Prof. Dr. Kai Carstensen and Prof. Dr. T. Wollmershäuser
New Keynesian Macroeconomics Chapter 5: Monetary Policy Design in the
New Keynesian Model
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 2
Monetary Policy Design
Literature (textbooks) Galí, J. (2008), “Monetary Policy, Inflation, and the Business
Cycle”, Princeton University Press, Chapters 4 and 5
Walsh, C. (2003), “Monetary Theory and Policy”, The MIT Press, Chapter 11
Woodford, M. (2003), “Interest and Prices”, Princeton University Press, Chapters 6, 7 and 8
How should monetary policy be conducted? Optimal policy rules
• response to unobservable variables required
Simple policy rules
• response to a small set of observable variables
Welfare function
• evaluation of different policy rules
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 3
Monetary Policy Design
The NK model (summary of Chapter 4)
NKPC:
DIS:
where and
1 t t t tE y
1 1
1
n
t t t t t t ty i E r E y
1 1ln
11
1
n n n n
t t t ya t t ya a t t
a t
r E y E a a a
a
1
1
0
t t t t tY i A N i A N i
n
t t ty y y
1 1 1
1
1ln ln a
t t a t ta A A
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 4
Monetary Policy Design
Lucas critique
“Given that the structure of an econometric model consists of
optimal decision rules of economic agents, and that optimal
decision rules vary systematically with changes in the
structure of series relevant to the decision maker, it follows
that any change in policy will systematically alter the structure
of econometric models.”
• Lucas, R. (1976), “Econometric Policy Evaluation: A Critique”, in:
Brunner, K. and A. Meltzer, “The Phillips Curve and Labor
Markets”, Carnegie-Rochester Conference Series on Public
Policy, 1, New York: American Elsevier, pp. 19–46.
The introduction of rational expectations in macroeconomics
in the middle of the 1970s represented a serious challenge
for large-scale, backward-looking econometric models that
were used for policy analysis.
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 5
Monetary Policy Design
Lucas critique
The observable reduced form of the economy can be
represented by 𝑌𝑡+1 = 𝐹 𝑌𝑡, 𝑋𝑡, 𝜃, 𝑢𝑡 , where 𝑌𝑡 is a vector of
economic variables, 𝑋𝑡 is a vector of policy instruments, 𝜃 is a
parameter vector, and 𝑢𝑡 represents random shocks.
A policy rule for setting the policy instrument can be given by
𝑋𝑡 = 𝐺 𝑌𝑡, 𝑔, 𝜉𝑡 , where 𝑔 is a vector of policy rule coefficients,
and 𝜉𝑡 is a random shock.
Lucas argued that “scientific, quantitative policy evaluation”
required comparing alternative policy rules, that is, examining
changes in 𝑔 while taking into account agents‟ expectations
of future policy actions.
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 6
Monetary Policy Design
Lucas critique
He stressed that, “A change in policy [in 𝑔] affects the
behavior of the system in two ways: first by altering the time
series behavior of [𝑋𝑡]; second by leading to modification of
the behavioral parameters [𝜃(𝑔)] governing the rest of the
system.”
• The first effect is the obvious direct influence of the change in the
policy rule on the dynamics of the system.
• The second expectational effect captures the fact that changes in
the policy rule should alter agents‟ expectations of the future and,
hence, change the reduced-form dynamics of the economy.
This sensitivity of the reduced form to the expectational
effects of structural policy changes is the essence of the
Lucas critique.
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 7
Monetary Policy Design
Lucas critique
Because the parameters of reduced-form models are not
structural, i.e. not policy-invariant, they would necessarily
change whenever policy was changed.
• Lucas argued that changes in policy have an immediate effect on
agents‟ decision rules since they are inherently forward-looking
and adapt to the effects of the new policy regime.
• An important corollary of this argument is that any policy
evaluation based on backward-looking macroeconomic models is
misleading whenever such policy shifts occur.
The Lucas critique suggests that if we want to predict the
effect of a policy experiment, we should model and estimate
the “deep parameters” (relating to preferences, technology
and resource constraints) that govern individual behavior.
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 8
Monetary Policy Design
Lucas critique
We can then predict what individuals will do, taking into
account the change in policy, and then aggregate the
individual decisions to calculate the macroeconomic effects of
the policy change.
• Changes in policy will have effects on the reduced-form dynamics
of the model.
• This is important for normative policy analysis.
The Lucas critique encouraged macroeconomists to build
microfoundations for their models.
• Macro models of the New Keynesian type are microfounded.
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 9
Monetary Policy Design
Contents of Chapter 5
a. Sources of suboptimality
b. Optimal monetary policy I
c. Simple policy rules I
d. The welfare function
e. Monetary policy tradeoffs
f. Optimal monetary policy II (discretion versus commitment)
g. Simple policy rules II
h. Monetary policy under uncertainty
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 10
Monetary Policy Design
Sources of suboptimality
The New Keynesian model departs from a classical monetary
model, which corresponds to a conventional RBC model with
a monetary sector, in two important aspects:
1. Monopolistic competition
• Prices are set by firms that optimize their profit.
• They are not determined by an anonymous Walrasian
auctioneer who is trying to clear competitive markets at once.
2. Nominal rigidities (sticky prices)
• Firms are subject to some constraints on the frequency with
which they can adjust their prices.
In the event of macroeconomic disturbances both features
together lead to deviations of the adjustment process from
the optimal path that is implied by the frictionless classical
monetary economy.
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 11
Sources of suboptimality
Efficient allocation (first-best, classical monetary
model)
perfect competition
Flexible price equilibrium (second-best)
monopolistic competition
flexible prices
Staggered price equilibrium
monopolistic competition
staggered prices
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 12
Sources of suboptimality
Efficient allocation (first-best)
consumer optimality condition (utility maximization,
consolidated foc)
firms optimality condition under perfect competition (profit
maximization)
equivalently
,
,
n t tt t t t t
c t t
U WMRS w p y n
U P
1 ln 1tt t t t t t t
t
WMPN A N w p a n
P
1
n t tt t
tt t
n
t t tt
t t
W WMC P
MPNA N
MC W PMC
P MPN
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 13
Sources of suboptimality
consumer optimality condition
the marginal rate of substitution between consumption and labor is the amount of consumption needed to compensate a household if it is asked to supply one more unit of labor, so that in the end it as happy as before (stays on the same indifference curve)
N
C
NM
PN
indifference curve
NM
RS
N
W/P
budget constraint
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 14
consumer optimality condition
the marginal rate of substitution between consumption and labor is the amount of consumption needed to compensate a household if it is asked to supply one more unit of labor, so that in the end it as happy as before (stays on the same indifference curve)
labor supply curve is upward
sloping: household are willing
to work more, the higher the
real wage
Sources of suboptimality M
RS
N W
/P
N
given real wage
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 15
Sources of suboptimality
firms optimality condition
profit maximization: a firm will hire an additional unit of labor
as long as the value of the extra output produced by a worker
is greater than (or just equal to) the cost of the additional unit
of labor.
N
Y
NM
PN
t t tP MPN W
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 16
Sources of suboptimality
firms optimality condition
profit maximization: a firm will hire an additional unit of labor
as long as the value of the extra output produced by a worker
is greater than (or just equal to) the cost of the additional unit
of labor.
t t tP MPN W
N
MP
N
NW
/P
labor demand curve is
downward sloping: firms
want to hire less labor, the
higher the real wage
given real wage
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 17
Sources of suboptimality
Efficient allocation (first-best)
in a symmetric equilibrium the economy‟s average marginal
product of labor coincides with the marginal product for each
individual firm
it is optimal to produce and consume the same quantity of all
goods and to allocate the same amount of labor to all firms
marginal rate of substitution between consumption and labor
is equal
first-best (efficient) output
,
,
n t tt t
c t t
U WMRS MPN
U P
ln 1
1 ln 1
t t t t
e e
t t t
y n a n
y a n
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 18
Sources of suboptimality
Efficient allocation (first-best)
using the production function
1 1 1
1 ln 1 1
1 1
t t t t t t
e
t t
y a n n y a
y a
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 19
Sources of suboptimality
Flexible price equilibrium (second-best)
monopolistic competition implies that firms have some market
power, which allows them to set prices above marginal costs
in the symmetric equilibrium all firms set the same prices
according to the following rule
where the gross mark-up is given by
the optimality condition can be stated as
n tt t
t
WP MC
MPN
11
,
,
n t t tt
c t t
U W MPNMRS
U P
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 20
Sources of suboptimality
Flexible price equilibrium (second-best)
in log-levels
the presence of a markup distortion leads to an inefficiently
low level of output
while both the efficient and the natural level of output vary
over time, the gap between both remains constant
ln 1
1 ln 1
1 ln 1 1
1 1
1
1
t t t t
n n
t t t
n
t t
n e
t t
y n a n
y a n
y a
y y
ln
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 21
Sources of suboptimality
Flexible price equilibrium (second-best)
this inefficiency, which is related to the existence of market
power, can easily be eliminated by an employment subsidy
• costs of labor are subsidized by a rate which effectively lowers
the wages paid by the firms
employment subsidy is financed by means of a lump-sum tax
• a fixed (constant) amount of tax paid by all workers
• which has no impact on the optimality condition (enters the
households‟ budget constraint as a constant term and drops out
when taking first derivatives)
in the symmetric equilibrium all firms set the same prices
according to the following rule
1 tn
t t
t
WP MC
MPN
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 22
Sources of suboptimality
Flexible price equilibrium (second-best)
the optimality condition can be stated as
if the flexible price equilibrium
coincides with the efficient equilibrium
thus, the flexible price equilibrium is efficient
,
, 1
n t t tt
c t t
U W MPNMRS
U P
1 1 1
ln 1 ln 1t t t t
n e
t t
y n a n
y y
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 23
Sources of suboptimality
Staggered price equilibrium
price decisions are staggered à la Calvo
accordingly inflation is positive when firms expect the
deviations of marginal costs from their steady state to be
positive in the future
the steady-state of the model is given by the flexible price
equilibrium, in which prices are set as a markup over nominal
marginal cost and real marginal cost is constant
or in log levels
1
0
kt t kt t t t
k
E mc E mc
1n
t t tP MC MC MC
mc
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 24
Sources of suboptimality
Staggered price equilibrium
the price adjustment equation can be written as
since firms cannot adjust their prices continuously, the
economy‟s average markup
• will vary over time in response to shocks
• will generally differ from the constant frictionless markup
under flexible prices
under sticky prices
1 1 1tt t t t t t t t tE mc E mc mc E mc
1
1
flextflex n t
t t
t t t
W PP MC
MPN W MPN
1
1
tn tt t t t t
t t t
W PP MC
MPN W MPN
prices are not
flexible enough
to keep this ratio
constant
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 25
Sources of suboptimality
Staggered price equilibrium
the economy‟s average markup fluctuates around its steady
state (the frictionless markup)
using (i.e. the subsidy exactly offsets the
monopolistic distortion), we get
and the optimality condition becomes
thus and the efficiency condition is violated
unless ; this leads to short-run fluctuations of output
around its natural level
1
t tt
t t t t
P P
W MPN W MPN
1 1
,
,
n t tt t
c t t t
U WMRS MPN
U P
t tMRS MPN
t
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 26
Sources of suboptimality
Staggered price equilibrium
an alternative interpretation to the inefficiency related to the
staggered price equilibrium: relative price distortions
all agents have the same (symmetric) preferences and
technologies, yet relative prices differ because of a-
synchronized price setting: some firms will adjust prices in
response to new technology, some will not
thus, Pt(i)Pt(j) for any pair of goods (i,j) whose prices have
not been adjusted in the same period
but also Yt(i)Yt(j) and Nt(i)Nt(j) for some pair of goods (i,j)
• recall demand function implying that in
equilibrium t t t tC i P i P C
t t t tY i C i C Y
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 27
Optimal monetary policy I
Can policy contribute to minimize or even eliminate
these fluctuations?
Optimal policy requires that
• the average markup be stabilized at its frictionless level
• prices and quantities (and hence marginal costs) are equalized
across goods
Assumptions
• optimal employment subsidy is in place (the flex-price equilibrium
is efficient)
• no inherited relative price distortions (Pt-1(i)=Pt-1 for all i)
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 28
Optimal monetary policy I
Can policy contribute to minimize or even eliminate
these fluctuations?
The policy implemented
• should promise to always stabilize marginal costs at a level
consistent with firms‟ desired markup
• should be expected to be in place indefinitely
Then, no firm has an incentive to change its price because
• it is currently charging its optimal markup
• it expects to keep doing so in the future without having to change
its price
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 29
Optimal monetary policy I
How does such a policy look like?
we need a policy that guarantees that
in this case inflation would always be zero (at its steady state)
marginal cost are related to the output gap: independent of
the nature of price setting, average real marginal cost can be
expressed as
1 1tt t t t t tE mc E mc
tmc
ln 1
1ln 1
1 1
tt
t
t t t t t t t t
t t
MRSMC
MPN
mc w p mpn y n a n
y a
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 30
Optimal monetary policy I
How does such a policy look like?
under flexible prices and
thus can be expressed as
and the inflation adjustment equation can be written as
where
mc n
t ty y
1
ln 11 1
n
t tmc y a
tmc
1 1
nt t t t tmc mc y y y
1 t t t tE y
1
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 31
Optimal monetary policy I
How does such a policy look like?
thus, a policy that guarantees that is equivalent to a
policy that guarantees that the output gap is
closed
a closed output gap for all t and an inflation rate equal to zero
for all t can be attained if for all t:
in other words, the nominal interest rate (which equals the
real interest rate if inflation is zero) must be equal to the
natural real interest rate
tmc n
t t ty y y
1 1
1
n
t t t t t t ty i E r E y
n
t ti r
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 32
Optimal monetary policy I
Implications of this optimal policy
1. stabilizing inflation is equivalent to stabilizing the output gap
since stabilizing the
output gap is in turn equivalent to stabilizing the
welfare-relevant gap between output and its first-best level
from this follows that a policy that stabilizes inflation around
its zero-inflation steady state is equivalent to stabilizing the
welfare relevant distance of output from first best
• Blanchard, O. and J. Galí (2007), “Real Wage Rigidities and the
New Keynesian Model”, Journal of Money, Credit and Banking,
Supplement to Vol. 39(1), 35-65, refer to this equivalence as the
divine coincidence, which results from the absence of any
disturbances leading to stochastic variations in the gap between
first- and second-best output
1
n
t t t t tE y y
1 1 .e n
t ty y const n
t ty ye
t ty y
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 33
Optimal monetary policy I
Implications of this optimal policy
2. stabilizing output is not desirable
instead, output should vary one for one with the natural level
of output for all t:
the natural level of output is positively correlated with
productivity shocks:
or, if a subsidy offsets the monopolistic distortion:
the nominal interest rate is negatively correlated with at
n
t ty y
1 ln 1 1
1 1
n
t ty a
1 ln 1 1
1 1
n e
t t ty y a
1
1ln 1
1
n n
t t t a tr E y a
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 34
Optimal monetary policy I
Implications of this optimal policy
3. price stability is a feature of optimal policy, even though the
policymaker does not attach any weight to such an objective
price stability rather is a consequence of the efficient
allocation
when prices are sticky the central bank‟s objective is to make
all firms content with their existing prices
• the assumed constraints on the adjustment of those prices are
effectively nonbinding
• distortions due to relative price dispersion are eliminated (i.e.
Pt(i)/Pt(j)=1)
price stability follows as a consequence of no firm willing to
adjust its price
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 35
Optimal interest rate rules
Targeting the natural rate of interest
the central bank adjusts the nominal interest rate one for one
with the natural real interest rate:
• the nominal interest rate is a function of purely exogenous state
variables
such a policy is able to produce the efficient allocation:
however, the solution of the dynamical system associated
with this rule is not unique
• in addition to the efficient allocation, a multiplicity of other
equilibria exist
• it cannot be guaranteed that the efficient allocation will be the one
that will emerge in equilibrium
• interpretation: lack of a nominal anchor
n
t ti r
0t ty t
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 36
Optimal interest rate rules
Targeting the natural rate of interest
some remarks about determinacy
• cashless economy in which money plays no role
– neither as a means of transaction
– nor can the stock of money be used to determine the price
level
• if it is not money that determines the price level, something else
must be in the model
• in a neo-Wicksellian framework the fundamental determinants of
the price level are
– the real factors that determine the equilibrium real rate of
interest
– and the systematic relation between interest rates and prices
established by the central bank‟s policy rule
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 37
Optimal interest rate rules
Targeting the natural rate of interest
uniqueness proof: system of difference equations
in matrix notation
1 1
1 1
1
1
1
1
n
t t t t t t t
t t t t t
t t t t
nt t t t
t t
y E y i E r
y E y EE y
E yi r
1 1 1
11 1
1 -1
1
11 1t t t t t t t t
t t tt t t t t
t t t
t t t
y E y E y E y
EE y E
E y y
E
0
0
A
A
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 38
Optimal interest rate rules
Targeting the natural rate of interest
Blanchard-Kahn condition for determinacy: A dynamic system
has a unique, stationary solution if and only if the number of
eigenvalues of A outside (A0 inside) the unit circle is equal to
the number of forward-looking (non-predetermined) variables
it can be shown that only one eigenvalue of matrix A (A0) is
outside (inside) the unit circle
calculation of the eigenvalues Λ of A0 by solving the
characteristic polynomial
1
-11 1 11
1det
1 1
1
0
0
A AA
2 tr detp 0 0A A
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 39
Optimal interest rate rules
Targeting the natural rate of interest
where
and
both eigenvalues 1 and 2 of A0 are inside the unit circle if
and only if both of the following conditions hold (see LaSalle,
1986, “The Stability and Control of Discrete Processes”, p. 28):
tr
0A det 0A
1 det 1 1 0A
2 tr 1 det 0 0 0A A
always satisfied
determinacy only for a negative
slope of the Phillips curve
1 1 1
1 1
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 40
Optimal interest rate rules
Targeting the natural rate of interest
if the equilibrium is indeterminate, then
• then an infinite number of different possible equilibrium responses
of the endogenous variables to real disturbances exist
• some of these equilibria will lead to fluctuations in inflation and
output which are disproportionately large relative to the size of the
change in the “fundamentals” (technology shock) that has
occurred
• some of these equilibria will lead to fluctuations in inflation and
output in response to random events with no fundamental
significance whatsoever
Lubik, T. A., and F. Schorfheide (2003), “Computing Sunspot
Equilibria in Linear Rational Expectations Models”, in: Journal
of Economic Dynamics and Control, Vol. 28(2), 273-285.
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 41
Optimal interest rate rules
On the solution of rational expectations models • see lecture notes on Numerical Methods, Chapter 1, by Fabrice
Collard (http://fabcol.free.fr) for more details and Matlab codes
the case of |a| < 1
1
1 1 2 1 2 1
2 3
where is an exogenous variable to be specified
solving by forward iteration and
by applying the law of iterated expectations:
t t t t
t
t t t t t t t t t t t
t t t t
y aE y bx
x
E y E aE y bE x aE y bE x
E y aE y bE
2
1
1
0
lim lim
t t
kk i
t t t k t t ik k
i
x
y a E y b a E x
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 42
Optimal interest rate rules
On the solution of rational expectations models
the case of |a| < 1
1
1
t=0
1 for lim to be non-explosive
we impose that lim
this so-called terminal condition excludes the existence of bubbles
as it bounds the sequence of to a stationary process
given
k
t t kk
tt
t
a E y
y
y
1
1
0
that 1 we have
lim 0
and the solution of the model reduces to
lim
k
t t kk
ki
t t t ik
i
a
a E y
y b a E x
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 43
Optimal interest rate rules
On the solution of rational expectations models
the case of |a| < 1
0
1
2
2 for lim to converge
must evolve at a rate lower than
1 lim 1 1 11
let's assume that is an exogenous white-noise shock process
0,
which implies that
ki
t t ik
i
t t i
ki
ki
t
t x
b a E x
E x a
aa a
a
x
x N
E
0 for 1
then the solution takes the form
t t i
t t
x i
y bx
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 44
Optimal interest rate rules
On the solution of rational expectations models
in the case in which |a| < 1, when we impose the terminal
non-explosion condition, we can obtain a unique and
determinate solution of the rational expectations model that
only depends on fundamental shocks:
in the case in which |a| > 1, the solution technique applied so
far, the sum involved in the forward solution is unlikely to
converge:
therefore, the solution must be computed in an alternative
way
1
1
0
lim limk
k i
t t t k t t ik k
i
y a E y b a E x
t ty bx
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 45
Optimal interest rate rules
On the solution of rational expectations models
the case of |a| > 1
1
2
1
1 1 1
introduce the expectational error which is white noise ( 0, ),
and uncorrelated with the information set , , 0, ,
the expectational difference equation
t t t t
t
t i t i
t t t t
y aE y bx
N
y x i
y E y
1 1
1 1
then rewrites
which can be transformed in a backward-looking system
1
since >1 this equation is stable
t t t t
t t t t
y a y bx
by y x
a a
a
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 46
Optimal interest rate rules
On the solution of rational expectations models
the case of |a| > 1
1 1
0
1
the problem with this equation is, however, that for a given
realization of the fundamental shock there is an infinite number
of solutions for the process
WHY?
because there
t t t t
t
t t
by y x
a a
x
y
is an infinite number of specifications of the process
for the expectational error
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 47
Optimal interest rate rules
On the solution of rational expectations models
in the case in which |a| > 1 we can obtain an infinite number
of solutions of the rational expectations model
in this case the solution is said to be indeterminate
note
• that these solutions are stable, implying that the economy always
converges to its long-run solution
• that the dynamics of the economy depends on the volatility of the
non-fundamental shock (the expectational error)
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 48
Optimal interest rate rules
evolution of yt for a
calibrated model:
to get some more “regular”
dynamics, the fundamental
shock is assumed to follow
a stationary AR(1) process:
Matlab: backward.m
0 100 200-1
0
1
Time
=0.0
0 100 200
-1
0
1
Time
=0.1
0 100 200-2
0
2
Time
=0.5
0 100 200-4
-2
0
2
4
Time
=1
On the solution of rational expectations models
1.8, 1, 0.1xa b
1
2
0.95
0,
x
t t t
x
t x
x x
N
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 49
Optimal interest rate rules
Response to endogenous variables
optimal policy not only reacts to variations in the natural real
interest rate, but also to some endogenous target variables:
system of difference equations
n
t t t y ti r y
1 1
1 11
1
1 1
1
1
1
11
1
n
t t t t t t ty
t t t t t t tt t t t
nt t t tt t t y t
t t t t t
y
t t t y
y
y E y i E r
y E y E yE y
E yi r y
y E y E
E y
1t tE
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 50
Optimal interest rate rules
Response to endogenous variables
in matrix notation
since all exogenous variables drop out of the system (due to
the interest rate rule) the efficient allocation is
always a solution to the system
in order to guarantee the uniqueness of the efficient solution,
the Blanchard/Kahn condition must be satisfied (i.e. both
eigenvalues of A0 should be inside the unit circle
the characteristic polynomial of A0 is given by
1
1
t t t
t t t
y E y
E
0A
0t ty t
11
yy
0A
2 tr detp 0 0A A
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 51
Optimal interest rate rules
Response to endogenous variables
where
and
both eigenvalues 1 and 2 of A0 are inside the unit circle if
and only if both of the following conditions hold (see LaSalle,
1986, “The Stability and Control of Discrete Processes”, p. 28):
try
y
0A det
y
0A
0 since 0 1
1 det 1 1y
0A
2 tr 1 det 1 1 0y 0 0A A
always satisfied, even if
response coefficients to
endogenous variables
are equal to zero
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 52
Optimal interest rate rules
Response to endogenous variables
a closer look at the second condition
• if both response coefficients are equal to zero, the condition is
violated and at least one of the eigenvalues of A0 is outside the
unit circle and there is no unique solution
• if the central bank reacts sufficiently strongly to deviations of
inflation and output from their target levels, the associated
equilibrium is likely to be determinate (determinate equilibrium =
locally unique and non-explosive equilibrium)
• a sufficient condition is , that is a sufficiently strong policy
response to inflation (Taylor principle)
the graph on the next slide illustrates the second condition by
plotting the region of determinacy for a calibrated model
1 1 0y
1, 1 0.1275
2 3, 0.99, 1 3, 6 0.0425
1
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 53
Optimal interest rate rules
Response to endogenous variables
determinacy and
indeterminacy
regions
taken from Galí
(2008)
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 54
Optimal interest rate rules
Response to endogenous variables
the unique equilibrium involves
from this follows that the interest rate rule effectively
collapses to
thus the existence of a credible policy threat to react
sufficiently strongly to inflation-output gap developments is
sufficient to prevent any movements in these variables!
moreover, such a rule provides the cashless economy with a
nominal anchor that determines the equilibrium path of
interest rates and prices
0t ty t
n
t ti r t
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 55
Optimal interest rate rules
Response to endogenous variables
a forward-looking policy rule
since all exogenous variables drop out of the system the
efficient allocation is always a solution to the system
for both eigenvalues of A0 to be within the unit circle the
following two conditions must hold
1 1
n
t t t t y t ti r E E y
1
1
t t t
t t t
y E y
E
0A
1 1
1 1
1
1
y
y
0A
1 1 0
1 1 2 1
y
y
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 56
Optimal interest rate rules
Response to endogenous variables
determinacy and
indeterminacy
regions
taken from Galí
(2008)
1 1 0
11
1 1 2 1
2 1 1
y
y
y
y
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 57
Simple policy rules I
Optimal interest rate rules
require that the exogenous state variables of the economy
are observable
here
• the natural rate of interest
• the natural level of output (to compute the output gap)
unrealistic, since it requires exact knowledge of the
economy‟s model, the parameter values, and the realized
value of the shocks
Simple interest rate rules
defined as interest rate rules that makes the policy instrument
a function of observable variables only
do not require the above knowledge
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 58
Simple policy rules I
Taylor-type interest rate rule
the policy instrument is set according to
and is hence a function
• the steady state neutral real rate (a constant, e.g. the average
interest rate over a period)
• the actual inflation rate
• the deviation of actual output from steady state output, which is
typically calculated as some smooth trend of actual output
this reaction function can be formulated in terms of the
welfare-relevant output gap
ˆt t y ti y
t t y t ti y
ˆn n
t y t y ty y y
ˆt ty y y
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 59
Simple policy rules I
Taylor-type interest rate rule
the dynamics of the system can be written as
1 1
1
1 1
1
1 1
1
ˆ
1ˆ1
1
n
t t t t t t t
t t t t
t t y t t
n
y t t t t t t t t
t t t t
n
t t t t t t t
y
t t
y
y E y i E r
E y
i y
y E y E r
E y
y E y E r
E
1 1ˆn
t y t t t ty E r
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 60
Simple policy rules I
Taylor-type interest rate rule
where
this exogenous term is a function of the technology shock
ˆn n
t tr r
ˆn n n
t t t y tr r y y
11
1
1 ln 1 1
1 1
1 ln 1
1
1 1ˆ
1
n
t a t
n
t t
a yn
t t t t
r a
y a
y
r a a
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 61
Simple policy rules I
Taylor-type interest rate rule
in matrix notation (composite error for simplicity)
determinacy
• apart from the constant term, the A0 matrix is identical with the A0
matrix under optimal policy (response to contemporaneous
endogenous)
• thus, the following condition must hold
1
1
ˆt t t n
t t
t t t
y E yr
E
0 0A B
11
11
yy
y
0
0
A
B
1 1 0y
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 62
Simple policy rules I
Taylor-type interest rate rule
guess that the solution of a rational expectations model
with
takes the following form
accordingly
applying the method of undetermined coefficients
which is easy to solve if
1t t t tz E z u 0 0A B
t tz u
1t t tE z u 0C
1t t tu u 0C
t t tu u u
0 0 0
0 0 0
A C B
A C B
00C
0B
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 63
Simple policy rules I
Taylor-type interest rate rule
guess that the solution takes the following form
accordingly
plugging those guesses into the system
where
ˆ ˆn n
t y t t t t ty r r ,
1 1ˆ ˆn n
t t y a t t t t a t tE y r E r ,
1 1
1 1
ˆ1
ˆ
n
t t t t t t t
n
t t t y t t t t
y E y E r
E y E r
1
y
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 64
Simple policy rules I
Taylor-type interest rate rule
solving the first equation for
plugging this into the second equation and solving for
using this to finally obtain
1 1y y a a
y a y a
1 11
y a
a
1
1 1a
a y a a
1y a a
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 65
Simple policy rules I
Taylor-type interest rate rule
the solution of the rational expectations model is given by
the law of motion of the endogenous variables depends on
the policy rule implemented by the central bank
• once the central bank alters its policy (i.e. 𝜙𝜋 and 𝜙𝑦), the
reduced-form dynamics of the economy will change
• this is the essence of the Lucas critique
ˆ1 1
ˆ
n
t a a t t a a t
n
t a t t a t
y r a
r a
1
1 1
1 1
1
a
a y a a
a y
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 66
Simple policy rules I
Taylor-type interest rate rule
the solution of the rational expectations model is given by
simulation of a calibrated model
• calibration
• law of motion of endogenous variables
ˆ1 1
ˆ
n
t a a t t a a t
n
t a t t a t
y r a
r a
2
2 3, 0.99, 1 3, 6 0.0425, 1, 1 0.1275
0.9, 1a
1 ,t a a t t a t
y a a
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 67
Simple policy rules I
Taylor-type interest rate rule
simulation of a calibrated model
• law of motion of additional variables
1
ˆ
ˆ
ˆ
1ˆ ˆ
1
ˆˆ
1ˆ 1
1
t t
t t y t
t t
n
t t t
n n
t t t t t t t
t t t t
n
t a t
i i
i y
y y y
y y y
y y y y y y y a
r i E
r a
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 68
Simple policy rules I
Taylor-type interest rate rule
impulse responses to a technology shock 𝜙𝜋 = 1.5, 𝜙𝑦 = 0.5/4
Matlab: NKM_simplepolicy_solved_IRFs.m
0 5 10-2
-1
0Inflation
0 5 10-0.4
-0.2
0Output gap (tilde)
0 5 10-2
-1
0Nominal interest rate
0 5 100
0.5
1Output gap (hat)
0 5 10-1
-0.5
0Real interest rate
0 5 10-0.1
-0.05
0Efficient real rate
the model is
written down in
quarterly rates;
the IRFs for
inflation and
interest rates are
annualized (i.e.
multiplied by 4)
original
(empirical) Taylor
rule parameter;
since in empirical
work inflation and
interest rates are
expressed in
annualized terms,
the output gap
parameter is
divided by 4
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 69
Simple policy rules I
Taylor-type interest rate rule
variance of the endogenous variables is larger than zero
• variance of the autoregressive technology shock 𝑎𝑡 = 𝜌𝑎𝑎𝑡−1 + 𝜀𝑡𝑎
is given by 𝑣𝑎𝑟 𝑎𝑡 = 𝜌𝑎2𝑣𝑎𝑟(𝑎𝑡−1) + 𝜍𝜀
2, which can be solved for
𝑣𝑎𝑟 𝑎𝑡 =𝜍𝜀
2
1−𝜌𝑎2
• variance of the endogenous variables can be calculated as
𝑣𝑎𝑟 𝑦 𝑡 = ((1 − 𝛽𝜌𝑎)Λ𝑎Ψ)2𝑣𝑎𝑟 𝑎𝑡 =((1−𝛽𝜌𝑎)Λ𝑎Ψ)2
1−𝜌𝑎2 𝜍𝜀
2 and
𝑣𝑎𝑟 𝜋𝑡 = (𝜅Λ𝑎Ψ)2𝑣𝑎𝑟 𝑎𝑡 =(𝜅Λ𝑎Ψ)2
1−𝜌𝑎2 𝜍𝜀
2
(note that the variance of inflation has to be multiplied by 42, if
inflation is expressed in annualized terms)
𝑣𝑎𝑟 𝜋𝑡 = 2.60𝑝𝑝 2
𝑣𝑎𝑟 𝑦 𝑡 = 0.55𝑝𝑝 2
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 70
Simple policy rules I
Taylor-type interest rate rule
variance of the endogenous variables can also be deduced
from the impulse response functions
𝑣𝑎𝑟 𝑦 𝑡 = 𝑦 𝑡+𝑖2∞
𝑖=0 and 𝑣𝑎𝑟 𝜋𝑡 = 𝜋𝑡+𝑖2∞
𝑖=0
0 10 20 30 40 50-0.25
-0.2
-0.15
-0.1
-0.05
0Output gap (tilde)
0 10 20 30 40 500
0.01
0.02
0.03
0.04
0.05
0.06Squared output gap (tilde)
0 10 20 30 40 500
0.1
0.2
0.3
0.4CumSum of squared output gap (tilde)
0 10 20 30 40 500
0.2
0.4
0.6
0.8Square root of CumSum of squared output gap (tilde)
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 71
Simple policy rules I
Taylor-type interest rate rule
What happens to the variance of the endogenous variables if
the policy parameters are changed?
Exercise 1 (Matlab: NKM_simplepolicy_solved_variance_y.m):
• inflation response parameter is fixed at 𝜙𝜋 = 1.5
• output gap response parameter is varied −0.5 < 𝜙𝑦< 1
• determinacy is guaranteed since 𝜙𝑦 > 𝜅(1 − 𝜙𝜋) 1 − 𝛽 = −6.3
• the graphs show the variance of the welfare-relevant output gap
and inflation in percentage points to the square against 𝜙𝑦
-0.5 0 0.5 10
5
10Output gap (tilde)
-0.5 0 0.5 10
100
200
300Inflation
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 72
Simple policy rules I
Taylor-type interest rate rule
What happens to the variance of the endogenous variables if
the policy parameters are changed?
Exercise 1 (Matlab: NKM_simplepolicy_solved_variance_y.m):
• for positive output gap response parameters the variance of the
welfare-relevant output gap and inflation increases, the more
aggressively the central bank reacts to deviations of actual output
from its steady-state (its trend)
• if 𝜙𝑦 = −0.1 the variance of both variables becomes zero
-0.5 0 0.5 10
5
10Output gap (tilde)
-0.5 0 0.5 10
100
200
300Inflation
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 73
Simple policy rules I
Taylor-type interest rate rule
the solution of the rational expectations model is given by
in this example the efficient allocation can be
obtained if
this has some implications for the policy rule parameters
it can be shown that in this case the policy rule is equivalent
to
ˆ1 1
ˆ
n
t a a t t a a t
n
t a t t a t
y r a
r a
0t ty t
1 1
0 11
a y
y a
0
n
t t ti r
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 74
Simple policy rules I
Taylor-type interest rate rule
IRFs to a technology shock 𝜙𝜋 = 1.5, 𝜙𝑦 = −𝜍(1 − 𝜌𝑎)
Matlab: NKM_simplepolicy_solved_IRFs.m
0 5 10-1
0
1Inflation
0 5 10-1
0
1Output gap (tilde)
0 5 10-0.1
-0.05
0Nominal interest rate
0 5 100
0.5
1Output gap (hat)
0 5 10-0.1
-0.05
0Real interest rate
0 5 10-0.1
-0.05
0Efficient real rate
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 75
Simple policy rules I
Taylor-type interest rate rule
What happens to the variance of the endogenous variables if
the policy parameters are changed?
Exercise 2 (Matlab: NKM_simplepolicy_solved_variance_pi.m): • output gap response parameter is fixed at 𝜙𝑦 = 0
• inflation response parameter is varied 1 < 𝜙𝜋 < 4
• determinacy is guaranteed since 𝜙𝜋 > 1
• the graphs show the variance of the welfare-relevant output gap
and inflation in percentage points to the square against 𝜙𝜋
0 2 40
0.5
1
1.5Output gap (tilde)
0 2 40
10
20
30Inflation
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 76
Simple policy rules I
Taylor-type interest rate rule
What happens to the variance of the endogenous variables if
the policy parameters are changed?
Exercise 2 (Matlab: NKM_simplepolicy_solved_variance_pi.m): • fluctuations in the output gap and inflation become smaller as the
strength of the central bank„s response to inflation increases
• the efficient allocation (and hence optimal policy) can be
approximated by a simple Taylor-type rule that responds
aggressively to movements in inflation
0 2 40
0.5
1
1.5Output gap (tilde)
0 2 40
10
20
30Inflation
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