Energy and Capital in a New-Keynesian Framework

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

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

Outline 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.

Outline Goals Model Household Firms Government Estimation Impulse Response Functions

Model Structure Domestic Economy

Model Structure Domestic Economy Final Good Firm Household

Model Structure Domestic Economy Final Good Firm l.s taxes invests works Household consumes

Model Structure Domestic Economy Final Good Firm l.s taxes bonds capital invests works Household consumes

Model Structure Domestic Economy Final Good Firm l.s taxes bonds capital invests works Household consumes Final Goods Energy

Model Structure Domestic Economy Final Good Firm l.s taxes bonds capital invests works Household Final Goods consumes Energy produces

Model Structure Domestic Economy Final Good Firm l.s taxes bonds capital invests works Household Intermediate Firms Final Goods consumes Energy produces

Model Structure Domestic Economy Final Good Firm l.s taxes bonds capital invests works Household Intermediate Firms Final Goods consumes Energy produces exo p. Energy Labor Capital

Model Structure Domestic Economy Final Good Firm l.s taxes bonds capital invests works Household profits Intermediate Firms exo p. Final Goods consumes Energy produces Energy Labor Capital

Model Structure Domestic Economy Final Good Firm l.s taxes bonds capital invests works Household profits Intermediate Firms exo p. Final Goods consumes Energy produces exogenous price Energy Labor Capital exo p. Foreign

Model Structure Government Taylor l.s taxes Domestic Economy Final Good Firm bonds capital invests works Household profits Intermediate Firms exo p. Final Goods consumes Energy produces exogenous price Energy Labor Capital exo p. Foreign

Outline Goals Model Household Firms Government Estimation Impulse Response Functions

Household Problem [ ] max E 0 β t U(C t, L t ), 0 < β < 1 t=0 s. t P e,t C e,t + P q,t C q,t + P k,t I t + B t + T t (1 + i t 1 )B t 1 + W t L t + D t + r k t P k,t K t

Household Problem Θ x := x x (1 x) (1 x) [ ] max E 0 β t U(C t, L t ), 0 < β < 1 t=0 C t := Θ x C x e,tc 1 x q,t s. t P e,t C e,t + P q,t C q,t + P k,t I t + B t + T t (1 + i t 1 )B t 1 + W t L t + D t + r k t P k,t K t

Household Problem Θ x := x x (1 x) (1 x) [ ] max E 0 β t U(C t, L t ), 0 < β < 1 t=0 C t := Θ x C x e,tc 1 x q,t s. t P e,t C e,t + P q,t C q,t + P k,t I t + B t + T t U(C t, L t ) = log(c t ) L1+φ t 1+φ (1 + i t 1 )B t 1 + W t L t + D t + r k t P k,t K t

Household Problem Θ x := x x (1 x) (1 x) [ ] max E 0 β t U(C t, L t ), 0 < β < 1 t=0 C t := Θ x C x e,tc 1 x q,t s. t P e,t C e,t + P q,t C q,t + P k,t I t + B t + T t U(C t, L t ) = log(c t ) L1+φ t 1+φ (1 + i t 1 )B t 1 + W t L t + D t + r k t P k,t K t C q,t := ( ) ɛ 1 0 C q,t(i) 1 1 ɛ 1 ɛ di

Household Problem Θ x := x x (1 x) (1 x) [ ] max E 0 β t U(C t, L t ), 0 < β < 1 t=0 C t := Θ x C x e,tc 1 x q,t s. t P e,t C e,t + P q,t C q,t + P k,t I t + B t + T t U(C t, L t ) = log(c t ) L1+φ t 1+φ (1 + i t 1 )B t 1 + W t L t + D t + r k t P k,t K t C q,t := ( ) ɛ 1 0 C q,t(i) 1 1 ɛ 1 ɛ di I t := K t+1 (1 δ)k t

Optimization Household s Optimal Expenditure Allocation

Optimization Household s Optimal Expenditure Allocation max C q,t,c e,t P c,t C t s. t P c,t C t = P e,t C e,t + P q,t C q,t C t = Θ x C x e,tc 1 x q,t

Optimization Household s Optimal Expenditure Allocation max C q,t,c e,t P c,t C t s. t P q,t C q,t = (1 x)p c,t C t P e,t C e,t = xp c,t C t P c,t = P x e,tp (1 x) q,t P c,t C t = P e,t C e,t + P q,t C q,t C t = Θ x C x e,tc 1 x q,t

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

Final Good Producers Final Good Firm

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

Final Good Producers Intermediate Good i [0, 1] Final Good Firm ( 1 Q t = 0 Q t(i) ɛ 1 ɛ ) ɛ ɛ 1 di

Final Good Producers Intermediate Good i [0, 1] Final Good Firm ( 1 Q t = 0 Q t(i) ɛ 1 ɛ ) ɛ ɛ 1 di ɛ: the elasticity of substitution among intermediate goods

Final Good Producer Problem Final Good Firm Profit Optimization max P q,tq t 1 Q 0 P q,t(i)q t (i)di t(i) s. t i demand final good price ( ) ɛ Pq,t (i) Q t (i) = Q t P q,t = P q,t ( ) 1 1 0 P q,t(i) 1 ɛ 1 ɛ di ( 1 Q t = 0 Q t(i) ɛ 1 ɛ ) ɛ ɛ 1 di

Intermediate Good Firms Intermediate Firms

Intermediate Good Firms Intermediate Firms Q t (i) = A t E t (i) αe L t (i) α l K t (i) α k α e, α l, α k 0, α e + α l + α k 1

Intermediate Good Firms Intermediate Firms Q t (i) = A t E t (i) αe L t (i) α l K t (i) α k α e, α l, α k 0, α e + α l + α k 1 strategy of firm i: Marginal cost pricing behavior FOC Given: P e,t, P k,t, W t and Q t (i) Choses: E t (i), L t (i) and K t (i)

Intermediate Good Firms Intermediate Firms Q t (i) = A t E t (i) αe L t (i) α l K t (i) α k α e, α l, α k 0, α e + α l + α k 1 strategy of firm i: Marginal cost pricing behavior FOC FOC Given: P e,t, P k,t, W t and Q t (i) Choses: E t (i), L t (i) and K t (i) Given: prices and quantities Choses: P q,t

Price Optimization Price Maximization (at each date t) (Calvo Price Setting) θ cannot change 1 θ can change P q,t (i) = P q,t 1 (i) P q,t (i) = P o q,t(i)

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

GDP and GDP Deflator Definition GDP (in value added) P y,t Y t = P q,t Q t P e,t E t

GDP and GDP Deflator Definition GDP (in value added) GDP Deflator P y,t Y t = P q,t Q t P e,t E t P y,t = P c,t

Government Government

Government Central Bank Government

Government Central Bank Government set ( ) 1 + i t = 1 β (Π φy q,t) φπ Y t εi,t Y

Government Central Bank Government set ( ) 1 + i t = 1 β (Π φy q,t) φπ Y t εi,t Y Π q,t := P q,t P q,t 1 ln(ε i,t) = ρ i ln(ε i,t 1) + e i,t

Government Central Bank set Government budget constraint ( ) 1 + i t = 1 β (Π φy q,t) φπ Y t εi,t Y (1 + i t 1 )B t 1 + G t = B t + T t Π q,t := P q,t P q,t 1 ln(ε i,t) = ρ i ln(ε i,t 1) + e i,t

Government ln(g r,t) = (1 ρ g )(ln(ωq)) + ρ g ln(g r,t 1) + ρ alk,g e alk,t + ρ ae,g e ae,t + e g,t Central Bank set Government spending function budget constraint ( ) 1 + i t = 1 β (Π φy q,t) φπ Y t εi,t Y (1 + i t 1 )B t 1 + G t = B t + T t Π q,t := P q,t P q,t 1 ln(ε i,t) = ρ i ln(ε i,t 1) + e i,t

Other Shocks Oil Price AR(1) S e,t := Pe,t P q,t log(s e,t) = ρ s,elog(s e,t 1) + e se,t

Other Shocks Oil Price Capital Price AR(1) S e,t := Pe,t P q,t S k,t := P k,t P q,t AR(1) log(s e,t) = ρ s,elog(s e,t 1) + e se,t log(s k,t ) = ρ s,k log(s k,t 1 ) + e sk,t

Other Shocks TFP AR(1) ln(a t ) = ρ a ln(a t 1 ) + e a,t

Other Shocks TFP AR(1) Price Markup ARMA(1,1) ln(a t ) = ρ a ln(a t 1 ) + e a,t ε p,t = ρ pε p,t 1 + e p,t ν pe p,t 1

Definition of Equilibrium Equilibrium

Definition of Equilibrium agents maximize its problems all markets clear Equilibrium Goverment budget const. fulfilled

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

Data Observed Variable invobs yobs labobs infobs Transformation ( ( ) ) PFI GDPDEF detrend ln LNSIndex 100 detrend ( ln ( ) ) GDPC09 LNSIndex 100 ln ( ) ( ( ) ) Averagehours CE16OVIndex LNSIndex 100 mean ln Averagehours CE16OVIndex LNSIndex 100 ( ln GDPDEF GDPDEF ( 1) ) ( 100 mean ln ( GDPDEF GDPDEF ( 1) ) ) 100 iobs eobs ( ln ( 1 + FEDFUND 400 ln ( TotalSAOil LNSIndex ) ( ( ))) mean ln 1 + FEDFUND 400 100 ) ( ( 100 mean ln TotalSAOil ) ) LNSIndex 100

Calibrated Parameters β δ ω x ɛ 0.99 0.025 0.18 0.023 8 Table : Calibrated Parameters

Estimation Results - θ estimated Parameter Prior distribution Posterior distribution Mode Mean 10% 90% θ estimated Capital elasticity α k IGamma(0.1,2) 0.3728 0.3599 0.3380 0.3822 Labor elasticity α l IGamma(0.4,2) 0.6424 0.6411 0.6111 0.6745 Oil elasticity α e 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 φ π Normal(1.2,0.1) 1.2235 1.2253 1.0686 1.3558 Taylor rule response to output φ y 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

Estimation Results - θ estimated Table : Prior and Posterior Distribution of Shock Parameters Parameter Prior distribution Posterior distribution Mode Mean 10% 90% Autoregressive parameters Technology ρ a Beta(0.5,0.2) 0.8619 0.8481 0.7960 0.8999 Real oil price ρ se Beta(0.5,0.2) 0.5761 0.5611 0.4629 0.6669 Real capital price ρ sk Beta(0.5,0.2) 0.7210 0.7080 0.6647 0.7524 Price markup1 ρ p Beta(0.5,0.2) 0.9418 0.9283 0.8955 0.9640 Price markup2 ν p Beta(0.5,0.2) 0.9796 0.9760 0.9610 0.9913 Government ρ g Beta(0.5,0.2) 0.9058 0.8995 0.8712 0.9258 Tech. in Gov. ρ ag Beta(0.5,0.2) 0.6904 0.6127 0.3549 0.9472 Monetary ρ i Beta(0.5,0.2) 0.9399 0.9308 0.9035 0.9581 Standard deviations Technology σ a IGamma(1,2) 0.4361 0.4435 0.3901 0.4942 Real oil price σ se IGamma(1,2) 2.0000 1.9373 1.8652 2.000 Real capital price σ sk IGamma(1,2) 0.7740 0.7675 0.6379 0.8781 Price markup σ p IGamma(1,2) 0.1814 0.1854 0.1615 0.2094 Government σ g IGamma(1,2) 2.0000 1.7921 1.5508 1.9998 Monetary σ i IGamma(1,2) 0.5410 0.4566 0.3859 0.5205

Estimation Results - θ calibrated Parameter Prior distribution Posterior distribution Mode Mean 10% 90% θ calibrated Capital elasticity α k IGamma(0.2,2) 0.3918 0.3809 0.3624 0.3989 Labor elasticity α l IGamma(0.4,2) 0.5947 0.5966 0.5622 0.6305 Oil elasticity α e 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 φ π Normal(1.2,0.1) 1.5236 1.5307 1.3883 1.6722 Taylor rule response to output φ y Normal(0.5,0.1) 0.0265 0.0214 0.0001 0.0402 Table : Prior and Posterior Distribution of Structural Parameters

Estimation Results - θ calibrated Table : Prior and Posterior Distribution of Shock Parameters Parameter Prior distribution Posterior distribution Mode Mean 10% 90% Autoregressive parameters Technology ρ a Beta(0.5,0.2) 0.9605 0.9401 0.9033 0.9774 Real oil price ρ se Beta(0.5,0.2) 0.9934 0.9872 0.9754 0.9977 Real capital price ρ sk Beta(0.5,0.2) 0.8940 0.8924 0.8483 0.9314 Price markup1 ρ p Beta(0.5,0.2) 0.9839 0.9621 0.9299 0.9971 Price markup2 ν p Beta(0.5,0.2) 0.1652 0.1711 0.0593 0.2758 Government ρ g Beta(0.5,0.2) 0.9373 0.9312 0.9061 0.9560 Tech. in Gov. ρ ag Beta(0.5,0.2) 0.7129 0.6589 0.3808 0.9541 Monetary ρ i Beta(0.5,0.2) 0.1914 0.2104 0.1249 0.2856 Standard deviations Technology σ a IGamma(1,2) 0.4538 0.4542 0.3981 0.5078 Real oil price σ se IGamma(1,2) 2.0000 1.9475 1.8842 2.000 Real capital price σ sk IGamma(1,2) 0.5459 0.5750 0.4722 0.6714 Price markup σ p IGamma(1,2) 0.4235 0.4645 0.2868 0.6602 Government σ g IGamma(1,2) 2.0000 1.8359 1.6425 2.000 Monetary σ i IGamma(1,2) 0.4778 0.4769 0.4062 0.54555

Outline Goals Model Household Firms Government Estimation Impulse Response Functions

0.02 0.015 0.01 0.005 Dom. Inflation 0.01 0 0.01 0.02 0.03 0.04 0.05 Consump. 0.2 0.15 0.1 0.05 Real Wages 0 0.2 0.4 0.6 0.8 1 Oil 0 1 3 5 7 9 11 13 15 17 19 0.06 1 3 5 7 9 11 13 15 17 19 0 1 3 5 7 9 11 13 15 17 19 1.2 1 3 5 7 9 11 13 15 17 19 % Change 0.4 0.3 0.2 0.1 Labor 0.01 0.008 0.006 0.004 0.002 Capital 0.2 0.15 0.1 0.05 Investment 0.1 0.08 0.06 0.04 0.02 Dom.Output 0 1 3 5 7 9 11 13 15 17 19 0 1 3 5 7 9 11 13 15 17 19 0 1 3 5 7 9 11 13 15 17 19 0 1 3 5 7 9 11 13 15 17 19 0.5 1 1.5 2 0 x 10 3 GDP 0.02 0.015 0.01 0.005 Int. Rate 0.6 0.5 0.4 0.3 0.2 0.1 rk 0.5 0.4 0.3 0.2 0.1 Marg. Cost 2.5 1 3 5 7 9 11 13 15 17 19 Quarters 0 1 3 5 7 9 11 13 15 17 19 Quarters 0 1 3 5 7 9 11 13 15 17 19 Quarters 0 1 3 5 7 9 11 13 15 17 19 Quarters IRF to a Real Oil Price Shock. Case: θ Estimated

20 x 10 3 Dom. Inflation 0 Consump. 0 Real Wages 0 Oil 15 0.1 0.1 0.5 10 0.2 0.2 1 5 0.3 0.3 1.5 0 1 6 11 16 21 26 31 36 41 46 0.4 1 6 11 16 21 26 31 36 41 46 0.4 1 6 11 16 21 26 31 36 41 46 2 1 6 11 16 21 26 31 36 41 46 % Change 0.04 0.02 0 0.02 Labor 0 0.05 0.1 0.15 0.2 0.25 Capital 0 0.1 0.2 0.3 0.4 0.5 0.6 Investment 0 0.05 0.1 0.15 0.2 0.25 0.3 Dom.Output 1 6 11 16 21 26 31 36 41 46 1 6 11 16 21 26 31 36 41 46 1 6 11 16 21 26 31 36 41 46 1 6 11 16 21 26 31 36 41 46 GDP 0 0.05 0.1 0.15 0.2 0.25 0.3 1 6 11 16 21 26 31 36 41 46 Quarters 0.02 0.015 0.01 0.005 0 0.005 Int. Rate 0.01 1 6 11 16 21 26 31 36 41 46 Quarters rk 0.05 0 0.05 0.1 0.15 0.2 0.25 1 6 11 16 21 26 31 36 41 46 Quarters 0 5 10 15 x 10 4 Marg. Cost 20 1 6 11 16 21 26 31 36 41 46 Quarters IRF to a Real Oil Price Shock. Case: θ Calibrated

Optimization 1 = βe t [(1 + i t ) Ct C t+1 ] P c,t P c,t+1 Euler First Order Conditions competive labor supply sch. W t P c,t = C t L φ t Fisher 1 = βe t [ C t C t+1 P c,t P c,t+1 P k,t+1 P k,t (r k t+1 + 1 δ) ]

No Ponzi Scheme Transversality condition (no Ponzi Scheme) lim k E t t+k 1 s=0 B t+k 0, (1 + i s 1 ) t.

Stochastic Discount Factor 1. from date t to date t + 1 d t,t+1 := βu C (C t+1, L t+1 ) U C (C t, L t ) P c,t P c,t+1, i.e, 1 1 + i t = E t (d t,t+1 ). 2. from date t to date t + k d t,t+k := t+k 1 s=t s+1 P c,t s, then, d t,t+k := βk U C (C t+k, L t+k ). U C (C t, L t ) P c,t+k

Cost Minimization Cost minimization F.O.C mc t (i) := Wt α l Qt (i) Lt (i) = r k t P i,t α k Qt (i) Kt (i) = Pe,t α e Qt (i) Et (i) mc t (i) = F t Q t (i) 1 αe +α l +α 1 k cost(q t (i)) = (α e + α l + α k )F t Q t (i) 1 αe +α l +α k F t := ( Aα αe e α α l l αα k k P αe e,t W α l t (rt k P i,t) α k ) 1 αe +α l +α k

Price Optimization Price Maximization (at each date t) Flexible Price Setting Calvo Price Setting µ p = ɛ ɛ 1 P q,t = µ p mc t max P q,t(i)q t (i) cost(q t (i)) P q,t(i) s.t ( ) ɛ Pq,t (i) Q t (i) = Q t P q,t

Calvo Price Setting Calvo Price Setting θ cannot change 1 θ can change P q,t (i) = P q,t 1 (i) P q,t (i) = P o q,t(i) P q,t = ( θp 1 ɛ q,t 1 + (1 θ)(po q,t) 1 ɛ) 1 1 ɛ

Calvo Price Setting Calvo Price Setting Problem [ max E [ t θ k d t,t+k Pq,t (i)q t,t+k (i) cost(q t,t+k (i)) ]] P q,t(i) k=0 s.t ( ) ɛ Pq,t (i) Q t,t+k (i) = Q t+k, k 0 P q,t+k

Calvo Price Setting Calvo Price Setting Solution E t [ k=0 θ k d t,t+k Q o t,t+k ( P o q,t µ p mc o t,t+k) ] = 0 mc o t,t+k := F t+k(q o t,t+k ) 1 αe +α l +α k 1 ( P o ) ɛ Qt,t+k o = q,t Q t+k P q,t+k