Lecture 1: The basic New Keynesian model fileLecture 1: The basic New Keynesian model Prof. Michael Weber University of Chicago Booth School of Business September 15, 2015 1)

Lecture 1:The basic New Keynesian model

Prof. Michael WeberUniversity of Chicago Booth School of Business

September 15, 2015

1 )

Lecture Outline

I Introduction

I Derivation of Calvo Model

I Interpretation and Solution

2 )

Introduction

3 )

About Me

I Joined Booth in 2014I Master in Business Economics from University of MannheimI PhD from Haas School of Business, UC Berkeley

I Research interests:I Asset pricing, Macroeconomics, Household Finance,

International FinanceI Specific research topics:

1. Downside Risk across Asset Classes2. Effects of firms’ inability to adjust output prices to

macroeconomic shocks (confidential BLS microdata)3. Microfoundations of price ridigities4. Distrust in finance and stock market participation5. Term structure of equity returns (anomalies)6. Production networks, real effects of monetary shocks

→ http://faculty.chicagobooth.edu/michael.weberI Work experience

I Investment bankingI Consulting

4 )

Contact Information

Professor Michael Weber

Email: [email protected]

Tel. +1 510 725 9033

Office hours

I Wednesday 3.30 pm – 5.00 pm or by appointment

5 )

Derivation of Calvo Model

6 )

Household’s Problem

I The household solves the following optimization problem:

max E∑

βtU(Ct , Lt) = E∑

βt(C 1−ψ

t

1− ψ− L

1+1/ηt

1 + 1/η

)where Ct is consumption index (Dixit-Stiglitz aggregator):

Ct =(∫ 1

0Ct(i)

σ−1σ di

) σσ−1

Ct(i): variety i ∈ [0, 1] from the continuum of goods

ψ: inverse of the intertemporal elasticity of substitution (IES)

η: Frisch labor supply elasticity

7 )

Household’s Problem cont.

I The budget constraint is:

∫ 1

0Pt(i)Ct(i)di + QtBt = Bt−1 + WtLt

Qt : price of zero coupon bond Bt

I One can show that the optimal consumption of good i is:

Ct(i) = Ct

(Pt(i)

Pt

)−σ

8 )


I The price of the optimal consumption (also the price index) is:

Pt ≡(∫ 1

0Pt(i)

1−σdi

) 11−σ

I Then the price of the optimal bundle is:

(∫ 1

0Pt(i)Ct(i)di

)= PtCt

I This allows us to simplify the budget constraint.

9 )


I The FOCs are:

−U ′LU ′C

=Wt

Pt(1)

Qt = βEt

[U ′C ,t+1

U ′C ,t

Pt

Pt+1

](2)

I Equation 2: Euler equation governing intertemporal allocation

I Equation 1: intratemporal condition, can be rewritten as:

L1/ηt

C−ψt

=Wt

Pt⇒ Wt − Pt = ψCt + 1/ηLt

10 )


I Let Qt ≡ 1/ exp(it)⇒ Qt = −it

I Define inflation πt+1 = Pt+1−Pt

Pt≈ log(Pt+1/Pt) ≈ Pt+1 − Pt

I We assume zero steady state inflation rate

I After log-linearizing the Euler equation, we have:

Ct = Et [Ct+1]− 1

ψ(it − Et [πt+1]) (3)

I If ψ is low, consumption is sensitive to the real interest rate

11 )

Pricing Assumption

I The key ingredient of Calvo’s model is the pricing assumption:

1. Each firm can reset its price only with probability 1− θ2. Independence from time since last price adjustment

I Assumption of independent price adjustments is critical

I Allows simple aggregation of prices

I Straightforward dynamics of the aggregate price level

I Assumption also critical problem of the model

I Adjustment function of deviation from optimal price in data

12 )

Pricing Assumption cont.

I Probability of price non-adjustment is θ

I Average duration of the price is 11−θ

I We can interpret θ as an index of price stickiness.

13 )


I Aggregate price index dynamics in a symmetric equilibrium:

Pt =

(∫ 1

0Pt(i)

1−σdi

) 11−σ

=

(∫ 1−θ

0P∗t (i)1−σdi +

∫ 1

1−θPt−1(i)1−σdi

) 11−σ

=((1− θ)(P∗t )1−σ + θ(Pt−1)1−σ

) 11−σ

I We used the factsI Identical reset price P∗

t as no firm-specific state variable

I Price adjustment is random

I Same composition of prices in∫ 1

1−θPt−1(i)1−σdi and∫ 1

0Pt−1(i)1−σdi .

14 )


I Now divide both sides of this equation by Pt−1 to get:

( Pt

Pt−1

)1−σ= θ + (1− θ)

( P∗tPt−1

)1−σ

I Log-linearize around zero inflation steady-state:

πt = Pt − Pt−1 = (1− θ)(P∗t − Pt−1)

I Inflation rate is the combination of two margins:I extensive margin = (1− θ)

I intensive margin = (P∗t − Pt−1)

I The extensive margin is fixed in this model15 )

Firm’s Problem

We’ll make the following assumptions:

I Continuum of firms, i ∈ [0, 1]

I Each firm i is producing a fixed variety

I Production function is Yt(i) = ZtLt(i)1−α

I Zt is level of technology

I α > 0 yields an upward-sloping marginal cost

I No capital

I Labor is homogenous across firms

I All firms face the same demand curve: Yt(i) = Yt

(Pt(i)Pt

)−σ16 )

Firm’s Problem cont.

I Firms’ pricing problem is dynamic

I Won’t be able to change future prices

I Optimal reset price maximizes present value of profits:

maxP∗t

Et

+∞∑k=0

θk[Qt,t+k(P∗t Yt+k|t − Σt+k(Yt+k|t))

]Σt+k : cost function in period t + k

Yt+k|t : output in t + k of firm resetting price at t

Qt,t+k : stochastic discount factor (SDF)

17 )


I SDF obtained from the consumer’s Euler equation:

Qt,t+k = βk(Ct+k

Ct

)−ψ Pt

Pt+k

I The firm is subject to the demand constraint:

Yt+k|t = Yt+k

( P∗tPt+k

)−σ

18 )


I First order condition for P∗t :

Et

+∞∑k=0

θk[Qt,t+kYt+k|t(P

∗t −

σ

σ − 1Σ′t+k))

]= 0 (4)

µ = σσ−1 : desired markup over cost

Σ′t+k marginal cost at time t + k .

I This yields:

P∗t =Et∑θk(Qt,t+kYt+k|t

)σσ−1Σ′t+k

Et∑θkQt,t+kYt+k|t

19 )


I For θ = 0 we find the familiar static condition:

P∗t =σ

σ − 1×MC

I Divide equation (4) by Pt−1 to get:

Et

+∞∑k=0

θk[Qt,t+kYt+k|t

( P∗tPt−1

− σ

σ − 1

Σ′t+k

Pt+k

Pt+k

Pt−1

)]= 0

20 )


I Define real marginal cost:

MCt+k|t =Σ′t+k

Pt+k

I Log-linearize around the zero inflation steady state:

P∗t − Pt−1 = (1− βθ)+∞∑k=0

(βθ)kEt

(MC t+k|t + (Pt+k − Pt−1)

)

21 )


I Rewrite:

Pt∗

= (1− βθ)+∞∑k=0

(βθ)kEt

(MC t+k|t + Pt+k)

)

I Pt∗: weighted average of desired future markups

I Weights depend on θ, which governs the effective horizon

I Low θ ⇒ future does not matter as firms can reset frequently

22 )


I Equilibrium in the goods market implies:

Yt(i) = Ct(i)(∫ 1

0Yt(i)

σ−1σ di

) σσ−1 ≡ Yt = Ct

Yt = Ct

I Equilibrium in the labor market:

Lt =

(∫ 1

0Lt(i)di

)=

(∫ 1

0

(Yt(i)

Zt

) 11−α

di

)=

(Yt

Zt

) 11−α

(∫ 1

0

(Pt(i)

Pt

) −σ1−α

di

)⇒

(1− α)Lt = Yt − Zt + dt

23 )


I term dt arises from price dispersion(∫ 1

0 (Pt(i)Pt

)−σ1−α di

)I We will ignore term as zero inflation in steady state

I See Gali’s textbook for more rigorous treatment of this term

I Aggregate marginal cost is:

MCt = (Wt − Pt)− ˇMPLt

= (Wt − Pt)− (Zt − αLt)

= (Wt − Pt)−1

1− α(Zt − αYt)

ˇMPLt : average marginal product in the economy

24 )


I Therefore,

MC t+k|t = (Wt+k − Pt+k)− ˇMPLt+k|t

= (Wt+k − Pt+k)− 1

1− α(Zt+k − αYt+k|t)

= MC t+k +α

1− α(Yt+k|t − Yt+k)

= MC t+k −ασ

1− α(P∗t − Pt+k)

I We used expressions for demand and marginal cost

I With CRS, MC does not depend on production level

25 )


I We can now compute the optimal reset price as:

P∗t − Pt−1 = (1− βθ)+∞∑k=0

(βθ)kEt

(ΘMC t+k + (Pt+k − Pt−1)

)Θ = 1−α

1−α+ασ

26 )


P∗t − Pt−1 = (1 − βθ)ΘMC t + (1 − βθ)Et(Pt − Pt−1)

+(1 − βθ)+∞∑k=1

(βθ)kEt

(ΘMC t+k + (Pt+k − Pt−1)

)P∗t − Pt−1 = (1 − βθ)ΘMC t + (1 − βθ)Et(Pt − Pt−1)

+(1 − βθ)βθ+∞∑k=0

(βθ)kEt

(ΘMC t+k+1 + (Pt+k+1 − Pt−1)

)P∗t − Pt−1 = (1 − βθ)ΘMC t + (1 − βθ)πt + (1 − βθ)βθ

+∞∑k=0

(βθ)k(Pt − Pt−1)

+(1 − βθ)βθ+∞∑k=0

(βθ)kEt

(ΘMC t+k+1 + (Pt+k+1 − Pt)

)P∗t − Pt−1 = (1 − βθ)ΘMC t + πt + βθEt [P

∗t+1 − Pt ]

27 )


I Use πt = (1− θ)(P∗t − Pt−1) to get:

πt = βEt πt+1 + κMCt

κ = (1−θ)(1−βθ)θ Θ.

I Iterate forward to find:

πt = κ

∞∑s=t

βs−tEtMCs

I Inflation is present value of future marginal costs

I Inflation should have low inertia in the model

I Inflation is aggregate consequence of purposeful optimizingprice-setting decisions of firms

28 )


I Relate MC to measures of aggregate economic activity:

MCt = (Wt − Pt)− ˇMPLt

= (ψYt + 1/ηLt)− (Yt − Lt)

=(ψ +

1/η + α

1− α

)Yt −

1 + 1/η

1− αZt ,

I Use “aggregate” production function to compute ˇMPLt

I FOC for labor supply to replace (Wt − Pt)

29 )


I Denote the level of output under flexible prices as Y N

I In a frictionless world: Pi = P, Yi = Y , θ = 0

I From FOC for optimal reset price we have 1− σσ−1MCN = 0

⇒ hence MCN = σ−1σ = 1/µ

I Marginal cost in the frictionless world MCN fixed

I Replace output with the natural rate of output to get

MCNt = (ψ +

1/η + α

1− α)Y N

t −1 + 1/η

1− αZt

30 )


I MCN does not vary over time, MCN

= 0 and hence:

Y Nt =

1 + 1/η

(1− α)ψ + 1/η + αZt ⇒

MCt =(ψ +

1/η + α

1− α

)(Yt − Yt

N)

Yt − YtN

: output gap (∆ btw actual and flex-price output)

I Let κ∗ = κ(ψ + 1/η+α1−α )

I New Keynesian Phillips Curve (NKPC):

πt = βEtπt + κ∗(Yt − YtN

)

31 )


I NKPC key equation: links nominal and real variables

I Coefficient κ∗ intimately related to amount of real rigidity

I Lower values correspond to higher levels of real rigidity

I Modify Euler equation (3) to express it in terms of output gap:

(Yt − YtN

) = − 1

ψ(it − Etπt+1 − RN

t ) + Et(Yt+1 − Y Nt+1)

RNt ≡ ψEt∆Y N

t+1 = ψ(1+1/η)ψ(1−α)+1/η+αEt∆Zt+1

Y Nt = Et(Y

Nt+1)− 1

ψRNt

32 )


I This gives IS curve:

(Yt − YtN

) = −Et1

ψ(+∞∑k=0

(Rt+k − RNt+k)) = − 1

ψ(Rt − RN

t ) + Et(Yt+1 − Y Nt+1)

Rt = it − Etπt+1: real interest rate

I Output gap (Yt − YtN

) depends on path of future short rates

I Central bankers control only the short-term rate

I Can stabilize output if can promise path of interest rates

33 )


I To close the model, we have several options:

I Taylor rule: it = φππt + φy (Yt − Y Nt ) + vt

vt : monetary policy shock

I Money demand: (Yt − Y Nt )− ζ it = (Mt − Pt)− Y N

t

Mt − Pt : real money balances (or liquidity)

∆Mt : money shock

I The Taylor rule approximates central banks’ behavior well

I In summary, the basic New Keynesian model has 3 equation:

I IS curve

I NKPC curve

I Policy reaction function such as the Taylor rule

34 )

Interpretation and Solution

35 )

Interpretation and solution

I 3 equation New Keynesian macroeconomic model:

πt = βEtπt+1 + κ∗Xt (5)

Xt = − 1

ψ(it − Etπt+1 − RN

t ) + EtXt+1 (6)

it = φππt + φy Xt + vt (7)

I Equation (5) NKPC (supply side)

I Equation (6) IS curve (demand side)

I Equation (7) describes monetary policy rule

36 )


I Advantages compared to old Keynesian formulations:

I All curves are results of optimization problems

I Structural parameters: we can do policy experiments

I Fundamental role of forward looking behavior

37 )


I Eliminate the interest rate equation to get:

[Xt

πt

]= AEt

[Xt+1

πt+1

]+ B(RN

t − vt) (8)

A = Ω

[ψ 1− βφπψκ∗ κ∗ + β(ψ + φy )

](9)

Ω =1

ψ + φy + κ∗φπ(10)

B = Ω

[1κ∗

](11)

38 )


I Unique stable REE: # of stable eigenvalues = # ofpredetermined variables:

Et

[Xt+1

πt+1

]= Π0

[Xt

πt

](12)

(Blanchard and Kahn (1980))

I Here: all variables are jump variables (non-predetermined)

⇒ all eigenvalues must be unstable

I Necessary and sufficient condition for this to hold is:

(1− β)φy + κ∗(φπ − 1) > 0 (13)

39 )


I Monetary authority should respond sufficiently stronglyI Suppose that inflation increases permanently by ∆π:

∆i = φπ∆π + φy∆X

= φπ∆π +φy (1− β)

κ∗∆π

=(φπ +

φy (1− β)

κ∗

)∆π

I Equation (13) is equivalent to:

φπ +φy (1− β)

κ∗> 1 (14)

I Condition often called the (modified) Taylor principle40 )


I Taylor principle: higher inflation π > 0 ⇒ higher real rate

I Nominal rate increases so much that real rate increases

I Higher real interest rate depresses economic activity

I Output gap falls, inflation pressure cools down

I Policy keeps inflation in check

I Unique REE requires sufficiently aggressive stance on inflation

41 )


I To study dynamics of this model, suppose:

vt = ρvvt−1 + εvt (15)

(monetary policy shock)

I Innovation εvt > 0 interpreted as monetary contraction

I Only predetermined variable is vtI We guess a solution:

Xt = cxvvt

πt = cπvvt

42 )


I Use method of undetermined coefficients to find:

Xt = −(1− βρv )Γvvt

πt = −κ∗Γvvt

Γv = (1− βρv ) [ψ(1− ρv ) + φy ] + κ∗ [φπ − ρv ]−1

I If Taylor principle satisfied: Γv > 0

I Expression for the nominal and real interest rate:

Rt = ψ(1− βρv )(1− ρv )Γvvt

it = Rt + Etπt+1 = [ψ(1− βρv )(1− ρv )− ρvκ∗] Γvvt

I If ρv is sufficiently large: nominal rate can decrease

I Direct effect of vt offset by declines in inflation and output gap

43 )

Documents

Lecture 1: The basic New Keynesian model fileLecture 1: The basic New Keynesian model Prof. Michael Weber University of Chicago Booth School of Business September 15, 2015 1)