new keynesian macroeconomics - chapter 4: the new ... · more unit of nominal income et. one unit...

New Keynesian MacroeconomicsChapter 4: The New Keynesian Baseline Model

Prof. Dr. Kai Carstensen

Ifo Institute for Economic Research and LMU Munich

May 21, 2012

Prof. Dr. Kai Carstensen (LMU Munich) New Keynesian Macroeconomics May 21, 2012 1 / 86

Basic Concepts of New Keynesian Macroeconomics

Start from the RBC model but acknowledge that markets are imperfect, e.g., due to

Incomplete price and wage adjustments: contract duration, adjustment costs, imperfect(non-rational) expectations

Market structure: monopolistic competition / mark up pricing

Capital markets: credit restrictions

Information asymmetries / information costs


Methods used in the baseline model

Microeconomic foundations

Intertemporal optimization

Rational expectations

From this: dynamic stochastic general equilibrium (DSGE) model

⇒ Analogous to the RBC theory, plus some market imperfections. Therefore also called “NewNeoclassical Synthesis”.


Results

Market imperfections, rigidities etc. are quantitatively important!

Even small rigidities on the micro level can have big effects on the macro level.

Monetary impulses can have effects on the business cycle (e.g., on output).

Monetary policy shocks are probably more relevant to explain business cycle fluctuationsthan technology shocks. (However, not every economist would agree here!)

In the long run, there exist the same assumptions as in the RBC world: in the long run,monetary policy shocks are neutral with respect to the real variables like output.


Literature

Textbook: J. Gali (2008), “Monetary Policy, Inflation, and the Business Cycle”, PrincetonUniversity Press.


Outline of this Chapter

A simple New Keynesian baseline model:

real “friction”: monopolistic competition (=departure from perfect competition)

nominal rigidity: staggered price setting at the goods market

apart from this very simple: no population growth; no capital; the labor market clears

no money (cashless economy), but nonetheless monetary policy(!) – it would easily bepossible to add money as a separable part of the utility function (“Money in the UtilityFunction”, MIU), which however would not change the results.


Preface: Monopolistic Competition


Monopolistic Competition (1)

Why monopolistic competition in this model?

We want to allow firms to keep their price unchanged even though other firm adjust, e.g.,after a shock.

Perfect competition: rule of “one price”, hence incomplete or staggered price adjustmentthat implies not all firms set the same price would be impossible.

Here monopolistic competition: many firms, which produce a differentiated consumptiongood because households “love variety”, i.e., even if the price of product A is much higherthan the price for a similar product B, household will still buy A.

Hence, firms that do not adjust their price for a while after, say, a negative demand shocklose some but not all demand. Hence, there is room for staggered or otherwise incompleteprice adjustment.

Formally, the elasticity of substitution between the goods is finite, hence firms have pricesetting power.

As a ‘by-product”, even in equilibrium/steady state the firms set their selling prices abovemarginal costs and thus make profits.



Formally: There is an indefinite number of firms i . Each produces a differentiated consumptiongood Ct(i) and sells it at price Pt(i). Instead of counting the firms as indefinite discrete sequence1, 2, 3,..., we define a continuum of firms i ∈ [0, 1], i ∈ IR.

Let us use the utility function

Ut = U

[(∫ 1

0Ct(i)

ε−1ε di

) εε−1

]which has a constant elasticity of substitution (CES) between any two goods of size −ε.1

Later on, we are interested in aggregate outcomes. Therefore, define the CES consumption index:

Ct =

(∫ 1

0Ct(i)

ε−1ε di

) εε−1

. (1)

This allows us to write down a utility function in aggregate consumption:

Ut = U [Ct ] .

1Note: Elasticity of substitution in the consumption optimum = relative change in the ratio ofdemand for two goods due to relative changes of the relative price, i.e., for goods i and k:

−d(

Ct (k)Ct (i)

)/(

Ct (k)Ct (i)

)d(

Pt (i)Pt (k)

)/(

Pt (i)Pt (k)

) .



Next, define the price index:

Pt =(∫ 1

0 Pt(i)1−εdi) 1

1−ε.

We will prove that this definition implies that the sum of all good-specific expenditures equals theproduct of price index and consumption index:

PtCt =∫ 1

0 Pt(i)Ct(i)di .



Utility maximization of the household yields the demand equation

Ct (k)Ct (i)

=(

Pt (i)Pt (k)

)ε, i 6= k.

The relative demand depends on the inverse price ratio, where the price sensitivity is expressed bythe parameter ε. Hence, −ε is the price elasticity of demand. From this: perfect competitionincluded in the model as “corner” solution ε→∞.

By aggregation, demand for consumption good i can also be expressed as a function of overalldemand:

Ct(i)

Ct=

PεtPt(i)ε

⇒ Ct(i) = CtPεt

Pt(i)ε(2)

From this: two stage maximization problem of the households.Stage 1: find overall demand Ct at given price level Pt .Stage 2: find disaggregate consumption structure given relative prices and overall consumption.

In the following we are only interested in overall demand. Thus we will neglect the second stage.


Monopolistic Competition: Derivation of the Results (1)

Utility function:

U [Ct ] = U

[(∫ 10 Ct(i)

ε−1ε di

) εε−1

].

Budget constraint (given nominal income Et):

Et =∫ 1

0 Pt(i)Ct(i)di .

Lagrange function:

Lt = U

[(∫ 10 Ct(i)

ε−1ε di

) εε−1

]− λ

(∫ 10 Pt(i)Ct(i)di − Et

).

First derivative of the Lagrange function with respect to k:

∂L∂Ct (k)

= ∂U∂Ct

∂Ct∂Ct (k)

− λ ∂∫ 1

0Pt (i)Ct (i)di

∂Ct (k)= 0.



To find:

∂∫ 1

0Pt (i)Ct (i)di

∂Ct (k).

When taking derivatives, the integrals have to be interpreted as the sum over an indefinite numberof addends. As we only take the derivatives with respect to one addend, the other addends vanish.

From this follows

∂∫ 1

0Pt (i)Ct (i)di

∂Ct (k)=

∂∫i 6=k

Pt (i)Ct (i)di

∂Ct (k)+ ∂Pt (k)Ct (k)

∂Ct (k)= Pt(k).



To find:

∂Ct∂Ct (k)

.

Again one has to pay attention to the differentiation of the integrals:

∂Ct

∂Ct(k)=

∂(∫ 1

0 Ct(i)ε−1ε di

) εε−1

∂Ct(k)

=ε

ε− 1

(∫ 1

0Ct(i)

ε−1ε di

) εε−1−1

︸︷︷︸outer derivative

ε− 1

εCt(k)

ε−1ε−1︸︷︷︸

inner derivative

=

(∫ 1

0Ct(i)

ε−1ε di

) 1ε−1

Ct(k)−1ε .



Remember

Ct =(∫ 1

0 Ct(i)ε−1ε di

) εε−1 ⇒ C

1εt =

(∫ 10 Ct(i)

ε−1ε di

) 1ε−1

.

From this follows

∂Ct∂Ct (i)

=(∫ 1

0 Ct(i)ε−1ε di

) 1ε−1

Ct(k)−1ε = C

1εt Ct(k)−

1ε .

Substitute this back into the first order condition yields:

∂L∂Ct (k)

= ∂U∂Ct

C1εt Ct(k)−

1ε − λPt(k) = 0.



Now that we have the first order condition, let us think about the interpretation of λ and thedefinition of the price index which will turn out to be similar questions.

Intuitively, λ is the shadow price of a marginal change in the constraint function. Hence, λanswers the question by how much utility would increase if the endow the household with onemore unit of nominal income Et .

One unit of income buys 1/Pt more goods (where Pt is the average price level of theconsumption bundle) which give ∂U

∂Ctmore utility. Hence, we can use

λ =∂U

∂Ct

1

Pt

to define the price index.



So what is the price index Pt?

Substituting the expression for λ into the FOC and solving for Ct(k) yields

Ct(k) = CtPεt Pt(k)−ε.

Substitute this into the definition of the consumption basket:

Ct =

(∫ 1

0Ct(k)

ε−1ε dk

) εε−1

=

(∫ 1

0

(CtP

εt Pt(k)−ε

) ε−1ε dk

) εε−1

= CtPεt

(∫ 1

0Pt(k)1−εdk

) εε−1

Now solve for Pt which gives us the desired expression:

Pt =

(∫ 1

0Pt(k)1−εdk

) 11−ε

.



Now let us use the FOC to derive the relative demand for two goods i and k.

For a good k holds:

∂L

∂Ct(k)=

∂U

∂CtC

1εt Ct(k)−

1ε − λPt(k) = 0.

Analogically, for a different good i holds:

∂L

∂Ct(i)=

∂U

∂CtC

1εt Ct(i)

− 1ε − λPt(i) = 0.

Solving these two equations for λ and substitution yields:

Ct(k)

Ct(i)=

(Pt(i)

Pt(k)

)εor

Ct(i) = Ct(k)Pt(k)εPt(i)−ε.



Now let us derive the demand for good k in relationship to overall demand.

This relationship was already derived when discussing λ. Nevertheless, let us do it in a slightlydifferent but straightforward way here. Simply substitute the relative demand function

Ct(i) = Ct(k)Pt(k)εPt(i)−ε

into the definition of the consumption index to get an aggregated result:

Ct =

(∫ 1

0Ct(i)

ε−1ε di

) εε−1

=

(∫ 1

0

(Ct(k)Pt(k)εPt(i)

−ε) ε−1ε di

) εε−1

= Ct(k)Pt(k)ε(∫ 1

0Pt(i)

1−εdi

) 11−ε·(−ε)

Using the definition of the price index, this simplifies to:

Ct = Ct(k)Pt(k)εP−εt .

or

Ct(k) = CtPεt

Pt(k)ε.


The New Keynesian Baseline Model


Part 1: Households


Households

Representative immortal household: utility function

Et

∞∑s=t

βs−tU (Cs , Ns) = Et

∞∑s=t

βs−t

(C1−σs

1− σ−

N1+ϕs

1 + ϕ

)

and sequence of budget constraints:

WsNs + Bs−1 + Ts = PsCs + QsBs , s = t, ...,∞,

where

Cs is the consumption index,

Ns is the amount of labor provided (hours of work),

Ws is the nominal wage,

Bs is the amount of riskless securities, which are bought in period s at price Qs and are duein the following period at redemption price 1, and

Ts is representing other (general) income components, such as for example firm profits,minus a lump sum tax,

and Ponzi games are ruled out by lims→∞Et [Bs ] ≥ 0 ∀t.


Utility Function

Figure: Utility function with respect to consumption for different σ’s — For readability shown isC

1−σs1−σ − 11−σ

1−σ , σ > 0

-10

-8

-6

-4

-2

0

2

4

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9 1

1,1

1,2

1,3

1,4

1,5

1,6

1,7

1,8

1,9 2

2,1

2,2

2,3

2,4

2,5

2,6

2,7

2,8

2,9 3

3,1

3,2

3,3

3,4

3,5

3,6

3,7

3,8

3,9 4 sigma=0.2

sigma=0.4

sigma=0.6

sigma=0.8

sigma=1.0

sigma=1.2

sigma=1.4

sigma=1.6

sigma=1.8

sigma=2.0


Utility Maximization

Lt = Et

∞∑s=t

βs−t

[C1−σs

1− σ−

N1+ϕs

1 + ϕ+ λs

(WsNs + Bs−1 + Ts − PsCs − QsBs

)]. (3)

First order conditions (FOCs)

∂Lt

∂Ct=∂Ut

∂Ct− λtPt = C−σt − λtPt

!= 0 ⇒ C−σt = λtPt (4)

∂Lt

∂Ct+1= βEt

[∂Ut+1

∂Ct+1− λt+1Pt+1

]= βEt

[C−σt+1 − λt+1Pt+1

]!

= 0

⇒ Et

[C−σt+1

]= Et [λt+1Pt+1] (5)

∂Lt

∂Bt= −λtQt + βEt [λt+1]

!= 0 ⇒ λtQt = βEt [λt+1] (6)

∂Lt

∂Nt=∂Ut

∂Nt+ λtWt = −Nϕt + λtWt

!= 0 ⇒ Nϕt = λtWt (7)


Interpretation of the FOC’s (1)

Combining (4) and (7) to eliminate λt yields the well-known condition that the marginal rate ofsubstitution between leisure (= total time - labor) and consumption equals the relative price:

MRSt = −∂Ut/∂Nt

∂Ut/∂Ct= Wt/Pt . (8)


Interpretation of the FOC’s (2)

Combining (4), (5) and (6) to eliminate λt and λt+1 yields the consumption Euler equation:

∂Ut

∂Ct= βEt

[∂Ut+1

∂Ct+1·

Pt

Pt+1·

1

Qt

](9)

This equation can be used to define the so-called stochastic discount factor which is the factor bywhich households discount future nominal income. To this end, note that the Euler equationmakes a statement about today’s value Qt of tomorrow’s income of 1 (the redemption value ofthe safe bond). By rearranging the Euler equation, this becomes clear

Qt

1= βEt

[∂Ut+1/∂Ct+1

∂Ut/∂Ct·

Pt

Pt+1

]In general, one may define the expected k-period stochastic discount factor as

Et [Qt,t+k ] = βkEt

[∂Ut+k/∂Ct+k

∂Ut/∂Ct·

Pt

Pt+k

]= βkEt

[C−σt+k

C−σt

·Pt

Pt+k

]


Taking the Natural Logarithm of an Expected Value (1)

Next step: Make the system of equations linear. This is done by applying the natural logarithmbecause (a) the equations are multiplicative and (b) we will study log differences to a baselinesolution (steady state) that can be conveniently interpreted as percent deviations.

However, when log-linearizing an equation that contains expectations the problem arises that

lnEt [xt+1] 6= Et [ln xt+1].


Taking the Natural Logarithm of an Expected Value (2)

To solve this problem, assume xt+1 is log-normally distributed with constant variance,xt+1 ∼ lnN(m, s2). Note that this implies

Et [xt+1] = em+0.5s2.

Given the distribution for xt+1, one can show that ln xt+1 is normally distributed,ln xt+1 ∼ N(m, s2). This of course implies

Et [ln xt+1] = m.

Hence, we have that

lnEt [xt+1]− Et [ln xt+1] = ln em+0.5s2−m = 0.5s2.

Assuming that second moments are time-invariant, this implies that

lnEt [xt+1] = Et [ln xt+1] + constant.

If one is interested in the deviations from the steady state (or from another baseline solution),this constant vanishes. Therefore, it will be ignored in the following.


Log-Linearization

Denote variables in logs by lower-case letters. FOCs:

−σct = lnλt + pt (10)

−σEt [ct+1] = Et [lnλt+1] + Et [pt+1] (11)

lnλt + qt = lnβ + Et [lnλt+1] (12)

ϕnt = lnλt + wt (13)

Eliminating the Lagrange multiplier yields the log-linearized versions of (a) the relationshipbetween the marginal rate of substitution between labor and consumption and the real wage (8),and (b) of the consumption Euler equation (9):

ϕnt + σct = wt − pt

−σct = lnβ − σEtct+1 + pt − Etpt+1 − qt

ATTENTION: these are not deviations from the steady state!


Interest Rate and Inflation

Inflation rate: 1 + πt+1 = Pt+1/Pt

From this follows: lnPt+1 − lnPt = pt+1 − pt = ln(1 + πt+1)

Log-linear approximation: ln(1 + πt+1) ≈ πt+1 ⇒ pt+1 − pt ≈ πt+1

Interest rate: 1 + it = 1/Qt

Log-linear approximation: ln(1 + it) ≈ it ⇒ it ≈ ln (1/Qt) = −qt

Euler equation:

−σct = lnβ − σEtct+1 − Et [pt+1 − pt ]− qt

Substitution:

−σct = lnβ − σEtct+1 − Etπt+1 + it

⇒ ct = Etct+1 −1

σ(it − Etπt+1 + lnβ) .


Time Preference Rate and Steady State Real Interest Rate

Individual discount factor for future utility: β.

Time preference rate = the “individual interest rate” for discounting future utility. Let us call it ρ.

From this: β = 1/(1 + ρ) ⇒ lnβ = − ln(1 + ρ) ≈ −ρ.

Substitution into the Euler equation yields:

ct = Etct+1 − 1σ

(it − Etπt+1 − ρ) .

In the non-stochastic steady state (where we neglect growth!), households prefer a flatconsumption profile: ct = ct+1 = Etct+1. Hence, the steady state real interest rate is

rSS = ρ.


Summary: Household Behavior

Utility maximization by the representative household yielded two important equations thatdescribe optimal household behavior:

ct = Etct+1 −1

σ(it − Etπt+1 − ρ)

ϕnt + σct = wt − pt

Moreover, we defined the k-period stochastic discount factor as

Qt,t+k = βkEt

[C−σt+k

C−σt

·Pt

Pt+k

]


Part 2: Government


Government Consumption and Total Demand

Assume a share τt (you can think of this as a real tax rate) of the overall demand for each good,Y dt (i), is consumed by the government:

Gt(i) = τtYdt (i).

From this, the overall demand for good i is given by

Y dt (i) = Ct(i) + τtY

dt (i)

⇒ Y dt (i) = Ct(i)(1− τt)−1. (14)

Using the same CES aggregator for demand as for consumption yields total demand

Y dt = Ct(1− τt)−1.


Government Demand Shock

Taking logs of the total demand equation yields

ydt = ct − ln(1− τt).

Define gt = − ln(1− τt) ≈ τt as a stationary government demand shock with

gt = ρggt−1 + εgt , ρg ∈ [0, 1).

Then the log of total demand can be written as the sum of log private consumption and thegovernment demand shock:

ydt = ct + gt .


Part 3: Firms’ Behavior under Flexible Prices


Production Technology

There is a continuum of companies, which supply a differentiated good Yt(i), i ∈ [0, 1], producedby the technology

Yt(i) = AtNt(i)1−α.

The level of productivity At is for all companies identical. It is assumed that at = ln(At) follows astationary autoregressive process (hence we ignore technological progress)

at = ρaat−1 + εat , ρa ∈ [0, 1).

The marginal product of labor, MPNt(i), is thus

MPNt(i) = ∂Yt(i)/∂Nt(i) = (1− α)AtNt(i)−α = (1− α)Yt(i)

−α1−α A

11−αt

or, after taking logs,

mpnt(i) = ln(1− α) + at − αnt(i) = ln(1− α)−α

1− αyt(i) +

1

1− αat .


Cost Function

For later use, let us define the cost function that corresponds to the production technology. Givenlabor is the only input, nominal costs are WtNt(i). To express costs as a function of output usethe production function to substitute output for labor:

Ψt (Yt(i)) = WtA−1

1−αt Yt(i)

11−α .

Hence, nominal marginal costs are

Ψ′t (Yt(i)) = Wt∂Nt(i)

∂Yt(i)= WtMPNt(i)

−1 =1

1− αWtA

−11−αt Yt(i)

α1−α ∀ i ∈ [0, 1].

Also, define the real marginal cost function as

MCt(i) = Ψ′t (Yt(i)) /Pt =Wt

PtMPNt(i)

−1

or, in logs

mct(i) = wt − pt −mpnt(i).


Profit Maximization with Flexible Prices (1)

As a reference case, let us consider the profit maximization problem of a firm under theassumption of perfectly flexible prices which implies that the firms can adjust prices every period.Hence, the firm sets its price so as to maximize the current period’s profit

G ft (i) = Pt(i)Y

dt (i)−Ψt (Yt(i)) .

When doing this, the firm takes the overall price level and the total demand as given but at thesame time recognizes the demand function of the household. Hence it knows that is has someprice setting power due to the market structure of monopolistic competition.

The first-order condition for a profit maximum then is

∂G ft (i)

∂Pt(i)= Y d

t (i) + Pt(i)∂Y d

t (i)

∂Pt(i)−Ψ′t (Yt(i))

∂Y dt (i)

∂Pt(i)= 0,

where we implicitly assume that the firm produces just as much as is demanded.



The FOC can be simplified by using the demand function to calculate the demand sensitivity toprice changes. The demand schedule faced by the firm is

Y dt (i) = Ct(i)(1− τt)−1

= CtPεt Pt(i)

−ε(1− τt)−1

= Y dt P

εt Pt(i)

−ε.

Hence, the demand sensitivity to price changes is

∂Y dt (i)

∂Pt(i)=∂[Y dt P

εt Pt(i)−ε

]∂Pt(i)

= −εPt(i)−1Y d

t Pεt Pt(i)

−ε = −εPt(i)−1Y d

t (i).

Substituting this back into the FOC yields

∂G ft (i)

∂Pt(i)= Y d

t (i)− εY dt (i) + εΨ′t (Yt(i))Pt(i)

−1Y dt (i)

= Y dt (i)

[1− ε+ εΨ′t (Yt(i))Pt(i)

−1]

= (1− ε)Y dt (i)

Pt(i)

[Pt(i)−MΨ′t (Yt(i))

]= 0, (15)

where M = εε−1

.



From the latter expression, it is obvious that the nontrivial solution for the profit-maximizing priceis

P∗t (i) =M Ψ′t (Yt(i))

Hence, firms set a price that is by the factor M above marginal costs. This markup depends onthe demand elasticity −ε. The more price elastic demand is, the smaller the markup. Perfectcompetition is the borderline case ε→∞ for which M = ε

ε−1→ 1.

In natural logarithms we have

p∗t (i) = lnM+ ln Ψ′t (Yt(i)) = µ+ ln Ψ′t (Yt(i)) ,

where µ = lnM.


Summary: Firms’ Behavior under Flexible Prices (1)

Firms use the production function

Yt(i) = AtNt(i)1−α.

They have the corresponding marginal cost function

Ψ′t (Yt(i)) =1

1− αWtA

−11−αt Yt(i)

α1−α .

Under price flexibility, they set the profit-maximizing price as a markup on nominal marginal cost

P∗t (i) =M Ψ′t (Yt(i)) .

In logs this implies

p∗t (i) = µ+ wt −1

1− α[at − αyt(i)]− ln(1− α).


Summary: Firms’ Behavior under Flexible Prices (2)

Now, note that in the case of flexible prices, all firms readjust their prices in every period. Theyall face the same demand structure and use the same technology. Hence, they all set the sameprice and produce the same quantity of goods with the same labor input. This implies that theoptimal price level P∗t (i) is the same for all firms and thus equals the average price level, Pt .Moreover, Yt = Yt(i)∀i and Nt = Nt(i)∀i . Consequently, the aggregate results are equal to thefirm-specific results.

The aggregate production function (in logs) then is

yt = at + (1− α)nt .

The real marginal cost function (in logs) is

mct = wt − pt +α

1− αyt −

1

1− αat − ln(1− α).

The average price level equals the profit-maximizing price (in logs)

pt = µ+ wt −1

1− α[at − αyt ]− ln(1− α).


Part 4: Equilibrium with Flexible Prices


Equilibrium of the Goods Market

Market clearing on the goods market requires

Yt(i) ≡ Y dt (i) ∀i ∈ [0, 1]

and thus

Yt(i) = Ct(i)(1− τt)−1 ∀i ∈ [0, 1].

Due to symmetry, under flexible prices aggregation over all goods i trivially yields

Yt = Ct(1− τt)−1

or, in logs, using gt = − ln(1− τt),

yt = ct + gt .


Equilibrium of the Labor Market (1)

Market clearing on the labor market requires that labor supply Nt equals the sum of the labordemands for each good

Nt =

∫ 1

0Nt(i)di .

Given symmetry (all firms have the same labor demand and produce the same output), thisimplies that the firm-specific production function also holds on average,

yt = at + (1− α)nt .


Model Equations with Flexible Prices

wt − pt = σct + ϕnt (16)

ct = −1

σ(it − Etπt+1 − ρ) + Etct+1 (17)

yt = ct + gt (18)

yt = at + (1− α)nt (19)

pt = µ+ wt −1

1− α[at − αyt ]− ln(1− α) (20)


Solution with Flexible Prices

Solving the model gives rise to the so-called “natural” values:

mcnt = −µ, (21)

ynt =

(1− α)(ln(1− α)− µ)

ω+

1 + ϕ

ωat +

σ(1− α)

ωgt , (22)

wnt − pnt =

(ϕ+ σ − σα)(ln(1− α)− µ)

ω+ϕ+ σ

ωat −

σα

ωgt , (23)

nnt =ln(1− α)− µ

ω+

1− σω

at +σ

ωgt , (24)

cnt =(1− α)(ln(1− α)− µ)

ω+

1 + ϕ

ωat −

ϕ+ α

ωgt , , (25)

rnt = ρ− σ1 + ϕ

ω(1− ρa)at + σ

ϕ+ α

ω(1− ρg )gt , (26)

where rt = it − Etπt+1 is the real interest rate and ω = α+ ϕ+ σ(1− α) > 0.

We obtain the non-stochastic steady state by setting gt = Et [gt+1] = 0 and at = Et [at+1] = 0.

IMPORTANT: The natural values are independent of monetary policy shocks!

Note that the nominal side of the economy (nominal interest rate it , price level pt , and wage levelwt) is not determined unless we specify, e.g., an appropriate interest rate rule for the central bankor add a money demand function and a money supply rule to the model.


Part 5: Firms’ Behavior under Staggered Prices


Staggered Price Adjustment: Calvo-Model

Empirically: prices do not adjust instantaneously (see,e.g., ECB Inflation Persistence Network,www.ecb.int/home/html/researcher ipn.en.html).

Staggered price adjustment (Calvo, 1983): each firm is allowed to adjust its prices with aprobability of 1− θ (“lottery”).

Hence, whether a firm adjusts its price is independent of (a) whether the competitors adjust theirprices, (b) how long ago the last price adjustment was, and (c) how profitable an adjustment is(i.e., whether the difference between the optimal price and the old price is large or small).

Macroeconomic implication: in each period a share θ of the prices remains unchanged, while theother share 1− θ of the prices will be adjusted. The aggregate price level follows approximatelythe difference equation

pt = θpt−1 + (1− θ)p∗t ,

or, equivalently,

πt = (1− θ)(p∗t − pt−1),

where πt = pt − pt−1 and p∗t is the optimal price chosen by the firms at time t. Because of theidentical production technology and the identical demand structure, this price is identical for allcompanies.


Profit Maximization Problem with Staggered Price Adjustment (1)

(a) Company i is not allowed to adjust → uses old price set in period t − k: Pt−1(i) = P∗t−k (i).

Profit in t: G ft (i) = Pt−1(i)Yt|t−k (i)−Ψt(Yt|t−k (i))

(b) Company i is allowed to adjust → new price P∗t (i).

Profit in t: G ft (i) = P∗t (i) Yt|t(i)−Ψt(Yt|t(i))

Profit in t + 1 with prob. θ: G ft+1(i) = P∗t (i) Yt+1|t(i)−Ψt+1(Yt+1|t(i))

Profit in t + 2 with prob. θ2: G ft+2(i) = P∗t (i) Yt+2|t(i)−Ψt+2(Yt+2|t(i))

Profit in t + 3 with prob. θ3: G ft+3(i) = P∗t (i) Yt+3|t(i)−Ψt+3(Yt+3|t(i))

...

where Yt+k|t(i) is the output in period t + k of a firm i that last reset its price in period t. Notethat due to symmetry all firms that adjust their prices in period t will choose the same optimalprice P∗t (i) = P∗t . Hence, we can leave out the firm index i in other firm-specific variables such asoutput as long as we indicate in which period the last price adjustment has taken place:Yt+k|t(i) = Yt+k|t .


Profit Maximization with Staggered Price Adjustment (2)

Intertemporal profit function:

G st (i) = Et

∞∑k=0

θkQt,t+k

(P∗t Yt+k|k −Ψt+k (Yt+k|t)

)where the stochastic discount factor of the representative household, Qt,t+k , is used to discountfuture profits (at the end of the day, the household benefits from the profits).

Again, the firms know the demand functions

Yt+k|t =

(P∗tPt+k

)−εYt+k

and take them into account (taking the average price level Pt+k and average production Yt+k asgiven). Substituting this into the profit function yields

G st (i) = Et

∞∑k=0

θkQt,t+k

[P∗t

(P∗tPt+k

)−εYt+k −Ψt+k

((P∗tPt+k

)−εYt+k

)].

The first-order condition for a profit maximum (=derivative w.r.t. P∗t ) is

∂G st

∂P∗t=∞∑k=0

θkEtQt,t+kP

εt+kYt+k

[(1− ε)(P∗t )−ε + ε(P∗t )−ε−1Ψ′t+k (Yt+k|t)

] != 0



Noting that Pεt+kYt+k = (P∗t )εYt+k|t and rearranging yields the FOC

∞∑k=0

θkEt

Qt,t+kYt+k|t

[P∗t −

ε

ε− 1Ψ′t+k (Yt+k|t)

]= 0 (27)

To facilitate a linearization around the steady state, we write the FOC in terms of inflationinstead of the price level because the model does not determine the levels of nominal variables.To this end, let us define

Πt,t+k =Pt+k

Pt=

Pt+k

Pt+k−1· . . . ·

Pt+1

Pt.

Dividing (27) by Pt−1 yields

∞∑k=0

θkEt

Qt,t+kYt+k|t

[P∗tPt−1

−MΨ′t+k (Yt+k|t)

Pt+kΠt−1,t+k

]= 0

⇒∞∑k=0

θkEt

Qt,t+kYt+k|t

[P∗tPt−1

−MMCt+k|tΠt−1,t+k

]= 0 (28)

where we use M = εε−1

and MCt+k|t =Ψ′t+k (Yt+k|t )

Pt+k.



Solving the FOC for P∗t /Pt−1 yields

P∗tPt−1

=M∑∞

k=0 θkEt

[Qt,t+kYt+k|tMCt+k|tΠt−1,t+k

]∑∞k=0 θ

kEt[Qt,t+kYt+k|t

] .

Unfortunately, this equation is not multiplicative (=linear in logs). Therefore, to make ittractable, we use a first-order Taylor expansion around the zero inflation steady state:

p∗t − pt−1 = (1− βθ)∞∑k=0

(βθ)kEt[mct+k|t + (pt+k − pt−1)

], (29)

where mct+k|t = mct+k|t −mc denotes the log deviation of marginal cost from its steady statevalue mc = −µ, and µ = lnM.

By noting that∑∞

k=0(βθ)k = (1− βθ)−1, we can subtract pt−1 from both sides of the aboveequation and simplify the summation over the steady value −µ of the marginal costs to find

p∗t = µ+ (1− βθ)∞∑k=0

(βθ)kEt[mct+k|t + pt+k

]. (30)


Part 6: Equilibrium with Staggered Prices


Equilibrium in the Goods Market

Market clearing on the goods market requires for all t

Yt(i) ≡ Y dt (i) ∀i ∈ [0, 1]

and thus

Yt(i) = Ct(i)(1− τt)−1 ∀i ∈ [0, 1].

Aggregation over all goods i yields

(∫ 1

0Yt(i)

ε−1ε di

) εε−1

=

(∫ 1

0

[Ct(i)(1− τt)−1

] ε−1ε di

) εε−1

(∫ 1

0Yt(i)

ε−1ε di

) εε−1

︸︷︷︸Yt

=

(∫ 1

0Ct(i)

ε−1ε di

) εε−1

︸︷︷︸Ct

(1− τt)−1.

Taking the natural logarithm:

yt = ct + gt .


Equilibrium in the Labor Market (1)

Market clearing on the labor market requires that labor supply Nt equals the sum of the labordemands for each good

Nt =

∫ 1

0Nt(i)di .

To understand this condition, use the production function to substitute out Nt(i):

Nt =

∫ 1

0

(Yt(i)

At

) 11−α

di .

and replace Yt(i) using the demand function of the household:

Nt =

∫ 1

0

(YtPεt Pt(i)−ε

At

) 11−α

di =

(Yt

At

) 11−α

∫ 1

0

(Pt(i)−ε

P−εt

) 11−α

di


Equilibrium in the Labor Market (2)

Define Dt as

Dt =

∫ 1

0

(Pt(i)−ε

P−εt

) 11−α

di

1−α

Using this definition yields

Nt =

(YtDt

At

) 11−α

⇒ Yt = AtN1−αt D−1

t ,

which shows that the aggregate “production function” equals the firm-specific productionfunctions up to the term Dt . Obviously, without price dispersion, Pt(i) = Pt∀i and Dt = 1.Hence, Dt can be interpreted as a measure of price dispersion across firms.

Shown in the tutorial: in a neighborhood of zero inflation, Dt = 1 in a first-order Taylorexpansion. Since all our solution will base on linear approximations, we can thus neglect Dt in thefollowing. Hence we have (approximately)

Yt = AtN1−αt ⇒ yt = at + (1− α)nt .


Average Real Marginal Cost

Using the approximate average production function, the corresponding average real marginal costfunction in period t + k is (analogously to the firm-specific cost function)

mct+k = wt+k − pt+k −mpnt+k

= wt+k − pt+k −1

1− α(at − αyt+k )− ln(1− α). (31)

Similarly, the cost function in period t + k for a firm that last changed its price in period t is

mct+k|t = wt+k − pt+k −mpnt+k|t

= wt − pt −1

1− α(at − αyt+k|t)− ln(1− α). (32)

The difference of the two is

mct+k|t −mct+k =α

1− α(yt+k|t − yt+k ).

Noting that Yt+k|t/Yt+k = Pεt+k/(P∗t )ε and thus yt+k|t − yt+k = ε(pt+k − p∗t ), and substitutingthis in the above equation yields a relationship between the firm-specific and the economy-widereal marginal cost

mct+k|t = mct+k −αε

1− α(p∗t − pt+k ). (33)

or, in deviation from the steady state,

mct+k|t + µ︸︷︷︸mct+k|t

= mct+k + µ︸︷︷︸mct+k

−αε

1− α(p∗t − pt+k ). (34)


Model Equations with Staggered Prices

wt − pt = σct + ϕnt (35)

ct = −1

σ(it − Etπt+1 − ρ) + Etct+1 (36)

yt = ct + gt (37)

yt = at + (1− α)nt (38)

πt = (1− θ)(p∗t − pt−1) (39)

p∗t − pt−1 = (1− βθ)∞∑k=0

(βθ)kEt[mct+k|t + (pt+k − pt−1)

](40)

mct+k|t = mct+k −αε

1− α(p∗t − pt+k ) (41)


Part 7: The New Keynesian Phillips Curve


The New Keynesian Phillips Curve (1)

To solve the model, we want to construct an equation that—very much like a traditional Phillipscurve—describes the relationship between inflation and a cyclical real (activity) variable. Thetraditional Phillips curve assumed a relationship between inflation and unemployment but in theNew Keynesian baseline model there is no unemployment. Instead, equation (41) suggests thatthere might be a relationship between inflation and real marginal cost.

To analyze this, first substitute (41) into (40)

p∗t − pt−1 = (1− βθ)∞∑k=0

(βθ)kEt

[mct+k − αε

1−αp∗t +

(αε

1−α + 1)pt+k − pt−1

]and add αε

1−α (p∗t − pt−1) to both sides which yields

p∗t − pt−1

Θ= (1− βθ)

∞∑k=0

(βθ)kEt [mct+k ] + 1−βθΘ

∞∑k=0

(βθ)kEt [pt+k − pt−1]

p∗t − pt−1 = (1− βθ)Θ∞∑k=0

(βθ)kEt [mct+k ] + (1− βθ)∞∑k=0

(βθ)kEt [pt+k − pt−1]

︸︷︷︸H

(42)

where Θ = 1−α1−α+αε

.



Now consider the last part of the above equation, H. Remember that we are interested inestablishing a relationship between real marginal cost and inflation. So far, H is a function of theprice difference pt+k − pt−1. Due to

pt+k − pt−1 = (pt+k − pt+k−1) + (pt+k−1 − pt+k−2) + . . .+ (pt − pt−1) = πt+k + . . .+ πt ,

we have inflation in the equation and thus

H = (1− βθ)Et

∞∑k=0

(βθ)kk∑

i=0

πt+i .

By writing down all parts of this double sum, it is straightforward to verify that the coefficientsfor πt are 1 + βθ + (βθ)2 + . . ., for πt+1 are βθ + (βθ)2 + (βθ)3 + . . ., for πt+2 are(βθ)2 + (βθ)3 + (βθ)4 + . . ., and so on. This implies

H = (1− βθ)Et

∞∑k=0

πt+k

∞∑i=k

(βθ)i = (1− βθ)Et

∞∑k=0

πt+k (βθ)k∞∑i=k

(βθ)i−k

= (1− βθ)Et

∞∑k=0

πt+k (βθ)k∞∑j=0

(βθ)j

︸︷︷︸(1−βθ)−1

= Et

∞∑k=0

(βθ)kπt+k .



In the next step, substitute the expression for H back into (42):

p∗t − pt−1 = (1− βθ)Θ∞∑k=0

(βθ)kEt [mct+k ] + Et

∞∑k=0

(βθ)kπt+k

This equation can be simplified by noting that for a sum

st =∞∑k=0

(βθ)kπt+k

it holds that

st − βθst+1 =∞∑k=0

(βθ)kπt+k −∞∑k=0

(βθ)k+1πt+1+k

= πt +∞∑k=1

(βθ)kπt+k −∞∑k=1

(βθ)kπt+k = πt .

Hence, we have

p∗t − pt−1 − βθEt(p∗t+1 − pt) = (1− βθ)Θmct + πt

⇒ p∗t − pt−1 = βθEt(p∗t+1 − pt) + (1− βθ)Θmct + πt . (43)



Now remember equation (39), πt = (1− θ)(p∗t − pt−1) which relates inflation and the differencebetween the current optimal and last period’s price. Substitute this into

p∗t − pt−1 = βθEt(p∗t+1 − pt) + (1− βθ)Θmct + πt .

This leads to the relation between inflation and real marginal cost

πt = βEtπt+1 + λmct (44)

with the slope parameter

λ =(1− θ)(1− βθ)

θΘ =

(1− θ)(1− βθ)(1− α)

θ(1− α+ αε).

This implies that inflation reacts less strongly to fluctuations in real marginal costs as

the price elasticity of demand ε,

the measure of decreasing returns α, or

the parameter of price stickiness θ

become larger.



Solving

πt = βEtπt+1 + λmct

forward yields

πt = λ∞∑k=0

βkEtmct+k .

Hence, inflation today is the discounted sum of current and expected future deviations of realmarginal cost from steady state. If average real marginal cost is expected to be above its steadystate value, inflation already rises today. Put differently, inflation rises if the average markup isexpected to be below its steady state value. Thereby, firms can re-establish the optimal steadystate markup.

Why do firms react today to expected future events? Because those firms allowed to changeprices today (remember: it’s a lottery) know that they are perhaps not allowed to this againbefore the event happens.

IMPORTANT: This is an equilibrium phenomenon with rigid prices!


New Keynesian Phillips-Curve (6)

As a final step, note that the deviation of real marginal cost from its steady state value is not atypical measure of economy-wide real activity and it might be difficult to measure empirically. Insome variants of the traditional Phillips curve, there is a relationship between inflation and theoutput gap. To mimic this, let us establish a relationship between the deviation of real marginalcost from its steady state value and a appropriately defined output gap.

To this end, remember that the average production function (38), yt = at + (1− α)nt , implies aneconomy-wide marginal product of labor

mpnt = at − αnt − ln(1− α) = yt − nt + ln(1− α)

and an average marginal cost function

mct = wt − pt −mpnt = wt − pt − yt + nt − ln(1− α).

Using the household’s optimality condition (35), wt − pt = σct + ϕnt , to substitute for the realwage yields

mct = σct − yt + (1 + ϕ)nt − ln(1− α).

Accounting for government consumption according to (37), yt = ct + gt , yields

mct = (σ − 1)yt + (1 + ϕ)nt − σgt − ln(1− α).


New Keynesian Phillips-Curve (7)

Using the average production function solved for labor, nt = (yt − at)/(1− α), leads to

mct =

(σ +

α+ ϕ

1− α

)yt +

1 + ϕ

1− αat − σgt − ln(1− α).

Remember that real marginal cost are constant under flexible prices and we have

mcnt =

(σ +

α+ ϕ

1− α

)ynt +

1 + ϕ

1− αat − σgt − ln(1− α).

The difference of these two equations is

mct =

(σ +

α+ ϕ

1− α

)(yt − yn

t ) =

(σ +

α+ ϕ

1− α

)yt , (45)

where the difference between actual output yt and natural output ynt (i.e., the output level that

would prevail under flexible prices) is called output gap yt .

Substituting (45) into (44) yields the New Keynesian Phillips curve

πt = βEtπt+1 + κyt , (46)

where κ = λ(σ + α+ϕ

1−α

).


Part 8: The New Keynesian Dynamic IS Curve


New Keynesian IS-Curve (1)

In the next step, we want to find a relationship between the (real) interest rate and economicactivity in order to mimic the traditional IS curve.

Starting point is the Euler equation of the household, (36):

ct = −1

σ(it − Etπt+1 − ρ) + Etct+1.

Due to (37), ct = yt + gt , which yields

yt − gt = −1

σ(it − Etπt+1 − ρ) + Et [yt+1 − gt+1].

Using Etgt+1 = ρggt , this can be simplified to

yt = Etyt+1 −1

σ(it − Etπt+1 − ρ) + (1− ρg )gt .

Since inflation depends on the output gap and since we are interested in business cyclefluctuations, let us replace output, yt , by yt + yn

t which yields

yt + ynt = Et yt+1 + Ety

nt+1 −

1

σ(it − Etπt+1 − ρ) + (1− ρg )gt

⇒ yt = Et yt+1 −1

σ(it − Etπt+1 − ρ) + (1− ρg )gt + Ety

nt+1 − yn

t (47)



To reduce the equation, remember the natural output is, according to (22),

ynt =

(1− α)(ln(1− α)− µ)

ω+

1 + ϕ

ωat +

σ(1− α)

ωgt

and thus

Etynt+1 =

(1− α)(ln(1− α)− µ)

ω+

1 + ϕ

ωρaat +

σ(1− α)

ωρggt .

For the difference we therefore have

Etynt+1 − yn

t = −1 + ϕ

ω(1− ρa)at −

σ(1− α)

ω(1− ρg )gt .

Substituting this into the above Euler-type equation (47) yields

yt = Et yt+1 −1

σ(it − Etπt+1 − ρ) + (1− ρg )gt −

1 + ϕ

ω(1− ρa)at −

σ(1− α)

ω(1− ρg )gt

= Et yt+1 −1

σ(it − Etπt+1 − ρ)−

1 + ϕ

ω(1− ρa)at +

ϕ+ α

ω(1− ρg )gt . (48)



Finally, we want to relate the real interest rate rt = it − Etπt+1 to the natural real interest rate rntinstead of the steady state real interest rate ρ. To this end, remember that, according to (26)


ω(1− ρa)at + σ

ϕ+ α

ω(1− ρg )gt

and thus

(rnt − ρ)/σ = −1 + ϕ

ω(1− ρa)at +

ϕ+ α

ω(1− ρg )gt

We can use this to replace 1+ϕω

(1− ρa)at + ϕ+αω

(1− ρg )gt from the rhs of (48) which yields

yt = Et yt+1 −1

σ(it − Etπt+1 − ρ) +

1

σ(rnt − ρ)

or, after re-arranging terms,

yt = Et yt+1 −1

σ(it − Etπt+1 − rnt ) .

This last equation is called the (New Keynesian) dynamic IS curve.



Note that the New Keynesian IS curve can be written as

yt = Et yt+1 −1

σ(rt − rnt )

and can be solved forward which yields

yt = −1

σ

∞∑k=0

Et(rt+k − rnt+k

).


Part 9: Monetary Policy


Why Monetary Policy Is Needed

So far, the New Keynesian baseline model can be summarized by equations for the output gap,

yt = Et yt+1 −1

σ(it − Etπt+1 − rnt ) .

inflation

πt = βEtπt+1 + κyt

and the natural real interest rate


ω(1− ρa)at + σ

ϕ+ α

ω(1− ρg )gt

which altogether constitute the non-policy block of the model. (All other real variables likeemployment and consumption can be calculated from these equations but do not feed back intothem.)To close the model, we need to determine the nominal interest rate. Hence, we need to introducemonetary policy. The inability to determine the equilibrium path of real variables independently ofthe nominal interest rate implies that monetary policy has real effects as long as prices have notfully adjusted.In the following, we assume that monetary policy follows a simple interest rate rule.


Central Bank Behavior (1)

What do central banks do?

In earlier times: money supply targeting (German Bundesbank), nonborrowed reserves(Volcker)

Today: direct inflation targeting / inflation expectations targeting

Example ECB: inflation should be close but below 2% over the medium term

Logic of the New Keynesian model:

NKPC: Inflation rises with high demand / positive output gap

This implies: to control inflation the CB has to control demand or, more precisely, theoutput gap

DIS: output gap depends on the real interest rate

Hence, the central bank has to influence the real interest rate


Central Bank Behaviour (2)

Thought experiment: starting from a steady state with π0 = 0, i0 = 2% and r0 = ρ = 2%,the inflation rate increases in period 1 to π1 = 1% due to some shock. As a result, inflationexpectations will probably also rise for future periods because not all firms can adjust theirprice immediately. Let us assume, for simplicity, E1π2 = 1%. Consequently, the real interestrate decreases to r1 = i1 − E1π2 = 1%. If the central bank wants to bring down inflation, ithas to increase the nominal interest rate by more than 1 percentage point to bring down thereal interest rate. Otherwise, the real interest rate would still have an expansionary effect.

Result: The central bank has to change the nominal interest by an amount that is higherthan the change in the inflation rate. This is the so-called Taylor principle.

Simple monetary policy rule (replacing inflation expectations by observable inflation):

it − πt = ρ+ φπ(πt − π∗), φπ > 0,

⇒ it = ρ− φππ∗ + φππt , φπ = 1 + φπ > 1,

where π∗ is the inflation target of the central bank and ρ is the real interest rate chosen ifinflation equals the target (steady state).


Central Bank Behaviour (3)

Additional consideration of the output gap: inflation indicator and/or business cyclesmoothing

Leads to the Taylor rule

it = ρ− φππ∗ + φππt + φy yt , φπ > 1, φy > 0.

Taylor (1999): an interest rate rule with the parameters φπ = 1.5 and φy = 0.5 describes thebehavior of the Fed well.

Here: inflation target π∗ = 0, therefore

it = ρ+ φππt + φy yt , φπ > 1, φy > 0.


Central Bank Behaviour(4)

In general, we may think of a forward looking interest rate rule

it − Etπt+1︸︷︷︸real interest rate

= rSSt︸︷︷︸steady state

+ (φπ − 1)︸︷︷︸>0

(Etπt+1 − πtargett+1 )︸︷︷︸

inflation expectations

+ φy yt︸︷︷︸output gap

+ vt︸︷︷︸MP shock

.

However, following Gali (2008), we will work with the simple interest rate rule

it = ρ+ φππt + φy yt + vt , φπ > 1, φy > 0.


Monetary Policy Shocks

vt : unsystematic interest rate setting behaviour of the central bank (can not be explained byother variables).

Assumption: autoregressive process

vt = ρvvt−1 + εvt , 0 ≤ ρv < 1.

εvt : monetary policy shock = interest rate change, which can not be predicted/forecasted.

Possible reasons for monetary policy shocks:

Mistakes of the central bank (e.g. when estimating of the output gap)

Internal disagreements (monetary hawks vs. doves)

Exceptional circumstances (e.g. oil price shocks)

Better information than the public.


Part 10: Solution of the Model


Model equations (1)

The model equations are:

yt = −1

σ(it − Etπt+1 − rnt ) + Et yt+1


it = ρ+ φππt + φy yt + vt


ω(1− ρa)at + σ

ϕ+ α

ω(1− ρg )gt


Model equations (2)

Substitution of the natural real interest rate:

yt = Et yt+1 −1

σ(it − Etπt+1 − ρ)−

1 + ϕ

ω(1− ρa)at +

ϕ+ α

ω(1− ρg )gt


it = ρ+ φππt + φy yt + vt

Substitution of the nominal interest rate:

(1 +

φy

σ

)yt = Et yt+1 −

φπ

σπt +

1

σEtπt+1 −

1 + ϕ

ω(1− ρa)at +

ϕ+ α

ω(1− ρg )gt −

1

σvt



Model equations (3)

Write this in matrix form as

A0

(ytπt

)= A1

(Et yt+1

Etπt+1

)+ A2

atgtvt

(49)

where

A0 =

[ σ+φy

σφπσ

−κ 1

]

A1 =

[1 1

σ0 β

]

A2 =

[− (1+ϕ)(1−ρa)

ω

(ϕ+α)(1−ρg )

ω− 1σ

0 0 0

]


Model equations (4)

Pre-multiplying the system of equations by the inverse of A0 yields

(ytπt

)= AT

(Et yt+1

Etπt+1

)+ BT

atgtvt

, (50)

where

Ω = (σ + φy + κφπ)−1

AT = A−10 A1 = Ω

[σ 1− βφπσκ κ+ β(σ + φy )

]

BT = A−10 A2 = Ω

[−σ (1+ϕ)(1−ρa)

ωσ

(ϕ+α)(1−ρg )

ω−1

−κσ (1+ϕ)(1−ρa)ω

κσ(ϕ+α)(1−ρg )

ω−κ

]


Stability of the Model

Note that both yt and πt depend on future expectations and are thus jump variables. Thisimplies that they require a forward solution of the model. A stable and unique solution exists ifthe eigenvalues of AT are absolutely smaller than 1.

To check, under which parameter restrictions this is the case, calculate the eigenvalues ξaccording to

|AT − ξI | =

∣∣∣∣ σΩ− ξ (1− βφπ)ΩσκΩ κΩ + β(σ + φy )Ω− ξ

∣∣∣∣ = 0.

This yields the quadratic equation

ξ2 −σ + κ+ βσ + βφy

σ + φy + κφπξ +

σβ

σ + φy + κφπ= 0.

It is straightforward to show that the two eigenvalues are absolutely smaller than 1 if

σ + φy + κφπ > 0 and

κ(φπ − 1) + (1− β)φy > 0.

Under conventional parameterizations, κ > 0 and σ > 0. Hence, the conditions are satisfied if thecentral bank reaction function satisfies the Taylor principle φπ > 1, and φy > 0.


new keynesian macroeconomics - chapter 4: the new ... · more unit of nominal income et. one unit...

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