Advanced Macroeconomics II. Real Business Cycle Models. Jordi Galí. Universitat Pompeu Fabra Spring 2018

Transcription

1 Advanced Macroeconomics II Real Business Cycle Models Jordi Galí Universitat Pompeu Fabra Spring 2018

2 Assumptions Optimization by consumers and rms Perfect competition General equilibrium Absence of a monetary sector or nominal variables Outline: Basic RCB model without capital RBC model with capital accumulation Fiscal policy

3 Basic RBC Model without Capital Households Preferences E 0 1 X t=0 t U(C t ; N t ) where [0; 1]; U c > 0, U n < 0, U cc 0, and U nn 0 Budget constraint Optimality conditions - intratemporal - intertemporal C t + B t = W t N t + (1 + r t 1 )B t 1 + D t W t = U n;t U c;t MRS t U c;t = (1 + r t )E t fu c;t+1 g

4 Example: t 1 U(C t ; N t ) = C1 1 N 1+' t = log C t 1 + ' N 1+' t 1 + ' if 6= 1 if = 1 Optimality conditions: W t = Ct N ' t 1 = (1 + r t )E t (Ct+1 =C t ) Log-linear version: w t = c t + 'n t 1 c t = E t fc t+1 g (r t )

5 Labor Supply w σ c + φ n n

6 Labor Supply w σ c 1 + φ n σ c 0 + φ n n

7 Firms Technology Y t = A t F (N t ) (1) where F n > 0, F nn 0 and A t expfa t g is a technology parameter that evolves according to the AR(1) process: a t = a a t 1 + " a t where a 2 [0; 1), and f" a t g is white noise. Firm s problem max Y t W t N t subject to (1). Optimality condition W t = A t F n;t MP N t

8 Example Y t = A t N 1 t Optimality condition W t = (1 )A t Nt = (1 )(Y t =N t ) Log-linear version: w t = a t n t + log(1 )

9 Labor Demand w a α n + log(1-α) n

10 Labor Demand w a 1 α n + log(1-α) a 0 α n + log(1-α) n

11 Equilibrium Goods markets Labor market Asset market Y t = C t C t N ' t = W t = (1 )A t N t B t = 0 1 = (1 + r t )E t (Ct+1 =C t ) all for all t

12 Equilibrium values (in logs and ignoring constant terms): Predictions vs Evidence 1 n t = (1 ) + ' + a t 1 + ' y t = (1 ) + ' + a t + ' w t = (1 ) + ' + a t (1 + ')(1 r t = a ) (1 ) + ' + a t

13 Labor Market Equilibrium w σ c 0 + φ n w 0 a 0 α n + log(1-α) n 0 n

14 Effects of a Technology Shock (σ < 1) w σ c 1 + φ n σ c 0 + φ n w 1 w 0 a 1 α n + log(1-α) a 0 α n + log(1-α) n 0 n 1 n

15 Effects of a Technology Shock (σ > 1) w σ c 1 + φ n w 1 σ c 0 + φ n w 0 a 1 α n + log(1-α) a 0 α n + log(1-α) n 1 n 0 n

16 Effects of a Technology Shock (σ = 1) w σ c 1 + φ n w 1 σ c 0 + φ n w 0 a 1 α n + log(1-α) a 0 α n + log(1-α) n 0 =n 1 n

17 E cient Allocation: The Social Planner s Problem subject to max U(C t ; N t ) C t = A t F (N t ) Optimality condition: Example: U n;t U c;t = A t F n;t C t N ' t = (1 )A t N t =) Equivalence with competitive equilibrium allocation =) Any observed uctuations are optimal =) Stabilization policies are not justi ed

18 The Basic RBC Model with Capital Households Preferences E 0 1 X t=0 t U(C t ; N t ) where [0; 1]; U c > 0, U n < 0, U cc 0, and U nn 0 Budget constraint and capital accumulation equation C t + I t + B t = W t N t + R k t K t + (1 + r t 1 )B t 1 + D t K t+1 = (1 )K t + I t

19 Optimality conditions - intratemporal - intertemporal W t = U n;t U c;t MRS t U c;t = (1 + r t )E t fu c;t+1 g U c;t = E t fu c;t+1 (1 + R k t+1)g Example: t 1 U(C t ; N t ) = C1 1 Optimality conditions: N 1+' t 1 + ' W t = C t N ' t 1 = (1 + r t )E t (Ct+1 =C t ) 1 = E t (Ct+1 =C t ) (1 + R k t+1)

20 Firms Technology Y t = A t F (K t ; N t ) (2) where F k > 0, F n > 0, F kk 0, and F nn 0. De ning a t log A t, we assume a t = a a t 1 + " a t where a 2 [0; 1), and f" a t g is white noise. Firm s problem subject to (2). max Y t W t N t R k t K t Optimality conditions W t = A t F n (K t ; N t ) MP N t Rt k = A t F k (K t ; N t ) MP K t

21 Example (Cobb-Douglas) Y t = A t K t N 1 t Optimality conditions: Rt k = A t (K t =N t ) (1 ) W t = (1 )A t (K t =N t )

22 Equilibrium Goods market Labor market Y t = C t + I t (3) K t+1 = (1 )K t + A t Kt Nt 1 C t (4) C t N ' t = W t = (1 )A t (K t =N t ) (5) Asset market B t = 0 (6) o 1 = E t n(c t+1 =C t ) (1 + A t+1 (K t+1 =N t+1 ) (1 ) ) (7) 1 = (1 + r t )E t (Ct+1 =C t ) (8)

23 An Example with an Exact Solution Long and Plosser, JPE 1983 Complete depreciation ( = 1) + logarithmic utility ( = 1). Equilibrium conditions Conjecture: (1 )(Y t =N t ) = C t N ' t 1 = E t f(c t =C t+1 ) (Y t+1 =K t+1 )g K t+1 + C t = Y t Implications: K t+1 = Y t C t = (1 = )Y t N t = ((1 )(Y t =C t )) 1 1+' 1 = ' N

24 Equilibrium dynamics (in logs) y t = k t + a t + const: c t = y t + const: = k t + a t + const: k t+1 = y t + const: = k t + a t + const: Discussion: - dynamic e ects of a technology shock - "intrinsic" persistence - limitations: constant employment, uniform volatility,...

25 General Case Step 1: determination of steady state Step 2: approximate equilibrium conditions around the steady state ("log-linearization") Step 3: calibration Step 4: simulation of calibrated model

26 Determination of Steady State Steady state: equilibrium A t = A, C t = C, K t = K, N t = N,... Evaluating (7) at the steady state: K=N = A (9) Evaluating (4) at the steady state (dividing by N), C=N = A A + Evaluating (5) at the steady state: N +' = (1 1 A )(C=N) A (K=N) Given N, use (9) and (10) to determine K and C, etc. Exercise: What determines long run labor productivity Y =N? (10)

27 Log-linearization of Equilibrium Conditions around Steady State Goods market b k t+1 = b k t +(1 (1 ))((1 ) bn t +a t ) (1 +(1 ))bc t Labor market bc t + ( + ')bn t = a t + b k t Capital rental market bc t = E t fbc t+1 g+(1 (1 )) (1 )( b k t+1 E t fbn t+1 g) a a t where bx t log(x t =X). More compactly: bc t E t fbc t+1 g 4bn t 5 = A e 4E t fbn t+1 g5 + B e a t b kt b kt+1

28 Technical Note on the Solution to Dynamical Systems Dynamical System y t = AE t fy t+1 g + Bz t z t = Rz t 1 + " t y t = [x 0 t; k 0 t] 0 : vector (n 1) of endogenous variables x t : vector (n x 1) of non-predetermined endogenous variables k t : vector (n k 1) of predetermined endogenous variables z t : vector (n z 1) of exogenous variables " t : vector (n z 1) following a white noise process Solution ("state-space representation"): s t = Cs t x t = Ms t 1 + D" t where s t = [k 0 t; z 0 t] 0 is the vector of state variables.

29 In our RBC example: b kt = kk b kt 1 + ka a t 1 a t = a a t 1 + " a t and for any other endogenous variable bx t : bx t = xk b kt + xa a t

30 Calibration [ ]: R = 1 average return S&P500 = 6:5% =) = (1 + (0:065=4)) 1 ' 0:985 [ ]: 0:10=4 = 0:025. [ ]: W = (1 )(Y=N) =) = 1 S n;t S n;t W tn t Y t ' 2=3 S n;t W tn t Y t ' 2=3 =) = 1=3 []: (1 ) Y t N t = C t N ' t... balanced growth requirement =) = 1 [' ]: w t = c t + 'n t...=)... n t = ' 1 w t ' 1 c t =) ' 1 : labor supply elasticity ' 4 (controversial) Aside: King-Rebelo speci cation: U(C; L) = C1 t L1 t 1 1 with restriction N t + L t = 1 L Implied labor supply elasticity: N = 0:8 (1)(0:2) = 4 [ a ; 2 a ]: a t = y t k t (1 )n t Estimated AR(1) process for fa t g: a = 0:979; 2 a = (0:007) 2

31 Predictions vs Empirical Evidence (KR, Tables 1 and 3) Volatility: - the model accounts for 70 percent of observed output volatility - can explain relative volatility of consumption and investment - consumption and hours too little volatile relative to output Persistence: - accounts for high positive autocorrelation Cyclical patterns - accounts for procyclicality of consumption, investment and hours. - main limitation: predicts too high procyclicality of interest rate and wages Simulations (KR Figure 7) - correlation simulated and actual output ' 0:8 - weaker comovement for labor market variables

32 Source: King and Rebelo (1999)

33 Source: King and Rebelo (1999)

34 Source: King and Rebelo (1999)

35

36 Signi cance of Findings Role of technology: end of growth vs uctuations dychotomy. Fluctuations are not necessarily ine cient, given optimality of equilibrium allocation ) stabilization policies may be counterproductive Exercise: Social planner s problem: max E 0 1 X t=0 t U(C t ; N t ) subject to K t+1 = (1 )K t + A t F (K t ; N t ) C t Derive optimality conditions and check equivalence with decentralized equilibrium

37 Criticisms No role for monetary policy No involuntary unemployment What is a negative technology shock? Shortcomings of Solow residual as a measure of technology Evidence on the e ects of technology shocks (Galí (AER, 1999), Basu et al. (AER, 2006))

38 Source: Galí (1999)

39 Source: Basu, Fernald and Kimball (2006)