optimal policies with robust concerns Anmol Bhandari Jaroslav Borovička Paul Ho

optimal policies with robust concerns
Anmol Bhandari1
August 2015
1
University of Minnesota
2
New York University
3
Princeton University
Jaroslav Borovička2
Paul Ho3
inflation data
6
annual inflation (%)
4
2
0
−2
actual CPI index
1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014
time
1/32
inflation data
6
annual inflation (%)
4
2
0
−2
actual CPI index
Survey of Professional Forecasters
1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014
time
1/32
inflation data
6
annual inflation (%)
4
2
0
−2
actual CPI index
Survey of Professional Forecasters
Michigan Survey of Consumers
1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014
time
1/32
unemployment data
change in unemployment rate (%)
4
3
2
1
0
−1
−2
−3
actual change in unemployment
1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014
time
2/32
unemployment data
change in unemployment rate (%)
4
3
2
1
0
−1
−2
−3
actual change in unemployment
Survey of Professional Forecasters
1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014
time
2/32
unemployment data
change in unemployment rate (%)
4
3
2
1
0
−1
−2
−3
actual change in unemployment
Survey of Professional Forecasters
Michigan Survey of Consumers
1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014
time
2/32
questions
Systematic belief distortions in answers from household surveys
1. Can these belief distortions be rationalized by uncertainty aversion /
robustness?
∙ Which states are perceived as adverse?
3/32
questions
Systematic belief distortions in answers from household surveys
1. Can these belief distortions be rationalized by uncertainty aversion /
robustness?
∙ Which states are perceived as adverse?
2. Can survey data be used to guide modeling / estimation?
∙ Direct source of information disciplining beliefs.
3/32
questions
Systematic belief distortions in answers from household surveys
1. Can these belief distortions be rationalized by uncertainty aversion /
robustness?
∙ Which states are perceived as adverse?
2. Can survey data be used to guide modeling / estimation?
∙ Direct source of information disciplining beliefs.
3. How should optimal policy respond to such belief distortions?
∙ Promises of future actions can be loaded on states that are perceived as more
likely.
∙ Expectations management by the policy maker.
3/32
outline
1. Modeling belief distortions
∙ extending the concept of robust preferences
2. Equilibrium framework
∙ linear solution with time-varying impact of belief distortions
3. A simple monetary policy model
∙ information from survey data
4. Optimal policy
4/32
robust preferences
∙ preferences of an agent with concerns for robustness
Vt =
ut + βEt [
Vt+1 ]
5/32
robust preferences
∙ preferences of an agent with concerns for robustness
Vt =
ut + βEt [mt+1 Vt+1 ]
5/32
robust preferences
∙ preferences of an agent with concerns for robustness
Vt =
min ut + βEt [mt+1 Vt+1 ]
mt+1
E[mt+1 ]=1
5/32
robust preferences
∙ preferences of an agent with concerns for robustness
Vt =
1
min ut + βEt [mt+1 Vt+1 ] + β Et [mt+1 log mt+1 ]
mt+1
θ
E[mt+1 ]=1
∙ penalty parameter θ (rational expectations θ = 0)
5/32
robust preferences
∙ preferences of an agent with concerns for robustness
Vt =
1
min ut + βEt [mt+1 Vt+1 ] + β Et [mt+1 log mt+1 ]
mt+1
θ
E[mt+1 ]=1
∙ penalty parameter θ (rational expectations θ = 0)
∙ implied worst-case distortion
mt+1 =
exp (−θVt+1 )
Et [exp (−θVt+1 )]
5/32
robust preferences
∙ preferences of an agent with concerns for robustness
Vt =
1
min ut + βEt [mt+1 Vt+1 ] + β Et [mt+1 log mt+1 ]
mt+1
θ
E[mt+1 ]=1
∙ penalty parameter θ (rational expectations θ = 0)
∙ implied worst-case distortion
mt+1 =
exp (−θVt+1 )
Et [exp (−θVt+1 )]
∙ nonlinear continuation-value recursion
Vt = ut − β
1
log Et [exp (−θVt+1 )]
θ
5/32
time-varying concern for robustness
∙ consider an economy with equilibrium law of motion
xt+1 = ψ (xt , wt+1 )
6/32
time-varying concern for robustness
∙ consider an economy with equilibrium law of motion
xt+1 = ψ (xt , wt+1 )
∙ allow θ to depend vary over time
et
θt = θx
6/32
time-varying concern for robustness
∙ consider an economy with equilibrium law of motion
xt+1 = ψ (xt , wt+1 )
∙ allow θ to depend vary over time
et
θt = θx
∙ continuation value recursion
Vt = u (xt ) − β
1
log Et [exp (−θt Vt+1 )]
θt
6/32
time-varying concern for robustness
∙ consider an economy with equilibrium law of motion
xt+1 = ψ (xt , wt+1 )
∙ allow θ to depend vary over time
et
θt = θx
∙ continuation value recursion
Vt = u (xt ) − β
1
log Et [exp (−θt Vt+1 )]
θt
Examples
∙ exogenous belief fluctuations: θt follows an AR(1)
θt+1 = (1 − ρθ ) θ + ρθ θt + σθ wθt+1
∙ θt one element of xt , and θe = (1, 0, 0, . . .)
6/32
time-varying concern for robustness
∙ consider an economy with equilibrium law of motion
xt+1 = ψ (xt , wt+1 )
∙ allow θ to depend vary over time
et
θt = θx
∙ continuation value recursion
Vt = u (xt ) − β
1
log Et [exp (−θt Vt+1 )]
θt
Examples
∙ exogenous belief fluctuations: θt follows an AR(1)
θt+1 = (1 − ρθ ) θ + ρθ θt + σθ wθt+1
∙ θt one element of xt , and θe = (1, 0, 0, . . .)
∙ endogenous beliefs: θt explicitly depends on xt
6/32
equilibrium and a linear approximation
Vector of equilibrium conditions
Et [mt+1 g (xt+1 , xt , xt−1 , wt+1 , wt )] = 0
∙ mt+1 is a vector of belief distortions associated with individual equations
7/32
equilibrium and a linear approximation
Vector of equilibrium conditions
Et [mt+1 g (xt+1 , xt , xt−1 , wt+1 , wt )] = 0
∙ mt+1 is a vector of belief distortions associated with individual equations
Linear approximation and solution (Borovička and Hansen (2013))
∙ zero-th order dynamics — deterministic steady state
∙ first order dynamics — linear dynamics
∙ jointly scaling shock volatility and concern for robustness
7/32
equilibrium and a linear approximation
Series expansion method (Holmes (1995), Lombardo (2015))
xt+1 (q) = ψ (xt (q) , qwt+1 , q) ≈ x0t+1 + qx1t+1
8/32
equilibrium and a linear approximation
Series expansion method (Holmes (1995), Lombardo (2015))
xt+1 (q) = ψ (xt (q) , qwt+1 , q) ≈ x0t+1 + qx1t+1
∙ ‘derivatives’
x0t+1
=
ψ (x0t , 0, 0)
x1t+1
=
ψx x1t + ψw wt+1 + ψq
8/32
equilibrium and a linear approximation
Continuation value recursion
Vt (q)
=
u (xt (q)) − β
≈
V0t + qV1t
q
θe (x0t + x1t )
[
log Et
(
)]
θe (x0t + x1t )
exp −
Vt+1 (q)
≈
q
9/32
equilibrium and a linear approximation
Continuation value recursion
Vt (q)
=
u (xt (q)) − β
≈
V0t + qV1t
[
q
θe (x0t + x1t )
log Et
(
)]
θe (x0t + x1t )
exp −
Vt+1 (q)
≈
q
∙ solution
V1t = Vx x1t + Vq
with
Vx = ux −
β
Vx ψw ψw′ V′x θe + βVx ψx
2
9/32
equilibrium belief distortions
Linear dynamics distorted by time-varying beliefs
(
)
exp −θe (x0t + x1t ) Vx ψw wt+1
(
)]
M0t+1 = [
Et exp −θe (x0t + x1t ) Vx ψw wt+1
∙ under the agent’s worst-case measure, innovations are distributed as
(
)
wt+1 ∼ N −θe (x0t + x1t ) (Vx ψw )′ , Ik
10/32
equilibrium belief distortions
Linear dynamics distorted by time-varying beliefs
(
)
exp −θe (x0t + x1t ) Vx ψw wt+1
(
)]
M0t+1 = [
Et exp −θe (x0t + x1t ) Vx ψw wt+1
∙ under the agent’s worst-case measure, innovations are distributed as
(
)
wt+1 ∼ N −θe (x0t + x1t ) (Vx ψw )′ , Ik
Equilibrium determined by jointly solving
∙ system of equations for ψx , ψw (equilibrium conditions)
∙ continuation value recursion
∙ belief distortions
10/32
equilibrium belief distortions
Equilibrium law of motion
x1t+1 = ψx x1t + ψw wt+1 + ψq
∙ belief dynamics in θt alters the matrices ψx , ψw and ψq
∙ exogenous shocks to θt provide additional sources of uncertainty in
equilibrium
11/32
equilibrium belief distortions
Equilibrium law of motion
x1t+1 = ψx x1t + ψw wt+1 + ψq
∙ belief dynamics in θt alters the matrices ψx , ψw and ψq
∙ exogenous shocks to θt provide additional sources of uncertainty in
equilibrium
∙ when θt is an autonomous AR(1), we can write
x1t+1 = ψx x1t + ψxθ θt + ψw wt+1 + ψwθ wθt+1
11/32
a new-keynesian economy
∙ standard NK economy (Galí (2008))
Phillips curve
:
Euler equation
:
Taylor rule
:
πt = β e
Et [πt+1 ] + κyt
)
1 (
it − e
Et [πt+1 ] − rnt + e
Et [yt+1 ]
yt = −
σ
it = ρ + ϕπ πt + ϕy yt + vt
12/32
a new-keynesian economy
∙ standard NK economy (Galí (2008))
Phillips curve
:
Euler equation
:
Taylor rule
:
πt = β e
Et [πt+1 ] + κyt
)
1 (
it − e
Et [πt+1 ] − rnt + e
Et [yt+1 ]
yt = −
σ
it = ρ + ϕπ πt + ϕy yt + vt
∙ exogenous processes
beliefs
:
θt+1 = (1 − ρθ ) θ + ρθ θt + σθ wθt+1
technology
:
rnt
=
at+1 = ρa at + σa wat+1
n e
Et [at+1 − at ]
ρ + σψya
monetary policy
:
vt+1 = ρv vt + σv wvt+1
12/32
preferences
∙ period utility function
(
(
))
exp (1 − σ) e
yt + ynt
exp ((1 + χ) nt )
ut =
−
1−σ
1+χ
∙ production function
e
yt + ynt
=
at + (1 − α) nt
ynt
=
n
ψya
at + ϑny
13/32
state space representation
∙ Markov state
xst = (θt , at , vt )′
∙ State space representation xt = (xst , xot )
xst+1
=
xs + Axst + Bwt+1
xot+1
=
xo + Cxst
14/32
data
Inflation πt , output gap yt
∙ St. Louis Fed database
Survey data
∙ Survey of Consumers at the University of Michigan
∙ quarterly data since 1960
∙ survey data on unemployment and inflation forecasts
∙ construct e
Et [nt+1 ] and e
Et [πt+1 ]
∙ state space representation of the model implies Et [nt+1 ] and Et [πt+1 ]
∙ construct the belief wedges
e
Et [nt+1 ] − Et [nt+1 ]
and e
Et [πt+1 ] − Et [πt+1 ]
15/32
survey belief wedges
·10−2
1
employment
inflation
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
2010
2015
16/32
calibrated parameters
time preference
risk aversion
Frisch elasticity
elasticity of substitution between goods
Calvo fairy
Taylor rule coefficients
technology shock
monetary policy shock
parameter
β
σ
φ
ε
ϑ
ϕπ
ϕy
ρa
σa
ρv
σv
value
0.99
1.5
1
6
0.67
1.2
0.125
0.9
0.116
0.5
0.01
17/32
dynamics of θt
∙ Belief distortion process
θt+1 = (1 − ρθ ) θ + ρθ θt + σθ wθt+1
∙ θ̄ =⇒ mean wedge between surveys and rational expectations forecasts
∙ ρθ and σθ =⇒ persistence and volatility of the wedge
18/32
dynamics of θt
∙ Belief distortion process
θt+1 = (1 − ρθ ) θ + ρθ θt + σθ wθt+1
∙ θ̄ =⇒ mean wedge between surveys and rational expectations forecasts
∙ ρθ and σθ =⇒ persistence and volatility of the wedge
Preferred parameterization
θ
=
3.8
ρθ
=
0.3
σθ
=
5.7
∙ Target the wedge for unemployment forecast =⇒ study the implied
dynamics of the wedge for the inflation forecast.
∙ all subjective forecasts (survey answers) are jointly restricted by the same θt
process
18/32
the role of the belief shock
Phillips curve
:
Euler equation
:
Taylor rule
:
πt = β e
Et [πt+1 ] + κyt
)
1 (
yt = −
it − e
Et [πt+1 ] − rnt + e
Et [yt+1 ]
σ
it = ρ + ϕπ πt + ϕy yt + vt
An increase in robust concern: θt ↗
Et [wat+1 ] decreases
∙ innovation e
Et [wvt+1 ] increases, e
∙ e
Et [πt+1 ] increases
∙ e
Et [yt+1 ] decreases
∙ output gap yt falls, inflation πt increases
19/32
quantifying belief distortions
Mean distortions e
Et [zt+1 ] − Et [zt+1 ] for z ∈ {n, π}
data
model implied
employment
−0.0026
−0.0020
inflation
0.0015
0.0014
20/32
quantifying belief distortions
Mean distortions e
Et [zt+1 ] − Et [zt+1 ] for z ∈ {n, π}
data
model implied
employment
−0.0026
−0.0020
inflation
0.0015
0.0014
Volatility of the distortions
data
model implied (extracted shocks)
model implied (theoretical)
employment
0.0029
0.0029
0.0030
inflation
0.0024
0.0020
0.0024
20/32
survey and fitted inflation wedge
·10−2
1
data
fitted through the model
0.8
0.6
0.4
0.2
0
−0.2
−0.4
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
2010
2015
21/32
quantifying belief distortions
Correlation between distortions
data
data (last 20 years)
model implied
0.29
0.57
1.00
∙ In this model, the survey forecast wedge
e
Et [zt+1 ] − Et [zt+1 ]
is only a function of θt , not other shocks.
∙ All wedges in the model are perfectly correlated.
22/32
quantifying belief distortions
Correlation between distortions
data
data (last 20 years)
model implied
0.29
0.57
1.00
∙ In this model, the survey forecast wedge
e
Et [zt+1 ] − Et [zt+1 ]
is only a function of θt , not other shocks.
∙ All wedges in the model are perfectly correlated.
Generalization
et
θt = θx
∙ allows for imperfect correlation between wedges in the model.
22/32
dynamics under belief distortions
Model dynamics under the objective measure
x1t+1 = ψx x1t + ψw wt+1 + ψq
∙ Impulse response functions
irft = (ψx )t−1 ψw
23/32
dynamics under belief distortions
Model dynamics under the objective measure
x1t+1 = ψx x1t + ψw wt+1 + ψq
∙ Impulse response functions
irft = (ψx )t−1 ψw
Model dynamics under the agents’ beliefs
∙ Recall
(
)
wt+1 ∼ N −θe (x0t + x1t ) (Vx ψw )′ , Ik
∙ Dynamics
]
]
[
[
e 0t
x1t+1 = ψx − ψw ψw′ V′x θe x1t + ψw wt+1 + ψq − ψw ψw′ V′x θx
∙ Impulse response functions
[
]t−1
e = ψx − ψw ψw′ V′x θe
irf
ψx
t
23/32
impulse response functions
monetary policy shock vt
·10−4
belief θt
6
2
−0.01
4
0
−0.01
1
2
technology shock at
0
−0.02
0
10
20
quarters
30
0
0
10
20
quarters
30
−0.02
0
10
20
quarters
30
Blue line under correct beliefs, red line under agent’s subjective beliefs.
24/32
impulse response functions
·10−2
0
output yt + ynt
·10−2
0.5
output gap yt
·10−3
inflation πt
4
−0.5
0
−1
−0.5
−1.5
0
10
20
quarters
30
−1
2
0
10
20
quarters
30
0
0
10
20
quarters
30
Blue line under correct beliefs, red line under agent’s subjective beliefs.
25/32
impulse response functions
0
employment
belief wedge
·10−3
inflation belief wedge
·10−3
2
1.5
−1
1
−2
−3
0.5
0
10
20
quarters
30
0
0
10
20
quarters
30
Blue line under correct beliefs, red line under agent’s subjective beliefs.
26/32
optimal policy
We have not taken a position regarding the source of subjective beliefs
∙ concern for robustness
∙ distorted beliefs (in particular when θt is endogenous)
There may be scope for policy action by a paternalistic planner.
∙ Model with heterogeneous, endogenous beliefs.
∙ Ramsey planner exploits belief distortion to allocates commitment.
∙ Expectations management.
27/32
planner’s problem
∙ policymaker’s problem
max
min E0
{πt ,xt ,Vt+1 }t≥0 {mp }
t+1
∞
∑
t=0
[
]
1
β t Mpt upt + β p mpt+1 log mpt+1
θt
28/32
planner’s problem
∙ policymaker’s problem
max
min E0
{πt ,xt ,Vt+1 }t≥0 {mp }
t+1
∞
∑
t=0
[
]
1
β t Mpt upt + β p mpt+1 log mpt+1
θt
subject to
[λt ]
:
[µt ]
:
πt = βEt [mt+1 πt+1 ] + κyt + zt (zt ‘cost push’ shock)
1
Vt = ut − β log Et [exp (−θt Vt+1 )]
θt
respecting
mt+1 =
exp (−θt Vt+1 )
Et [exp (−θt Vt+1 )]
and the law of motion for worst-case distortion: Mpt+1 = Mpt mpt+1
28/32
optimality
∙ The first-order conditions lead to the following set of equations
(
)
[yt ] : − upyt + µt uyt = κλt
(
)
m
[πt ] : − upπt + µt uπt = λt−1 pt − λt
mt
| {z }
λ̃t
[ p ]
( p p )
p
mt+1
: mt+1 ∝ exp −θt Vt+1
m
1 ∂mt+1
[Vt+1 ] : µt+1 = µt t+1
+ πt+1 λt p
p
mt+1
mt+1 ∂Vt+1
where
(1)
(2)
(3)
(4)
∂mat+1
= −θt mt+1 (1 − mt+1 dPt+1 ) .
∂Vat+1
∙ plus the Phillips curve and definition of mt+1
29/32
interpretation — commitment to inflation
∙ Phillips curve
πt = βEt [mt+1 πt+1 ] + κyt + zt
∙ low zt =⇒ ↗ yt , ↘ πt and a commitment to inflate in the future
∙ encoded through a high λ̃t = λt−1 mmpt :
t
(
)
mt+1
λ̃t+1
=[
λ̃t
+ upπt + µt uπt ]
|{z}
|{z}
mpt+1
|
{z
}
| {z }
previous
future
delivered
belief
commitment
commitment
commitment
tilting
30/32
interpretation — commitment to inflation
∙ Phillips curve
πt = βEt [mt+1 πt+1 ] + κyt + zt
∙ low zt =⇒ ↗ yt , ↘ πt and a commitment to inflate in the future
∙ encoded through a high λ̃t = λt−1 mmpt :
t
(
)
mt+1
λ̃t+1
=[
λ̃t
+ upπt + µt uπt ]
|{z}
|{z}
mpt+1
|
{z
}
| {z }
previous
future
delivered
belief
commitment
commitment
commitment
tilting
∙ heterogeneous beliefs: provide more inflation in states with high
mt+1
p
mt+1
∙ inflation costly to the policymaker
∙ provided in states which are unlikely under policymaker’s beliefs
30/32
interpretation — commitment to inflation
∙ Phillips curve
πt = βEt [mt+1 πt+1 ] + κyt + zt
∙ low zt =⇒ ↗ yt , ↘ πt and a commitment to inflate in the future
∙ encoded through a high λ̃t = λt−1 mmpt :
t
(
)
mt+1
λ̃t+1
=[
λ̃t
+ upπt + µt uπt ]
|{z}
|{z}
mpt+1
|
{z
}
| {z }
previous
future
delivered
belief
commitment
commitment
commitment
tilting
∙ heterogeneous beliefs: provide more inflation in states with high
mt+1
p
mt+1
∙ inflation costly to the policymaker
∙ provided in states which are unlikely under policymaker’s beliefs
∙ time-variation in µt
∙ determines relative cost of fluctuations in πt and yt to the economy
30/32
interpretation — changes in weights
∙ Law of motion for the Lagrange multiplier µt
µt+1 = µt
mat+1
1 ∂mt+1
+ πt+1 λt p
mpt+1
mt+1 ∂Vt+1
∙ provision of utility in adverse states reduces the probability mt+1 of those states
∙ expectations management
31/32
interpretation — changes in weights
∙ Law of motion for the Lagrange multiplier µt
µt+1 = µt
mat+1
1 ∂mt+1
+ πt+1 λt p
mpt+1
mt+1 ∂Vt+1
∙ provision of utility in adverse states reduces the probability mt+1 of those states
∙ expectations management
∙ Special case: Exogenous heterogeneous beliefs
µt+1 = µt
mt+1
mpt+1
31/32
conclusion
∙ Household survey data indicate nontrivial systematic deviations of survey
answers from rational expectations.
∙ Introduce a framework that incorporates subjective beliefs.
∙ Two sources of discipline
∙ modeling framework: tie survey answers to the notion of a ‘worst-case’ model,
imposes restrictions on answers for different survey questions
∙ data on survey answers: discipline the belief shock process
∙ Belief dynamics have a nontrivial impact on the equilibrium dynamics of
the model.
∙ Support for a relationship across survey answers.
∙ Not perfect: A more flexible specification of beliefs may be necessary.
∙ What are the implications for optimal policy?
∙ Relationship to asset price dynamics?
32/32