1 =2)−Vt+1 (s t+1 =1) π(st+1 = 1|st ) + π(st+1 = 2|st ) exp − t+1 θ
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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2 = =20 =10
1.8
1.6
1.2
1
d
t+1
(s
t+1
t
=1|s )
1.4
0.8
0.6
0.4
0.2
0 -50
-40
-30
-20
-10 VH (s t+1
t+1
0
10
=2|st) - VH (s t+1
t+1
20
30
40
50
=1|st)
Figure: Equilibrium distortion in beliefs Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Competitive Equilibrium with Model Uncertainty (3)
Recursion for continuation values
VtH
= u ct s
t
+ v mt s
t
X − βθ log π (st+1 |st ) exp st+1
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
H −Vt+1 s t+1 θ
!
February 22, 2011
(3)
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Competitive Equilibrium with Model Uncertainty (4)
Budget constraint rewritten −xt s t = mt s t 1 − ht s t
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
(4)
February 22, 2011
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Competitive Equilibrium with Model Uncertainty
CEs =
o n m, x, h, d, VH ∈ M∞ × X∞ × Π∞ × D∞ × V∞ | (1)-(4) hold and s0 = s
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Ramsey Problem for a Paternalistic Planner
VsG0 =
max
∞ X
(m,x,h,d,VH )
t=0
s.t.
Anna Orlik (NYU Stern)
βt
X
π(s t ) u ct s t + v (mt s t )
st
m, x, h, d, VH ∈ CEs0
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Time Inconsistency of the Ramsey Plan
Ramsey policy: set taxes to zero at t = 0, and start deflating the economy from t = 1 on to bring the real money holdings towards the satiation level. For t ≥ 1, the planner would want to reset its policy and deflate at a slower pace to reduce the distortionary taxation associated with deflation.
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
23 / 42
Sequential Problem: Elements relevant public history: ht = (h0 , h1 , ..., ht ) and s t strategy for the government, σ G : G t t−1 G → Π for all t ≥ 1 {σtG }∞ t=0 such that σ0 : S → Π and σt : S × Π strategy for a representative household, σ H : H t t {σtH }∞ t=0 such that σt : S × Π → M for all t ≥ 0. AE strategy for the : alter ego, σ AE AE AE ∞ σt ≡ σt t=0 such that σ0 = d0 = 1 and σtAE : St × Πt → D for all t ≥ 1.
allocation rule for tax revenues, αx : ∞ αx ≡ {αtx }t=0 such that αtx : St × Πt → X for all t ≥ 0.
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Sequential Problem: Solution Concept
Definition (σ G , σ H , σ AE , αx ) are said to constitute a sustainable plan with model uncertainty (SP) if after any history s t and ht−1 (CE) given government strategy σ G , the continuation of σ G together with σ H , σ AE , and αx induce a competitive equilibrium sequence; (IC) given σ H , σ AE , and αx , there is no profitable deviation for the government.
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Sequential Problem: Solution Concept
Definition (σ G , σ H , σ AE , αx ) are said to constitute a sustainable plan with model uncertainty (SP) if after any history s t and ht−1 (CE) given government strategy σ G , the continuation of σ G together with σ H , σ AE , and αx induce a competitive equilibrium sequence; (IC) given σ H , σ AE , and αx , there is no profitable deviation for the government.
→ factorize the strategies
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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SP with Model Uncertainty: Factorization of Strategies
lifetime utility = period payoff + β ∗ expected continuation value | {z } action profile a
where a = (b h, m(h), x(h), d 0 (h))
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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SP with Model Uncertainty: Factorization of Strategies
lifetime utility = period payoff + β ∗ expected continuation value | {z } action profile a
where a = (b h, m(h), x(h), d 0 (h))
→ need recursive formulation of CE → need recursive formulation of IC
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Towards Recursive Formulation of CE Chang (1998) mt u 0 (f (xt )) − v 0 (mt ) = βµt+1 , ≤ if mt = m
This paper u 0 (f (xt s t , st )) − v 0 (mt s t ) = V H (s t+1 ) exp − t+1 θ µt+1 s t+1 , ≤ if mt = m H Vt+1 (s t+1 ) π(st+1 |st ) exp − θ
mt s t
β
X
π(st+1 |st )
st+1
Anna Orlik (NYU Stern)
P
st+1
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Towards Recursive Formulation of CE Chang (1998) mt u 0 (f (xt )) − v 0 (mt ) = βµt+1 , ≤ if mt = m
This paper u 0 (f (xt s t , st )) − v 0 (mt s t ) = V H (s t+1 ) exp − t+1 θ µt+1 s t+1 , ≤ if mt = m H Vt+1 (s t+1 ) π(st+1 |st ) exp − θ
mt s t
β
X
π(st+1 |st )
st+1
P
st+1
→ key state variables: µs , VsH
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Recursive Formulation of CE with Model Uncertainty
VsH
µs = u 0 [f (xs , s)] [xs + ms ] X −VsH0 0 = u (cs ) + v (ms ) − βθ log π (s 0 |s) exp θ 0 s
−xs = ms [1 − hs ] ms {u 0 (f (xs , s)) − v 0 (ms )} = β
X s0
Anna Orlik (NYU Stern)
V H0 exp − θs 0 V H0 µ0s 0 ≤ if m = m π(s0|s) P s0 0 s 0 π(s |s) exp − θ
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Recursive Formulation of IC Use one-period deviation principle
X 0 u f x b h ,s +v m b h +β π s 0 |s VsG0 b h ≥ s 0 ∈S X 0 π s 0 |s VsG0 (h) u (f (x (h) , s)) + v (m (h)) + β 0 s ∈S
∀h ∈ CEs0
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
29 / 42
Recursive Formulation of IC Use one-period deviation principle
X 0 u f x b h ,s +v m b h +β π s 0 |s VsG0 b h ≥ s 0 ∈S X 0 π s 0 |s VsG0 (h) u (f (x (h) , s)) + v (m (h)) + β 0 s ∈S
∀h ∈ CEs0
→ key state variable: VsG
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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SP with Model Uncertainty: Set of Equilibrium Values
Value correspondence Λ : S −→ R3
Set of values associated with SP with model uncertainty n o Λ(s0 ) = VsG0 , VsH0 , µs0 | there is a (m, x, h, d) associated with SP
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Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Admissibility
b (a, V G 0 (h), V H0 (h), µ0 (h)) is said to be Definition For any correspondence Z ⊂ G, admissible with respect to Z at state s if (i) a = (b h, m(h), x(h), d 0 (h)) ∈ Π × [0, m]Π × X Π × RΠ ; 0 0 (ii) (VsG0 0 (h), VsH0 0 (h), µs 0 (h)) ∈ Z (s )
∀ h ∈ CEs0 , s 0 ∈ S;
(iii) recursive CE; (iv) recursive IC.
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Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Recursive Formulation b : Gb −→ Gb Take Z ⊂ Gb and define B n b )(s) = co (VsG , VsH , µs )|∃ admissible (a, V G 0 (·), V H0 (·), µ0 (·)) w.r.t. Z at s with: B(Z
VsG
h, m(h), x(h), d 0 (h)) a = (b X = u(f (x(b h), s)) + v (m(b h)) + β π(s 0 |s)VsG0 0 (b h) s 0 ∈S
(
VsH
b V H0 0 (h) π(s |s) exp − s θ s 0 ∈S o µs = u(f (x(b h), s))(x(b h) + m(b h))
h), s)) + v (m(b h)) − βθ log = u(f (x(b
X
)
0
b n (Z0 )(s) → Abreu, Pearce and Stacchetti (1990): Λ(s) = limn→∞ B
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Computation Monotone Outer Hyperplane Approximation (Judd, Yeltekin, Conklin (2003))
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Computation Monotone Outer Hyperplane Approximation (Judd, Yeltekin, Conklin (2003))
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
33 / 42
Computation Monotone Outer Hyperplane Approximation (Judd, Yeltekin, Conklin (2003))
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
33 / 42
Computation Monotone Outer Hyperplane Approximation (Judd, Yeltekin, Conklin (2003))
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
33 / 42
Computation Monotone Outer Hyperplane Approximation (Judd, Yeltekin, Conklin (2003))
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
33 / 42
Computation Monotone Outer Hyperplane Approximation (Judd, Yeltekin, Conklin (2003))
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
33 / 42
Computation Monotone Outer Hyperplane Approximation (Judd, Yeltekin, Conklin (2003))
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
33 / 42
Computation Monotone Outer Hyperplane Approximation (Judd, Yeltekin, Conklin (2003))
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
33 / 42
Computation i+1 cl,s (m, h) = max gl1 VsG + gl2 VsH + gl3 µs X 0 VsG = u[f (xs , s] + v [ms ] + β π(s 0 |s)VsG0 s0
( VsG ≥ max min hs
ms
u[f (ms (hs − 1)), s] + v (ms ) + β
X
π(s 0 |s)VsG0
0
)
s0 0
µs = u [f (xs , s)] (ms + xs ) ( VsH = u[f (xs , s)] + v (ms ) − βθ log
X
π(s 0 |s) exp
−
s 0 ∈S
xs = ms (hs − 1) X
ms [u 0 (f (xs , s)) − v 0 (ms )] ≤ β
VsH0 0
)
θ
0 πs,s 0 µ0s 0 ds,s 0
s 0 ∈S 0 ds,s 0
exp(−
= P
g11 · gD1
Anna Orlik (NYU Stern)
g12 · gD2
g13 h 0 · VsG0 gD3
VsH0 0 ) θ V H0 0
π(s 0 |s) exp(− θs ) i c1,s 0 i H0 0 µs 0 ≤ · Vs 0 i cD,s 0
s 0 ∈S
Credible Monetary Policies with Model Uncertainty
∀s 0 ∈ S
February 22, 2011
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Functional Forms and Parameters
per-period utility of consumption u(c) = log (c) per-period utility from real money holdings v (m) = 0.0001[(mm − 0.5m2 )](1/2) income f (x, s) = 64 − (0.5 + 0.5s)(0.4x)2 action space [0, m] = [0, 30], Π = [0.9, 1.2] parameter values β = 0.9, θ = 5, θ = 108 probabilities π(s 0 = 1|s = 1) = π(s 0 = 2|s = 2) = 0.75.
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Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Value Sets Value Set (1)
Value Set (2)
0.8
0.6
0.6
0.4
0.4 2
1
0.8
0.2
0.2
0
0
-0.2 42 41.5
42 41.5
41
41
40.5 VH 1
-0.2 42 41.5
42 41
40.5 40
40
G
V1
41
40.5 VH 2
40
40
VG 2
Figure: Equilibrium values for θ = 5
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Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Implication 1: Sustainable Values Compared 42
41.9
41.9
41.8
41.8 VH s
VH s
42
41.7
41.7
41.6
41.6
41.5 41.5
41.6
41.7
41.8
41.9
42
41.5 41.5
41.6
VG s
41.7
41.8
41.9
42
VG s
Figure: Value sets for θ = ∞ (LHS) and θ = 5 (RHS)
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Implication 2: Transition Paths 24 22
mt
20 18 16 14
2
4
6
8 t
10
12
14
2
4
6
8 t
10
12
14
1.12 1.1
ht
1.08 1.06 1.04 1.02 1
Figure: Equilibrium paths for θ = 5 (red) and θ = ∞ (blue) Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Alternative Hypothesis: Two-Sided Model Uncertainty
VtG =
max
∞ X
min
βt
X
(m,x,h,d,VH ) (DG ,dG ) t=0
st
G
X
+θ β
π(s t )DtG (s t ){ u ct s t + v (mt s t )
G G π(st+1 |st )dt+1 (st+1 |s t ) log dt+1 (st+1 |s t )}
st+1
X
m, x, h, d, VH
∈
CEs0
G Dt+1 (s t+1 )
=
G dt+1 (st+1 |s t )DtG (s t )
G )dt+1 (st+1 |s t )
=
1
πt+1 (st+1 |s
t
st+1
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
39 / 42
Two-Sided Model Uncertainty cont’d
VtG =
max u ct s (m,x,h,d,VH )
t
+ v mt s
− βθG log
X
π (st+1 |st ) exp
st+1
Anna Orlik (NYU Stern)
t
! G (s t+1 ) −Vt+1 θ
m, x, h, d, VH ∈ CEs0
Credible Monetary Policies with Model Uncertainty
February 22, 2011
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Conclusion
Accounting for the fact that households’ models are imperfect leads to a novel channel of optimal monetary policymaking: the channel of management of expectations.
Tools developed in this paper can be used more broadly to discuss the design of such robust policies.
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
41 / 42
Future Research
Quantify the degree of disagreement between a central banker and private agents. How good are their models? Discuss the possibility of expectations management during the liquidity crisis.
Explore the role of this novel channel of optimal monetary policymaking in avoiding expectations traps.
Anna Orlik (NYU Stern)
Credible Monetary Policies with Model Uncertainty
February 22, 2011
42 / 42
