Advanced Macroeconomics II The RBC model with Capital

Transcription

1 Advanced Macroeconomics II The RBC model with Capital Lorenza Rossi (Spring 2014) University of Pavia Part of these slides are based on Jordi Galì slides for Macroeconomia Avanzada II.

2 Outline Real business cycle model with capital Matching real world data (calibration) Evaluation of the RBC approach: the debate over RBC theory

3 TRADITIONAL BUSINESS CYCLE THEORY: output trend Ȳ t evolves smoothly over time, Ȳ t = a + bt. Cycles are viewed as deviation from trends, i.e. Y t Ȳ t RBC THEORY: cycles can be explained also assuming that Ȳ t evolves according to a random walk, i.e., Ȳ t = b + Ȳ t 1 + u t. In this case much of the movements in Y t are due to movements in Ȳ t and rather then to trend deviations Y t Ȳ t

4 The Lucas Research Program (LRP) Macroeconomists should build so-called structural models, i.e. models that are based on microeconomics foundations, maximizing households and rms, exible prices/wages, market clearing, etc. The LRP is the logical outcome of the Rational Expectations Revolution of the 1970s. Kydland & Prescott (1982) accepted the challenge posed by Lucas: they built the rst Real Business Cycle (RBC) model.

5 Basic RBC model Outline of the RBC methodology: a discrete-time stochastic model of the economy populated by maximizing households and rms MAIN SOURCE OF FLUCTUATIONS: The erratic nature of technological progress

6 MAIN RESULT AND FIRST INTUITION There is only one nal good in the economy which is produced according to a constant return to scale (CRS) production function Y t = A t F (K t, N t ) where ln(a t /A) = a t is an exogenous process of technological progress (or total factor productivity TFP), which evolves according to: ln(a t /A) = ρ a ln(a t /A) + ε a,t, ε a,t WN 0, σ 2 a i.i.d. A positive shock to the TFP shifts rms labor demand and the AS curve Movements in employment and economic activity are seen as the e cient responses of a perfectly competitive economy to a productivity shock. =) Movements from a Walrasian equilibrium to another one.

7 POSITIVE TECHNOLOGY SHOCK

8 MAIN FEATURES of RBC MODELS All prices are exible, both in the long-run and in the short-run. =) All markets (of inputs and output) are Walrasian Rational expectations (RE) money short-run and long run neutrality =) classical dichotomy Fluctuations of all variables (output, consumption, employment, investment...) are the optimal responses to exogenous changes in the economic environment (technology shocks, government spending shocks) Shocks are not always desirable. But once they occur, uctuations in output, employment, consumption and other variables are the optimal responses to them!!

9 HOUSEHOLDS Intertemporal Utility function E 0 t=0 β t U(C t, N t ) with β 1 1+ρ 2 [0, 1], U c > 0, U n < 0, U cc 0, and U nn 0 Households budget constraint C t + I t + B t = w t N t + R k t K t + (1 + r t 1 )B t 1 + D t Law of motion of capital K t+1 = (1 δ)k t + I t

10 FIRMS There is only one nal good in the economy which is produced according to a constant return to scale (CRS) production function: Y t = A t F (K t, N t ) K t is a predetermined capital stock. N t is the labor input. ln(a t /A) = a t is an exogenous process of technological progress (or total factor productivity TFP), which evolves according to: log (A t /A) = ρ a log (A t 1 /A) + ε a,t, ε a,t WN 0, σ 2 a i.i.d.

11 HOW TO SOLVE THE MODEL - SEVEN STEPS 1 Find all the rst order necessary conditions 2 Calculate the economy steady state 3 Log-linearize the model around the steady state 4 Solve for the recursive law of motion 5 Calibrate the model and calculate the IRFs in response to di erent shocks 6 Calculate the moments: correlations, and standard deviations for the di erent variables both for the arti cial economy and for the actual economy. 7 Compare how well the model economy matches the actual economy s characteristics

12 STEP 1 Solve the Lagrangian max fc t,n t,k t+1,b t g E 0 Focs: β t t=0 8 >< >: λ t 2 4 U (C t, N t ) + w t N t + Rt k K t + (1 + r t 1 )B t 1 + D t C t B t (K t+1 (1 {z δ) K t ) + } I t >= >; C t : U c (t) = λ t N t : U n (t) = λ t w t K t+1 : λ t = β 1 δ + Rt+1 k E t λ t+1 B t : λ t = β (1 + r t ) E t λ t+1

13 STEP 1 - Household Problem Combining the three Focs: Lab. supply : w t = U n U C (1) and Euler eq. for C t : U c (t) = βe t [(1 + r t ) U c (t + 1)] (2) and i Euler eq. for K t+1 : U c (t) = βe t h(1 δ + Rt+1)U k c (t + 1) where r t = R k t+1 δ and β = 1 1+ρ. (3)

14 STEP 1 - Household Problem Example Focs imply: U(C t, N t ) = C t 1 σ 1 σ N 1+ϕ t 1 + ϕ w t = Ct σ N ϕ t n 1 = β(1 + r t )E t (C t+1 /C t ) σo o 1 = βe t n(c t+1 /C t ) σ (1 δ + Rt+1) k

15 STEP 1 - Firms Problem Competitive rms solve their pro t maximization problem, taking all prices w t and r t as given, max fn t,k t g Π = Y t w t N t R k t K t s.t. Y t = A t F (K t, N t )

16 STEP 1 - Firms Problem First order conditions of the rms problem are: w t = A t F N (K t, N t ) Rt k = A t F k (K t, N t ) Example (Cobb-Douglas) Y t = A t K α t N 1 t α Optimality conditions R k t = αa t (K t /N t ) (1 α) w t = (1 α)a t (K t /N t ) α

17 NOTICE: CRS production function implies zero pro ts This implies that A t F (γk t, γn t ) = γa t F (K t, N t ) F (K t, N t ) = F k (K t, N t ) K t + F N (K t, N t ) N t = αkt α 1 Nt 1 α K {z } t + (1 α) Kt α Nt α N t {z } F k (K t,n t ) F N (K t,n t ) = αf (K t, N t ) + (1 α) F (K t, N t ) = F (K t, N t ) Firms pro ts are: Π t = A t F (K t, N t ) w t N t R k t K t = A t [F k (K t, N t ) K t + F N (K t, N t ) N t ] + A t F N (K t, N t ) N t = 0 Pro ts are always equal to zero! A t F k (K t, N t ) K t

18 THE PLANNER PROBLEM RBC models do not consider any distortion or market imperfection, therefore the welfare theorems apply to these models: 1 the competitive equilibrium is pareto-optimal 2 a pareto-optimal allocation can be decentralized as a competitive equilibrium Try to solve the social planner problem and to verify the competitive equilibrium and the social planner equilibrium are identical and admit a unique solution!

19 Equilibrium Good Market Labor Market Y t = C t + I t (4) K t+1 = (1 δ)k t + I t = (1 δ)k t + A t K α t N 1 α t C t (5) C σ t N ϕ t = (1 α)a t (K t /N t ) α (6) Assets markets B t = 0 (7) o 1 = βe t n(c t+1 /C t ) σ (1 δ + αa t+1 (K t+1 /N t+1 ) (1 α) ) (8) 1 = β(1 + r t )E t n(c t+1 /C t ) σo (9) Tecnology Y t = A t Kt α Nt 1 α (10)

20 STEP 2 - Steady state De nition: equilibrium with A t = A, C t = C, K t = K, N t = N,... Equation (8) evaluated in SS implies: (K /N) (1 α) = αa ρ + δ (11) Notice: since Y = AN 1 α K α =) Labor productivity is Y /N = A (K /N) α = α αa 1 α ρ + δ (12) Equation (5) evaluated in SS implies: (divided by N), α αa 1 α C /N = A ρ + δ 1 αa 1 α δ ρ + δ (13)

21 STEP 2 - Steady state Equation (6) evaluated in SS implies (multiplying both LHS and RHS by N σ ): N σ+ϕ = (1 α)(c /N) σ A (K /N) α Notice: once we assign a value to A and to the parameters, it is possible to nd N and then to nd also the value of C, K, Y,...In particular notice that K K = N, I = δk From Euler we also know that N C C = N, Y = N w = (1 α)a (K /N) α Y N N 1 + r = 1 β = 1 + ρ, RK δ = r

22 STEP 3 - Log-linearization around the steady state Replace the dynamic non linear equations with dynamic linear equations Equations are linear in percentage deviations from the steady state De ne ˆx t = log Xt X, i.e. the log-deviations of X t from its steady state X. Thus 100*ˆx t is approximately the percentage deviation of X t from its steady state. Take the rst order approximation of X t around X t = X (meaning around ˆx t = 0) X t = Xe ˆx t Xe ˆx t ˆxt =0 + Xe ˆx t ˆxt =0 (ˆx t 0) X (1 + ˆx t ) notice that it implies that X t EXAMPLE 1: Y t = f (X t ) : using that X t X = X ˆx t Y (1 + by t ) = f (X ) + f x (X ) (X X t ) X = X ˆx t and simplifying by t = f x (X ) Y X ˆx t

23 EXAMPLE 2 Similarly: If Z t = f (X t, Y t ) : 0 f (X t, Y t ) = f (X, Y ) 1 + f x (X, Y ) f (X, Y ) X {z } elast. of f. w.r.t. X 1 ˆx t + f y (X, Y ) f (X, Y ) Y ŷ t C A and ẑ t = f x (X, Y ) f (X, Y ) X ˆx t + f y (X, Y ) f (X, Y ) Y ŷ t

24 Example with exact Solution (Long and Plosser, JPE 1983) Depreciation of capital (δ = 1) +log-utility in consumption (σ = 1). Equilibrium (1 α)(y t /N t ) = C t N ϕ t 1 = αβe t f(c t /C t+1 ) (Y t+1 /K t+1 )g K t+1 + C t = Y t Conjecture: K t+1 = λy t C t = (1 λ)y t

25 Example with exact Solution (Long and Plosser, JPE 1983) Steady state: (1 α)(y /N) = CN ϕ 1 = αβ f(y /K )g =) K Y = αβ K Y = 1 Steady state of the conjecture: thus and C Y K Y = λ C Y = (1 λ) λ = αβ N t = ((1 α)(y t /C t )) 1+ϕ 1 1 =) N t = 1 1+ϕ α 1 αβ

26 Example with exact Solution (Long and Plosser, JPE 1983) Dynamics (log-deviations from the steady state): Dynamics of Capital in log-terms k t = log λ + y t 1 = log λ + (1 α)n + αk t 1 + a t 1 In log-deviation from the steady state: bk t = by t 1 = αbk t 1 + a t 1 Production function and consumption by t = αbk t + a t = αby t 1 + a t bc t = by t = αbc t 1 + a t Discussion: - "intrinsic persistence" - drawback: constant employment, volatility,...

27 STEP 3 - GENERAL CASE. Model log-linearization Good Market αβbk t+1 = αbk t + (1 (1 δ)β)((1 α) bn t + a t ) + (14) (1 β + βδ(1 α))bc t Labor Market σbc t + (α + ϕ)bn t = a t + αbk t (15) Capital σbc t = σe t fbc t+1 g + (1 (1 δ)β) (1 α)(bk t+1 E t fbn t+1 g) ρ a a t (16) Matrix Form 2 3 bc t 4bn t 5 = ea bk t 2 3 E t fbc t+1 g 4E t fbn t+1 g5 + eb a t bk t+1

28 STEP 4 - Solving the model Dynamic System y t = AE t fy t+1 g + Bz t z t = Rz t 1 + ε t y t = [x 0 t, k 0 t] 0 : vector (n 1) of endogenous variables x t : vector (n x 1) of endogenous variables non predeterminates k t : vector (n k 1) of endogenous variables predeterminates z t : vector (n z 1) of exogenous variables ε t : vector (n z 1) white noise process (0,Ω) Form of the solution ("state-space representation"): s t = Cs t 1 + Dε t x t = Ms t on s t = [k 0 t, z 0 t] 0 is a vector of "state variables"

29 STEP 4 - Solving the RBC model for the recursive law of motion (state-space representation). State variables: bk t = ψ kk bk t 1 + ψ ka a t 1 a t = ρ a a t 1 + ε a t other endogenous variables: bx t = ψ xk bk t + ψ xa a t

30 STEP 5 - Calibracion [β ]: βr = 1 S&P500 = 6.5% =) β = (1 + (0.065/4)) 1 ' [δ ]: 0.10/4 = [α ]: w = (1 α)(y /N) =) α = 1 S n,t on S n,t W t N t Y t ' 2/3 =) α = 1/3 [σ]: (1 α) Y t N t = C σ t N ϕ t...=) balanced growth σ = 1 [ϕ ]: w t = σc t + ϕn t...=)... n t = ϕ 1 w t σϕ 1 c t =) ϕ 1 : Frisch Elasticity of Labor Supply ' 4 (controversial) King-Rebelo: U(C, L) = C t 1 σ 1 σ + θ L1 η t 1 η with restriction on N t + L t = 1 Elasticity = L ηn = 0.8 (1)(0.2) = 4 [ρ a, σ 2 a ]: a t = y t αk t (1 α)n t Estimation AR(1): ρ a = 0.979, σ 2 a = (0.007) 2

31 STEP 5: IRFs to a positive technology shock 2 y 0.8 c 8 i n 0.8 w 0.8 k r 1.5 a

32 STEP 5: IRFs to a positive technology shock A technology shock determines a shift in labor demand. Part of the increase in labor demand is accommodated by an increase in labor supply by agents. Real wage increases. The e ect on the real wage is stronger the lower is the elasticity of labor supply. In the extreme case of xed labor supply all the increase in labor demand would translate into an increase in the real wage. Hours increase. As a consequence the marginal product of capital increases, determining a positive response of investment. Both the increase in hours and the stock of capital contributes to the increase in output. Households postpone consumption because the interest rate is expected to be falling in the future. For this reason consumption is characterized by a hump shaped response.

33 Persistence of a technology shock ρ a = 0 (solid line) vs. ρ a = 0.97 (dashed line) 2 output 1 Consumption 10 i n 1 w 1.5 k r 1.5 a

34 NOTICE The variables inherit the dynamic properties of the shock. Since with ρ a = 0 technology is higher just for a single period, the marginal products of factors are higher just in the rst period, when the shock hits the economy. The household works and save more just in that period. In the second period the e ects of shock disappear and variables return quickly to initial level. With ρ a = 0.97, the shock is persistent and therefore the e ects on the variables become more persistent over time. Weakness of basic RBC model: the lack of endogenous persistence. Transitory shocks have just a transitory e ect on output. Persistent e ects on output can be obtained uniquely by imposing a highly persistent shock.

35 STEP second moments - US Data (King and Rebelo 1999)

36 STEP second moments RBC model (King and Rebelo 1999)

37 STEP second moments and comparisons with the data King and Rebelo 1999, provides you with summary statistics. Compare them with those in the data. In particular we notice that: output volatility is quite lower but similar to that in the data. consumption volatility is lower than that in the data. investment are much more volatile than output as found in the data volatility of labor is too low wrt to the data. there is too much comovements of variables with output. real wage is procyclical while it is almost acyclical in the data. real rate is procyclical while is countercyclical in the data.

38 STEP comparisons with the data (King and Rebelo 1999)

39 STEP comparisons with the data (King and Rebelo 1999)

40 Some questions In RBC setup labor market is Walrasian and therefore involuntary unemployment is absent. The question is: Do changes in employment always re ect voluntary changes in employment? Is the economy hit by large and exogenous productivity shocks in the short run? In other word, are technology shocks the main sources of economic uctuations? Is money really neutral in the short run? Are wages and price fully exible in the short run?

41 SHORTCOMINGS LABOR MARKET Intertemporal substitution of labor supply: households reallocate labor over time in response to changes in w t versus w t+1. in RBC models: technology shocks cause uctuations in the intertemporal relative wage and then workers responds adjusting their labor supply ) employment and output uctuate Contrary to RBC results, empirical evidence suggests that the long-run labor supply is independent of real wages. CRITICS: - labor supply does not depend that much on the intertemporal real wage; - high unemployment is mainly involuntary

42 Solow residual and technology shocks TFP is TFP = A A = Y α K Y K (1 α) N N the Solow residual should be highly correlated with output

43 Criticism Solow residual and technology shocks Need for highly persistent technological shocks: why? and above all: what are they? Solow residual is highly correlated with output

44 RBC View: Critics: RBC theory argue that the strong correlation between output growth and Solow residuals is the evidence that productivity shocks are an important source of economic uctuations. Are economic uctuations really caused by productivity shocks? Expansions arise because of increases in productivity!... What does that mean about recessions? (Summers 1986) Does it mean that recessions are periods of technical regress! Less implausible if supply shock considered more broadly (OPEC, strikes etc.)

45 Are wages and prices exible? Critics: RBC theory assumes that wages and prices are completely exible, so markets always clear. RBC proponents argue that the degree of price stickiness occurring in the real world is not important for understanding economic uctuations. They also assume exible prices to be consistent with microeconomic theory. Wage and price stickiness explains involuntary unemployment and the non-neutrality of money.

46 SUMMING UP Empirical signi cance of intertemporal labor substitution mechanism is doubtful. RBC theory implies that recessions are periods of technical regress! Money is neutral so does not explain positive correlation between prices and output; and that this can be recti ed by endogenizing the money supply. Wage and price rigidity can help to explain involuntary unemployment and non-neutrality of money