Energy and Capital in a New-Keynesian Framework

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1 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

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

3 Outline Goals Model Household Firms Government Estimation Impulse Response Functions

4 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.

5 Outline Goals Model Household Firms Government Estimation Impulse Response Functions

6 Model Structure Domestic Economy

7 Model Structure Domestic Economy Final Good Firm Household

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

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

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

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

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

13 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

14 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

15 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

16 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

17 Outline Goals Model Household Firms Government Estimation Impulse Response Functions

18 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

19 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

20 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

21 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

22 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

23 Optimization Household s Optimal Expenditure Allocation

24 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

25 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

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

27 Final Good Producers Final Good Firm

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

29 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

30 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

31 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 ( ) P q,t(i) 1 ɛ 1 ɛ di ( 1 Q t = 0 Q t(i) ɛ 1 ɛ ) ɛ ɛ 1 di

32 Intermediate Good Firms Intermediate Firms

33 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

34 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)

35 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

36 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)

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

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

39 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

40 Government Government

41 Government Central Bank Government

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

43 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

44 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

45 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

46 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

47 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

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

49 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

50 Definition of Equilibrium Equilibrium

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

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

53 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 ) ( ( 100 mean ln TotalSAOil ) ) LNSIndex 100

54 Calibrated Parameters β δ ω x ɛ Table : Calibrated Parameters

55 Estimation Results - θ estimated Parameter Prior distribution Posterior distribution Mode Mean 10% 90% θ estimated Capital elasticity α k IGamma(0.1,2) Labor elasticity α l IGamma(0.4,2) Oil elasticity α e IGamma(0.6,2) Inverse Frisch elasticity φ IGamma(1.17,0.5) Taylor rule response to inflation φ π Normal(1.2,0.1) Taylor rule response to output φ y Normal(0.5,0.1) Calvo price parameter θ Beta(0.5,0.1) Table : Prior and Posterior Distribution of Structural Parameters

56 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) Real oil price ρ se Beta(0.5,0.2) Real capital price ρ sk Beta(0.5,0.2) Price markup1 ρ p Beta(0.5,0.2) Price markup2 ν p Beta(0.5,0.2) Government ρ g Beta(0.5,0.2) Tech. in Gov. ρ ag Beta(0.5,0.2) Monetary ρ i Beta(0.5,0.2) Standard deviations Technology σ a IGamma(1,2) Real oil price σ se IGamma(1,2) Real capital price σ sk IGamma(1,2) Price markup σ p IGamma(1,2) Government σ g IGamma(1,2) Monetary σ i IGamma(1,2)

57 Estimation Results - θ calibrated Parameter Prior distribution Posterior distribution Mode Mean 10% 90% θ calibrated Capital elasticity α k IGamma(0.2,2) Labor elasticity α l IGamma(0.4,2) Oil elasticity α e IGamma(0.5,2) Inverse Frisch elasticity φ IGamma(1.17,0.5) Taylor rule response to inflation φ π Normal(1.2,0.1) Taylor rule response to output φ y Normal(0.5,0.1) Table : Prior and Posterior Distribution of Structural Parameters

58 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) Real oil price ρ se Beta(0.5,0.2) Real capital price ρ sk Beta(0.5,0.2) Price markup1 ρ p Beta(0.5,0.2) Price markup2 ν p Beta(0.5,0.2) Government ρ g Beta(0.5,0.2) Tech. in Gov. ρ ag Beta(0.5,0.2) Monetary ρ i Beta(0.5,0.2) Standard deviations Technology σ a IGamma(1,2) Real oil price σ se IGamma(1,2) Real capital price σ sk IGamma(1,2) Price markup σ p IGamma(1,2) Government σ g IGamma(1,2) Monetary σ i IGamma(1,2)

59 Outline Goals Model Household Firms Government Estimation Impulse Response Functions

60 Dom. Inflation Consump Real Wages Oil % Change Labor Capital Investment Dom.Output x 10 3 GDP Int. Rate rk Marg. Cost Quarters Quarters Quarters Quarters IRF to a Real Oil Price Shock. Case: θ Estimated

61 20 x 10 3 Dom. Inflation 0 Consump. 0 Real Wages 0 Oil % Change Labor Capital Investment Dom.Output GDP Quarters Int. Rate Quarters rk Quarters x 10 4 Marg. Cost Quarters IRF to a Real Oil Price Shock. Case: θ Calibrated

62 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 δ) ]

63 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.

64 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, 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

65 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

66 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

67 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

68 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

69 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

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