Sticky Prices

Monetary Theory University of Bern

1/75

What is at stake?

• Flexible price model: Money is neutral because prices fully respond and fully absorb the change in the nominal dimension. (Vertical supply curve) • Break this by making price less responsive • How? Make prices sluggish

2/75

Modeling rigidities

• Focus on Price rigidities; • Needed: The firm must set prices =⇒ monopolistic competition • Various reasons for rigidities • It is costly to vary prices (price adjustment costs, menu costs) • Firms contract prices (Nominal price contracts)

• Still a problem: reasons for such rigidities (lack of micro-foundations)

3/75

Modeling rigidities

• State–dependent pricing: Firms choose when to change prices subject to “menu costs”. • Those changing prices are those with the most to gain from doing so. • Price changes may be bunched or staggered (depends on the current state)

• Time-dependent pricing: The timing of individual price changes is exogenous. • Firms set their price every n-th period (Taylor 1980) or • Firms are randomly selected to adjust their price (Calvo 1983). =⇒ Firms do not select when to change prices

4/75

Building Blocks of the Model: The Households • Representative household with preferences of the form [∞ ( )] ∑ h1+ν t+τ τ Et β log(Ct+τ ) − 1+ν τ =0

• Key assumption: • Time separability of preferences; • Complete markets; • Household have complete information;

• Unimportant assumption: • Functional forms

5/75

Building Blocks of the Model: The Households

• Budget Constraint Pt Ct + Bt ⩽ Rt−1 Bt−1 + Wt ht + Ωt

• Key Assumption: Households are price takers • Unimportant assumption: Riskless bond

6/75

Building Blocks of the Model: The Households

• Program of the household [ max

{Ct ,ht ,Bt }∞ t=0

Et

∞ ∑ s=0

βs

(

h1+ν log(Ct ) − t 1+ν

)]

subject to Pt Ct + Bt ⩽ Rt−1 Bt−1 + Wt ht + Ωt

7/75

Building Blocks of the Model: The Households

• First order conditions hνt =

Wt Pt Ct

[ ] 1 1 = βRt Et Pt C t Pt+1 Ct+1 lim β j

j→∞

Bt+j =0 Pt+j Ct+j

8/75

Building Blocks of the Model: Monetary Authorities

• Monetary authorities select the path of nominal GDP P t Y t = Mt =⇒ The model is only about prices and output, no more than that. • Will assume no steady state inflation (Not a big deal)

9/75

Building Blocks of the Model: Final Good Producers

• Final good market is perfectly competitive. • One representative, price taker, final good producer • Assemble intermediate goods, Yit , i ∈ (0, 1), to form a final good Yt (∫ Yt =

0

1

θ−1 θ

Yit

θ ) θ−1 di

with θ > 1. • Key Assumption: Homotheticity across varieties • Unimportant assumption: Dixit–Stiglitz

10/75

Building Blocks of the Model: Final Good Producers • Optimal demand for good i so as to maximize profits ∫ 1 max Pt Yt − Pit Yit di {Yit ;i∈(0,1)}

subject to

0

(∫ Yt =

1

0

• First Order Condition

( Yit =

θ−1 θ

Yit

Pit Pt

θ ) θ−1 di

)−θ Yt

• Zero profit condition implies (∫ Pt =

0

1

1 ) 1−θ

P1−θ it di

11/75

Building Blocks of the Model: Intermediate Good Firms

• Firms maximize profits. • Since final producers demand each good, each firm has local monopoly power =⇒ Firms are price setters (Key assumption!) • Technology: Yit = At hα it with α ∈ (0, 1). • Aggregate technology shock: At • Unimportant assumptions: • Labor is the only input • Degree of returns to scale

12/75

State Dependent Pricing: The Price Adjustment Cost Model

Price Adjustment Cost Model

• Have to take a stand on the modeling of the price rigidity. • First use Price adjustment costs (Rotemberg, 1982): • Every time a firm wants to change its price, it has to pay a cost • Goes back to menu costs à la Mankiw • reprint catalogs, change website, change labels in shop, loss of customers…

• The cost function takes the form: C (Pit , Pit−1 ) ≡

ϕ 2

(

Pit Pit−1

−1

)2

Yt

• Current profit: Πt (i) = Pit Yit − Pt Ψ(Yit ) − Pt C (Pit , Pit−1 ) where Ψ(Yit ) denotes the real cost function.

13/75

Price Adjustment Cost Model • Price setting problem: max Et Pit

[∞ ∑

( Φt,t+τ Pit+τ Yit+τ − Pt+τ Ψ(Yit+τ )

τ =0

ϕ − Pt+τ 2

(

Pit+τ −1 Pit+τ −1

)]

)2 Yt+τ

subject to ( Yit =

Pit Pt

)−θ Yt

where Φt,t+1 is the firms’ discount factor and Ψ(·) is the cost function of the firm. 14/75

Price Adjustment Cost Model

• First order condition

(

(1 − θ) −ϕ

Pt Pit−1

Pit Pt (

)−θ

( Yt + θsit )

Pit Pt

)−θ−1

Yt

[

Pt+1 Pit+1 Pit − 1 Yt + Et Φt,t+1 ϕ Pit−1 P2it

(

] ) Pit+1 − 1 Yt+1 = 0 Pit

′

where sit ≡ Ψ (Yit ) denotes the marginal cost. • Since all firms are identical (Pit = Pjt = Pt ) Pt (1 − θ)Yt + θst Yt − ϕ Pt−1

(

[ ] ) ( ) P2t+1 Pt Pt+1 − 1 Yt + Et Φt,t+1 2 ϕ − 1 Yt+1 = 0 Pt−1 Pt Pt

15/75

Price Adjustment Cost Model

• The discount factor Φt,t+1 is given by βPt Ct /Pt+1 Ct+1 (1 − θ)Yt + θst Yt ( ) [ ( ) ] Pt Ct Pt+1 Pt+1 Pt −ϕ − 1 Yt + βEt ϕ − 1 Yt+1 = 0 Pt−1 Pt−1 Ct+1 Pt Pt

• Set ϕ = 0, and recover the standard price setting behavior (1 − θ)Yt + θst Yt = 0 ⇐⇒ st =

θ−1 θ

16/75

Building Blocks of the Model: Intermediate Good Firms

• Labor demand • Use Sheppard’s Lemma Wt ∂Ψ(Yit ) = Ψ′ (Yit )αAt hα−1 = αAt sit hα−1 = it it Pt ∂hit • In a symmetric equilibrium

Wt = αAt st hα−1 t Pt

17/75

Price Adjustment Cost Model: General Equilibrium

• Aggregate hours

∫ ht =

1

0

hit di

• Aggregate output ∫ 0

1

Yit di =

∫ 1( 0

• Resource Constraint: Yt = Ct +

ϕ 2

Pit Pt (

)−θ

Pt Pt−1

di Yt = −1

)2

∫ 1( 0

Pt Pt

)−θ di Yt = Yt

Yt .

18/75

Price Adjustment Cost Model: General Equilibrium • Potentially Money may grow (if π > 1), we will get rid of the nominal growth component by deflating nominal variables: Pt Wt Mt πt = , wt = , mt = Pt−1 Pt Pt−1 • Deflated equilibrium hνt =

wt Ct

[ ] 1 1 = βRt Et Ct πt+1 Ct+1 Yt = At hα t

wt = αst At hα−1 t Yt = Ct +

ϕ (πt − 1)2 Yt 2

πt Yt = mt 19/75

Price Adjustment Cost Model: General Equilibrium

• Price setting [ (1 − θ) + θst − ϕπt (πt − 1) + βϕEt

] Ct Yt+1 πt+1 (πt+1 − 1) = 0 Ct+1 Yt

20/75

Price Adjustment Cost Model: General Equilibrium • Log–linear representation bt = w bt − b νh yt −b yt = brt − Et (b yt+1 + π bt+1 ) bt bt + α h b yt = a bt bt + b bt = a w st + (α − 1)h bt = b m yt + π bt bt ) where χ ≡ • Therefore: b st = χ(b yt − a

1+ν α

• The larger ν (the less elastic the labor supply) the larger χ. • The lower α (the lower the returns to scale) the larger χ. 21/75

Price Adjustment Cost Model: General Equilibrium • Price Setting π bt = κb st + βEt π bt+1 where κ ≡

θ−1 ϕ .

• This is the New Keynesian Phillips Curve. • Inflation reflects all future expected changes in the marginal cost [∞ ] ∑ kb π bt = κEt β st+k k=0

• Slope of the NKPC depends on • competition θ, • nominal rigidities ϕ. 22/75

Price Adjustment Cost Model: General Equilibrium Holding inflation expectations constant: πt

AS

AS

AS (High ϕ)

π0

y⋆

yt

23/75

Price Adjustment Cost Model: General Equilibrium • General equilibrium for output and prices reduces to bt ) + βEt π π bt = κχ(b yt − a bt+1

(AS)

bt − π b yt = m bt

(AD)

• Given that with flexible price, b yft = at , the NKPC rewrites π bt = κχb xt + βEt π bt+1 where b xt = b yt − b yft denotes the output gap. • Departures of Inflation from steady state reflect changes in the gap between the economy and the flexible price economy 24/75

Price Adjustment Cost Model: General Equilibrium

• In General Equilibrium: π bt = (1 − γ)

∞ ∑

b t+j − a bt+j ] (βγ)j Et [m

j=0

with γ ≡

1 1+κχ

=

αϕ αϕ+(1+ν)(θ−1)

∈ (0, 1).

• limϕ→∞ γ = 1 and π bt = 0 bt − a bt • limϕ→0 γ = 0 and π bt = m

25/75

Price Adjustment Cost Model: General Equilibrium b t ) and decreases with technology • Inflation increases with the nominal component (m bt ): Demand vs Supply analysis. (a • Since we have bt − π b yt = m bt b t − (1 − γ) =m

∞ ∑ j=0

j

b t+j ] + (1 − γ) (βγ) Et [m

∞ ∑

bt+j ] (βγ)j Et [a

j=0

• Output increases with technology, may increase with the nominal component. b t. • limϕ→∞ γ = 1 and b yt = m bt . • limϕ→0 γ = 0 and b yt = a

26/75

Price Adjustment Cost Model: General Equilibrium Holding inflation expectations constant: πt

AS

AS

π1 π2 π0

AS (High ϕ)

∆m > 0

AD′ AD y⋆

y1

y2

yt

27/75

Solution

• AR(1) case: b t = ρa a bt−1 + εat a b t = ρm m b t−1 + εmt m • Then 1−γ 1−γ bt − bt m a 1 − βγρm 1 − βγρa γ(1 − βρm ) 1−γ bt + bt b yt = m a 1 − βγρm 1 − βγρa

π bt =

• Output increases with technology, may increase with the nominal component.

28/75

Solution

• The nominal interest rate is bt = Etb R yt+1 − b yt + Et π bt+1 • Using the solution for b yt and π bt+1 1−γ bt = ρm − γ(1 − βρm (1 − ρm )) m bt − bt R a 1 − βγρm 1 − βγρa • Liquidity effect ⇐⇒ γ >

ρm 1−βρm (1−ρm )

29/75

Implications

Response to a Technology Shock Inflation Rate

Interest Rate 0.0

0.8

0.2

0.2

0.6

0.4

0.4

0.4

0.6

0.6

0.2

0.8 1.0

Output

1.0

0.0

0.8

0.0 5

10 15 Quarters

20

Flexible Prices (ϕ = 0),

5

10 15 Quarters

20

Sticky Prices (ϕ = 50),

1.0

5

10 15 Quarters

20

Sticky Prices (ϕ = ∞).

30/75

Implications

Response to a Nominal Shock Inflation Rate

1.0

Output

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0 5

10 Quarters

15

20

Flexible Prices (ϕ = 0),

5

10 Quarters

15

20

Sticky Prices (ϕ = 50),

1.0 0.8 0.6 0.4 0.2 0.0 0.2

Interest Rate

5

10 Quarters

15

20

Sticky Prices (ϕ = ∞).

31/75

Time Dependent Pricing: The Calvo Fairy

The Calvo Fairy

• Time dependent pricing • Dates back to Taylor (1980): Firms set their price for a fixed number of periods • Calvo (1983): Firms fix prices for a random number of periods (simpler) • Fixed probability of resetting their price in each period • Contract: • The firm holds the price constant • It must satisfy demand

32/75

The Calvo Fairy

ξ2

ξ

... Pit+2 = π 2 P⋆it

Pit+1 = πP⋆it Pit = P⋆it

ξ

ξ 1−ξ Pit+2 = P⋆it+2

1−ξ Pit+1 = P⋆it+1

t

t+1

t+2

time

33/75

The Calvo Fairy: Distribution of firms

• There exists a distribution of firms: 1 − ξ firms set their price today (1 − ξ)ξ firms set their price one period ago (1 − ξ)ξ 2 firms set their price two periods ago .. . (1 − ξ)ξ k firms set their price k periods ago

34/75

The Calvo Fairy: Distribution of firms

P.D.F.

0.5 0.4

0.8

0.3

0.6

0.2

0.4

0.1

0.2

0.0 0

10

C.D.F

1.0

20 30 40 Length of Contract N=2,

50

0.0 0 N=4,

10

20 30 40 Length of Contract

50

N=8

35/75

The Calvo Fairy: Towards the NKPC

• The problem of the firm is to select P⋆it so as to maximize [ ∞ ] ∑ τ τ ⋆ Et Φt,t+τ ξ (π Pit Yit+τ − Pt+τ Ψt+τ (Yit+τ )) τ =0

(

s.t. yit+τ =

π τ P⋆it Pt+τ

)−θ Yt+τ

• Assume again no steady state inflation π = 1. • F.O.C [ ( Et

∞ ∑

τ =0

Φt,t+τ ξ

τ

P⋆it −θ Pθt+τ Yt+τ

θ sit P⋆it −θ−1 P1+θ − t+τ Yt+τ θ−1

)] =0

with sit ≡ Ψ′ (Yit ).

36/75

The Calvo Fairy: Towards the NKPC • Note that we have a distribution of prices • Ought to matter a lot (it will at some point!) • To see that, consider the labor market equilibrium ∫ ht =

1

0

hit di =

∫ 1( 0

Yit At

) α1

− α1

∫

di = At

0

1

1

yitα di

• Using the demand function 1 α

1 α

At ht = Yt

∫ 1( |0

)− αθ Pit di Pt {z }

Distribution term (dt )

bt = 0) • Does not matter up to a 1–st order approximation (d 37/75

The Calvo Fairy: Towards the NKPC

• Using the definition of the discount factor, this rewrites [

∞ ∑

Pt+τ Ct+τ Et (βξ) P t+τ +1 Ct+τ +1 τ =0 τ

(

P⋆it −θ Pθt+τ Yt+τ

θ − sit+τ P⋆it −θ−1 P1+θ t+τ Yt+τ θ−1

)] =0

• Log–linearizing, this yields [ b⋆it = (1 − βξ)Et p

∞ ∑

] bt+τ ) (βξ)τ (b sit+τ + p

τ =0

• Problem: the marginal cost is still firm specific (due to decreasing returns to scale)

38/75

The Calvo Fairy: Towards the NKPC 1

α−1

• Recall that wt = αsit At hα−1 = αsit Atα Yit α , which leads to it b bt − sit = w

bt a 1−α b + yit α α

b bt − st = w

bt a 1−α b + yt α α

• Aggregating over firms

• Hence: b sit = b st +

1−α b α (yit

−b yt )

b⋆it − p bt ) • Using the demand function b yit − b yt = −θ(p [∞ ( )] ∑ 1−α ⋆ ⋆ τ bit = (1 − βξ)Et bit − p bt+τ ) + p bt+τ p (βξ) b st+τ − θ (p α τ =0

b⋆it = p b⋆t =⇒ Hence all firm that reset their price choose the same: p 39/75

The Calvo Fairy: Towards the NKPC • Then the optimal price writes [ b⋆t p

= (1 − βξ)Et

∞ ∑

( τ

(βξ)

τ =0

α b bt+τ st+τ + p α + θ(1 − α)

)]

bt+τ = p bt−1 + π • Using the fact that p bt + . . . + π bt+τ [∞ ] [∞ ] ∑ ∑ ⋆ τ τ bt − p bt−1 = (1 − βξ)ΘEt p (βξ) b st+τ + Et (βξ) π bt+τ τ =0

with Θ =

τ =0

α α+θ(1−α)

• Or, recursively b⋆t − p bt−1 = (1 − βξ)Θb b⋆t+1 − p bt ) p st + π bt + βξEt (p 40/75

The Calvo Fairy: Towards the NKPC • Aggregate Price Level (∫ Pt = =

1

0

(∞ ∑

P1−θ it di

1 ) 1−θ

(1 − ξ)ξ

τ

(

)1−θ π τ P⋆t−τ

1 ) 1−θ

τ =0

1 ( ) 1−θ = (1 − ξ)P⋆t 1−θ + ξ(πPt−1 )1−θ

• In log–linear form (using π = 1) bt = (1 − ξ)p b⋆t + ξ p bt−1 ⇐⇒ π b⋆t − p bt−1 ) p bt = (1 − ξ)(p 41/75

The Calvo Fairy: Towards the NKPC

• Combining the price setting behavior and the aggregate price level determination π bt+1 π bt = (1 − βξ)Θb st + π bt + βξEt 1−ξ 1−ξ which solves to π bt = κΘb st + βEt π bt+1 with κ ≡

(1−βξ)(1−ξ) ξ

• Similar to the Price Adjustment Model, but additional term Θ • Due to the distribution of prices (size matters due to DRS) • ξ controls the average length of the contracts: 1 − ξ =

1 N

42/75

The Calvo Fairy: Towards the NKPC

• Reduced log–linearization bt = κΘb st + βEt π bt+1 π 1+ν b bt ) st = (b yt − a α bt − π b yt = m bt bt = Etb R yt+1 − b yt + Et π bt+1

43/75

The Calvo Fairy: Towards the NKPC

• Rewrites NKPC as π bt = κΘχb xt + βEt π bt+1 where χ =

1+ν α

bt = b and b xt = b yt − a yt − b yft .

• Similar to the PAC model. • Due to the log–linear approximation • Inflation is again explained by all future changes in the marginal cost (or output gap) π bt = κΘ

∞ ∑ j=0

β j Etb st+j = κΘχ

∞ ∑

β j Etb xt+j

j=0

44/75

The New–Keynesian Phillips Curve • To wrap it up π bt = κΘχb xt + βEt π bt+1 where (1 − ξ)(1 − βξ) ξ α Θ= α + θ(1 − α) 1+ν χ= α κ=

• κ depends on frequency and discount rate =⇒ Dynamics. • Θ relates the firm marginal cost (MC) to the average MC. • χ relates the economy wide marginal cost to the output gap. 45/75

Dynamic Aspects: κ =

(1−ξ)(1−βξ) ξ

• Small κ =⇒ flat NKPC, monetary policy potency. • κ decreases with the discount rate: Place more weight on the future =⇒ cares not to be stuck with a too high (low) price for a long time. • κ decreases with ξ: The less probable it is to readjust the price, the less responsive is inflation. • ξ estimated between 0.66 and 0.75 =⇒ adjust prices every 3 to 4 months. • Can be even more rigid: • Klenow and Malin (2010): adjust for sales =⇒ 6.9 months • Eichenbaum, Jaimovich and Rebelo (2008): adjust for reference prices =⇒ 10.6 months

46/75

Aggregating Marginal Cost: Θ =

α α+θ(1−α)

• Θ increases with α: More diminishing returns to scale (low α) are associated to flat NKPC (low Θ) • Marginal cost is increasing with the production level (with DRS) • ϵMC/y increases as returns to scale diminish (lower α) • =⇒ MC increases more for low α.

Graph

• Θ decreases with θ: A high demand elasticity (θ) means that if the firm increases its price, then demand falls more and so do costs. So high demand elasticity mitigates Graph price increases.

47/75

Aggregate Equilibrium Marginal Cost: χ =

1+ν α

• χ decreases with labor supply elasticity (lower ν) • Elastic labor supply: an expansion in production, only requires a small change in the wage to obtain an increase in labor supply. • Small changes in wages translate in small change in the marginal cost and hence the Graph price.

• χ decreases as returns to scale increase (higher α): As α increases ϵMC/y decreases.

48/75

Does the previous decomposition matter?

• Do we care about this decomposition? What does matter? κ, Θ, χ or κΘχ? • Each component matters if it has different welfare implications. • 2 sources of distortions in the model • Imperfect competition leads to mis-allocation of resources: Shows up only in Θ • Variation of markups stems from the price rigidity: Shows up only in κ.

• The central bank can fix one but not the other • The decomposition matters!

49/75

The Calvo Fairy

• Key to the story: Inflation is positively related to the marginal cost • Marginal cost is procyclical • Recall that: Pt = µt MCt where µt is the markup rate • If prices are rigid (in the extreme case are fixed), then markups are countercyclical.

50/75

The Calvo Fairy: The General Equilibrium

bt − π • Using aggregate demand, b yt = m bt , we have bt − a bt ) + βγEt π π bt = (1 − γ)(m bt+1 where γ =

1 1+κΘχ

∈ (0, 1).

• Again, same as PAC! =⇒ Same solution but γ takes different forms.

51/75

Implications: Dynamics

Response to a Technology Shock Inflation Rate

Interest Rate 0.0

0.8

0.2

0.2

0.6

0.4

0.4

0.4

0.6

0.6

0.2

0.8 1.0

Output

1.0

0.0

0.8

0.0 5

10 15 Quarters

20

Flexible Prices (N = 0),

5

10 15 Quarters

20

Sticky Prices (N = 4),

1.0

5

10 15 Quarters

20

Sticky Prices (N = 8).

52/75

Implications: Dynamics

Response to a Nominal Shock Inflation Rate

1.0

Output

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0 5

10 Quarters

15

20

Flexible Prices (N = 0),

5

10 Quarters

15

20

Sticky Prices (N = 4),

1.0 0.8 0.6 0.4 0.2 0.0 0.2

Interest Rate

5

10 Quarters

15

20

Sticky Prices (N = 8).

53/75

Evaluation of the Calvo Fairy

• Impulse response analysis: • • • •

Monetary transmission mechanism ✓ Liquidity effect × Persistence × Hump shapes ×

54/75

Evaluating the NKPC

Evaluating the NKPC: Framework

• New Keynesian Phillips Curve in terms of output gap π bt = κΘχ(b y −b yft ) + βEt π bt+1 |{z} t κ e

• Alternative representation d ) + βEt π b t − mpl bt+1 π bt = |{z} κΘ (w t κ

d is the marginal product of labor where mpl • Inflation is created by the distortion created by imperfect competition and stickiness.

55/75

Evaluating the NKPC: Measurement

• Output Gap (xt ): detrended output (HP) vs unemployment • Since β ≈ 1 and Et π bt+1 = π bt+1 − επt+1 π bt+1 − π bt = −e κb xt + επt+1 • Increases in output gap yield decreases in inflation • Measurement of output gap: • Detrended output =⇒ decreasing relationship • Unemployment =⇒ increasing relationship

• Does it work?

56/75

Not such a great job! (US Data, 1960Q1:2008Q4)

(a) Unemployment

Inflation Rate (%)

1.5 1.0

1.0

0.5

0.5

0.0

0.0

0.5

0.5

1.0

1.0

1.5

2

4

6

8

Unemployment Rate (%)

(b) Detrended GDP

1.5

10

12

1.5

2

1

0

Output Gap (%)

1

2

57/75

Econometric Evaluation

• GMM estimation: Gertler and Galí (1999), Sbordone (2002) • US Data, 1960Q1–2008Q4, Inst: 4 lags of gap and inflation

Detrended Output Unemployment

κ

β

J–stat

-0.0419

0.9959

17.9

(0.0216)

(0.0401)

[0.01]

0.0025

0.9724

17.3

(0.0092)

(0.0349)

[0.01]

• Wrong signs for output gap • J–stat leads to model rejection

58/75

NKPC: Fitting the Detrended Output Model • Assume an AR(1) representation for detrended output: b yt = ρb yt−1 + εbt κ b • Solution: π bt = 1−βρ yt 2.0 1.5 1.0 0.5 0.0 0.5 1.0

Model,

20 05

00 20

95 19

19 90

85 19

19 80

75 19

19 70

65 19

19 60

1.5

Data. 59/75

KPC: Fitting the Unemployment Model b t = ρ1 u bt−1 + ρ2 u bt−2 + εbt • Assume an AR(2) representation for unemployment: u βρ2 κ κ bt + 1−βρ −β 2 ρ u bt−1 • Solution: π bt = 1−βρ −β 2 ρ u 1

2

1

2

2.0 1.5 1.0 0.5 0.0 0.5

Model,

05 20

00 20

95 19

19 90

85 19

19 80

75 19

19 70

65 19

19 60

1.0

Data. 60/75

A Hybrid Version of the NKPC

• Fuhrer and Moore (1995), Galí and Gertler (1999), Sbordone (2002) propose to consider the hybrid NKPC π bt = κb xt + ϕEt π bt+1 + (1 − ϕ)b πt−1 • US Data, 1960Q1–2008Q4, Inst: 4 lags of gap and inflation

Detrended Output Labor Share

κ

ϕ

J–stat

-0.0175

0.6350

14.4

(0.0152)

(0.1067)

[0.07]

0.0055

0.6510

10.4

(0.0100)

(0.0820)

[0.24]

61/75

Hybrid NKPC: Fitting the Detrended Output Model • Assume an AR(1) representation for detrended output: b yt = ρb yt−1 + εbt 2.0

Model Data

1.5 1.0 0.5 0.0 0.5

Model,

05 20

00 20

95 19

19 90

85 19

19 80

75 19

19 70

65 19

19 60

1.0

Data.

62/75

Hybrid NKPC: Fitting the labor share Model • Assume an AR(1) representation for labor share: b sw,t = ρb sw,t−1 + εbt 2.0 1.5 1.0 0.5 0.0 0.5

Model,

05 20

00 20

95 19

19 90

85 19

19 80

75 19

19 70

65 19

19 60

1.0

Data.

63/75

The Hybrid Model

The Hybrid NKPC

• How to legitimate the Hybrid NKPC? • Several avenues: • Galí and Gertler (1999): A fraction of the firms that reset prices just set P⋆t (i) = πP⋆t−1 . • CEE (2005): Prices that are not reset are indexed on past inflation.

• Will use the second (No big deal) • Keep in mind: No micro–foundations!!!

64/75

The Hybrid NKPC

• Assume constant returns to scale (α = 1) • The price are set like in the standard Calvo model • The prices that are not reset follow Pt = πt−1 Pt−1 • In other words

{ Pt+τ = Xt,τ Pt with Xt,τ =

1 πt × . . . × πt+τ −1

τ =0 τ ⩾1

65/75

The Hybrid NKPC

• Price setting Et P⋆t =

θ θ−1

∞ ∑

( ) 1+θ Φt,t+τ ξ τ X−θ t,τ st+τ yt+τ Pt+τ

τ =0 ∞ ∑

Et

(

Φt,t+τ ξ τ X1−θ yt+τ Pθt+τ t,τ

)

=

θ Pnt θ − 1 Pdt

τ =0

• Pnt and Pdt can be expressed as [ ] Pt Ct −θ n Pnt = st yt P1+θ + βξπ E P t t t Pt+1 Ct+1 t+1 ] [ Pt Ct Pdt = yt Pθt + βξπt1−θ Et Pdt+1 Pt+1 Ct+1

66/75

The Hybrid NKPC

• Aggregate Price Level 1 ( ) 1−θ Pt = (1 − ξ)P⋆t 1−θ + ξ(πt−1 Pt−1 )1−θ

• Distribution of prices )− 1 ( bt−1 )−θ θ bt = (1 − ξ)P⋆ −θ + ξ(πt−1 P P t

67/75

The Hybrid NKPC

• Log–linearized version of Price–setting π bt = κb st + with κ =

β 1 π bt−1 + Et π bt+1 1+β 1+β

(1−ξ)(1−βξ) ξ(1+β)

• Distribution bt = p bt−1 p bt−1 = 0 =⇒ p bt = 0. • Assumption: start from the steady state distribution, so p

68/75

The Hybrid NKPC

bt ) • Therefore b st = (1 + ν)(b yt − a • Such that bt ) + π bt = (1 + ν)κ(b yt − a

1 β π bt−1 + Et π bt+1 1+β 1+β

bt − π b yt = m bt • Inflation dynamics is then given by bt − a bt ) + π bt = (1 − γ)(m with γ =

βγ γ π bt−1 + Et π bt+1 1+β 1+β

1 1+κ(1+ν) .

69/75

The Hybrid NKPC: Dynamics

Basic vs Hybrid NKPC after a nominal shock Inflation Rate

1.0

Output

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0 5

10 Quarters

15

20

5

10 Quarters

Basic Model,

15

20

1.0 0.8 0.6 0.4 0.2 0.0 0.2

Interest Rate

5

10 Quarters

15

20

Hybrid Model.

70/75

The Hybrid NKPC: Dynamics

Response to a Nominal Shock Inflation Rate

1.0

Output

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0 5

10 Quarters

15

20

Flexible Prices (N = 0),

5

10 Quarters

15

20

Sticky Prices (N = 4),

1.0 0.8 0.6 0.4 0.2 0.0 0.2

Interest Rate

5

10 Quarters

15

20

Sticky Prices (N = 8).

71/75

The Hybrid NKPC

• Impulse response analysis: • • • •

Monetary transmission mechanism ✓ Liquidity effect × Persistence ✓× Hump shapes ✓×

• But very basic model, lacks monetary policy

72/75

Flatness of Phillips Curve α

yit

α

y y1

sα

sα

s

sit Back

73/75

Flatness of Phillips Curve Marginal Cost

Demand for Good yit

y

θ θ sit

sθ

sθ

p

p1

pit Back

74/75

Flatness of Phillips Curve ν

hit

ν h1 h

w

wν

wν

Wt /Pt Back

75/75