[PPT] - Energy and Capital in a New-Keynesian Framework Vernica Acurio PowerPoint Presentation

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Goals Model Household Firms Government Estimation Impulse Response Functions

Energy and Capital in a New-Keynesian Framework

Verónica Acurio Vásconez, Gaël Giraud, Florent Mc Isaac, Ngoc Sang Pham CES, PSE, University Paris I March 27, 2014

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Goals Model Household Firms Government Estimation Impulse Response Functions

Outline

Goals Model Household Firms The Final Good Firm Intermediate Good Firms Government GDP and GDP Deflator Estimation Setting Estimation Results Impulse Response Functions

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Goals Model Household Firms Government Estimation Impulse Response Functions

Outline

Goals Model Household Firms Government Estimation Impulse Response Functions

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Goals Model Household Firms Government Estimation Impulse Response Functions

Goals

This paper constructs a New-Keynesian model with oil in the

production function and in consumption.

The model’s parameters are estimated using Bayesian techniques.
We observe the impact of the oil shock in this economy.

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Goals Model Household Firms Government Estimation Impulse Response Functions

Outline

Goals Model Household Firms Government Estimation Impulse Response Functions

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Goals Model Household Firms Government Estimation Impulse Response Functions

Model Structure

Domestic Economy

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Goals Model Household Firms Government Estimation Impulse Response Functions

Model Structure

Domestic Economy Household Final Good Firm

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Goals Model Household Firms Government Estimation Impulse Response Functions

Model Structure

Domestic Economy Household Final Good Firm

invests works consumes l.s taxes

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Goals Model Household Firms Government Estimation Impulse Response Functions

Model Structure

Domestic Economy Household Final Good Firm

invests works consumes l.s taxes

capital bonds

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Goals Model Household Firms Government Estimation Impulse Response Functions

Model Structure

Domestic Economy Household Final Good Firm

invests works consumes l.s taxes

capital bonds Final Goods Energy

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Goals Model Household Firms Government Estimation Impulse Response Functions

Model Structure

Domestic Economy Household Final Good Firm

invests works consumes l.s taxes

capital bonds Final Goods Energy

produces

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Goals Model Household Firms Government Estimation Impulse Response Functions

Model Structure

Domestic Economy Household Final Good Firm

invests works consumes l.s taxes

capital bonds Final Goods Energy

produces

Intermediate Firms

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Goals Model Household Firms Government Estimation Impulse Response Functions

Model Structure

Domestic Economy Household Final Good Firm

invests works consumes l.s taxes

capital bonds Final Goods Energy

produces

Intermediate Firms

Energy Labor Capital

exo p.

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Goals Model Household Firms Government Estimation Impulse Response Functions

Model Structure

Domestic Economy Household Final Good Firm

invests works consumes l.s taxes

capital bonds Final Goods Energy

produces

Intermediate Firms

Energy Labor Capital

exo p.

profits

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Goals Model Household Firms Government Estimation Impulse Response Functions

Model Structure

Domestic Economy Household Final Good Firm

invests works consumes l.s taxes

capital bonds Final Goods Energy

produces

Intermediate Firms

Energy Labor Capital

exo p.

profits Foreign

exo p. exogenous price

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Goals Model Household Firms Government Estimation Impulse Response Functions

Model Structure

Domestic Economy Household Final Good Firm

invests works consumes l.s taxes

capital bonds Final Goods Energy

produces

Intermediate Firms

Energy Labor Capital

exo p.

profits Foreign

exo p. exogenous price

Government

Taylor

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Goals Model Household Firms Government Estimation Impulse Response Functions

Outline

Goals Model Household Firms Government Estimation Impulse Response Functions

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Goals Model Household Firms Government Estimation Impulse Response Functions

Household

Problem max E0 ∞

t=0

βtU(Ct, Lt)

,

0 < β < 1 Pe,tCe,t + Pq,tCq,t + Pk,tIt + Bt + Tt ≤ (1 + it−1)Bt−1 + WtLt + Dt + r k

t Pk,tKt

s. t

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Goals Model Household Firms Government Estimation Impulse Response Functions

Household

Problem max E0 ∞

t=0

βtU(Ct, Lt)

,

0 < β < 1 Pe,tCe,t + Pq,tCq,t + Pk,tIt + Bt + Tt ≤ (1 + it−1)Bt−1 + WtLt + Dt + r k

t Pk,tKt

s. t

Ct := ΘxC x

e,tC 1−x q,t

Θx := x−x(1 − x)−(1−x)

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Goals Model Household Firms Government Estimation Impulse Response Functions

Household

Problem max E0 ∞

t=0

βtU(Ct, Lt)

,

0 < β < 1 Pe,tCe,t + Pq,tCq,t + Pk,tIt + Bt + Tt ≤ (1 + it−1)Bt−1 + WtLt + Dt + r k

t Pk,tKt

s. t

Ct := ΘxC x

e,tC 1−x q,t

Θx := x−x(1 − x)−(1−x) U(Ct, Lt) = log(Ct) − L1+φ

t

1+φ

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Goals Model Household Firms Government Estimation Impulse Response Functions

Household

Problem max E0 ∞

t=0

βtU(Ct, Lt)

,

0 < β < 1 Pe,tCe,t + Pq,tCq,t + Pk,tIt + Bt + Tt ≤ (1 + it−1)Bt−1 + WtLt + Dt + r k

t Pk,tKt

s. t

Ct := ΘxC x

e,tC 1−x q,t

Θx := x−x(1 − x)−(1−x) U(Ct, Lt) = log(Ct) − L1+φ

t

1+φ

Cq,t := 1

0 Cq,t(i)1− 1

ǫ di

ǫ

ǫ−1

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Goals Model Household Firms Government Estimation Impulse Response Functions

Household

Problem max E0 ∞

t=0

βtU(Ct, Lt)

,

0 < β < 1 Pe,tCe,t + Pq,tCq,t + Pk,tIt + Bt + Tt ≤ (1 + it−1)Bt−1 + WtLt + Dt + r k

t Pk,tKt

s. t

Ct := ΘxC x

e,tC 1−x q,t

Θx := x−x(1 − x)−(1−x) U(Ct, Lt) = log(Ct) − L1+φ

t

1+φ

Cq,t := 1

0 Cq,t(i)1− 1

ǫ di

ǫ

ǫ−1

It := Kt+1 − (1 − δ)Kt

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Goals Model Household Firms Government Estimation Impulse Response Functions

Optimization

Household’s Optimal Expenditure Allocation

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Goals Model Household Firms Government Estimation Impulse Response Functions

Optimization

Household’s Optimal Expenditure Allocation max

Cq,t,Ce,t Pc,tCt

Pc,tCt = Pe,tCe,t + Pq,tCq,t Ct = ΘxC x

e,tC 1−x q,t

s. t

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Goals Model Household Firms Government Estimation Impulse Response Functions

Optimization

Household’s Optimal Expenditure Allocation max

Cq,t,Ce,t Pc,tCt

Pc,tCt = Pe,tCe,t + Pq,tCq,t Ct = ΘxC x

e,tC 1−x q,t

s. t

Pq,tCq,t = (1 − x)Pc,tCt Pe,tCe,t = xPc,tCt Pc,t = Px

e,tP(1−x) q,t

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Goals Model Household Firms Government Estimation Impulse Response Functions

Outline

Goals Model Household Firms The Final Good Firm Intermediate Good Firms Government Estimation Impulse Response Functions

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Goals Model Household Firms Government Estimation Impulse Response Functions

Final Good Producers

Final Good Firm

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Goals Model Household Firms Government Estimation Impulse Response Functions

Final Good Producers

Final Good Firm Intermediate Good i ∈ [0, 1]

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Goals Model Household Firms Government Estimation Impulse Response Functions

Final Good Producers

Final Good Firm Intermediate Good i ∈ [0, 1] Qt = 1

0 Qt(i)

ǫ−1 ǫ di

ǫ

ǫ−1

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Goals Model Household Firms Government Estimation Impulse Response Functions

Final Good Producers

Final Good Firm Intermediate Good i ∈ [0, 1] Qt = 1

0 Qt(i)

ǫ−1 ǫ di

ǫ

ǫ−1

ǫ: the elasticity of substitution among intermediate goods

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Goals Model Household Firms Government Estimation Impulse Response Functions

Final Good Producer Problem

Final Good Firm Profit Optimization max

Qt(i) Pq,tQt −

1

0 Pq,t(i)Qt(i)di

Qt = 1

0 Qt(i)

ǫ−1 ǫ di

ǫ

ǫ−1

s. t

Qt(i) = Pq,t(i) Pq,t −ǫ Qt Pq,t = 1

0 Pq,t(i)1−ǫdi

1

1−ǫ

i demand fi n a l g

d

p r i c e

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Goals Model Household Firms Government Estimation Impulse Response Functions

Intermediate Good Firms

Intermediate Firms

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Goals Model Household Firms Government Estimation Impulse Response Functions

Intermediate Good Firms

Intermediate Firms Qt(i) = AtEt(i)αeLt(i)αℓKt(i)αk αe, αℓ, αk ≥ 0, αe + αℓ + αk ≤ 1

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Goals Model Household Firms Government Estimation Impulse Response Functions

Intermediate Good Firms

Intermediate Firms Qt(i) = AtEt(i)αeLt(i)αℓKt(i)αk αe, αℓ, αk ≥ 0, αe + αℓ + αk ≤ 1 Given: Pe,t, Pk,t, Wt and Qt(i) Choses: Et(i), Lt(i) and Kt(i)

strategy of firm i: Marginal cost pricing behavior FOC

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Goals Model Household Firms Government Estimation Impulse Response Functions

Intermediate Good Firms

Intermediate Firms Qt(i) = AtEt(i)αeLt(i)αℓKt(i)αk αe, αℓ, αk ≥ 0, αe + αℓ + αk ≤ 1 Given: Pe,t, Pk,t, Wt and Qt(i) Choses: Et(i), Lt(i) and Kt(i)

strategy of firm i: Marginal cost pricing behavior FOC

Given: prices and quantities Choses: Pq,t

F O C

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Goals Model Household Firms Government Estimation Impulse Response Functions

Price Optimization

Price Maximization (at each date t) (Calvo Price Setting) Pq,t(i) = Pq,t−1(i) Pq,t(i) = Po

q,t(i)

θ cannot change 1 − θ c a n c h a n g e

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Goals Model Household Firms Government Estimation Impulse Response Functions

Outline

Goals Model Household Firms Government GDP and GDP Deflator Estimation Impulse Response Functions

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Goals Model Household Firms Government Estimation Impulse Response Functions

GDP and GDP Deflator Definition

GDP (in value added) Py,tYt = Pq,tQt − Pe,tEt

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Goals Model Household Firms Government Estimation Impulse Response Functions

GDP and GDP Deflator Definition

GDP (in value added) Py,tYt = Pq,tQt − Pe,tEt GDP Deflator Py,t = Pc,t

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Goals Model Household Firms Government Estimation Impulse Response Functions

Government

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Goals Model Household Firms Government Estimation Impulse Response Functions

Government

Government Central Bank

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Goals Model Household Firms Government Estimation Impulse Response Functions

Government

Government Central Bank 1 + it = 1

β (Πq,t)φπ

Yt

Y

φy εi,t

set

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Goals Model Household Firms Government Estimation Impulse Response Functions

Government

Government Central Bank 1 + it = 1

β (Πq,t)φπ

Yt

Y

φy εi,t

set

Πq,t := Pq,t Pq,t−1 ln(εi,t) = ρiln(εi,t−1) + ei,t

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Goals Model Household Firms Government Estimation Impulse Response Functions

Government

Government Central Bank 1 + it = 1

β (Πq,t)φπ

Yt

Y

φy εi,t

set

Πq,t := Pq,t Pq,t−1 ln(εi,t) = ρiln(εi,t−1) + ei,t (1 + it−1)Bt−1 + Gt = Bt + Tt

budget constraint

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Goals Model Household Firms Government Estimation Impulse Response Functions

Government

Government Central Bank 1 + it = 1

β (Πq,t)φπ

Yt

Y

φy εi,t

set

Πq,t := Pq,t Pq,t−1 ln(εi,t) = ρiln(εi,t−1) + ei,t (1 + it−1)Bt−1 + Gt = Bt + Tt

budget constraint ln(Gr,t) = (1 − ρg)(ln(ωQ)) + ρg ln(Gr,t−1) + ρalk,gealk,t + ρae,geae,t + eg,t spending function

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Goals Model Household Firms Government Estimation Impulse Response Functions

Other Shocks

Oil Price Se,t := Pe,t

Pq,t log(Se,t) = ρs,elog(Se,t−1) + ese,t A R ( 1 )

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Goals Model Household Firms Government Estimation Impulse Response Functions

Other Shocks

Oil Price Se,t := Pe,t

Pq,t log(Se,t) = ρs,elog(Se,t−1) + ese,t A R ( 1 )

Capital Price Sk,t := Pk,t

Pq,t log(Sk,t) = ρs,klog(Sk,t−1) + esk,t AR(1)

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Goals Model Household Firms Government Estimation Impulse Response Functions

Other Shocks

TFP ln(At) = ρaln(At−1) + ea,t

AR(1)

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Goals Model Household Firms Government Estimation Impulse Response Functions

Other Shocks

TFP ln(At) = ρaln(At−1) + ea,t

AR(1)

Price Markup

εp,t = ρpεp,t−1 + ep,t − νpep,t−1 ARMA(1,1)

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Goals Model Household Firms Government Estimation Impulse Response Functions

Definition of Equilibrium

Equilibrium

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Goals Model Household Firms Government Estimation Impulse Response Functions

Definition of Equilibrium

Equilibrium agents maximize its problems all markets clear

Goverment budget const. fulfilled

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Goals Model Household Firms Government Estimation Impulse Response Functions

Outline

Goals Model Household Firms Government Estimation Setting Estimation Results Impulse Response Functions

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Goals Model Household Firms Government Estimation Impulse Response Functions

Data

Observed Variable Transformation invobs detrend

ln
PFI

GDPDEF

LNSIndex

∗ 100
yobs

detrend

ln

GDPC09

LNSIndex

∗ 100
labobs

ln

Averagehours∗CE16OVIndex

LNSIndex

∗ 100 − mean
ln
Averagehours∗CE16OVIndex

LNSIndex

∗ 100
infobs

ln

GDPDEF

GDPDEF(−1)

∗ 100 − mean
ln
GDPDEF

GDPDEF(−1)

∗ 100
iobs
ln
1 + FEDFUND

400

− mean
ln
1 + FEDFUND

400

∗ 100

eobs ln TotalSAOil

LNSIndex

∗ 100 − mean
ln

TotalSAOil

LNSIndex

∗ 100

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Goals Model Household Firms Government Estimation Impulse Response Functions

Calibrated Parameters

β δ ω x ǫ 0.99 0.025 0.18 0.023 8

Table : Calibrated Parameters

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Goals Model Household Firms Government Estimation Impulse Response Functions

Estimation Results - θ estimated

Parameter Prior distribution Posterior distribution Mode Mean 10% 90% θ estimated Capital elasticity αk IGamma(0.1,2) 0.3728 0.3599 0.3380 0.3822 Labor elasticity αℓ IGamma(0.4,2) 0.6424 0.6411 0.6111 0.6745 Oil elasticity αe IGamma(0.6,2) 0.1234 0.1254 0.1051 0.1460 Inverse Frisch elasticity φ IGamma(1.17,0.5) 0.6209 0.6308 0.4736 0.8019 Taylor rule response to inflation φπ Normal(1.2,0.1) 1.2235 1.2253 1.0686 1.3558 Taylor rule response to output φy Normal(0.5,0.1) 0.8020 0.7882 0.6884 0.8876 Calvo price parameter θ Beta(0.5,0.1) 0.9812 0.9812 0.9380 0.9883

Table : Prior and Posterior Distribution of Structural Parameters

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Goals Model Household Firms Government Estimation Impulse Response Functions

Estimation Results - θ estimated

Table : Prior and Posterior Distribution of Shock Parameters

Parameter Prior distribution Posterior distribution Mode Mean 10% 90% Autoregressive parameters Technology ρa Beta(0.5,0.2) 0.8619 0.8481 0.7960 0.8999 Real oil price ρse Beta(0.5,0.2) 0.5761 0.5611 0.4629 0.6669 Real capital price ρsk Beta(0.5,0.2) 0.7210 0.7080 0.6647 0.7524 Price markup1 ρp Beta(0.5,0.2) 0.9418 0.9283 0.8955 0.9640 Price markup2 νp Beta(0.5,0.2) 0.9796 0.9760 0.9610 0.9913 Government ρg Beta(0.5,0.2) 0.9058 0.8995 0.8712 0.9258

Tech. in Gov.

ρag Beta(0.5,0.2) 0.6904 0.6127 0.3549 0.9472 Monetary ρi Beta(0.5,0.2) 0.9399 0.9308 0.9035 0.9581 Standard deviations Technology σa IGamma(1,2) 0.4361 0.4435 0.3901 0.4942 Real oil price σse IGamma(1,2) 2.0000 1.9373 1.8652 2.000 Real capital price σsk IGamma(1,2) 0.7740 0.7675 0.6379 0.8781 Price markup σp IGamma(1,2) 0.1814 0.1854 0.1615 0.2094 Government σg IGamma(1,2) 2.0000 1.7921 1.5508 1.9998 Monetary σi IGamma(1,2) 0.5410 0.4566 0.3859 0.5205

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Goals Model Household Firms Government Estimation Impulse Response Functions

Estimation Results - θ calibrated

Parameter Prior distribution Posterior distribution Mode Mean 10% 90% θ calibrated Capital elasticity αk IGamma(0.2,2) 0.3918 0.3809 0.3624 0.3989 Labor elasticity αℓ IGamma(0.4,2) 0.5947 0.5966 0.5622 0.6305 Oil elasticity αe IGamma(0.5,2) 0.1132 0.1177 0.0915 0.1434 Inverse Frisch elasticity φ IGamma(1.17,0.5) 1.2562 1.2625 0.9073 1.6069 Taylor rule response to inflation φπ Normal(1.2,0.1) 1.5236 1.5307 1.3883 1.6722 Taylor rule response to output φy Normal(0.5,0.1) 0.0265 0.0214 0.0001 0.0402

Table : Prior and Posterior Distribution of Structural Parameters

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Goals Model Household Firms Government Estimation Impulse Response Functions

Estimation Results - θ calibrated

Table : Prior and Posterior Distribution of Shock Parameters

Parameter Prior distribution Posterior distribution Mode Mean 10% 90% Autoregressive parameters Technology ρa Beta(0.5,0.2) 0.9605 0.9401 0.9033 0.9774 Real oil price ρse Beta(0.5,0.2) 0.9934 0.9872 0.9754 0.9977 Real capital price ρsk Beta(0.5,0.2) 0.8940 0.8924 0.8483 0.9314 Price markup1 ρp Beta(0.5,0.2) 0.9839 0.9621 0.9299 0.9971 Price markup2 νp Beta(0.5,0.2) 0.1652 0.1711 0.0593 0.2758 Government ρg Beta(0.5,0.2) 0.9373 0.9312 0.9061 0.9560

Tech. in Gov.

ρag Beta(0.5,0.2) 0.7129 0.6589 0.3808 0.9541 Monetary ρi Beta(0.5,0.2) 0.1914 0.2104 0.1249 0.2856 Standard deviations Technology σa IGamma(1,2) 0.4538 0.4542 0.3981 0.5078 Real oil price σse IGamma(1,2) 2.0000 1.9475 1.8842 2.000 Real capital price σsk IGamma(1,2) 0.5459 0.5750 0.4722 0.6714 Price markup σp IGamma(1,2) 0.4235 0.4645 0.2868 0.6602 Government σg IGamma(1,2) 2.0000 1.8359 1.6425 2.000 Monetary σi IGamma(1,2) 0.4778 0.4769 0.4062 0.54555

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Goals Model Household Firms Government Estimation Impulse Response Functions

Outline

Goals Model Household Firms Government Estimation Impulse Response Functions

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1 3 5 7 9 11 13 15 17 19 0.005 0.01 0.015 0.02

Dom. Inflation

1 3 5 7 9 11 13 15 17 19 −0.06 −0.05 −0.04 −0.03 −0.02 −0.01 0.01 Consump. 1 3 5 7 9 11 13 15 17 19 0.05 0.1 0.15 0.2 Real Wages 1 3 5 7 9 11 13 15 17 19 −1.2 −1 −0.8 −0.6 −0.4 −0.2 Oil 1 3 5 7 9 11 13 15 17 19 0.1 0.2 0.3 0.4 Labor % Change 1 3 5 7 9 11 13 15 17 19 0.002 0.004 0.006 0.008 0.01 Capital 1 3 5 7 9 11 13 15 17 19 0.05 0.1 0.15 0.2 Investment 1 3 5 7 9 11 13 15 17 19 0.02 0.04 0.06 0.08 0.1 Dom.Output 1 3 5 7 9 11 13 15 17 19 −2.5 −2 −1.5 −1 −0.5 x 10

−3

GDP Quarters 1 3 5 7 9 11 13 15 17 19 0.005 0.01 0.015 0.02

Int. Rate

Quarters 1 3 5 7 9 11 13 15 17 19 0.1 0.2 0.3 0.4 0.5 0.6 rk Quarters 1 3 5 7 9 11 13 15 17 19 0.1 0.2 0.3 0.4 0.5

Marg. Cost

Quarters

IRF to a Real Oil Price Shock. Case: θ Estimated

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1 6 11 16 21 26 31 36 41 46 5 10 15 20 x 10

−3

Dom. Inflation

1 6 11 16 21 26 31 36 41 46 −0.4 −0.3 −0.2 −0.1 Consump. 1 6 11 16 21 26 31 36 41 46 −0.4 −0.3 −0.2 −0.1 Real Wages 1 6 11 16 21 26 31 36 41 46 −2 −1.5 −1 −0.5 Oil 1 6 11 16 21 26 31 36 41 46 −0.02 0.02 0.04 Labor % Change 1 6 11 16 21 26 31 36 41 46 −0.25 −0.2 −0.15 −0.1 −0.05 Capital 1 6 11 16 21 26 31 36 41 46 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 Investment 1 6 11 16 21 26 31 36 41 46 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 Dom.Output 1 6 11 16 21 26 31 36 41 46 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 GDP Quarters 1 6 11 16 21 26 31 36 41 46 −0.01 −0.005 0.005 0.01 0.015 0.02

Int. Rate

Quarters 1 6 11 16 21 26 31 36 41 46 −0.25 −0.2 −0.15 −0.1 −0.05 0.05 rk Quarters 1 6 11 16 21 26 31 36 41 46 −20 −15 −10 −5 x 10

−4

Marg. Cost

Quarters

IRF to a Real Oil Price Shock. Case: θ Calibrated

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Goals Model Household Firms Government Estimation Impulse Response Functions

Optimization

First Order Conditions 1 = βEt

(1 + it) Ct

Ct+1 Pc,t Pc,t+1

Euler

1 = βEt

Ct

Ct+1 Pc,t Pc,t+1 Pk,t+1 Pk,t (r k t+1 + 1 − δ)

Fisher

Wt Pc,t = CtLφ t

competive labor supply sch.

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Goals Model Household Firms Government Estimation Impulse Response Functions

No Ponzi Scheme

Transversality condition (no Ponzi Scheme) lim

k→∞ Et

     Bt+k

t+k−1

s=0

(1 + is−1)      ≥ 0, ∀t.

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Goals Model Household Firms Government Estimation Impulse Response Functions

Stochastic Discount Factor

1. from date t to date t + 1

dt,t+1 := βUC(Ct+1, Lt+1) UC(Ct, Lt) Pc,t Pc,t+1 , i.e, 1 1 + it = Et(dt,t+1).

2. from date t to date t + k

dt,t+k :=

t+k−1

s=t

∆s+1

s

, then, dt,t+k := βkUC(Ct+k, Lt+k) UC(Ct, Lt) Pc,t Pc,t+k .

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Goals Model Household Firms Government Estimation Impulse Response Functions

Cost Minimization

Cost minimization mct(i) :=

Wt αℓ

Qt (i) Lt (i) =

r k

t Pi,t

αk

Qt (i) Kt (i) =

Pe,t αe

Qt (i) Et (i)

F.O.C

cost(Qt(i)) = (αe + αℓ + αk)FtQt(i)

1 αe +αℓ+αk

mct(i) = FtQt(i)

1 αe +αℓ+αk −1

Ft :=

Aααe

e

α

αℓ ℓ α αk k

Pαe

e,t W αℓ t

(r k

t Pi,t)αk

−1

αe +αℓ+αk

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Goals Model Household Firms Government Estimation Impulse Response Functions

Price Optimization

Price Maximization (at each date t) Flexible Price Setting Calvo Price Setting max

Pq,t(i) Pq,t(i)Qt(i) − cost(Qt(i))

Qt(i) = Pq,t(i) Pq,t −ǫ Qt

s.t

Pq,t = µpmct µp =

ǫ ǫ−1

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Goals Model Household Firms Government Estimation Impulse Response Functions

Calvo Price Setting

Calvo Price Setting Pq,t(i) = Pq,t−1(i) Pq,t(i) = Po

q,t(i)

θ c a n n

t

c h a n g e 1 − θ c a n c h a n g e

Pq,t =

θP1−ǫ

q,t−1 + (1 − θ)(Po q,t)1−ǫ

1 1−ǫ

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Goals Model Household Firms Government Estimation Impulse Response Functions

Calvo Price Setting

Calvo Price Setting Problem max

Pq,t(i) Et

∞

k=0

θkdt,t+k

Pq,t(i)Qt,t+k(i) − cost(Qt,t+k(i))
Qt,t+k(i) =

Pq,t(i) Pq,t+k −ǫ Qt+k, ∀k ≥ 0

s.t

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Goals Model Household Firms Government Estimation Impulse Response Functions

Calvo Price Setting

Calvo Price Setting Solution Et ∞

k=0

θkdt,t+kQo

t,t+k

Po

q,t − µpmco t,t+k

= 0

mco

t,t+k := Ft+k(Qo t,t+k)

1 αe +αℓ+αk −1

Qo

t,t+k =

Po

q,t