new keynesian macroeconomics - uni- · pdf filecarstensen / wollmershäuser, new keynesian...

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



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



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.



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.



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.



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.



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.



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



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.



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



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



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


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



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



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




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




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




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




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




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




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




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




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




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




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


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)



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



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




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




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



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




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




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


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




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




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




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




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




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.



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

http://fabcol.free.fr/



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




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




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




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




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




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)



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


1.8, 1, 0.1xa b

1

2

0.95

0,

x

t t t

x

t x

x x

N



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




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




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




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




determinacy and

indeterminacy

regions

taken from Galí

(2008)




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




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




determinacy and

indeterminacy

regions

taken from Galí

(2008)

1 1 0

11

1 1 2 1

2 1 1

y

y

y

y


Simple policy rules I


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



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




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




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




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




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




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




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




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





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




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




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




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




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)




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






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





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




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






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






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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