The RBC model. Micha l Brzoza-Brzezina. Warsaw School of Economics. Advanced Macro. MBB (SGH) RBC Advanced Macro 1 / 56

Transcription

1 The RBC model Micha l Brzoza-Brzezina Warsaw School of Economics Advanced Macro MBB (SGH) RBC Advanced Macro 1 / 56

2 8 Summary MBB (SGH) RBC Advanced Macro 2 / 56 Plan of the Presentation 1 Trend and cycle 2 Introduction to RBC 3 The model 4 Steady state 5 Log-linearisation 6 Solution 7 Simulations

3 What are business cycles? GDP per capita in developed economies since 19th century: Stable growth trend Long-run growth rate approximately constant Substantial deviations from the growth trend in the short run - business cycles Definition of business cycles: high to medium frequency fluctuations in economic activity (usually proxied by GDP) After every expansion comes a recession But timing and amplitude irregular Difficult to predict Business cycles are not restricted to GDP, they are typical for virtually all macrovariables MBB (SGH) RBC Advanced Macro 3 / 56

4 Extracting cyclical information from the data how do we define a trend? linear trend applied to logs (consistently with expotential growth) filters operating in the frequency domain: extracting components within a given frequency range the most popular proxy for trend in the business cycle literature: hp filter applied to logs owed to hodrick and prescott (1980) for any time series {y t } T t=1, the trend component {ȳ t} T t=1 minimizes: T t=1 T 1 (y t ȳ t ) 2 + λ [(ȳ t+1 ȳ t ) (ȳ t ȳ t 1 )] 2 t=2 where: λ 0 is a smoothing parameter λ penalizes variations in the trend slope: λ : linear trend λ 0: original data λ = 1600: standard value for quarterly data MBB (SGH) RBC Advanced Macro 4 / 56

5 8 Summary MBB (SGH) RBC Advanced Macro 5 / 56 Plan of the Presentation 1 Trend and cycle 2 Introduction to RBC 3 The model 4 Steady state 5 Log-linearisation 6 Solution 7 Simulations

6 Origins of the RBC revolution 1970s: dissatisfaction with the standard Keynesian macroeconomics The Lucas (1976) critique: Lack of microfoundations and expectations in the Keynesian model In particular: if firms and households actions depend on their expectations about future policy, it is impossible to analyze the effects of this policy in a model that does not include expectations in an explicit form Microfoundations and rational expectations important Kydland and Prescott (1982): a very stylized macromodel with microfoundations and rational expectations performs surprisingly well in terms of its fit to the data MBB (SGH) RBC Advanced Macro 6 / 56

7 RBC - some basics The basic RBC model assumes an economy featuring perfectly functioning competitive markets and rational expectations. It gives primacy to technology shocks as the source of economic fluctuations. It exhibits complete monetary neutrality and Ricardian equivalence (and sets role for monetary and fiscal policy) In a nutshell - the standard Ramsey model with: Endogenous (but constant in the long run, i.e. n = 0) labour supply Stochastic productivity (i.e. real shocks) MBB (SGH) RBC Advanced Macro 7 / 56

8 8 Summary MBB (SGH) RBC Advanced Macro 8 / 56 Plan of the Presentation 1 Trend and cycle 2 Introduction to RBC 3 The model 4 Steady state 5 Log-linearisation 6 Solution 7 Simulations

9 RBC model - basic features General equilibrium model No market imperfections or adjustment costs = allocations are Pareto optimal The only source of uncertainty: stochastic productivity shocks In a nutshell - the standard Ramsey model with: Endogenous (but constant in the long run, i.e. n = 0) labour supply Stochastic productivity (i.e. real shocks) MBB (SGH) RBC Advanced Macro 9 / 56

10 Households - the problem A representative household maximises lifetime utility [ ] c 1 σ E t β t+i t+i 1 σ l 1+ϕ t+i 1 + ϕ i=0 subject to the budget constraint and the capital accumulation rule c t + i t = w t l t + R k,t k t k t+1 = (1 δ)k t + i t The household rents labour and capital to firms and receives as compensation the real wage w t and the rental rate R k,t. Note that additionally we have a transversality condition (TVC) lim t k t s= r s = 0 MBB (SGH) RBC Advanced Macro 10 / 56

11 Lagrangean First, to simplify substitute for investment c t + k t+1 = w t l t + (1 + R k,t δ)k t denote r t R k,t δ as the real interest rate. The household chooses c t, l t and k t+1. Write down the Lagrangean: L t = E t i=0 [ ( c 1 σ β t+i t+i 1 σ l 1+ϕ ) t+i 1 + ϕ λ t+i ( ct+i + k t+1+i w t+i l t+i (1 + R k,t+i δ)k t+i ) ] MBB (SGH) RBC Advanced Macro 11 / 56

12 Expectations Important: Because of uncertainty (stochastic shocks), households do not make deterministic plans for consumption and labour supply Any decision at time t depends on expectations concerning the relevant variables in the next periods These expectations take into account all information available at the moment of decision making We denote these expectations using the expetations operator E t Stochastic shocks make households revise their expectations every period Expectations are rational, i.e. households know the structure of the economy and characteristics (but not realizations) of stochastic shocks MBB (SGH) RBC Advanced Macro 12 / 56

13 Expectations cont d Basic rules for expectations E t (X t + Y t ) = E t X t + E t Y t E t (kx t ) = ke t X t E t (k + X t ) = k + E t X t and, unless X t and Y t are independent: E t (X t Y t ) E t X t E t Y t MBB (SGH) RBC Advanced Macro 13 / 56

14 First order conditions Sequence of events: at time t the households, having k t units of capital, observe stochastic disturbance ɛ t (and so productivity z t ), form rational expectations about future productivity levels, make optimal work, consumption and investment decisions, which determine capital stock at period t + 1 etc. First order conditions: c t : Do it yourself :-) l t : Do it yourself :-) k t+1 : Do it yourself :-) MBB (SGH) RBC Advanced Macro 14 / 56

15 Equilibrium conditions - Euler equation λ t = βe t [λ t+1 (1 + R k,t+1 δ)] substitute for λ t to get c σ t = βe t [c σ t+1 (1 + R k,t+1 δ)] This is the Euler equation. It determines the household s intertemporal choice (how much to consume today, how much to save). In equilibrium the disutility from one unit less consumed today equals expected discounted utility of consuming (1 + R k,t+1 δ) units tomorrow. MBB (SGH) RBC Advanced Macro 15 / 56

16 Equilibrium conditions - consumption vs. leisure l ϕ t c σ t = w t This equation determines the household s intratemporal choice (how much to consume, how much to work). In equilibrium the utility of one unit more of work should be equal to the utility from consuming the compensation (real wage). MBB (SGH) RBC Advanced Macro 16 / 56

17 Firms - the problem A representative firm uses capital and labour hired from households to produce a unique good y t. Its objective is to maximise profits: Div t = y t w t l t r t k t subject to technology y t = z t k α t l 1 α t MBB (SGH) RBC Advanced Macro 17 / 56

18 Firms - equilibrium condition for labour Substitute for production to get: Div t = z t k α t l 1 α t w t l t R k,t k t First order condition for labour is: l t : δdivt δl t = (1 α)z t k α t l α t w t = 0 In equilibrium the marginal product of labour equals the real wage To simplify things substitute from production funtion: (1 α) yt l t = w t MBB (SGH) RBC Advanced Macro 18 / 56

19 Firms - equilibrium condition for capital k t : do it yourself In equilibrium the marginal product of capital equals the real rental rate of capital. To simplify things substitute from production funtion: α yt k t = R k,t In the RBC setting profits are zero (perfect competition). To see this substitute R k,t and w t into the profit function. MBB (SGH) RBC Advanced Macro 19 / 56

20 Productivity and market clearing It is assumed that productivity z t follows an AR(1) process: z t = exp(ɛ t )z ρ t 1 ẑ t = ρẑ t 1 + ɛ t where ɛ t is a productivity shock This is the only stochastic process in the basic RBC model Together with the internal persistence of the model it generates the business cycle The goods market clears: c t + i t = y t MBB (SGH) RBC Advanced Macro 20 / 56

21 Equilibrium conditions - summary Now we have a system of 8 equations with 8 endogeneous variables (c, r, l, w, k, i, y, z): c σ t = βe t [c σ t+1 (1 + R k,t+1 δ)] (1) l ϕ t c σ t = w t (2) k t+1 = (1 δ)k t + i t (3) c t + i t = y t (4) MBB (SGH) RBC Advanced Macro 21 / 56

22 Equilibrium conditions - cont d y t = z t k α t l 1 α t (5) (1 α) y t l t = w t (6) α y t k t = r t (7) z t = exp(ɛ t )z ρ t 1 (8) MBB (SGH) RBC Advanced Macro 22 / 56

23 Simplified version Note that the system can be simplified to 3 equations c σ t { ( = βe t c σ t+1 αzt+1 kt+1 α 1 l t+1 1 α + 1 δ)} l ϕ t c σ t = (1 α)z t k α t l α t k t+1 = (1 δ)k t + z t k α t L 1 α t c t MBB (SGH) RBC Advanced Macro 23 / 56

24 8 Summary MBB (SGH) RBC Advanced Macro 24 / 56 Plan of the Presentation 1 Trend and cycle 2 Introduction to RBC 3 The model 4 Steady state 5 Log-linearisation 6 Solution 7 Simulations

25 Interest rate and productivity In the deterministic steady state uncertainty disappears and all variables are constant, for example c t = c t+1 = c ss. We assume that in the steady state z is constant and From (1) we get c σ t z ss = 1 (9) = βe t [c σ t+1 (1 + R k,t+1 δ)] Simplifying (c ss ) σ = β(c ss ) σ (1 + R ss k δ) R ss k = β 1 (1 δ) (10) MBB (SGH) RBC Advanced Macro 25 / 56

26 Capital-labour and investment-capital ratios Substituting form the production function (5) for y t into (7) we get α z tkt α lt 1 α k t = r t In the steady state R ss k ( k = α(k ss ) α 1 (l ss ) 1 α ss ) α 1 = α l ss Rearranging we can obtain formula for capital-labour ratio where r ss is given by (10). k ss l ss = ( R ss ) 1 k α 1 α (11) MBB (SGH) RBC Advanced Macro 26 / 56

27 Wage and investment-capital ratio Substituting form the production function (5) for y t into (6) we get w t = (1 α) z tkt α lt 1 α l t = (1 α)z t k α t l α t which in the steady state becomes where k ss /l ss is given by (11). From (3) in the steady state we have ( k w ss ss ) α = (1 α) l ss (12) k ss = (1 δ)k ss + i ss which simplifies to i ss = δ (13) kss MBB (SGH) RBC Advanced Macro 27 / 56

28 Consumption-labour ratio Substituting from (5) for y t into (4) we get which in the steady state becomes c t + i t = z t k α t l 1 α t ( k c ss + i ss = (k ss ) α (l ss ) 1 α ss ) αl ss = l ss Dividing by l ss c ss i ss ( k ss ) α + l ss l ss = l ss Substituting for i ss from (13) and rearranging c ss l ss where k ss /l ss is given by (11). = ( k ss l ss ) α δ k ss l ss (14) MBB (SGH) RBC Advanced Macro 28 / 56

29 Labour From (2) we have lt ϕ ct σ = w t which in the steady state it becomes (l ss ) ϕ (c ss ) σ = w ss Substituting for w ss from (12) and for c ss from (14) (l ss ) ϕ[( k ss ) α k ss ] σ(l ( k δ ss l ss l ss ) σ ss ) α = (1 α) l ss Solving with respect to l ss we get l ss = where k ss /l ss is given by (11). [ ( ) (1 α) k ss α ] 1 l [( ss ϕ+σ ) k ss α ] σ l δ k ss ss l ss (15) MBB (SGH) RBC Advanced Macro 29 / 56

30 Remaining variables To obtain c ss, note c ss = css l ss l ss (16) where c ss /l ss is given by (14) and l ss is given by (15). We can obtain capital k ss from k ss = kss l ss l ss (17) where k ss /l ss is given by (11) and l ss is given by (15). To obtain output we use the production function y ss = (k ss ) α (l ss ) 1 α where k ss is given by (17) and l ss is given by (15). and we can obtain investment form (13) where k ss is given by (17). i ss = δk ss MBB (SGH) RBC Advanced Macro 30 / 56

31 8 Summary MBB (SGH) RBC Advanced Macro 31 / 56 Plan of the Presentation 1 Trend and cycle 2 Introduction to RBC 3 The model 4 Steady state 5 Log-linearisation 6 Solution 7 Simulations

32 Non-linear vs. linear models We have a system of non-linear difference equations This does not have a closed form solution Standard solution technique: log-linear approximation of the model equations around the (non-stochastic) steady-state applying an algorithm for solving linear rational expectations models (e.g. Blanchard-Kahn algorithm) Two options 1 take it as it is and let your software (e.g. Dynare) linearise it 2 linearise by hand Linearising on your own is tedious but has some advantages: 1 some parameters may disappear 2 the system is easier to understand 3 solution easier (steady state is known) MBB (SGH) RBC Advanced Macro 32 / 56

33 Log-linearisation Log-linearisation allows to change non-linear equations into linear equations This is a valid approximation in the viscinity of a given point (usualy the steady state)!!! Two steps: Express variables as log deviation from steady state using the identity: x t = x ss x t x ss = x ss exp(ln x t ln x ss ) = x ss exp ˆx t MBB (SGH) RBC Advanced Macro 33 / 56

34 Useful tricks Apply first-order series expansion w.r.t. ˆx t around steady state, i.e. around ˆx ss = 0 This yields x t = x ss exp( ˆx t ) x ss exp(ˆx ss ) + x ss exp(ˆx ss )(ˆx t ˆx ss ) = x ss (1 + ˆx t ) x t y t = x ss exp( ˆx t )y ss exp(ŷ t ) x ss y ss (1 + ˆx t + ŷ t ) xt a (x ss ) a (1 + ax ˆ t ) xt a yt b (x ss ) a (y ss ) b (1 + aˆx t + bŷ t ) MBB (SGH) RBC Advanced Macro 34 / 56

35 Log-linearise labour - consumption choice l ϕ t c σ t = w t In the steady state: (l ss ) ϕ (c ss t ) σ = w ss Let s log-linearise: (l ss ) ϕ (1 + ϕˆl (c ss ) σ t + σc ˆ t ) = w ss (1 + ŵ t ) Divide by steady state: ϕˆl t + σĉ t = ŵ t MBB (SGH) RBC Advanced Macro 35 / 56

36 Log-linearise Euler c σ t = βe t [c σ t+1 (1 + R k,t+1 δ)] In the steady state: 1 = β(1 + R ss k δ) So let s linearise: ] (c ss ) σ (1 σĉ t ) = β (c ss ) σ E t [(1 σĉ t+1 )(1 δ + Rk ss(1 + ˆR k,t+1 )) MBB (SGH) RBC Advanced Macro 36 / 56

37 Log-linearise Euler cont d Substitute for Rk ss: ] 1 σĉ t = βe t [(1 σĉ t+1 )(1 δ + ( 1 β 1 + δ)(1 + ˆR k,t+1 )) ] 1 σĉ t = E t [(1 σĉ t+1 )(β βδ + (1 β + βδ)(1 + ˆR k,t+1 )) ] 1 σĉ t = E t [(1 σĉ t+1 )(1 + (1 β(1 δ))ˆr k,t+1 ) Multiply and drop higher order terms: 1 σĉ t = 1 σe t ĉ t+1 + (1 β(1 δ))e t ˆR k,t+1 Rearange terms: σ(e t ĉ t+1 ĉ t ) = (1 β(1 δ))e t ˆR k,t+1 MBB (SGH) RBC Advanced Macro 37 / 56

38 Log-linearise market clearing condition c t + i t = y t Do it yourself :-) MBB (SGH) RBC Advanced Macro 38 / 56

39 Log-linearise firm s equilibrium conditions For labour (1 α) yt l t = w t Do it yourself :-) For capital α yt k t = r t Do it yourself :-) MBB (SGH) RBC Advanced Macro 39 / 56

40 Log-linearise production function y t = z t k α t l 1 α t Do it yourself :-) MBB (SGH) RBC Advanced Macro 40 / 56

41 Log-linearise capital accumulation equation k t+1 = (1 δ)k t + i t Do it yourself :-) MBB (SGH) RBC Advanced Macro 41 / 56

42 Log-linearise shock process z t = exp(ɛ t )z ρ t 1 ẑ t = ρẑ t 1 + ɛ t MBB (SGH) RBC Advanced Macro 42 / 56

43 Steady state values Our equations contain steady state ratio i ss y ss. These are determined by our parameters From the Euler equation: R ss k = β 1 (1 δ) and from the equilibrium condition for capital: Rk ss k ss = αy ss k ss y ss = α Rk ss = α β 1 (1 δ) MBB (SGH) RBC Advanced Macro 43 / 56

44 Steady state values cont d From the capital accumulation equation: δk ss = i ss thus i ss kss = δ y ss y ss = αδ β 1 (1 δ) MBB (SGH) RBC Advanced Macro 44 / 56

45 Log-linearised system We now have a system of 8 linear (difference) equations and 8 variables Have to solve it MBB (SGH) RBC Advanced Macro 45 / 56

46 8 Summary MBB (SGH) RBC Advanced Macro 46 / 56 Plan of the Presentation 1 Trend and cycle 2 Introduction to RBC 3 The model 4 Steady state 5 Log-linearisation 6 Solution 7 Simulations

47 Solving linear DSGE models Solving a DSGE model means changing the system of forward looking diference equations that we have into a VAR system There are several techniques for solving such systems E.g. Blanchard & Kahn (1980), Sims (2002) MBB (SGH) RBC Advanced Macro 47 / 56

48 Blanchard & Kahn solution & stability condition where: One very important thing is the stability condition Write the system in state space form: A 1 [ Xt+1 E t P t+1 ] = A 0 [ Xt P t ] + γz t+1 X t : vector n 1 of state variables (backward-looking) P t : vector m 1 of jumpers (forward-looking) Z t : wektor k 1 of shocks (with mean equal to 0 every period) A 1, A 0 : (n + m) (n + m) matrices γ: (n + m) k matrix MBB (SGH) RBC Advanced Macro 48 / 56

49 Blanchard & Kahn solution method where: Assume A 1 is invertible and rewrite the system as: [ Xt+1 E t P t+1 ] [ Xt = A P t ] + RZ t+1 A A 1 1 A 0 R A 1 1 γ and apply Jordan decomposition A = CΛC 1 where Λ is a diagonal matrix of eigenvalues and C is matrix of eigenvectors. MBB (SGH) RBC Advanced Macro 49 / 56

50 Blanchard & Kahn solution method cont d Then (under some conditions stated below) P t = C 1 22 C 21X t X t+1 = ( A 11 A 12 C 1 22 C 21) Xt + R 1 Z t+1 Where the matices A, C and R have been divided into blocks whose size is determined by the number of stable eigenvalues Λ (lying within the unit circle) This is the recursive solution of the DSGE model Alternative solution method is based on the QZ decomposition of A 0 and A 1. Does not require invertibility of A 1 (Sims 2002). MBB (SGH) RBC Advanced Macro 50 / 56

51 Blanchard & Kahn stability conditions Blanchard & Kahn stated two necessary conditions for obtaining a unique solution: C 22 is of full rank the number of eigenvalues of A lying outside the unit circle (unstable roots) must equal the number of forward looking variables (jumpers) If too many unstable roots - no solution If too few - infinite number of solutions MBB (SGH) RBC Advanced Macro 51 / 56

52 8 Summary MBB (SGH) RBC Advanced Macro 52 / 56 Plan of the Presentation 1 Trend and cycle 2 Introduction to RBC 3 The model 4 Steady state 5 Log-linearisation 6 Solution 7 Simulations

53 Convergence Notes: grey line - exogenous labour (Ramsey model); black line - endogenous labour choice (RBC model); time unit - quarters; all variables expressed as percentage deviations from their steady-state values 0 GDP 0.4 Labour input Consumption Investment Capital stock MBB (SGH) RBC Advanced Macro 53 / 56

54 Impulse responses - technology shock Notes: black line - baseline (ρ = 0.976); dark grey line - permanent productivity shock (ρ = 1); light grey line - no producitity shock inertia (ρ = 0); time unit - quarters; all variables expressed as percentage deviations from their steady-state values GDP Consumption Capital stock Real interest rate Labour input Investment Real wage Productivity MBB (SGH) RBC Advanced Macro 54 / 56

55 Moments Std. deviation Autocorrelation Corr. with GDP Data Model Data Model Data Model GDP Consumption Investment Hours worked Capital Wages Interest rate Notes: Calculations based on HP-detrended variables MBB (SGH) RBC Advanced Macro 55 / 56

56 RBC model - summary Very simple dynamic stochastic general equilibrium model (DSGE) Fluctuations driven by productivity shocks only No market imperfections or adjustment costs = decentralized allocations Pareto optimal, attempts to smooth business cycles harmful Remarkable success: Surprisingly good fit to the data (in terms of moment matching) Methodological approach adopted by the New Keynesian school - great neoclassical synthesis Criticism: Estimation rather than moment matching Other shocks Some counterfactual implications or questionable calibration choices Response to criticism and extensions: Unprecedented development program - numerous refinements and extensions Important extension: monopolistic competition and sticky prices (the New Keynesian school) MBB (SGH) RBC Advanced Macro 56 / 56