econ 4325 monetary policy lecture 5 - universitetet i oslo · econ 4325 monetary policy lecture 5...

ECON 4325Monetary Policy

Lecture 5

Martin Blomhoff Holm

Outline

1. Brief review of lecture 4.

2. Equilibrium in the New-Keynesian model.

Holm Monetary Policy, Lecture 5 1 / 33

Part I: Brief review of lecture 4.


Let’s check what you remember from last time.


Part II: Equilibrium in the New-Keynesian model.


Competitive Equilibrium in the NKM

1. The household maximizes welfare.

2. Firms maximize profits.

3. All markets clear (goods and labor markets).


I. The household maximizes welfare

Household problem:

ωt = σct + φnt

ct = Etct+1 −1

σ(it − Etπt+1 − ρ)

Ct(i) =Pt(i)

Pt

−εCt


II. Firms maximize profits

Firm problem:

Yt(i) = AtNt(i)1−α

p∗t = µ+ (1− βθ)∞∑k=0

(βθ)kEt{ψt+k|t}


III. All markets clear

Goods market:Yt = Ct ⇒ yt = ct

Labor market:

nt =1

1− α(yt − at)

Definition of aggregate inflation:

πt = (1− θ)(p∗t − pt−1)


Equilibrium

ωt = σct + φnt (1)

ct = Etct+1 −1

σ(it − Etπt+1 − ρ) (2)

Ct(i) =Pt(i)

Pt

−εCt (3)

Yt(i) = AtNt(i)1−α (4)

p∗t = µ+ (1− βθ)∞∑k=0

(βθ)kEt{ψt+k|t} (5)

yt = ct (6)

nt =1

1− α(yt − at) (7)

πt = (1− θ)(p∗t − pt−1) (8)


Outline

1. Equilibrium behavior of inflation

2. Introducing the output gap

3. Finding the New Keynesian Phillips Curve

4. Finding the Dynamic IS equation

Goal: Reduce eqs (1) to (8) to two equations.


Equilibrium behavior of inflation I

We will now work with equation (1) - (8) to say something about inflationdynamics. First, combine (2) and (6) to get

yt = Etyt+1 −1

σ(it − Etπt+1 − ρ) + zt︸︷︷︸

preference shock

and solve it forward

yt = zt −1

σ

∞∑k=0

(it+k − Etπt+1+k − ρ) + limT→∞

EtyT

Thus, an exogenous shock only affects output if

I It shifts the equation directly (’preference’ shock)

I It has a permanent effect on the level of output

I It leads to a deviation of the real interest rate from the discount rate,either now or at any point in the future.


Equilibrium behavior of inflation II

Next, individual firm’s marginal product of labor can be found fromequation (4):

MPNt(i) = (1− α)AtNt(i)−α ⇒ mpnt(i) = log(1− α) + at − αnt(i)

Since all firms face the same technology and wages, average nominalmarginal costs is

Ψt =Wt

MPNt⇒ ψt = wt −mpnt = wt − log(1− α)− (at − αnt) (9)

and average real marginal costs (divide nominal by Pt is the same assubtracting pt from the log-expression) is

mct = wt − pt − log(1− α)− (at − αnt)


Equilibrium behavior of inflation III

Use expression (7) (nt = yt1−α −

at1−α) to replace for nt in equation (9) to

obtain

ψt = wt − log(1− α)− 1

1− α(at − αyt)

For firms that last sat prices in period t, marginal costs equal:

ψt+k|t = wt+k − log(1− α)− 1

1− α(at+k − αyt+k|t)

= wt+k − log(1− α)− 1

1− α(at+k − αyt+k + αyt+k − αyt+k|t)

= ψt+k +α

1− α(yt+k|t − yt+k) (10)

Interpretation:


Equilibrium behavior of inflation IVLet’s develop the expression for marginal costs a bit further. Taking logs ofequation (3), and inserting for the market clearing condition (6) yields:

ct(i) = −ε(pt(i)− pt) + ct

= −ε(pt(i)− pt) + yt

⇒ yt = yt(i) + ε(pt(i)− pt)

In period t + k, this becomes

yt+k = yt+k(i) + ε(pt+k(i)− pt+k)

yt+k = yt+k|t + ε(p∗t − pt+k)

⇒ yt+k|t − yt+k = −ε(p∗t − pt+k)

Inserting this into equation (10) yields:

ψt+k|t = ψt+k −αε

1− α(p∗t − pt+k) (11)


Equilibrium behavior of inflation V

Recall equation (5): p∗t = µ+ (1− βθ)∑∞

k=0(βθ)kEt{ψt+k|t} and thatψt+k = mct+k + pt+k .

Now insert from equation (11) in equation (5):

p∗t = µ+ (1− βθ)∞∑k=0

(βθ)kEt{ψt+k −αε

1− α(p∗t − pt+k)}

= µ− (1− βθ)∞∑k=0

(βθ)kαε

1− αp∗t

+ (1− βθ)∞∑k=0

(βθ)kEt{ψt+k +αε

1− αpt+k}

⇒ p∗t

(1− α + αε

1− α

)= µ+ (1− βθ)

∞∑k=0

(βθ)kEt{ψt+k +αε

1− αpt+k}


Equilibrium behavior of inflation VI

p∗t

(1− α + αε

1− α

)= µ+ (1− βθ)

∞∑k=0

(βθ)kEt{mct+k +1− α + αε

1− αpt+k}

p∗t = Θ(1−βθ)∞∑k=0

(βθ)kEt{µ+ mct+k +1− α + αε

1− αpt+k}

where Θ =(

1−α1−α+αε

)≤ 1. Furthermore, we know that

µ+ Et{mct+k} = µ− Et{µt+k} = −Et{µt+k} sinceµt = pt − ψt = pt −mct − pt = −mct . Then

p∗t = (1− βθ)∞∑k=0

(βθ)kEt{pt+k} −Θ(1− βθ)∞∑k=0

(βθ)kEt{µt+k} (12)


Equilibrium behavior of inflation VIISubtract pt−1 from both sides:

p∗t−pt−1 = (1−βθ)∞∑k=0

(βθ)kEt{pt+k−pt−1}−Θ(1−βθ)∞∑k=0

(βθ)kEt{µt+k}

Then notice that

(1− βθ)∞∑k=0

(βθ)kEt{pt+k − pt−1} = (1− βθ)πt

+ (1− βθ)βθ(Et{πt+1}+ πt)

+ ...

which is equal to

= (1− βθ)[πt + βθEt{πt+1}+ ...]

+ (1− βθ)βθ[πt + βθEt{πt+1}+ ...] + ...

=∞∑k=0

(βθ)kEt{πt+k}


Equilibrium behavior of inflation VIII

Hence,

p∗t − pt−1 =∞∑k=0

(βθ)kEt{πt+k} −Θ(1− βθ)∞∑k=0

(βθ)kEt{µt+k} (13)

Now, move this one time-period step forward

Et{p∗t+1 − pt} =∞∑k=0

(βθ)kEt{πt+1+k} −Θ(1− βθ)∞∑k=0

(βθ)kEt{µt+1+k}

and subtract it (βθEt{p∗t+1 − pt}) from (13) to obtain

p∗t − pt−1 = πt + βθEt{p∗t+1 − pt} −Θ(1− βθ)µt

Almost there...


Equilibrium behavior of inflation IX

Now, recall that aggregate price dynamics are given by

πt = (1− θ)(p∗t − pt−1)⇒ p∗t − pt−1 =1

1− θπt

Use this to replace for all p∗ − p expressions to obtain

1

1− θπt = πt + βθEt{

1

1− θπt+1} −Θ(1− βθ)µt

Now, solve for πt to obtain

πt = βEt{πt+1} −(1− θ)(1− βθ)

θΘ︸︷︷︸

λ

µt (14)

Interpretation?


Equilibrium behavior of inflation X

Solve (14) forward:

πt = βEt{βEt{πt+2} − λEt{µt+1}} − λµt...

πt = −λ

[ ∞∑k=0

βkEt{µt+k}

]

Interpretation?


Summing up this far

We have now found the inflation dynamics in the NKM.

RBC/Classical Monetary Model: Inflation is a result of movements inthe aggregate price level created by the monetary policy rule. Noexplanation of the mechanisms going on.

NKM: Inflation is a result of carefully reasoned price-setting decisionsmade by forward-looking firms.


Outline






Introducing the output gap I

2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020

–5

–4

–3

–2

–1

0

1

2

3

4

5

–5

–4

–3

–2

–1

0

1

2

3

4

5

30% 50% 70% 90%

Chart 1.1b Projected output gap1)

with fan chart2)

. Percent.2010 Q1 – 2020 Q4

1) The output gap measures the percentage deviation between mainland GDP and projected potential mainland GDP. 2) The fan charts are based on historical experience and stochastic simulations in Norges Bank’s mainmacroeconomic model, NEMO. Source: Norges Bank

Projections MPR 4/17

Projections MPR 3/17


Introducing the output gap IILet’s consider the relationship between µt and aggregate economicactivity. Recall that µt = pt − ψt and use equation (9) to replace for ψt .

µt = pt − ψt

= pt −(wt −

1

1− α(at − αyt)− log(1− α)

)= −ωt +

1

1− α(at − αyt) + log(1− α)

Now, insert the equilibrium real wage from the household problem (1):

µt = −(σyt + φnt) +1

1− α(at − αyt) + log(1− α)

and replace for nt from equation (7) (nt = 11−α(yt − at)):

µt = −(σ +

φ+ α

1− α

)yt +

1 + φ

1− αat + log(1− α)


Introducing the output gap III

I Now, under flexible prices, average mark-up is equal to the desiredmark-up, i.e. µ = −mc .

I Define ynt as the natural level of output corresponding to thefrictionless mark-up.

The natural level of output is then

µ = −(σ +

φ+ α

1− α

)ynt +

1 + φ

1− αat + log(1− α)

⇒ ynt =1 + φ

σ(1− α) + φ+ αat −

(1− α)(µ− log(1− α))

σ(1− α) + φ+ α

Recognize something from lecture 3?


Introducing the output gap IV

Subtracting µ from µt then yields

µt − µ = −(σ +

φ+ α

1− α

)yt +

1 + φ

1− αat + log(1− α)

− (−(σ +

φ+ α

1− α

)ynt +

1 + φ

1− αat + log(1− α))

= −(σ +

φ+ α

1− α

)(yt − ynt )

Interpretation?


Outline






The New Keynesian Phillips Curve

Now, insert the solution for log-deviations of mark-ups

µt − µ = −(σ +

φ+ α

1− α

)(yt − ynt )

into the expression for inflation dynamics

πt = βEt{πt+1} − λ(µt − µ)

to get

πt = βEt{πt+1}+ λ

(σ +

φ+ α

1− α

)︸︷︷︸

κ

(yt − ynt )

orπt = βEt{πt+1}+ κyt

This is the New Keynesian Phillips curve!(We have used all but equation (2) on slide 10)


Outline






The dynamic IS equation INow, we are going to solve for the dynamics IS equation. First, take theeuler-equation defined in terms of logs and add+subtract the naturaloutput.

yt + ynt − ynt = Et{yt+1}+ Et{ynt+1} − Et{ynt+1} −1

σ(it − Et{πt+1} − ρ)

⇒ yt = Et{yt+1}+ Et{∆ynt+1} −1

σ(it − Et{πt+1} − ρ)

Now, make use of the equilibrium real interest rate we found in lecture 3:

rnt = ρ+ σψyaEt{∆at+1} = ρ+ σEt{∆ynt+1}⇒ ρ = rnt − σEt{∆ynt+1}

and replace for ρ in the DIS-expression

yt = Et{yt+1}+ Et{∆ynt+1} −1

σ(it − Et{πt+1} − rnt + σEt{∆ynt+1})

yt = Et{yt+1} −1

σ(it − Et{πt+1} − rnt )


Summing up so far

πt = βEt{πt+1}+ κyt (15)

yt = Et{yt+1} −1

σ(it − Et{πt+1} − rnt ) (16)

rnt = ρ+ σEt{∆ynt+1} (17)

I Can say something about equilibrium behavior of, not only real, butalso nominal variables!

I Mechanisms?


Summing up so far

The Classical Monetary Model:

I Unique solution for the equilibrium dynamics of real variables.

I Had to introduce monetary policy to say something about nominalvariable - no monetary policy better than any other.

New Keynesian Model:

I Need a description of nominal interest rate to close the model.

I Path of real variables indeterminate without the nominal interest ratepath.

I Solution: must introduce monetary policy.

I Result: Non-neutrality of monetary policy in the short run.

I The two models have equal properties in the long run.


Next week

I Monetary Policy in the NKM


econ 4325 monetary policy lecture 5 - universitetet i oslo · econ 4325 monetary policy lecture 5...

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