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Monetary and Fiscal Policy under
Imperfect KnowledgeMonash University Mini-course Part III
Bruce Preston
July 16, 2012
Motivation
Macroeconomic models depend on household and Örm expectations
Policy design
ñ Not enough to observe some history of beliefs
ñ Need a theory of the determination of beliefs: rational expectations
Rational expectations policy design
ñ Heavy reliance on managing expectations through announced commitments
ñ What are the consequences of imprecise control of beliefs?
Motivation II
Reasonable to suppose agents have limited information about policy regime
ñ US Financial Crisis witnessed dramatic change in the nature and conduct ofstabilization policy
ñ Does imperfect knowledge limit the e¢cacy of policy?
For example, Bernanke (2004):
ì[...] most private-sector borrowing and investment decisions depend not onthe funds rate but on longer-term yields, such as mortgage rates and corporatebond rates, and on the prices of long-lived assets, such as housing and equities.Moreover, the link between these longer-term yields and asset prices and thecurrent setting of the federal funds rate can be quite loose at times.î
The Agenda
Part 3 of this mini-course proceeds as follows
ñ Lecture 1: Tools and conceptual issues
ñ Lecture 2: Consequences of learning for the choice of monetary policy rule
ñ Lecture 3: Monetary and Öscal interactions
Introducing Learning Dynamics
Two foundations of rational expectations equilibrium analysis
ñ Optimization
ñ Mutual consistency of beliefs
Strong knowledge assumptions
ñ Agents know more than the modeler
ScientiÖc Method
Principle components
ñ Maintained theories
ñ Procedures for collecting data
ñ Procedures for confronting theory with data
ñ Procedures for updating theory given discrepancy between theory and data
Bounded Rationality: Learning Dynamics
Attempts to place the agent and modeler on equal footing
Agents might proceed as:
ñ Classical econometrician: know model but unsure about parameters
ñ Bayesian econometrician: unsure about model and parameters but understandhow they are unsure
Learning Dynamics
Analyze broad class of model that relaxes the second pillar of REE analysis
ñ Retain optimization
ñ Permit non-rational expectations
Beware of theorists bearing free parameters!
ñ Constrain the analysis by requiring the procedure to nest rational expectations
Why learning?
Permits analysis of a broader class of problems
May resolve/elucidate
ñ Questions of indeterminacy
ñ Questions of robustness
ñ Questions of dynamics
ñ Questions of regime change/regime uncertainty
Today
Mathematical foundations
ñ Learning renders model dynamics self-referential
ñ Requires new set of tools to analyze model properties
Conceptual issues in modeling multi-period decision problems
A Simple Model
Consider a variant of the Cobweb model
pt = + Et1pt + wt1 + t (1)
where ; a; > 0; wt and exogenous process; and t a bounded iid disturbance
Unique bounded solution
pt = a+bwt1 + t
where
a = (1 )1 and b = (1 )1
Statistical Learning
Suppose agents forecast pt as a linear function
Et1pt = at1 + bt1wt1 (2)
where fat1; bt1g are currently maintained beliefs
Price dynamics then
pt = (+ at1) + ( + bt1)wt1 + t
ñ Self-referential system
ñ Data generating process is non-stationary
Least Squares Learning
Beliefs fat1; bt1g estimates using data fpi; wigi=t1i=0
Ordinary least squares implies
at1bt1
=
0
@t1X
i=1
zi1z0i1
1
A10
@t1X
i=1
zi1pi
1
A (3)
where zi = [1 wi]
ñ Fully speciÖed system: relations (1) - (3)
Question
Under what conditions do at ! a and bt ! b as t!1?
Theorem
Consider the model (1) - (3). If < 1 then (at; bt)!a;b
with probability one.
If > 1 then convergence occurs with probability zero.
ñ Property < 1 referred to as expectational stability
ñ E-Stability principle governed by mapping between agent beliefs and true modelcoe¢cients
E-Stability principle
Recall
Et1pt = at1 + bt1wt1 (4)
and
pt = (+ at1) + ( + bt1)wt1 + t
Latter expression implies optimal rational forecast
Et1pt = (+ at1) + ( + bt1)wt1
Mapping DeÖned
Agentsí beliefs and optimal forecast deÖne the mapping
T
a
b
!
=
+ a
+ b
!
ñ An REE is a Öxed point of this mapping
ñ Application of method of undetermined coe¢cients
E-Stability principle II
DeÖne the associated ordinary di§erential equation
d
d
a
b
!
= T
a
b
!
a
b
!
where is notional time
ñ Local stability properties govern E-Stability principle
E-Stability principle III
In particular, the REE is E-Stable if and only if the ODE is local stable ata;b
Henceda
d= + ( 1) a
db
d= + ( 1) b
requires < 1
Recursive Least Squares
Ordinary least squares regression can be written as
t = t1 + t1Rtzt1pt 0t1zt1
Rt = Rt1 + t1zt1z
0t1 Rt1
where t = [at bt]0
Recursive Least Squares II
Since
pt = (+ at1) + ( + bt1)wt1 + t
= Tt1
0zt1 + t
we have
t = t1 + t1R1t zt1z0t1
Tt1
t1
+ t
Rt = Rt1 + t1zt1z
0t1 Rt1
ñ Question: does this system converge??
Stochastic Approximation Methods
Provide results characterizing convergence of such systems
ñ Ljung (1977), Marcet and Sargent (1989)
Consider stochastic recursive algorithm
t = t1 + tQ (t; t1; Xt)
where
t - parameter estimates (at; bt; Rt)
Xt - state vector (e§ects of pt, zt and t)
t - deterministic sequence of gains (t1)
Stochastic approximation approach associates the ODE
d
d= h ( ())
where
h () limt!1
EQ (t; ;Xt)
ñ E denotes the expectation of Q (t; ;Xt) with respect to the invariant distrib-ution of Xt for Öxed
Stochastic Approximation Results
Under suitable assumptions:
ñ If is a locally stable equilibrium point of the ODE, then is a possible pointof convergence of the SRA
ñ If is not a locally stable equilibrium point of the ODE, then is not a possiblepoint of convergence of the SRA. That is t ! with probability zero.
Technical Assumptions
Stochastic approximation results rely on assumptions regarding
ñ Regularity conditions on Q
ñ Conditions on the rate at which t ! 0
ñ Assumptions on the properties of Xt
Evans and Honkapohja (2001, Chapter 6 & 7)
Example Revisited
To write our RLS algorithm in SRA form deÖne St1 Rt and write the systemas
t = t1 + t1S1t1zt1z0t1
Tt1
t1
+ t
St = St1 + t1
t
t+ 1
ztz
0t St1
so that
t = vec (t; St)
Xt = [1 wt wt1 t]0
t = t1
Example Revisited II
Hence
Q (t; t1; Xt) = S1t1zt1z0t1
Tt1
t1
+ t
QS (t; t1; Xt) = vec
t
t+ 1
ztz
0t St1
ñ To do: compute the associated ODE!
ñ Fix t and compute expectation over Xt
Example Revisited III
Fixing (; S) implies
h (; S) = limt!1
ES1zt1z0t1 (T () ) + t
hS (; S) = limt!1
E
t
t+ 1
ztz
0t S
Since
Eztz0t = Ezt1z
0t1 =
"1 00
#
M
where = Ewtw
0t
; Ezt1t = 0 and limt!1
tt+1 = 1
h (; S) = S1M (T () )
hS (; S) = M S
The Associated ODE is thend
d= S1M (T () )
dS
d= M S
ñ The latter is globally stable for any initial S: S !M
ñ Therefore S1M ! I
Henced
d= T ()
ñ Intuition?
Summary
Have motivated the concept of E-Stability
ñ Governed by eigenvalues of the associated ODE
Remainder
ñ Microfoundations and modeling issues
ñ Subsequent lectures: Applications in policy design
Modeling Learning Dynamics: Some Issues
Intertemporal optimization implies multi-period forecasts
ñ Fundamental to macroeconomic analysis
Transversality conditions central to this theory
ñ Partial Equilibrium: Marcet and Sargent (1989)
ñ General Equilibrium: Preston (2005)
Yet most of the macro learning literature ignores this requirement of optimal-ity
Beliefs
Under rational expectations:
1. Agents optimize given beliefs
2. The probabilities they assign to future state variables coincide with the predictionsof the model
This paper retains (1) and replaces (2) with
20. Future state variables outside agentís control are forecasted using an econometricmodel.
Knowledge
Know own preferences and constraints
Do not know true economic model of determination of variables outside their con-trol.
ñ Uncertain about the equilibrium mechanisms determining prices and policy vari-ables
ñ E.g. even if Y it = Yjt = Yt in EQ ; agents know Y it = Y
jt
Observe aggregate variables and disturbances
Forecast variables outside their control using an atheoretical VAR
ñ Takes the minimum state variable form
Townsend Investment Problem
Consider a representative Örm solving
maxfkTg
Et
1X
T=t
TtpTkT
2
kT kT1
2
where 0 < < 1 and ; > 0: The price of output, kt, is determined oncompetitive markets according to
pt = Kt + ut
ut = ut1 + "t
where 0 < < 1 and Kt the aggregate capital stock. Equilibrium requires:
kt = Kt =
1Z
0
kt (i) di
ñ Assumption: Örms do not know market mechanism determining prices
Townsend Investment Problem II
To solve this problem a Örm requires forecasts of future capital prices and statevariables
ñ Prices are exogenous to their decision problem
ñ As are the aggregate capital stock and stochastic disturbance
Denoting the set of variable beyond the Örmís control as
zt fpt;Kt; utg
assume forecasting model of the form
zt = at + btzt1 + t
ñ Use data available until period t 1
Forecasts are then constructed according to
EtzT = (I3 )1I3 bTtt1
at1 + bTtt1 zt
ñ Note that it is assumed that when making decisions over the forecast horizonbeliefs are held Öxed
ñ Formally we solve an anticipated utility problem in the sense of Kreps andSargent
ñ That agents do not account for subsequent revisions in their beliefs when makingdecisions in the current period represents the non-rationality of agents decisions
Townsend Investment Problem III
The Örst-order conditions of the Örmís problem are
kt kt1 = Et (kt+1 kt) +
pt
and
limT!1
EtTkT = 0
Solving the Örst condition forward and applying the second gives the optimal in-vestment rule
kt = kt1 +
Et
1X
T=t
TtpT
ñ This is the optimal rule conditional on arbitrary beliefs: holds under rationalexpectations
Rational Expectations Solution and Learning
Straightforward to show capital is determined by
kt = Kt1 +
( )ut
where 0 < < 1
ñ Under learning dynamics, optimal decisions converge to this allocation withprobability 1
Alternative Approaches
The predominant approach to modeling learning in macroeconomics is to commencewith Euler equations implied by a rational expectations analysis.
In the context of the Townsend problem, commence with
kt kt1 = Et (kt+1 kt) +
pt
as the primitive behavioral assumption
Requires a forecasting model
kt = ~at +~btKt1 + ~ctut1 + t
so that
Etkt+1 = ~at1 + ~bt1Kt + ~ct1ut
Alternative Approaches II
This second approach is fundamentally distinct from the Örst. By ignoring thetransversality condition agents make suboptimal decisions
Represents a basic confusion of the economics: Örms forecast their own endogenousdecisions variable using an arbitrary model
ñ Represents a contradiction of what agents know to be true about their decisionproblem
ñ Forecasts of endogenous variable induces by optimal decision rules and primitiveassumptions about forecasting variables beyond the control of the decision maker
It is not that the Euler equation is irrelevant. Note that quasi di§erencing theoptimal decision rule yields
kt = kt1 +
Et
1X
T=t
TtpT
= kt1 +
pt +
EtEt+1
1X
T=t+1
TtpT
= kt1 +
pt +
Et (kt+1 kt)
which is the Euler equation.
Optimality requires capital to be forecast in a particular way. SpeciÖcally
Etkt+1 = kt +
Et
1X
T=t+1
TtpT
ñ In general this will not be equal to
Etkt+1 = ~at +~btKt + ~ctut
Model Agents
Households
Firms
Monetary authority
Fiscal authority
Model Features
Money or cashless limit
Monopolistic competition/nominal rigidities
Incomplete asset markets
No capital
Non-rational expectations
Beliefs
Under rational expectations:
1. Agents optimize given beliefs
2. The probabilities they assign to future state variables coincide with the predictionsof the model
This paper retains (1) and replaces (2) with
20. Future state variables outside agentís control are forecasted using an econometricmodel.
Knowledge
Know own preferences and constraints
Do not know true economic model of determination of variables outside their con-trol.
ñ E.g. even if Y it = Yjt = Yt in EQ ; agents know Y it = Y
jt
Observe aggregate variables and disturbances
Forecast variables outside their control using an atheoretical VAR
ñ Takes the minimum state variable form
Household Problem
Households seek to maximize
Eit
1X
T=t
TthUCiT ; T
v
hiT ; T
i
subject to
Bit = (1 + it1)Bit1 + wth
it +
Z 1
0t (j) dj Tt PtC
it
= (1 + it1)Bit1 + PY it Tt PtC
it
ñ Beliefs: a-theoretical VAR of exogenous variables ó nests MSV REE
ñ Assume agents forecast period income
First-order conditions
Log-linear approximation yields
Cit = EitCit+1
it Eitt+1 gt + Eitgt+1
and
Eit
1X
T=t
TtCiT = Ait + Eit
1X
T=t
TtY iT
where
Ait = Bit=PtY ; gt = tuc=uc; 1 = ucc Y =uc
ñ Assume government debt in zero net supply
Deriving the Consumption Decision Rule
Solving the Euler equation backwards recursively from date T to date t and takingexpectations at that time gives
EitCiT = Cit gt + Eit
2
4gT + T1X
T=t
({t t+1)
3
5
ñ Use this to eliminate future expected consumption in the intertemporal budgetconstraint
Optimal decision rule
First-order conditions imply
Cit = (1 )Ait +
Eit
1X
T=t
Tth(1 ) Y iT
{T T+1
+
gT gT+1
i
ñ This is an example of permanent income theory
ñ Forecasts about prices into the indeÖnite future matter!
ñ A particularly important source of uncertainty are anticipated future interestrates
Comparison to Existing Approaches
Are decisions implied by the optimal decision equivalent to those implied by
Cit = EitCit+1
it Eitt+1 gt + Etgt+1
ñ Where forecasts determined by the model
zt = at1 + bt1zt1 + "t
with zt =hCit t it gt
i0and zt =
hYt t it gt
i0
Comparison to Existing Approaches II
Are decisions implied by the optimal decision equivalent to those implied by
Cit = EitCit+1
it Eitt+1 gt + Etgt+1
ñ Where forecasts determined by the model
zt = at1 + bt1zt1 + "t
with zt =hCit t it gt
i0and zt =
hYt t it gt
i0
In general - No!
Optimal decision rule if expectations treated correctly
ñ i.e. the conditional expectation of consumption induced by model primitivesand optimality
Forecasting Consumption
The optimal forecast
EitCit+1 = (1 ) EitA
it+1 +
Eit
1X
T=t+1
Tth(1 ) Y iT
{T T+1
+
gT gT+1
i
Not the same as
Eitzt+1 = at1 + bt1zt
ñ Conditional expectations of endogenous variables induced by policy function andforecasts about variable the are exogenous to the agentís decision problem
Commentary
Market clearing conditions are part of the REE
One-period-expectations approach does not deliver optimal decision rule
ñ Confuses endogenous decision variables and exogenous state variables
ñ The correct distribution with respect to which expectations are taken is inducedfrom the decision problem and properties of exogenous variables
ñ In general intertemporal budget constraint will be violated in this formulation
Expected future consumption plans are inconsistent with what agents knowto be true about their own future behavior given their own decision problem
Aggregate Consumption Dynamics
Aggregating over the continuum provides
Ct = Et
1X
T=t
Tth(1 ) YT
{T T+1
+
gT gT+1
i
where
Ct ZCitdi; Et
ZEitdi; 0
ZAitdi
ñ For any variable Xt, EtEt+1Xt+1 6= EtXt+1
Summary
Under rational expectations equilibrium probability laws by construction satisfy allrelevant constraints
ñ In particular: Transversality conditions
Under learning dynamics expectations about future endogenous decision variablesare taken with respected to a distribution
ñ Induced by beliefs about exogenous states
ñ Induced by optimal decisions
Firms
Continuum of Örms maximize
Ejt
1X
T=t
TtQt;ThjT (pt(j))
i
where
jT (p) = YTP
Tp1 wiTf
1YTP
Tp=AT
yit = Atfhit
Optimal Price Setting
Log-linear approximation implies
pjt = E
jt
1X
T=t
()Tt
2
4(1 )
! + 1
1 + !xT + T
3
5
ñ InÖnite horizon forecasts matter
ñ xt = Yt Y nt is the output gap
Model Summary
Aggregate demand and supply:
xt = Et
1X
T=t
Tt(1 )xT+1
iT T+1 rT
t = xt + Et
1X
T=t
()TtxT+1 + (1 )T+1 + ut
where
rt = rt1 + "t and 0 < < 1
ut = ut1 + t and 0 < < 1
Rational Expectations
Under this assumption EtEt+1Xt+1 = EtXt+1. Hence
xt = Et [(1 )xt+1 (it t+1 rt)] +
Et
1X
T=t+1
Tt(1 )xT+1
iT T+1 rT
= Et(1 )xT+1 (it t+1 rt)
+ Etxt+1
= Etxt+1 (it Ett+1 rt)
Rational expectations analysis gives
xt = Etxt+1 (it Ett+1 rt)
t = kxt + Ett+1
Bullard and Mitra (2002, JME) and Evans and Honkapohja (2003, ReStud)
xt = Etxt+1 it Ett+1 rt
t = kxt + Ett+1
Does any of this matter?
Objectives
Implications of Learning for the choice of monetary policy rule
The role of communication in policy design
Benchmark Model: Optimal Decisions
Aggregate demand and supply:
xt = Et
1X
T=t
Tt(1 )xT+1
iT T+1 rT
t = xt + Et
1X
T=t
()TtxT+1 + (1 )T+1 + ut
where
rt = rt1 + "t
ut = ut1 + t
are known exogenous processes with 0 < ; < 1 and "t, t iid disturbances
Implicitly assumes Ricardian Öscal policy
ñ Agents need not forecast debt or taxes
Alternative Model: ìOne-Period-Ahead Forecastsî
Interested in comparing policy emerging under the anticipated utility approach tothe more familiar approach given by
xt = Etxt+1 it Ett+1 rt
t = kxt + Ett+1
ñ Bullard and Mitra (JME, 2002), Evans and Honkapohja (ReStud, 2003)
Issues in Policy Design
Early literature concerned with policy robustness and equilibrium selection
ñ Given that there are a multiplicity of rules consistent with implementing a givenequilibrium, can impose additional desiderata
ñ Robustness to small expectational errors
Models with learning also permit addressing critiques of policy
ñ Friedman (1968) ó Monetary policy procedures based on interest-rate rulesprone to Wicksellian cumulative process
An Interest-Rate Peg: Optimal Decisions
Consider the policy it = rrt where rt is iid and no ut
ñ Indeterminacy of rational expectations equilibrium: Sargent and Wallace (1975)
Agent forecasts given by
EtzT = at1
for zt = [t xt it]0 : Combined with structural model gives mapping
da
d= T (a) a
db
d= T (b) b
ñ E-Stability determined by the eigenvalues of the Jacobian of these ordinarydi§erential equation
Conditions on Eigenvalues
Consider the matrix A with dimension (3 3) : From jA Ij = 0 the character-istic equation is
3 c12 + c2 c3 = 0
where c1 =Trace(A), c2 is the sum of all second-order principal minors of A andc3 = jAj. The following restrictions on the coe¢cients ci must be satisÖed for alleigenvalues to have negative real parts:
c1 < 0
c3 c1c2 > 0
c3 < 0:
For a matrix A with dimension (2 2) ; jA Ij = 0 implies the characteristicequation is
2 c1+ c2 = 0
where c1 =Trace(A) and c2 = jAj. For both eigenvalues to have negative realparts c1 < 0 and c2 > 0 must be satisÖed
Stability Analysis
Straightforward to show
T 0 (a) =
2
6664
(1)1 +
11 +
11 1
10 0 0
3
7775
and
det(T 0 (a) I) =
1 +
(1 ) (1 )> 0
ñ Expectations driven áuctuations
ñ What about the Euler equation approach: Evans and Honkapohja (2003)
An Interest Rate Peg: One-Period-Ahead Expectations
Consider the policy it = rrt with rt i.i.d.
Agent forecasts given by
Etzt+1 = at1
for zt = [t xt]0 : Combined with structural model gives mapping
da
d= T (a) a
db
d= T (b) b
An Interest Rate Peg II
The true data generating process is"xtt
#
=
"1
( + )
# "ax;t1a;t1
#
" (r 1) (r 1)
#
rt
which imply forecasts
Et
"xt+1t+1
#
=
"1
( + )
# "ax;t1a;t1
#
DeÖning the T-map
T
axa
!
=
"1
( + )
# "axa
#
Expectations stability requires the real parts of the eigenvalues of T 0 (a) I to benegative
ñ That is: detT 0 (a) I
> 0 and trace
T 0 (a) I
< 0
The determinant is
detT 0 (a) I
= < 0
ñ The minimum-state-variable solution is not learnable
Taylor Rule
McCallum (1983): conditioning on endogenous variables resolves indeterminacy
Consider the rule
it = t + xxt
ñ Requires
( 1) + (1 )x > 1
for determinacy under REE and stability under learning
ñ Bullard and Mitra (2002), Preston (2005)
Emerging di§erences
Policy actions that depend on misspeciÖed beliefs can lead to self-fulÖlling expec-tations
Investigate a range of policies that are consistent with implementing the optimalcommitment equilibrium under rational expectations
ñ Implementation of such policy involve a dependence of interest-rate policy onexpectations
ñ This will reveal some sharp di§erences across the anticipated utility approachand the one-period-ahead expectations approach
Optimal Commitment Policy under Rational Expectations
Central bank is assumed to minimize the loss function
Et0
1X
t=t0
tt02t + x2t
(5)
where 0 < < 1 for some weight > 0 subject to
t = kxt + Ett+1 + ut
t0 = t0 = (1 )
xt1 +
1 ut
ñ Optimality from the ìtimeless perspectiveî: Woodford (2003, chap. 6) andGiannoni and Woodford (2002, 2012)
Optimal Commitment Solution
Standard methods show that the optimal state-contingent paths of ft; xt; itg aregiven by the following relations for all t t:
t = (1 )
xt1 +
1 ut
xt = xt1
1 ut
and
it =
(1 ) xt1 +
(+ 1)1
ut +1
rnt
where 0 < < 1 is the modelís only eigenvalue within the unit circle.
Optimal Commitment Solution II
Useful to represent this solution in terms of a particular exogenous state variable
Solving the output gap relation backwards recursively gives
xt =
1
1X
j=0
jutj:
DeÖning the exogenous state variable
t 1X
j=0
jutj;
which satisÖes the process t = t1 + ut
Allows the optimal solution to be written as
t =
1 [ut (1 ) t1]
xt =
1 [t1 + ut]
it =
1 [ut (1 ) (t1 + ut)] +
1
rnt
where denotes the optimal solution when expressed as a linear function of
ft1; ut; rtg :
ñ Note the question of what policy implements the optimal commitment equilib-rium is yet to be answered
ñ We have only described what that equilibrium looks like
Candidate Policy Rules
Consider the following instrument rule as a means to implement the optimal equi-librium:
it = it + x(Ecbt xt+1 Etx
t+1) + (E
cbt t+1 Et
t+1)
= {ft + xE
cbt xt+1 + E
cbt t+1
where Ecbt denotes the forecasts that the central bank intends to respond to and
{lt = it xEtxt+1 Et
t+1
collects exogenous terms and Etxt+1 and Ett+1
ñ Recall ft ; xt ; i
tg denote the optimal paths for each endogenous variable in
the optimal commitment equilibrium when written as a linear function of theexogenous state variables ft1; ut; rtg.
Candidate Policy Rules II
Expectations based rules argued to be desirable and given empirical support
ñ Batini and Haldane (1999), Levin, Wieland and Williams (2003), Clarida, Galiand Gertler (1998, 2000)
Assume Ecbt = Et where the forecasts are described below
The policy parameters ; x determine the response to deviations of the actualpath of the ináation rate and output gap expectations from that path consistentwith the optimal equilibrium
ñ In the desired equilibrium it is clear that Ecbt t+1 = Ett+1, E
cbt xt+1 =
Etxt+1 and it = it
Forecasts
Agents therefore estimate the linear model
zt = at + bt zt1 + ct ut + dt rt + t (6)
where zt = (t; xt; it;t)0, t is the usual error-vector term, fat; bt; ct; dtg are
parameters to be estimated of the form
at =
2
6664
a;tax;tai;ta;t
3
7775 ; ct =
2
6664
c;tcx;tci;tc;t
3
7775 ; dt =
2
6664
d;tdx;tdi;td;t
3
7775
and
bt
2
6664
0 0 0 b;t0 0 0 bx;t0 0 0 bi;t0 0 0 b;t
3
7775
Forecasts II
Believes revised according to recursive least squares formulation
t = t1 + t1R1t wt1(zt1 w0t1t1)
Rt = Rt1 + t1(wt1w0t1 Rt1)
where t = (at; b4t; ct; dt)and wt f1;t1; ut; rtg0.
Given homogeneity of beliefs, average forecasts can then be constructed by solving(9) backward and taking expectations to give
EtzT = (I4 bt1)1 (I4 bTtt1 )at1 + bTtt1 zt
+ut (I4 bt1)1TtI4 bTtt1
ct
+rt (I4 bt1)1TtI4 bTtt1
dt
for T t, where I4 is a (4 4) identity matrix:
Properties
A unique bounded rational expectations equilibrium will obtain i§
0 < ( 1) + (1 ) x < 2 (1 + )
Under learning a necessary condition for E-Stability is
+ x
>
1
1 2
(1 ):
ñ Clearly violates the Taylor principle; consider the limit case of ! 1
ñ Di¢culty: by having central bank behavior depend on ìmisspeciÖedî privatebeliefs makes self-fulÖlling expectations more likely
Note central regulator of how shifting expectations impact current outcomes
Properties II
What about the one-period-ahead expectations models?
Can show that under the same rule expectation stability obtains if and only if
( 1) + (1 ) x > 1
ñ That is ó the Taylor principle holds
ñ Fundamentally di§erent predictions: arises from the one-period-ahead expecta-tions approach gutting the model of economics. For example, expectations offuture interest are irrelevant to dynamics in that approach
Raises important questions
What is the appropriate use of forecasts in policy design?
Can instability be mitigated?
ñ Optimal policy/targeting rules imply a particular kind of dependency of interestrate decisions on forecasts
ñ The role of communication
ñ The role of debt policy
Di§erent statistical properties
ñ Forecasts, dynamics and policy evaluation
Optimal Internal Forecasts
Consider a sophisticated central bank that understands private-sector expectationsformation
Substituting the private forecasts into the structural relations implies the outputgap and ináation to be given by
t = a +bt1 + cut + drt + eit (7)
xt = ax +bxt1 + cxut + dxrt + exit (8)
where the coe¢cientsai;bi; ci; di; ei
for i 2 f; xg are functions of the coe¢-
cients (at; bt; ct; dt)
These equations give the period t ináation rate and output gap realizations, condi-tional on the current choice of the interest rate.
Can then compute ìoptimalî forecasts as
Ecbt t+1 = a +bt + cut + drt + eEcbt it+1
Ecbt xt+1 = ax +bxt + cxut + dxrt + exEcbt it+1:
Substitution of these relations into
it = {ft + xE
cbt xt+1 + E
cbt t+1
yields an equation of the form
it = ( a + xax) + ( b + xbx)t + ( c + xcx)ut
+( d + xdx)rt + ( e + xex)E
cbt it+1
ñ Has unique bounded solution if j e + xexj < 1
ñ Determines an interest-rate decision that is a function of agentsí learning be-havior
Optimal Internal Forecasts II
If the monetary authority constructs optimal internal forecasts then the Taylor prin-ciple
( 1) + (1 ) x > 0
is necessary and su¢cient for stability under learning dynamics.
ñ Underscores an important point: understanding the details/mechanics of expec-tations formation critical
ñ Policy needs to depend on expectations in a very speciÖc way to ensure stability
Implementing Optimal Policy using Target Criteria
Recent monetary economics literature has proposed implementing policy using tar-get criteria
ñ See, for example, Svensson and Woodford (2005), Svensson (2003), Giannoniand Woodford (2002, 2012) and Woodford (2003, chp. 7)
The approach speciÖes a criterion that must be checked each time an interest-ratedecision is made
ñ For example: Bank of Englandís ináation forecast targeting
ñ Requires assumption about what model the central bank has when implementingpolicy
ñ The approach gives very speciÖc guidance on how forecasts should be incorpo-rated in policy decisions
Deriving the Target Criterion
Recall the optimal commitment equilibrium implies the dynamics
t = (1 )
xt1 +
1 ut
xt = xt1
1 ut
it =
(1 ) xt1 +
(+ 1)1
ut +1
rnt
Deriving the Target Criterion II
Giannoni and Woodford (2002) and Woodford (2003, chap 7) show
t =
(xt xt1)
fully characterizes the optimal equilibrium from the timeless perspective in the sensethat the previous dynamics hold for all t t0 if and only if the above target criterionholds
ñ Take this as the desired target criterion: under rational expectations impliesdeterminacy for all parameter values
ñ It is a restriction on endogenous variables ó central bank requires a model toimplement
ñ History dependence of optimal commitment captured by the lagged output gap
Forecasts
Agents therefore estimate the linear model
zt = at + bt zt1 + ct ut + dt rt + t (9)
where zt = (t; xt; it)0, t is the usual error-vector term, fat; bt; ct; dtg are para-
meters to be estimated of the form
at =
2
64a;tax;tai;t
3
75 ; ct =
2
64c;tcx;tci;t
3
75 ; dt =
2
64d;tdx;tdi;t
3
75
and
bt
2
640 b;t 00 bx;t 00 bi;t 0
3
75
Forecasts II
Believes revised according to recursive least squares formulation
t = t1 + t1R1t wt1(zt1 w0t1t1)
Rt = Rt1 + t1(wt1w0t1 Rt1)
where t = (at; b2t; ct; dt)and wt f1; xt1; ut; rtg0.
Given homogeneity of beliefs, average forecasts can then be constructed by solving(9) backward and taking expectations to give
EtzT = (I3 bt1)1 (I3 bTtt1 )at1 + bTtt1 zt
+ut (I3 bt1)1TtI3 bTtt1
ct
+rt (I3 bt1)1TtI3 bTtt1
dt
for T t, where I3 is a (3 3) identity matrix:
The Central Bankís Model
A number of assumptions could be made. The central bank could:
ñ Make projections based on minimum-state-variable rational expectations equi-librium
ñ Make projections based on the structural equations implied by the learning model
ñ Make projections based on the assumption that the Euler equations implied byrational expectations are valid.
Projecting under Rational Expectations
Suppose the central bank mistakenly assumes agents have rational expectations
If it projects the evolution of the economy assuming the minimum-state-variableequilibrium the target criterion will be satisÖed under the interest-rate policy
it =
(1 ) xt1 +
(+ 1)1
ut +1
rnt
ñ This is not an exogenous instrument rule ó depends on lagged output gap
ñ Concerns of Sargent and Wallace (1975) need not be valid
ñ Result: > su¢cient for instability under learning
Projecting under the True Model
Suppose the central bank understands the true structural model ó that is, theaggregate demand and supply equations, and the precise details of agents beliefformation
The central bank can then guarantee satisfaction of the target criterion by settinginterest rates according to
it = 1
xt +
1
Et
1X
T=t
Tt(1 )xT+1 ( iT+1 T+1) + rT
where
xt =
+ 2xt1
+ 2Et
1X
T=t
()Tt xT+1 + (1 ) T+1 + uT
:
ñ Second relation is the value of the output gap that jointly satisÖed the aggregatesupply curve and the target criterion
This approach to implementing policy guarantees satisfaction of the target criterionregardless of the nature of expectations formation
ñ Requires an interest-rate setting that depends upon expectations into the indef-inite future
ñ Is consistent with implementing the optimal commitment equilibrium if agentshave rational expectations
Under learning dynamics implies the underlying rational expectations equilibrium isE-Stable for all parameter values
Projecting using Euler Equations
Suppose the central bank thinks the true model is
xt = Etxt+1 it Ett+1 rt
t = kxt + Ett+1
Then the target criterion implies the instrument rule
it =1
Etxt+1
+ 2xt1 +
+ 2+
Ett+1 +
+ 2ut + rnt
What are the stability properties?
Final Remark on Target Criteria
Under rational expectations the target criterion
pt =
xt
implies the same state-contingent responses to disturbances as
t =
(xt xt1)
Under learning they imply di§erent outcomes ó that is, out of rational expectationsequilibrium they imply di§erent policy responses when the central bank Önds itstarget criterion is not met because expectations evolved in a way that was notanticipated
ñ It turns out the price-level targeting is more conducive to stability
Communication as Stabilization Policy
The discussion so far emphasizes that policy must be conditioned on expectationsin certain ways
An important source of instability is that agents do not know how interest rates aredetermined
ñ This is one of the many equilibrium restrictions that they are attempting tolearn
ñ Agents beliefs need not be consistent with monetary policy strategy; coherentstatement of ìunanchoredî expectations
ñ But what if agents understood some details of policy? Does this improve thestability properties of forecast based instrument rules?
Communication as Stabilization Policy II
Suppose the central bank declares that it interest rate decisions are determined by
it = Ett+1
Agents can use this information to restrict their forecasting model. SpeciÖcally,policy-consistent forecasts of the nominal interest rate can be determined as
EtiT = EtT+1
ñ Such ináation and interest-rate forecasts are consistent with monetary policystrategy
ñ They ensure projections satisfy the Taylor principle
Communication as Stabilization Policy III
Policy consistent forecasts imply aggregate demand is
xt = Et
1X
T=t
Tth(1 )xT+1
EtT+1 T+1 rT
i
ñ Agents need only forecast ináation and the output gap
ñ Can show that E-Stability obtains so long as the Taylor principle holds
ñ Communication about future policy intentions can resolve important uncertainty
Taken up in detail by Eusepi and Preston (2010, AEJM)
Now consider simpler analysis by Orphanides and Williams (2005)
Part III: Monetary and Fiscal Interactions
The analysis so far has pushed Öscal policy to the back ground
The remainder of the lecture will consider the consequences of debt managementpolicy for the implementation of monetary policy
ñ Interested in the role of scale and composition of the public debt
ñ What kind of constraints do Öscal policy impose on monetary policy?
Motivation
US Financial Crisis witnessed dramatic change in the nature and conduct of stabi-lization policy
ñ Large-scale asset purchases intended to ináuence long interest rates
ñ Under what conditions can this be e§ective?
Standard models satisfy an irrelevance proposition
ñ Ricardian equivalence; more generally state-contingent consumption indepen-dent of government liabilities
What we do
Simple model of output gap and ináation determination
One informational friction:
ñ Agents have an incomplete knowledge about the economy
ñ Implication: departures from Ricardian Equivalence
Explore constraints imposed on stabilization policy by choice of Öscal policy
ñ SpeciÖcally: Scale and composition of Debt
ñ Are there additional consequences of shortening the maturity structure of debt,over and above the presumed stimulus of lower interest rates
Literature
Monetary and Öscal interactions:
ñ Unpleasant monetary arithmetic: Sargent and Wallace (1980)
ñ FTPL: Leeper, Sims and Woodford (early 90s)
Learning: Marcet and Sargent (1989), Evans and Honkapohja (2001)
Extensive literature on stability of expectations in NK models
ñ Howitt (1992), Bullard and Mitra (2002)
ñ Evans and Honkapohja (2011): survey of recent literature
Model Agents
Households
Firms
Monetary authority
Fiscal authority
Beliefs
Under rational expectations:
1. Agents optimize given beliefs
2. The probabilities they assign to future state variables coincide with the predictionsof the model
This paper retains (1) and replaces (2) with
20. Future state variables outside agentís control are forecasted using an econometricmodel.
Knowledge
Know own preferences and constraints
Do not know true economic model determining variables outside their control.
ñ E.g. even if Y it = Yjt = Yt in EQ ; agents know Y it = Y
jt
Observe aggregate variables and disturbances
Forecast variables outside their control using an atheoretical VAR
ñ Takes the minimum state variable form
Asset Structure and the Fiscal Authority
Exogenous purchases of Gt per period
Issue two kinds of debt
ñ Bst : One period debt in zero net supply with price Pst = (1 + it)
1
ñ Bmt : An asset in positive supply that has the payo§ structure
T(t+1) for T t+ 1
Let Pmt denote the price of this second asset. Asset has the properties:
ñ Price in period t+ 1 of debt issued in period t is Pmt+1
ñ Average maturity of the debt is (1 )1
Monetary and Fiscal Authorities
Flow budget constraint
Pmt Bmt = Bmt1 (1 + Pmt ) PtSt
Fiscal policy maintains intertemporal solvency (ëPassiveí)
it = i Bmt1Bm
!il
; il > ;il
Monetary policy controls ináation (ëActiveí)
1 + it
1 +{=t
; > 1
Under rational expectations: standard view of monetary policy
Household Problem
Saving and work decision. No capital in the model, only gov. bonds.
Householdsí preferences
U(Ct;Ht) = (1 )1
Ct
1 + CtH
1+t
!1, > 0
where
ñ = 0! GHH; = 1! KPR preferences
ñ 1 ! IES 1; Ct;Ht complements
Household Problem
Household i maximizes
(1 )1 E(i)t
1X
T=t
Tt
Ct(i)
1 + Ct(i)H
1+t (i)
!1
subject to
Pst Bst (i) + Pmt B
mt (i) (1 + Pmt )B
mt1 (i) +Bst1 (i) +
1 Wt
WtHt (i)
+Ptt LSt PtCt (i)
and No-Ponzi condition
limT!1
E(i)t Qit;T
h(1 + PmT )B
mT1 (i) +BsT1 (i)
i=PT 0
Key Equation 1: Consumption Decision
Combining Euler equations, labor supply, budget constraint to log-linear approxi-mation provides
C(i)t (1 )H
(i)t = s1C (; )
Pm Bm
P Y
!
1 1
0
@bm;(i)t1 t + Pmt E(i)t
1X
T=t
TthTLS
LST + TW
WT
{T T+1
i1
A
| {z }Gov. Debt and Taxes
+s1C (; ) (1 ) E(i)t
1X
T=t
TtIWwT + IT
+
1E(i)t1X
T=t
Tt{T T+1
Key Equation 1: Consumption Decision
Combining Euler equations, labor supply, budget constraint to log-linear approx.provides
C(i)t (1 )H
(i)t = s1C (; )
Pm Bm
P Y
!
1 1
0
@bm;(i)t1 t + Pmt E(i)t
1X
T=t
TthTLS
LST + TW
WT
{T T+1
i1
A
| {z }Gov. Debt and Taxes
+s1C (; ) (1 ) E(i)t
1X
T=t
TtIWwT + IT
+
1E(i)t1X
T=t
Tt{T T+1
Key Equation 1: Consumption Decision
Combining Euler equations., labor supply, budget constraint to log-linear approx.provides
C(i)t (1 )H
(i)t = s1C (; )
Pm Bm
P Y
!
1 1
0
@bm;(i)t1 t + Pmt E(i)t
1X
T=t
TthTLS
LST + TW
WT
{T T+1
i1
A
| {z }Gov. Debt and Taxes
+s1C (; ) (1 ) E(i)t
1X
T=t
TtIWwT + IT
+
1E(i)t1X
T=t
Tt{T T+1
Key Equation 1: Consumption Decision
Combining Euler equations., labor supply, budget constraint to log-linear approx.provides
C(i)t (1 )H
(i)t = s1C (; )
Pm Bm
P Y
!
1 1
0
@bm;(i)t1 t + Pmt E(i)t
1X
T=t
TthTLS
LST + TW
WT
{T T+1
i1
A
| {z }Gov. Debt and Taxes
+s1C (; ) (1 ) E(i)t
1X
T=t
TtIWwT + IT
+
1E(i)t1X
T=t
Tt{T T+1
Key Equation 2: Public Debt
Price of government debt
Pmt = Et1X
T=t
()Tt {T| {z }Expectation Hypothesis
Evolution of public Debt
bmt = 1bmt1 t
+ (1 ) {t
1 1
st
+(1 ) Et
1X
T=t
()Tt {T+1
Firms
Monopolistic competition + nominal rigidities (Rotemberg)
Firm j has produces according to
Yt (j) = Ht (j) = (pt (j) =Pt) Yt
pt (j) chosen to maximize expected proÖts, where
t (j) = [pt (j)Wt] (pt (j) =Pt) Yt [pt (j) =pt1 (j) 1]2
Firms
Monopolistic competition + nominal rigidities (Rotemberg)
In aggregate, the Phillips curve
t =
+YC
!
Yt +
+Et
1X
T=t
()Tt wT+1 + (1 )T+1
Learning and Expectations (use ináation as an example)
Rational Expectations (RE)
EREt T+1 = 0 + REb bmt1, where T > t
Learning
EtT+1 = Lc;t1 +
Lb;t1b
mt1 + lagged variables,
Convergence to RE? E-Stability
ñ Use methods of Marcet and Sargent (1989), Evans and Honkapohja (2001)
E-Stability and Learning
Actual Law of Motion:
t = T (Lc;t1;Lb;t1)
"1
bmt1
#
+ :::
Convergence under least-squares learning IFF the associated ODE:
d
d(c;b) = T (Lc ;
Lb )| {z }
Actual Law of Motion
(Lc ;Lb )| {z }
Perceived Law of Motion
is locally stable at the REE of interest: T (REc ;REb ) = (REc ;REb ).
Source of instability: economy as a self-referential system
E-Stability
Numerical study: model with LUMP SUM taxation
Key parameters: 1 = 1=4,and Pm Bm
4 P Y= 1:5
Others:
ñ Preferences and technology: = 0:99; = 0:8 (Price stickiness)
ñ Fiscal Policy: LSl = 1:5
ñ Steady State: C= Y = 0:78
0 2 4 6 8 10 12 14 16 18 201
1.1
1.2
1.3
1.4
1.5
1.6
φ π
4- 1*(1- βρ)- 1
E-stability: KPR vs. GHH preferences
KPR preferences
GHH preferences
Figure 1: E-stability properties in monetary policy ñ debt maturity space. Instabilityregiones lie above the contours.
Aggregate Demand: Increase in Ináation Expectations
Ricardian Households
Ct (1 )Ht =
1Et1X
T=t
TtT T+1
+
+s1C (; )Et
1X
T=t
TtIT
ñ we have substituted for the bond price equation and the monetary policy rule,{t = t
Aggregate. Demand: Increase in Ináation Expectations
Baseline
Ct (1 )Ht =
1 s1C (; )SY
!
Et
1X
T=t
TtT T+1
+
s1C (; )SY Et
1X
T=t
()Tt (T )+
+s1C (; )SYbmt1 + s
1C (; ) (1 ) Et
1X
T=t
Tt (IT TT ) :
Aggregate. Demand: Increase in Ináation Expectations
Baseline
Ct (1 )Ht =
1 s1C (; )SY
!
Et
1X
T=t
TtT T+1
+
s1C (; )SY Et
1X
T=t
()Tt (T )+
+s1C (; )SYbmt1 + s
1C (; ) (1 ) Et
1X
T=t
Tt (IT TT ) :
Aggregate. Demand: Increase in Ináation Expectations
Baseline
Ct (1 )Ht =
1 s1C (; )SY
!
Et
1X
T=t
TtT T+1
+
s1C (; )SY Et
1X
T=t
()Tt (T )+
+s1C (; )SYbmt1 + s
1C (; ) (1 ) Et
1X
T=t
Tt (IT TT ) :
Government Debt: Increase in Ináation Expectations
Using the bond price equation and the monetary policy rule, {t = t, permitsthe evolution of real debt to be written as
bmt = 1bmt1 t
+ (1 )t
1 1
st
+(1 ) Et
1X
T=t
()Tt T+1
which depends on the maturity structure
Government Debt: Increase in Ináation Expectations
Using the bond price equation and the monetary policy rule, {t = t, permitsthe evolution of real debt to be written as
bmt = 1bmt1 t
+ (1 )t
1 1
st
+(1 ) Et
1X
T=t
()Tt T+1
which depends on the maturity structure through (with exp. constant ináation)
(1 ) Et
1X
T=t
()Tt T+1 =(1 )
1 Et
ñ For = 0:99 the right hand side peaks ' 0:9: average maturity of 2 years
ñ For = 1: dynamics independent of its own price.
0 2 4 6 8 10 12 14 16 18 201
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
φ π
4- 1*(1- βρ)- 1
E-stability: KPR preferences with σ = 4
B/4Y = .7B/4Y = 1.5B/4Y = 2.2
Figure 2: E-stability properties in monetary policy ñ debt maturity space. Instabilityregiones lie above the contours which are indexed by average level of indebtedness..
2 3 4 5 6 7 81
1.5
2
2.5
3
3.5
4
4.5
φ π
σ
E-stability: KPR preferences
B/4Y = .7B/4Y = 1.5B/4Y = 2.2
Figure 3: E-stability properties in monetary policy ñ elasticity of substitution space.Instability regiones lie above the contours which are indexed by average level of indebt-edness.
1 2 3 4 5 6 7 80
0.2
0.4
0.6
0.8
1
1.2
1.4
σ
1/sKPRC
1/sGHHC
1/σ- δ/sKPRC
1/σ- δ/sGHHC
Figure 4: Decomposing wealth and subtitution e§ects. The black lines plot s1C , whichindexes the scale of wealth e§ects from holdings of the public debt, for KPR and GHHpreferences. The blue lines give the elasiticy of substitution of consumption deemandwith respect to the real interest rate. Both are plotted as function of .
Forward-looking policy rules
Alternative policy rules that responds to expectations
{t = Ett+1
Here we assume that agents know the policy rule
0 2 4 6 8 10 12 14 16 18 201
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
φ π
4- 1*(1- βρ)- 1
E-stability: forward-looking vs. baseline monetary policy rules
Baseline rule
Forward-looking rule
Figure 5: E-stability properties in monetary policy ñ debt maturity space for a standardTaylor rule and an expectations-based Taylor rule. Instability regions lie above thecontours.
Intuition: Debt Dynamics
Using the bond price equation and the monetary policy rule, {t = Ett+1, per-mits the evolution of real debt to be written as
bmt = 1bmt1 t
+ (1 )Ett+1
1 1
st
+(1 ) Et
1X
T=t
()Tt T+2
ñ Debt depends on expectations regardless of
ñ Destabilizing e§ects on consumption and debt decline monotonically with
Distortionary taxation
Consider the model labor taxes (wt )...
...and assume now: 1 = 1=2
0 5 10 15 20 25 301.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
φ π
4- 1*(1- βρ)- 1
E-stability: distortionary taxes
B/4Y = .7B/4Y = 1.5B/4Y = 2.2
Figure 6: E-stability properties in monetary policy debt maturity space. Instabilityregions lie above the contours indexed by average indebtedness.
Intuition: Distortionary taxation
Same mechanism as before but with a ëtwistí
Phillips curve has an extra term
t =
"
+YC
!
Yt +w
(1 w)wt
#
+
Et
1X
T=t
()Tt wT+1 + (1 )T+1
Changes in tax rates as cost-push ìshocksî, reinforcing the drift in expectations
Conclusion
Uncertainty about policy regime can induce drift in expectations
High debt levels and short to medium maturity debt induce instability
ñ Instability generated through wealth e§ects
Instability of expectations as ëÖscal limití
Fundamentally changes the nature of household and Örm responses to shocks óeven if expectations stable in the long-run
Motivation Revisited
Macroeconomic models depend on household and Örm expectations
Policy design
ñ Not enough to observe some history of beliefs
ñ Need a theory of the determination of beliefs: rational expectations
Rational expectations policy design
ñ Heavy reliance on managing expectations through announced commitments
ñ What are the consequences of imprecise control of beliefs?
Motivation II
For example, Bernanke (2004):
ì[...] most private-sector borrowing and investment decisions depend not onthe funds rate but on longer-term yields, such as mortgage rates and corporatebond rates, and on the prices of long-lived assets, such as housing and equities.Moreover, the link between these longer-term yields and asset prices and thecurrent setting of the federal funds rate can be quite loose at times.î
The Agenda
Simple model of output gap and ináation determination
One informational friction:
ñ Agents have an incomplete knowledge about the economy
Explore constraints imposed on stabilization policy by Önancial market expectations
ñ What if the expectations hypothesis of the yield curve does not hold?
ñ Is the zero lower bound on nominal interest rates an important constraint?
Asset Structure and the Fiscal Authority
Exogenous purchases of Gt per period
Issue two kinds of debt
ñ Bst : One period debt in zero net supply with price Pst = (1 + it)
1
ñ Bmt : An asset in positive supply that has the payo§ structure
T(t+1) for T t+ 1
Let Pmt denote the price of this second asset. Asset has the properties:
ñ Price in period t+ 1 of debt issued in period t is Pmt+1
ñ Average maturity of the debt is (1 )1
Asset Pricing: Theory I
Log-linear approximation of Euler equations for each kind of bond holdings implies
{t = P st = Eit
Pmt Pmt+1
ñ Restriction on asset price movements
ñ Combined with transversality
Pmt = Et1X
T=t
()Tt {T
ñ Asset price is a function of fundamentals: the expectations hypothesis of theterm structure holds; does not imply interest-rate expectations consistent withmonetary policy strategy
ñ Call: Anchored Financial Market Expectations
Optimal Spending Plans I
Assume agents understand Öscal policy is Ricardian
When the expectations hypothesis of the term structure holds ó aggregate demandrelation of the form
Cit = Eit1X
T=t
TtiT T+1
(10)
+s1C (1 ) Eit
1X
T=t
Tt" 1
!1 + 1
wT +
1T
#
ñ Example of permanent income theory
ñ Independent of the average maturity structure of debt; Isomorphic to an econ-omy with one-period debt ó i.e. = 0
ñ Independent on long-debt-price expectations
Asset Pricing: Theory II
Under non-rational expectations and incomplete markets: exist alternative ways toimpose no-arbitrage
Consider pricing the asset directly
{t = EitPmt Pmt+1
in all periods t
ñ Given monetary policy and expectations, asset price determined. No-arbitrageconsistent forecasts of the short-rate can then be determined by
Eit{T = Eit
PmT PmT+1
ñ Because interest rate projections are not directly related to current interest ratesó expectations hypothesis need not hold
Demand then determined by
Cit = Eit1X
T=t
TthPmT+1 PmT
T+1
i(11)
+s1C (1 ) Eit
1X
T=t
Tt" 1
!1 + 1
wT +
1T
#
ñ Nest anchored expectations model when = 0; since Eit{T = EitPT
ñ In this case forecasting bond prices and forecasting interest rates equivalent
As maturity increases this divergence increases
Beliefs Formation
Agents construct forecasts according to
EitXt+T = aXt1
where X =n; w; ; {; Pm
ofor any T > 0.
ñ In period t forecasts are predetermined.
ñ Beliefs are updated according to the constant gain algorithm
aXt = (1 g) aXt1 + gXt
where g > 0
ñ With i.i.d. shocks and zero debt nests the REE
Learning only about the constant
Beliefs Formation II
Under anchored expectations, based on interest-rate data:
Eit{T = ait1
Under unanchored expectations, based on bond-price data:
Et{T = EtPmT PmT+1
= (1 ) aPm
t1
ñ In general they are not equal
ñ Equivalent when = 0
Calibration
Discount factor is = 0:99
Labor supply elasticity 1 = 2
Nominal rigidities = 0:75
Elasticity of demand across di§erentiated goods = 8
Unanchored Expectations and Simple Rules
Consider the case of a simple Taylor rule
{t = t + yxt
Use notion of ìrobust stabilityî
ñ Dynamics converge for a given gain coe¢cient
ñ Distinct from E-Stability
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
10
20
30
40
50
60Stability with a Taylor Rule
ρ (debt duration)
φ π (p
olic
y re
sp. t
o in
flatio
n)φx = 0
φx = .5/4
φx = .5
Figure 7: Robust stability regions for di§erent maturity structures. The three contourscorrespond to di§erent Taylor distinguished by their respond to the current output gap.
Intuition
Aggregate demand can be written
xt = {t + EtPmt+1 + Et
1X
T=t
Tth (1 ) PmT+1 + T+1
i At
+s1C (1 ) Et
1X
T=t
Tt" 1
!1 + 1
wT+1 +
1T+1
#
:
ñ For inÖnite-period debt = 1: depends only on EtPmt+1
ñ Requires aggressive adjustment of current nominal interest rates
Key mechanism: increasing debt maturity implies arbitrage restrictions in the ex-pectations hypothesis are weaker
Optimal Policy under Rational Expectations
Consider target criterion
t = 1xt
Optimal Policy under Rational Expectations II
Target criterion approach implies
ñ Interest-rate policy su¢ciently aggressive to guarantee satisfaction of the targetcriterion
ñ Adjustment of policy in response to asset prices
Both might be thought to confer stabilization advantages
0 0.05 0.1 0.15 0.2 0.25 0.3 0.350
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Stability with optimal Targeting rule under RE
g (constant gain)
ρ (d
ebt d
urat
ion)
InstabilityRegion
StabilityRegion
Figure 8: Stability regions in gain-maturity space for the optimal rational expectationstarget criterion under discretion.
Intuition
Second distinct mechanism at play
ñ For a given average maturity of debt ó higher gains imply greater volatility inexpectations and therefore require more aggressive adjustment in interest rateswhich has destabilizing feedback e§ects
ñ As the average maturity rises, arbitrage relationship deÖning term structureweakens ó implies more aggressive policy feasible
Shifting interest-rate expectations represent an additional constraint on policy
ñ Limits scope to react to current macroeconomic developments
Optimal Policy
Central Bank seeks to minimize
minxt;t;it;P
mt ;a
t ;a
Yt ;a
wt ;a
t
ERE0
1X
t=0
th2t + x (xt x)2 + i (it i)
i
where x; i 0 and xt the output gap and Y = fPm; ig. Minimization subjectto the constraints:
ñ Aggregate demand and supply
ñ No-arbitrage equation
ñ Beliefs
ñ Disturbances: technology and cost-push shocks
Properties of Optimal Policy
Proposition 1 The Örst-order conditions representing a solution to the minimizationof the loss subject to i) the aggregate demand, supply and arbitrage equations;the no-
arbitrage condition; and ii) the law of motion for the beliefs at ; aYt ; a
wt ; a
t have a
unique bounded rational expectations solution for all parameter values. In particular,
model dynamics are unique for all possible gains.
First-order conditions constitute a linear rational expectations model
ñ Can be solved using standard methods
ñ Does not imply that learning is irrelevant for policy outcomes
Special Case: No uncertainty
Two limiting results of interest
ñ When g ! 0 and < 1 then
limT!1
EtT =xx
2 + x (1 )
ñ When g > 0 and ! 1 then
limT!1
EtT = 0
Patient Central Banker replicates the optimal commitment policy under rationalexpectations
ñ Price stability is optimal in the long run
Impulse Response Functions: Reintroducing Uncertainty
Assume i = 0 and x = i = 0
ñ Compare with dynamics under rational expectations
ñ Cost-push shock
0 5 10 15-0.5
0
0.5
1Inflation
0 5 10 15-4
-2
0
2Output Gap
0 5 10 15-2
0
2
4Short-term Interest Rate
0 5 10 15-1
0
1
2
3Interest Rate Spread
Figure 9: Impulse response functions in response to a cost-push shock. Gain = 0.15.Rational expectations: red dotted line; learning with anchored expectations: blue solidline; and learning with unanchored expectations: green dashed line.
0 5 10 15-0.5
0
0.5Inflation
0 5 10 15-4
-2
0
2Output Gap
0 5 10 15-2
0
2
4Short-term Interest Rate
0 5 10 150
0.2
0.4
0.6
0.8
1Interest Rate Spread
Figure 10: Impulse response functions in response to a cost-push shock. Gain = 0.005.Rational expectations: red dotted line; learning with anchored expectations: blue solidline; and learning with unanchored expectations: green dashed line.
Implications II
Anchored expectations:
ñ Expectations hypothesis holds which imposes a constraint on current interest-rate movements
ñ Ináation and output more volatile
Unanchored expectations:
ñ Current interest-rate policy divorced from interest-rate expectations
ñ Ináation and output less volatile; interest rates more volatile
E¢cient Policy Frontiers
Compute
L = V [] + xV [x] + iV [i]
where V [] denotes unconditional variance
Study variation in
V [] + xV [x]
as tolerated variance in interest-rates varies: that is V [i]
ñ Equivalent to studying the original loss function as i varies
0 2 4 6 8 10 12 14 16 18
0.4
0.5
0.6
0.7
0.8
0.9
1
V[ i ]
V[π
] +λ xV[
x ]
Policy Frontiers: cost-push shock
Low debt duration:ρ ∈ [0,.20]
High debt duration:ρ∈ [.90,.98]
RationalExpectations
Figure 11: Policy frontiers as weight on interest rate stability is increased. Exogenousdisturbance is a cost-push shock.
The Zero Lower Bound on Nominal Interest Rates
Compute the unconditional probability of being at the zero lower bound in eachmodel
Requires two additional parameter assumptions.
ñ The average level of the short-term nominal interest rate: 5:4% (annualized),
corresponds to the average rate of the US 3-month T-bill for the period1954Q3-2011Q3
ñ Under RE and i = 0:08 calibrate the volatility of the technology shocks todeliver a standard deviation of output of 1:5% (in log-deviations from its steadystate) and probability of ZLB being 3.5%
0 0.05 0.1 0.15 0.20
20
40Optimal Policy RE
λi
P(i t <
= 0)
0 0.05 0.1 0.15 0.20
0.2
0.4Optimal Policy anchored expectations
λi
P(i t <
= 0)
0 0.05 0.1 0.15 0.20
20
40Optimal Policy unanchored expectations
λi
P(i t <
= 0)
Figure 12: The Ögure shows the unconditional probability of being at the ZLB as afunction of i for the three di§erent models.
Implications
If Önancial market expectations unanchored
ñ Zero lower bound likely to a relevant constraint on monetary policy
ñ True even if substantial weight placed on losses from such variation
Contrasts markedly with claims that ìZero lower bound on nominal interest rate isof little quantitative relevance in standard New Keynesian modelsî
ñ Chung et. al. 2010, Schmitt-GrohË and Uribe 2007
Conclusion
Failure of beliefs to satisfy the expectations hypothesis of the term structure limitsthe e¢cacy of monetary policy
ñ The pricing of public debt places constraints also on optimal monetary policy
ñ Can make the zero lower bound constraint on nominal interest rates more severe
