advmacro Takeki Sunakawa NK

Basic New Keynesian Model

Takeki Sunakawa

Advanced Macroeconomics at Tohoku University

Introduction

The classical monetary economy (with perfect competition and fully flexible prices in all markets) provides a reference benchmark.

The model to be considered has two departures from the assumptions of the classical monetary model:

Imperfect competition in the goods market: Each firm produces a differentiated good for which it set the price.

Staggered price setting: Only a fraction of firms can reset their prices in any given period (c.f., Calvo, 1983).

The resulting framework is referred to as thebasic New Keynesian model.

Households

The representative household maximizes

E0

X∞ t=0

β^tU (Ct, Nt; Zt),

where C_tis nowa consumption indexgiven by

Ct≡

✂ 1 0

Ct(i)^ε−1ε

_ε−1^ε ,

and Ct(i) denotes the quantity of consumption good i ∈ [0, 1]. This is called Dixit-Stiglitz (1977) aggregator.

Household’s budget constraint

Maximization is subject to a sequence of budget constraints:

✂ 1 0

Pt(i)Ct(i)di + QtBt≤ Bt−1+ WtNt+ Dt,

where Pt(i) is the price of good i, Wt is nominal wage, Bt is one-period riskless discount bonds with its price Qt, and Dtrepresents dividends.

Household’s cost minimization

The household minimizes the cost of purchasing goods,

✂ 1 0

Pt(i)Ct(i)di, subject to

Ct≡

✂ 1 0

Ct(i)^ε−1ε

_ε−1^ε ,

for any given level of Ct. [Note the duality of the problem; see Appendix 3.1 in the text]

Aggregate price index and consumption

After some algebra, we havea downward-sloping demand schedule

Ct(i) =^{ Pt(i)} Pt

−ε

Ct, for all i ∈ [0, 1].

Furthermore, we have

✂ 1 0

Pt(i)Ct(i)di = PtCt. Pluging this into the budget constraint yields

PtCt+ QtBt≤ Bt−1+ WtNt+ Dt.

Household’s optimality conditions

Recap: The optimality conditions become Wt

Pt

= C_t^σN_t^ϕ,

Qt= βEt

( Ct

Ct+1

−σ

Zt+1

Zt

Pt

Pt+1

) .

The log-linearized version of the optimality conditions are wt− pt= σct+ ϕnt,

ct= Etct+1− σ⁻¹(it− Etπt+1− ρ) + (1 − ρz)zt. Also, an ad-hoc log-linear money demand equation is given by

mt− pt= yt− ηit.

Firms

A continuum of firms indexed by i ∈ [0, 1]. Each firm produces a differentiated good, but access to the same technology of production

Yt(i) = AtNt(i),

where At is the level of technology and at≡ log Atfollows at= ρaat−1+ ε^a_t.

All firms face the same demand schedule, and take P_t and C_tas given. [For simplicity, we consider the special case of α = 1 in Gali.]

Two-step problem

Each period the firm chooses {Yt(i), Nt(i), Pt(i)} so as to maximize Πt(i) = PtYt(i) − WtNt(i)

subject to the production and demand function. We decompose this proboem into two steps:

1 Firm minimizes the labor cost given the level of production min

Nt(i)^{WtNt(i), s.t. Yt(i) = AtNt(i).}

Real marginal cost ˜Ψt(i) is obtained as the shadow price.

2 Instead of the original problem, we consider the following maximization problem given Ψt(i) at hand:

maxPt(i)^{(Pt(i) − Ψt(i)) Yt(i), s.t. Yt(i) =}

 Pt(i) Pt

−ε

Yt.

Firm’s cost minimization

The firm minimizes the cost of production. Lagrangean is Lt = WtNt(i) − Ψt(i) (AtNt(i) − Yt(i)) , which yields the real marginal cost ˜Ψt(i) = Ψt(i)/Ptas

Ψ˜t(i) = ^Wt Pt

1 At

.

Real marginal cost multiplied by the marginal product of labor = real wage. Thanks to the CRS technology, the real marginal cost is common across firms, i.e., ˜Ψt(i) = ˜Ψt.

Aggregate price dynamics

Each firm may adjust its price only with prob. (1 − θ) in any given period, independent of the time elapsed from the last adjustment.

A measure of (1 − θ) producers reset their prices, whereas θ producers keep their price unchenged. Then we have

P_t^1−ε =

✂ 1 0

Pt(i)^1−εdi,

=

✂

S(t)

Pt−1(i)^1−εdi + (1 − θ)(P_t^∗)^1−ε,

= θP_t−1^1−ε+ (1 − θ)(P_t^∗)^1−ε,

where S(t) is the set of firms not adjusting their price in period t and P_t^∗ is the optimal price chosen by firms adjusting their price in period t.

Firm’s price setting

A reoptimizing firm chooses the price P_t^∗ so as to maximize X∞

k=0

θ^kEt



Λt,t+k^{ P}

∗ t

Pt+k

− ˜Ψt+k

 Yt+k|t

 ,

where Λt,t+k= β^k(Ct+k/Ct)^−σ is the stochastic discount factor, subject to the sequence of demand constraints

Yt+k|t=^{ P}

t∗

Pt+k

−ε

Yt+k.

Firm’s price setting, cont’d

The optimality condition takes the form

X∞ k=0

θ^kEt











Λt,t+kYt+k|t





 P_t^∗ Pt+k

− ^ε

ε − 1

| {z }

M

Ψ˜t+k

















= 0. (1)

Note that when θ = 0, the above equation collapses to P_t^∗= MΨt.

That is, the optimal price is the nomial marginal cost multiplied by the optimal markup.

Firm’s price setting, cont’d

The log-linearized version of (1) is given by

p^∗_t = µ + (1 − βθ) X∞ k=0

(βθ)^kEtψt+k, (2)

where µ := log M is the logged optimal markup and ψt+k:= log Ψt+kis the logged nominal marginal cost.

The log-linearized version of the aggregate dynamics is given by πt= (1 − θ)(p^∗_t − pt−1).

Inflation dynamics

Eq. (2) can be rewritten recursively

p^∗_t = βθEtp^∗_t+1+ (1 − βθ)^pt+ µ + ˜ψt+k

 . Combinining it with the aggregate dynamics, we have

πt= βEtπt+1−(1 − βθ)(1 − θ)

| {zθ }

=:λ

ˆ µt,

where ˆµt≡ µt− µ is the deviation between the average and desired markups and µt≡ pt− ψt= − ˜ψt.

Under flexible prices, ˆµt= 0 and µt= µ hold.

Equilibrium

In equilibrium, Yt(i) = Ct(i) and Yt= Ctholds. Also, aggregate employment is given by

Nt =

✂ 1 0

Nt(i)di,

=

✂ 1 0

 Yt(i) At

 di,

= ^{ Yt} At

 ✂ 1 0

 Pt(i) Pt

−ε

di

| {z }

=:Dt≥1

.

Note that YtDt= AtNtholds. Staggered price setting takes real resources.

Equilibrium, cont’d

Taking logs yields

nt= yt− at+ dt.

dtis a measure of price dispersion, which is equal to zero up to a first-order approximation. [See Appendix 3.4 in the text.]

Then the average price markup is given by µt = pt− ψt

= pt− wt+ at,

= −(σyt+ ϕnt) + at,

= −(σ + ϕ)yt+ (1 + ϕ)at.

Note that we have used ψt= wt− atand wt− pt= σyt+ ϕnt.

Solving for the output gap

Definethe natural level of outputy_tⁿ under flexible prices, µ = −(σ + ϕ)yⁿ_t + (1 + ϕ)at, which implies

y_tⁿ= ^{1 + ϕ} σ + ϕ

| {z }

ψya

at+ ^−µ σ + ϕ

| {z }

ψy

.

This is only function of technology shock. Also we have

ˆ

µt≡ µt− µ = −(σ + ϕ)(yt− yⁿ_t).

That is, the markup gap is proportional to the output gap, xt= yt− yⁿ_t.

Key equations

Using the definition of the output gap, we have the following key equations: New Keynesian Phillips curve (NKPC)

πt= βEtπt+1+ κxt,

where xt= yt− y_tⁿ, y_tⁿ= ψyaat+ ψy, and κ = (1−βθ)(1−θ)(σ+ϕ)

θ ^.

Dynamic IS curve

xt= Etxt+1− σ⁻¹(it− Etπt+1− r_tⁿ) , wherethe narutal rate of interestis given by

rⁿ_t = ρ − σ(1 − ρa)ψyaat+ (1 − ρz)zt.

In order to close the model, one or more equations determining the nominal interest rate it are needed.

Key equations, cont’d

Consider a simple interest rate rule (akaTaylor rule) it= ρ + φππt+ φy(yt− y) + νt, where φ_π and φ_y are chosen by the monetary authority.

There are 6 variables {xt, πt, it, r_tⁿ, yt, y_tⁿ} and 6 equations, so we can solve the model.

Be careful of the steady state.

Equilibrium under a simple interest rate rule

Combining the Taylor rule with NKPC and IS curve, we have

 xt

πt



= Ω

 σ 1 − βφπ

σκ κ + β(σ + φy)



| {z }

A

 Etxt+1

Etπt+1

 + Ω

 1 κ



| {z }

B

ut,

where Ω = (σ + φy+ κφπ)⁻¹ and

ut = (rⁿ_t − ρ) − φy(yⁿ_t − ψy) − νt,

= −ψya(φy+ σ(1 − ρa))at+ (1 − ρz)zt− νt. We will consider each shock {at, zt, νt} at a time.

Stability condition

When A (A⁻¹) has both eigenvalues inside (outside) the unit circle, the solution is locally unique. A necessary and sufficient condition for uniqueness is given by

κ(φπ− 1) + (1 − β)φy> 0

which is assumed to hold. [See Bullard and Mitra (2002) for a proof.]

The effects of a monetary policy shock

Assume that at= zt= 0 and νt= −utfollows an AR(1) process: νt+1= ρννt+ ε^ν_t+1.

Also conjecture the solution is of the form:

x_t= ψ_xν_t, π_t= ψ_πν_t. Then we obtain

ψx = Ω[σψx+ (1 − βφπ)ψπ]ρu+ Ω,

ψπ = Ω[σκψx+ (κ + β(σ + φy))ψπ]ρu+ Ωκ.

The effects of a monetary policy shock, cont’d

After some algebra, we have the decision rules xt = −(1 − βρν)Λννt, πt = −κΛννt,

where Λ_ν = _(1−βρ ¹

ν)[σ(1−ρν)+φy]+κ(φπ−ρν)^.

Also,

rt = ρ + σ(1 − ρν)(1 − βρν)Λννt, it = rt+ Etπt+1,

ρ + [ρ + σ(1 − ρν)(1 − βρν) − ρνκ]Λννt.

We immediately know that xt, πtand ˆrt= rt− ρ respond negatively to a positive shock to νt. The response of ˆit= it− ρ is unsigned and depends on

The effects of a technology shock

Similarly we have

xt = −ψya(φy+ σ(1 − ρa))(1 − βρa)Λaat, πt = −ψya(φy+ σ(1 − ρa))κΛaat,

where Λa =_(1−βρ ¹

a)[σ(1−ρa)+φy]+κ(φπ−ρa)^.

Also,

yt = ψyaκ(φπ− ρa)Λaat, nt = yt− at

=  (1 − σ)κ(φπ− ρa)

σ + ϕ ^{− (φy+ σ(1 − ρa))(1 − βρa)}

 Λaat. xt and πtrespond negatively to a positive shock to at. ytalso responds positively whereas the response of ntis unsigned.

Numerical exercises

We set: β = 0.99, σ = 1, ϕ = 5, ǫ = 9, θ = 3/4, φπ= 1.5, φy= 0.5/4, and η = 4.

We consider:

An increase of 25 basis points in ε^νt with ρν_{= 0.5.}

An increase of 100 basis points in ε^at with ρa= 0.9.

Impulse responses to monetary policy shock

0 5 10 15

-0.2 -0.1

0 ^{Output gap}

0 5 10 15

-1 -0.5

0 ^Inflation

0 5 10 15

-0.2 -0.1

0 ^Output

0 5 10 15

-0.2 -0.1

0 ^Employment

0 5 10 15

-1 -0.5

0 ^{Real wage}

0 5 10 15

-0.4 -0.2

0 Price level

0 5 10 15

-0.01 -0.005

0 Nominal rate

0 5 10 15

0 0.2

0.4 ^{Real rate}

0 5 10 15

-0.31 -0.305

-0.3 Money supply

0 5 10 15

0 0.2

0.4 ^v

Impulse responses to technology shock

0 5 10 15

-0.1 -0.05

0 ^{Output gap}

0 5 10 15

-2 -1

0 ^Inflation

0 5 10 15

0 0.5

1 ^Output

0 5 10 15

-0.1 -0.05

0 ^Employment

0 5 10 15

0 0.5

1 ^{Real wage}

0 5 10 15

-4 -2

0 Price level

0 5 10 15

-2 -1

0 Nominal rate

0 5 10 15

-0.4 -0.2

0 ^{Real rate}

5 Money supply

1.5 ^a

Price dispersion

The dispersion term can be written as

D_t =

✂ 1 0

 Pt(i) Pt

−ε

di

= (1 − θ)^{ P}

t∗

Pt

−ε

+ θ(1 − θ)^{ P}

t−1∗

Pt

−ε

+ θ²(1 − θ)^{ P}

t−2∗

Pt

−ε

+ · · ·

= (1 − θ)^{ P}

t∗

Pt

−ε

+ θDt−1.

Note that (1 − θ) 1 + θ + θ²+ · · ·
= 1.

Firm’s optimality condition

Also, the optimality condition can be written recursively

P_t^∗ Pt

= ^ε

ε − 1 P∞

k=0^(βθ)kEt

nC_t+k^1−σP_P^t+k

t

ε

Ψ˜t+k

o P∞

k=0^(βθ)kEt



C_t+k^1−σP_P^t+k

t

ε−1^{ ,}

= ^St Ft

, where

St = εC_t^1−σΨ^˜t+ βθEt

Π^ε_t+1St+1

, F_t = (ε − 1)C_t^1−σ+ βθE_t^Π^ε−1_t+1Ft+1 .

and Πt= Pt/Pt−1. (See, for example, "Optimal Operational Monetary

Nonlinear equilibrium conditions

Note that Ct= Ytand Q⁻¹_t = exp(it) = Rt. We have the following equilibrium conditions:

Wt

Pt

= C_t^σN_t^ϕ,

1 = βRtEt

(Ct+1

Ct

−σ

Zt+1

Zt

Pt

Pt+1

) , CtDt= AtNt,

Ψ˜t= ^Wt P_t

1 A_t^, Dt= (1 − θ)^{ P}

∗ t

Pt

−ε

+ θDt−1,

Nonlinear equilibrium conditions, cont’d

and

P_t^∗ Pt

=^St Ft

,

St= εC_t^1−σΨ^˜t+ βθEt

Π^ε_t+1St+1

, Ft= (ε − 1)C_t^1−σ+ βθEt

Π^ε−1_t+1Ft+1 , P_t^1−ε= θP_t−1^1−ε+ (1 − θ)(P_t^∗)^1−ε.

There are 10 variables {C_t, N_t, D_t, ˜Ψ_t, W_t, P_t, P_t^∗, S_t, F_t, R_t}. (Note that Πt= Pt/Pt−1.)

Also, a nonlinear version of the Taylor rule is given by

Rt= RΠ^φ_t^{π Yt} Y

φy

exp(νt).

Assignment #5 (due: January 6)

1 Derive the downward-sloping demand curve by solving the cost minimization problem.

2 Derive the firm’s optimality condition (1) and its log-linear counterpart (2).

3 Calculate the impulse responses to a decrease of 50 basis points in discount rate shock with ρz= 0.5. Discuss the difference from the impulse responses to a monetary policy shock shown in this slide.