business cycles

New Keynesian EconomicsIII. The Basic New Keynesian Model

Dr. Michael Paetz

IWWT

Department of EconomicsHamburg University

April 2011

Fakultät Wirtschafts- und

Sozialwissenschaften

Households Firms Equilibrium References Appendix

New Keynesian Economics - III. The New Keynesian ModelMotivation

Key advantages of the New Keynesian approach

Short-run non-neutrality of monetary policy:

1. persistent real effects,

2. sluggish price adjustment,

3. liquidity effect.

Dr. Michael Paetz New Keynesian Economics 04/11 2 / 81


New Keynesian Economics - III. The New Keynesian ModelOutline

1 Households

2 FirmsAggregate price dynamicsOptimal price setting

3 EquilibriumMarket clearing and log-linearizationEquilibrium under an interest rate ruleEquilibrium under an exogenous money supply

4 Appendix



New Keynesian Economics - III. The New Keynesian ModelHouseholds

Optimization problem

(1) maxCt ,Nt ,Bt

E0

∞

∑k=0

βtU (Ct ,Nt) ,

where the consumption index is given by

(2) Ct ≡

(∫ 1

0Ct (i)

ε−1

ε di

) εε−1

,

subject to

(3)

∫ 1

0Pt (i)Ct (i) di +QtBt ≤ Bt−1 +WtNt + Tt , ∀t.




The consumption index Ct ≡(∫ 1

0 Ct (i)ε−1

ε di) ε

ε−1

The continuum of consumption goods is normalized on the interval[0, 1] and the consumption index is a Dixit-Stiglitz aggregate of allsingle goods Ct (i).

ε measures the intratemporal elasticity of substitution between thedifferentiated goods, which is equal to the price elasticity of demand(− ∂Ct (j)/∂Pt (i)

Ct (j)/Pt (i)

).




Demand for a single good i

In addition to the optimization problem of the last chapter, householdsallocate their consumption expenditures optimally for any given

expenditure level Zt :

(4) maxCt (i)

(∫ 1

0Ct (i)

ε−1

ε di

) εε−1

− λ

(∫ 1

0Pt (i)Ct (i) di − Zt

).




Demand for a single good i

Optimization leads to a set of demand equations (see Appendix 3.1 fordetails):

(5) Ct (i) =

(Pt (i)

Pt

)−ε

Ct ,

where Pt ≡[∫ 1

0 Pt (i)1−ε

di] 1

1−εis the aggregate price index.

Moreover, the total consumption expenditures have to fulfill∫ 10 Pt (i)Ct (i) di = PtCt



New Keynesian Economics - III. The New Keynesian ModelFirms

Production of a single differentiated good

The continuum of firms is also normalized on [0, 1]. Each firm produces asingle good, but uses the same technology:

(6) Yt (i) = AtNt (i)1−α

,

where Nt (i) represents the labor input of a single firm i .Firms maximize profits, taking as given the demand for a single good i ,given by (5), and the aggregate price level and consumption index.




Sticky prices á la Calvo (1983)

Due to menu costs, firms do not adjust prices in every period. Sincewe assume a monopolistic competition framework, prices include amark-up over marginal costs (as we will see soon), and firmscontinue to make profits even when marginal costs increase, butprices are not adjusted immediately.

We assume that in each period a randomly selected fraction of firms(1− θ) adjust prices, while the remaining fraction of θ keeps pricesunchanged.⇒ θ is a measure of price rigidity/stickiness.

This implies an average duration of prices of ∑∞k=0 θk → 1

1−θ .




Aggregate price dynamics

Defining P∗t as the price set by those firms which adjust in period t, the

aggregate price level is given by:

(7) Pt =[θP1−ε

t−1 + (1− θ) (P∗t )

1−ε] 1

1−ε.

(Recall, that adjusting firms are randomly selected, so that the pricelevel of non-adjusting firms on average should be equal to Pt−1.)

Dividing by P1−εt−1 yields

(8) Π1−εt = θ + (1− θ)

(P∗

t

Pt−1

)1−ε

.




Log-linearization around a zero-inflation steady state (Πt = 1)

1. Rewrite the equation in logs:

e(1−ε)πt = e log θ + (1− θ) e(1−ε) log

(P∗t

Pt−1

)

.

2. Apply a first-order Taylor approximation:

e(1−ε)π + (1− ε) e(1−ε)π (πt − π)

= e log θ + (1− θ) e(1−ε) log

(P∗P

)

+ (1− θ) (1− ε) e(1−ε) log

(P∗P

) (log

(P∗

t

Pt−1

)− log

(P∗

P

)).




Log-linearization around a zero-inflation steady state (Πt = 1)

3. Subtract the equation evaluated at the steady state,

e(1−ε)π = e log θ + (1− θ) e(1−ε) log P∗P , yields

(1− ε) e(1−ε)π (πt − π)

= (1− θ) (1− ε) e(1−ε) log

(P∗P

) (log

(P∗

t

Pt−1

)− log

(P∗

P

))(9)

4. As in a zero inflation equilibrium π = 0 and P∗ = P, we derive

(10) πt = (1− θ) (p∗t − pt−1)




Optimal price setting

In opposition to the classical model, markets are not perfect and firmshave some monopoly power.





Those firms, which adjust prices in period t, face the followingoptimization problem (for convenience, we cancel out the i):

(11) maxP∗

t

∞

∑k=0

θkEt

[Qt,t+k

(P∗

t Yt+k |t − Ψt+k

(Yt+k |t

))],

where θk is the probability of not-adjusting prices for the next k periods,

Ψt+k

(Yt+k |t

)represents the cost function, Yt+k |t is output in t + k

for a firm, which reset prices in t

(⇒ P∗t Yt+k |t − Ψt+k

(Yt+k |t

)represents nominal payoffs in t + k),

and Qt,t+k is the stochastic discount factor.Firms maximize the sum of their expected discounted future profits!





The stochastic discount factor is defined as

(12) Qt,t+k ≡1

1+ it+k= βk

(Ct+k

Ct

)−σ (Pt

Pt+k

),

where the last equality follows from the households optimization.Moreover, firms face a sequence of constraints, given by the demandequations:

(13) Yt+k |t =

(P∗

t

Pt+k

)−ε

Ct+k .




Optimal price settingUsing the discount factor and the demand equations, the optimizationproblem becomes

maxP∗

t

∞

∑k=0

θkEt

βk

(Ct+k

Ct

)−σ (Pt

Pt+k

)×

[P∗

t

(P∗

t

Pt+k

)−ε

Ct+k − Ψt+k

((P∗

t

Pt+k

)−ε

Ct+k

)],(14)

implying a first order condition of

(15)∞

∑k=0

θkEt

[Qt,t+kYt+k |t (P

∗t −Mψt+k )

]= 0,

where ψt+k ≡ Ψ′t+k

(Yt+k |t

)are marginal costs, and M ≡ ε

ε−1 .




Flexible prices

In the absence of price rigidities (θ = 0) the optimization problem is

simply given by: maxP∗t

P∗t Yt+k |t − Ψt+k

(Yt+k |t

), implying

(16) P∗t = Mψt |t .

⇒ Under monopolistic competition optimal prices include a markup overmarginal costs, even under perfectly flexible prices! Henceforth, (M) iscalled the frictionless gross markup.

If ε → ∞ this markup would vanish and prices would equal marginal costsas in the classical model.




Log-linearizing the price-setting equation

For a log-linearization of (15) we postulate a zero-inflation steadystate, implying P∗

t /Pt−1 = 1 and Πt−1,t+k ≡ Pt+k/Pt−1 = 1.

Moreover, constant prices imply P∗t = Pt+k in the steady state, and

consequently Yt+k |t = Y , as all firms face the same problem andwill produce the same output.

Defining MCt+k |t ≡ ψt+k |t/Pt+k as real marginal costs, it alsofollows that MCt+k |t = MC = 1/M.

Since prices and consumption stay constant, it follows thatQt,t+k = βk .




Log-linearizing the price-setting equation

Dividing (15) by Pt−1 a first-order Taylor approximation leads to (see thetextbook for more details):

(17) p∗t = µ + (1− βθ)∞

∑k=0

(βθ)k Et

mct+k |t + pt+k

,

where µ = logM is the log of the frictionless gross markup. Sincemc = −µ = log (1/M), the equation can also be written in terms ofdeviations from equilibrium:

(18) p∗t = (1− βθ)∞

∑k=0

(βθ)k Et

mct+k |t + pt+k

,




Interpretation

(19) p∗t = µ + (1− βθ)∞

∑k=0

(βθ)k Et

mct+k |t + pt+k

,

According to (19) the optimal price equals the frictionless markup over aweighted average of current and future nominal marginal costs, adjustedfor the probability of the price remaining at each horizon (θk ).



New Keynesian Economics - III. The New Keynesian ModelEquilibrium

Market clearing

Goods market clearing requires that all produced goods are consumed:Yt (i) = Ct (i). Defining aggregate output by a Dixit-Stiglitz aggregator,

Yt =(∫ 1

0 Yt (i)ε−1

ε di) ε

ε−1

, it follows that Yt = Ct . By combining goods

market clearing with the Euler equation we derive the same log-linearizedEuler equation as in the classical model:

(20) yt = Et yt+1 − σ−1 (it − Et πt+1 − ρ) .




Market clearing

Labor market clearing requires the equality of labor supply (of

households) and labor demand (of firms): Nt =∫ 10 Nt (i) di . Using the

production function and the demand equations yields:

Nt =∫ 1

0

(Yt (i)

At

) 1

1−α

di

=

(Yt

At

) 1

1−α∫ 1

0

(Pt (i)

Pt

)− ε1−α

di ,(21)




Market clearing

This equation can be approximated by

(22) (1− α) nt = yt − at + dt ,

where dt ≡ (1− α) log∫ 10 (Pt (i) /Pt)

− ε1−α di is a measure of price

dispersion across firms.Since dt is equal to zero up to a first-order approximation (see appendix3.3) labor market clearing is given by

(23) yt = at + (1− α) nt .




Real marginal costs

Real marginal costs are defined as the real wage divided by the marginal

product of labor(

WtPt

/MPNt

), where:

(24) MPNt ≡∂Yt

∂Nt= (1− α)AtN

−αt .

Taking logs and substituting the labor demand schedulent =

11−α (yt − at), real marginal costs are given by

(25) mct = wt − pt −1

1− α(at − αyt)− log (1− α)




Real marginal costs

Real marginal costs in t + k for a firm that reset prices in t are given bymct+k |t = wt+k − pt+k −mpnt+k |t . Following the same algebra asbefore, we derive:

mct+k |t = wt+k − pt+k −1

1− α

(at+k − αyt+k |t

)− log (1− α)

= mct+k +α

1− α

(yt+k |t − yt+k

),(26)

where we used real marginal costs mct+k , given by (25).




Real marginal costs

Combining the goods market clearing with the demand equations yields

Yt+k |t =(

P∗t

Pt+k

)−εYt+k , and a log-linearization gives:

(27) yt+k |t = −ε (p∗t − pt+k ) + yt+k .

Plugging this into into the previous equation implies:

(28) mct+k |t = mct+k −αε

1− α(p∗t − pt+k ) .




Real marginal costs

Using this in the log-linearized price-setting equation leads to

p∗t − pt−1 = (1− βθ)∞

∑k=0

(βθ)k Et Θmct+k + (pt+k − pt−1)

= (1− βθ)Θ∞

∑k=0

(βθ)k Et mct+k

+∞

∑k=0

(βθ)k Et πt+k ,(29)

where Θ ≡ 1−α1−α+αε , and we used the fact that

(1− βθ)∑∞0 (βθ)k Et pt+k − pt−1 = ∑

∞k=0 (βθ)k Et πt+k.




Real marginal costs

The previous equation can be rearranged:

(1− βθ)Θ∞

∑k=0

(βθ)k Et mct+k+∞

∑k=0

(βθ)k Et πt+k

= (1− βθ)mct + πt

+ (1− βθ)Θ∞

∑k=1


∑k=1

(βθ)k Et πt+k




Real marginal costs

Moreover,

(1− βθ)Θ∞

∑k=1


∑k=1

(βθ)k Et πt+k

= (1− βθ)Θ∞

∑k=0

(βθ)k+1Et mct+k+1

+∞

∑k=0

(βθ)k+1Et πt+k+1

= βθEt

p∗t+1 − pt

.




The New Keynesian Phillips Curve

Hence, the linearized price-setting equation (29) can be written as

(30) p∗t − pt−1 = βθEt

p∗t+1 − pt

+ (1− βθ)Θmct + πt .

Combining this with the linearized aggregate price indexπt = (1− θ) (p∗t − pt−1), we finally derive the New Keynesian PhillipsCurve (NKPC):

(31) πt = βEt πt+1+ λmct ,

where λ ≡ (1−θ)(1−βθ)θ Θ.




Interpretation of the New Keynesian Phillips Curve

Iterating forward, the NKPC can be expressed as the discounted sumcurrent and expected future deviations of real marginal costs from steadystate

(32) πt = λ∞

∑k=0

βkEt mct+k .

Whenever firms expect average markups (µt = −mct) to be below theirsteady state they would choose higher prices to realign the frictionlessmarkup.




Interpretation of the New Keynesian Phillips Curve

(33) πt = βEt πt+1+ λmct ,λ ≡(1− θ) (1− βθ)

θΘ

When the discount factor β increases, firms give more weight tofuture expected profits, and λ declines,

an increase in price rigidity θ reduces λ, since with less opportunitiesto adjust, firms give more weight to future marginal costs,

if α or ε increase, returns are decreasing stronger, and currentmarginal costs become less important.




The NKPC vs. the classical model

In opposition to the classical model, there is an explicit mechanism in theNew Keynesian model, showing that inflation results from price-settingdecision of profit maximizing firms with some monopoly power.




Relating inflation to a measure of outputTraditional Phillips Curves (as in the IS-PC-LM model) depend on theoutput gap. In order to to relate inflation to an output gap measure, wederive a relationship between real marginal costs and the so called NewKeynesian Output Gap.




Relating inflation to a measure of output

Using the optimal labor supply decision of households and the aggregateproduction function, real marginal costs can be expressed as

mct = wt − pt −mpnt

= (σyt + ϕnt)− (yt − nt)− log (1− α)

=

(σ +

ϕ + α

1− α

)yt −

1+ ϕ

1− αat − log (1− α) .(34)




Relating inflation to a measure of output

Defining the natural level of output ynt as the equilibrium level of output

under flexible prices (mc = −µ) gives

mc =

(σ +

ϕ + α

1− α

)ynt −

1+ ϕ

1− αat − log (1− α)

⇔ ynt = ψn

yaat + ϑny ,(35)

where ϑny = − (1−α)(µ−log(1−α))

σ(1−α)+ϕ+α> 0 and ψn

ya = 1+ϕσ(1−α)+ϕ+α

.

Note: For µ = 0 the natural level of output would be equal to the onederived in the classical model! Due to the market power of firms outputis kept lower than under perfect markets.




The New Keynesian Output Gap

By subtracting mc from mct we derive

(36) mct =

(σ +

ϕ + α

1− α

)yt ,

where yt ≡ yt − ynt represents the New Keynesian Output Gap, defined

as the difference between actual output and the natural level.




The New Keynesian Output Gap

Substituting mct in the NKPC yields:

(37) πt = βEt πt+1+ κyt ,

where κ ≡ λ(

σ + ϕ+α1−α

).

The NKPC in this form is one key equation in the New Keynesianframework, representing the supply side of the model.




The New Keynesian IS equation

Adding Et

∆yn

t+1

to the log linearized Euler equation gives

yt + Et ∆ynt+1 = Et yt+1 − σ−1 (it − Et πt+1 − ρ) + Et ∆yn

t+1

⇔ yt + Et ynt+1 = Et yt+1 − σ−1 (it − Et πt+1 − ρ) + Et ∆yn

t+1

⇔ yt = Et yt+1 − σ−1 (it − Et πt+1 − ρ) + Et ∆ynt+1

⇔ yt = Et yt+1 − σ−1 (it − Et πt+1 − rnt ) ,

where rnt is called the natural rate of interest:

rnt ≡ ρ + σEt

∆yn

t+1

= ρ + σψnyaEt ∆at+1(38)




The New Keynesian IS equation

Solving the New Keynesian IS curve forward yields

(39) yt = −1

σEt

∞

∑k=0

(rt+k − rn

t+k

),

where rt ≡ it − Et πt+1 represents the real interest rate.

According to (39) the output gap depends on current and expectedfuture deviations of the real interest rate from its natural level. Ifmonetary policy is able to affect the real interest rate, current andexpected policy actions affect aggregate demand!




The New Keynesian model

The non-policy block of the New Keynesian model is given by the NKPC,the New Keynesian IS equation, and the path for the natural rate:

The NKPC determines inflation for a given output gap path,

the IS Curve gives the output gap for a given path of the naturaland the actual real interest rate,

the real natural rate depends on real exogenous forces (heretechnology).

In opposition to the classical model, we need a description of monetarypolicy to close the model!




Equilibrium under an interest rate rule

We start with assuming a simple Taylor-type interest rate rule:

(40) it = ρ + φππt + φy yt + νt ,

where νt represents a monetary policy shock.




Equilibrium under an interest rate rule

Combining the non-policy block with this rule leads to

(41)

[yt

πt

]= AT

[Et yt+1Et πt+1

]+Bt (r

nt − νt) ,

where rnt ≡ rn

t − ρ, and

(42) AT ≡ Ω

[σ 1− βφπ

σκ κ + β (σ + φy )

];BT ≡ Ω

[1κ

],

where Ω ≡ 1σ+φy+κφy

.




The Blanchard-Kahn condition

Uniqueness of a solution to such a system of difference equations is givenif and only if the matrix AT has as many Eigenvalues inside the unitcircle as forward looking-variables (see Blanchard and Kahn (1980)).This condition is called the Blanchard-Kahn condition and will be veryimportant in the simulation exercises, using DYNARE.

Since yt and πt are both forward-looking, both Eigenvalues of AT mustlie inside the unit circle. A necessary and sufficient condition foruniqueness is given by

(43) κ (φπ − 1) + (1− β) φy > 0.




A monetary policy shock

Suppose an AR(1)-process for the monetary policy shock:

(44) νt = ρννt−1 + ενt ,

where ενt ∼ N

(0, σ2

ν

).

Hence, a positive realization of ενt can be interpreted as a contractionary

monetary policy shock, implying an increase in the interest rate for agiven inflation and output gap.




Solving for equilibrium dynamics

As the real interest rate is unaffected by the monetary policy shock, wecan set rn

t = 0. To solve for the equilibrium dynamics of yt and πt weuse the method of undetermined coefficients (MUC).

The MUC is often used to solve equations including expectations, whenthere are no explicit informations about the formation of expectations. Itnormally consists of a three-step procedure.




The method of undetermined coefficients (MUC)

1. Suppose a guess solution. Most often the simplest function is used,which is a linear function in all possibly relevant variables. Forexample, if you are looking for a money demand function, you woulduse a linear function like md

t = a0 + a1yt + a2it + a3pt .

2. Use the guess solution with the equation(s) you would like to solve.Often you can get rid of terms including expectations by using theexpectation operator on the guess solution.

3. Solve for the parameters of the guess solution. (Sometimes helpful:Parameters of two polynomials of equal size have to be equal.)




Solving for the equilibrium dynamics

Since the only driving force for the case of a monetary policy shock is νt

we postulate guess solutions of the form

(45) yt = ψyννt , and πt = ψπννt .

Note: In the following we will extensively use

Et yt+1 = ψyνρννt , and Et πt+1 = ψπνρννt ,

which follows from Et+1 νt+1 = ρννt .





Using the guess solutions in the NKPC gives

ψπννt︸︷︷︸πt

= β ψπνρννt︸︷︷︸Etπt+1

+κ ψyννt︸︷︷︸yt

⇔ (1− βρν)ψπν = κψyν

⇔ ψπν =κ

(1− βρν)ψyν(46)





Plugging the interest rate rule into the New Keynesian IS Curve gives

yt = Et yt+1 − σ−1 (φππt + φy yt + νt − Et πt+1)

⇔

(1+

φy

σ

)yt = Et yt+1 − σ−1 (φππt + νt − Et πt+1)

⇔ yt =

(σ

σ + φy

) [Et yt+1 − σ−1 (φππt + νt − Et πt+1)

].





Using the guess solutions leads to

ψyννt︸︷︷︸yt

=

(σ

σ + φy

)ψyνρννt︸︷︷︸Etyt+1

−

(1

σ + φy

)φπ ψπννt︸︷︷︸

πt

+νt − ψπνρννt︸︷︷︸Etπt+1

⇔ ψyν =

(σρν

σ + φy

)ψyν −

(φπ − ρν

σ + φy

)ψπν −

(1

σ + φy

).





Rearranging gives:

(1−

σρν

σ + φy

)ψyν = −

(φπ − ρν

σ + φy

)ψπν −

1

σ + φy

ψyν = −

(φπ − ρν

σ (1− ρν) + φy

)ψπν

−1

σ (1− ρν) + φy(47)





Using the solution from the NKPC (46) in (47) finally determines ψyν

ψyν = −(

φπ−ρν

σ(1−ρν)−φy

) κ

(1− βρν)ψyν

︸︷︷︸ψπν

− 1σ(1−ρν)+φy

⇔(1+ κ(φπ−ρν)

[σ(1−ρν)+φy ](1−βρν)

)ψyν = − 1

σ(1−ρν)+φy

⇔ ψyν = − (1− βρν)Λν,

where Λν ≡ 1[σ(1−ρν)+φy ](1−βρν)+κ(φπ−ρν)

.





Using this in (46) determines ψπν:

ψπν = − (1− βρν)Λν︸︷︷︸ψyν

κ

(1− βρν)

= −κΛν.





Hence, the dynamics of inflation and the output gap are given by

yt = − (1− βρν)Λννt(48)

πt = −κΛννt ,(49)

with Λν ≡ 1[σ(1−ρν)+φy ](1−βρν)+κ(φπ−ρν)

.

Note: If the Blanchard-Kahn condition is satisfied, Λν > 0.⇒ An increase in the interest rate leads to a fall in inflation and theoutput gap!





Using the equilibrium dynamics for yt and πt in the New Keynesian ISCurve, we can solve for the real interest rate response:

(50) rt = it − Et πt+1 = σ (1− ρν) (1− βρν)Λννt

The real interest rate increases in response to an contractionary monetarypolicy.





The dynamics of the nominal interest rate combine the direct effect ofthe shock with the indirect effect due to variations in yt and πt :

(51) it = rt + Et πt+1 = [σ (1− ρν) (1− βρν)− ρνκ]Λννt

Note: For very persistent monetary policy shocks (high values of ρν) thenominal rate will decline due to the fall in inflation!(Inflation depends on future expected inflation. The higher ρν, thestronger will be the downturn of the inflation rate.)





By postulating a money demand equation,

(52) mt − pt = yt − ηit ,

we can also derive the impact response of the money supply:

∂mt

∂ενt

=∂pt

∂ενt

+∂yt

∂ενt

− η∂it

∂ενt

= −κΛν − (1− βρν)Λν − η [σ (1− ρν) (1− βρν)− ρνκ]Λν

= −Λ [(1− βρν) (1+ ησ (1− ρν)) + κ (1− ηρν)] .





(53)∂mt

∂ενt

= −Λ [(1− βρν) (1+ ησ (1− ρν)) + κ (1− ηρν)] .

There is the possibility of a countercyclical reaction (an increase onimpact in the money supply in response to a monetary tightening), if thepolicy shock leads to a decline in output and prices high enough toinduce a fall in the nominal interest rate.

However, for ∂it/∂ενt > 0, which seems to be the most relevant case, the

money supply falls in response to a monetary policy shock (liquidityeffect).





Note: (53) does not describe the equilibrium dynamics of the moneysupply, but the impact reaction in the first period!

The following figure shows the responses to an increase of 25 basis pointsin the monetary policy shock for β = 0.99, σ = 1, φ = 1, α = 1/3,ε = 6, η = 4, θ = 2/3, φπ = 1.5, φy = 0.5/4 and ρν = 0.5. The figuresare expressed in annual terms.



New Keynesian Economics - III. The New Keynesian ModelImpulse responses, monetary policy shock




Interpretation

In line with the empirical evidence from chapter one, a monetary policyshock leads to

a fall in inflation and output (the output gap),

an increase in the nominal interest rate,

a fall in the money supply (liquidity effect).




Interpretation

Moreover,

the increase in the interest rate is smaller than the increase in νt dueto the fall in inflation and output,

and the increase in the real rate is higher than the increase in thenominal rate due to the fall in expected inflation.




The effects of a technology shock

Next, we derive the response to a technology shock of the form

(54) at = ρaat−1 + εat ,

where εat ∼ N(0, σ2

a

).

According to (38) the response of the natural rate is given by

(55) rnt = −σψn

ya (1− ρa) at .

In the following, we set νt = 0.





Note, that rt enters the New Keynesian IS Curve in an analogous way asthe monetary policy shock did with an opposite sign. Hence,

yt = (1− βρa)Λa rnt

= −σψnya (1− ρa) (1− βρa)Λaat ,(56)

πt = κΛa rnt

= −σψnya (1− ρa) κΛaat .(57)

where Λa =≡ 1[σ(1−ρa)+φy ](1−βρa)+κ(φπ−ρa)

> 0.

Homework: For practicing, solve for the equilibrium dynamics using themethod of undetermined coefficients!





Since yt = yt − ynt , and yn

t = ψnyaat + ϑn

y output dynamics are given by

yt = ynt + yt

= ϑny + ψn

ya (1− σ (1− ρa) (1− βρa)Λa) at ,(58)

and employment is given by

(1− α) nt = yt − at

= ϑny +

[(ψn

ya − 1)− σψn

ya (1− ρa) (1− βρa)Λa

]at .(59)

⇒ The sign of the response of output and employment is ambiguous!



New Keynesian Economics - III. The New Keynesian ModelImpulse responses, technology shock (ρa = 0.9)




Interpretation

An increase in technology leads to an increase in the natural level ofoutput and hence a decline in the output gap,

the fall in the output gap, leads to a decline in inflation, andconsequently a fall in the nominal and real interest rate,

for the baseline calibration, output increases and employment falls,

Due to the liquidity effect, the money supply has to increase in orderto decrease the nominal interest rate.




Money demand

Subtracting the natural level of output from a money demand equationgives

mt − pt − ynt = yt − ηit − yn

t

⇔ lt − ynt = yt − ηit ,(60)

where lt ≡ mt − pt represents real balances.Solving for it and substituting the result into the New Keynesian ISCurve yields

(61) (1+ ση) yt = σηEt yt+1+ lt + ηEt πt+1+ ηrnt − yn

t .




Stability

Using lt−1 = lt + πt − ∆mt , we derive

(62) AM,0

yt

πt

lt−1

= AM,1

Et ˜yt+1Et πt+1

lt

+BM

rnt

ynt

∆mt

,

AM,0 ≡

1+ ση 0 0−κ 1 00 −1 1

;AM,1 ≡

ση η 10 β 00 0 1

;BM ≡

η −1 00 0 00 0 −1

.

Since two variables are forward looking, two Eigenvalues ofAM ≡ A−1

M,0AM,1 have to lie inside the unit circle.




A monetary policy shock

Next we assume an AR(1)-process for money growth:

(63) ∆mt = ρm∆mt−1 + εmt ,

where εmt ∼ N(0, σ2

m

).

The following figure shows the Impulse responses to a monetary policyshock for ρm = 0.5.(We will discuss the simulation exercise in detail in the DYNARE lessonnext week.)



New Keynesian Economics - III. The New Keynesian ModelImpulse responses, monetary policy shock




Interpretation

An increase in the money growth rate implies

an increase in real balances due to the sluggish price adjustment,⇒ money market clearing requires an increase in output or a declinein the interest rate. (For baseline parametrization, output and theinterest rate increase (no liquidity effect under exogenous moneygrowth).)

a fall in the real rate, causing a rise in aggregate demand andinflation.

Note: The emergence of a liquidity effect depends on the calibration ofσ.




A technology shock

Next we turn to the response of a technology shock under an exogenouspath for the money supply.The following figure show the responses to a one percent increase intechnology and ∆mt = 0.




InterpretationAn increase in technology leads to

an increase in the natural level of output and hence a decline in theoutput gap, implying a fall in inflation,

an increase in the real interest rate, since the money supply is keptunchanged, implying a much stronger contractionary effect as underan interest rate rule, leading to a strong downturn in employment.



New Keynesian Economics - III. The New Keynesian ModelReferences

Key papers

Blanchard, O. and C. M. Kahn (1980): “The Solution of Linear DifferenceModels under Rational Expectations,” Econometrica, 48, 1305–1311.

Calvo, G. (1983): “Staggered Prices in Utility-Maximizing Framework,”Journal of Monetary Economics, 12, 983–998.



New Keynesian Economics - III. The New Keynesian ModelAppendix: Linearized optimal price setting

Dividing (15) by Pt−1, and using MCt+k |t ≡ ψt+k |t/Pt+k ,

Qt,t+k ≡ 11+it+k

= βk(

Ct+k

Ct

)−σ (Pt

Pt+k

)and

Yt+k |t =(

P∗t

Pt+k

)−εCt+k , we can and rewrite the equation:

∞

∑k=0

(βθ)k(P∗

t )1−ε

Pt−1

C1−σt+k

P1−εt+k

=∞

∑k=0

(βθ)k (P∗t )

−ε C1−σt+k

P1−εt+k

MCt+k |tΠt−1,t+k

⇔∞

∑k=0

(βθ)kP∗

t

Pt−1

C1−σt+k

P1−εt+k

=∞

∑k=0

(βθ)kC1−σ

t+k

P1−εt+k

MCt+k |tΠt−1,t+k



New Keynesian Economics - III. The New Keynesian ModelAppendix: Linearized optimal price setting

Next, we apply a Taylor-approximation (recall that X ≈ X (1+ x)):

∞

∑k=0

(βθ)k X [1+ (p∗t − pt−1) + (1− σ) ct+k − (1− ε) pt+k ]

=∞

∑k=0

X[1+ (1− σ) ct+k − (1− ε) pt+k + mct+k |t + πt−1,t+k

]

⇔∞

∑k=0

(βθ)k (p∗t − pt−1) =

∞

∑k=0

(mct+k |t + πt−1,t+k

)

where X represents the equilibrium value of the equation. Note, that inequilibrium p∗ = p, and hence (p∗t − pt−1) are already deviations fromequilibrium. As the left hand side converges to 1

1−βθ (p∗t − pt−1), and

mct+k |t ≡ mct+k |t − µ, equation (17) follows.


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