3 The Standard Real Business Cycle (RBC) Model. Optimal growth model + Labor decisions


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1 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 36 3 The Standard Real Business Cycle (RBC) Model Perfectly competitive economy Optimal growth model + Labor decisions 2 types of agents Households Firms Shocks to productivity Pareto optimal economy
2 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 37 Can be solved using a Social Planner program or solving for a competitive equilibrium We will solve for the equilibrium
3 Franck Portier TSE Macro II Chapter 3 Real Business Cycles The Household Mass of agents = 1 (no population growth) Identical agents + All face the same aggregate shocks (no idiosyncratic uncertainty) Representative agents
4 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 39 Infinitely lived rational agent with intertemporal utility β (, 1): discount factor, E t s= β s U t+s Preferences over a consumption bundle leisure U t = U(C t, l t ) with U(, ) class C 2, strictly increasing, concave and satisfy Inada con
5 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 4 ditions compatible with balanced growth [more below]: U(C t, l t ) = { C 1 σ t 1 σ v(l t) if σ R + \{1} log(c t ) + v(l t ) if σ = 1
6 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 41 Preferences are therefore given by E t β s U(C t+s, l t+s ) s= Household faces two constraints Time constraint h t+s + l t+s T = 1 (for convenience T=1)
7 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 42 Budget constraint B t+s } {{ } Bond purchases (1 + r t+s 1 )B t+s 1 } {{ } Bond revenus Capital Accumulation + C t+s + I t+s } {{ } Good purchases + W t+s h t+s } {{ } W ages + z t+s K t+s } {{ } Capital revenus δ (, 1): Depreciation rate K t+s+1 = I t+s + (1 δ)k t+s
8 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 43 The household decides on consumption, labor, leisure, investment, bond holdings and capital formation maximizing utility constraint, taking the constraints into account max {C t+s,h t+s,l t+s,i t+s,k t+s+1,b t+s } t= E t subject to the sequence of constraints β s U(C t+s, l t+s ) h t+s + l t+s 1 B t+s + C t+s + I t+s (1 + r t+s 1 )B t+s 1 + W t+s h t+s + z t+s K t+s K t+s+1 = I t+s + (1 δ)k t+s K t, B t 1 given s=
9 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 44 subject to max {C t+s,h t+s,k t+1,b t+s } t= E t s= β s U(C t+s, 1 h t+s ) B t+s +C t+s +K t+s+1 (1+r t+s 1 )B t+s 1 +W t+s h t+s +(z t+s +1 δ)k t+s Write the Lagrangian L t = E t s= β s U(C t+s, 1 h t+s ) + Λ t+s ((1 + r t+s 1 )B t+s 1 + ) +W t+s h t+s + (z t+s + 1 δ)k t+s C t+s B t+s K t+s+1
10 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 45 First order conditions ( s ) C t+s : E t U c (C t+s, 1 h t+s ) = E t Λ t+s h t+s : E t U l (C t+s, 1 h t+s ) = E t (Λ t+s W t+s ) B t+s : E t Λ t+s = βe t ((1 + r t+s )Λ t+s+1 ) K t+s+1 : E t Λ t+s = βe t (Λ t+s+1 (z t+s δ)) and the transversality condition lim s + βs E t Λ t+s (B t+s + K t+s+1 ) =
11 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 46 Rearranging terms: K t+s+1 : E t U c (C t+s, 1 h t+s ) = βe t (U c (C t+s+1, 1 h t+s+1 )(z t+s h t+s : E t U l (C t+s, 1 h t+s ) = E t U c (C t+s, 1 h t+s )W t+s B t+s : E t U c (C t+s, 1 h t+s ) = βe t ((1 + r t+s )E t U c (C t+s+1, 1 h t+s+1 ) and the transversality condition lim s + βs E t U c (C t+s, 1 h t+s )(B t+s + K t+s+1 ) =
12 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 47 Simple example : Assume U(C t, l t ) = log(c t ) + θ log(1 h t ) h t+s : E 1 W t 1 h t+s = E t+s t C t+s B t+s : E 1 t C t+s = βe t (1 + r t+s ) 1 C t+s+1 K t+s+1 : E 1 t C t+s = βe 1 t C (z t+s+1 t+s δ) and the transversality condition lim s + E tβ sk t+s+1 + B t+s C t+s =
13 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 48 Remark :  It is convenient to write and interpret FOC for s = : h t : U l (C t, 1 h t ) = E t U c (C t, 1 h t )W t B t : U c (C t, 1 h t ) = βe t ((1 + r t )E t U c (C t+1, 1 h t+1 )) K t+1 : U c (C t, 1 h t ) = βe t (U c (C t+1, 1 h t+1 )(z t δ)) and the transversality condition lim s + E tβ sk t+s+1 + B t+s C t+s =
14 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 49 We have consumption smoothing and We have labor smoothing 3.2 The Firm θ W t (1 h t ) = β(1 + r t)e t W t+1 (1 h t+1 ) Mass of firms = 1 Identical firms + All face the same aggregate shocks (no idiosyncratic uncertainty) Representative firm θ
15 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 5 Produce an homogenous good that is consumed or invested by means of capital and labor Constant returns to scale technology (important) Y t = A t F(K t, Γ t h t ) Γ t = γγ t 1 Harrod neutral technological progress (γ 1), A t stationary (does not explain growth)
16 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 51  Remark: one could introduce long run technical progress in three different ways: Y t = Γ t F( Γ t K t, Γ t h t )  Γ t is Hicks Neutral, Γ t is Solow neutral
17 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 52
18 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 53  Harrod neutral technical progress and the preferences specified above are needed for the existence of a Balanced Growth Path that replicates Kaldor Stylized Facts: 1. The shares of national income received by labor and capital are roughly constant over long periods of time 2. The rate of growth of the capital stock is roughly constant over long periods of time 3. The rate of growth of output per worker is roughly constant over long periods of time 4. The capital/output ratio is roughly constant over long periods of time 5. The rate of return on investment is roughly constant over long periods of time 6. The real wage grows over time  End of the remark
19 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 54 Y t = A t F(K t, Γ t h t ) Γ t = γγ t 1 Harrod neutral technological progress (γ 1), A t stationary (does not explain growth) A t are shocks to technology. AR(1) exogenous process with ε t N (, σ 2 ). log(a t ) = ρ log(a t 1 ) + (1 ρ) log(a) + ε t
20 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 55 The firm decides on production plan maximizing profits First order conditions: max {K t,h t } A tf(k t, Γ t h t ) W t h t z t K t K t : A t F K (K t, Γ t h t ) = z t h t : A t F h (K t, Γ t h t ) = W t Simple Example: Cobb Douglas production function First order conditions Y t = A t K α t (Γ t h t ) 1 α K t : αy t /K t = z t h t : (1 α)y t /h t = W t
21 Franck Portier TSE Macro II Chapter 3 Real Business Cycles Equilibrium The (RBC) Model Equilibrium is given by the following equations ( t ): 1. Exogenous Processes : log(a t ) = ρ log(a t 1 )+(1 ρ) log(a)+ε t and Γ t = γγ t 1 2. Law of motion of Capital : K t+1 = I t + (1 δ)k t 3. Bond market equilibrium : B t = 4. Good Markets equilibrium : Y t = C t + I t
22 Franck Portier TSE Macro II Chapter 3 Real Business Cycles Labor market equilibrium : U l(c t,1 h t ) U c (C t,l t ) = A t F h (K t, Γ t h t ) 6. Consumption/saving decision + Capital market equilibrium : U c (C t, 1 h t ) = βe t [U c (C t+1, 1 h t+1 )(A t+1 F K (K t+1, Γ t+1 h t+1 ) + 1 δ)] 7. Financial markets : 1 + r t = E t [U c (C t+1, 1 h t+1 )(A t+1 F K (K t+1, Γ t+1 h t+1 ) + 1 δ)] E t U c (C t+1, 1 h t+1 )
23 Franck Portier TSE Macro II Chapter 3 Real Business Cycles An Analytical Example U(C t, l t ) = log(c t ) + θ log(l t ), Y t = A t K α t (Γ th t ) 1 α Equilibrium θc t = (1 α) Y t 1 h t [ ( h t 1 1 Yt+1 = βe t + 1 δ C t K t+1 C t+1 K t+1 = Y t C t + (1 δ)k t )] Y t = A t K α t (Γ t h t ) 1 α lim s βs E t [ Kt+1+s C t+s ] =
24 Franck Portier TSE Macro II Chapter 3 Real Business Cycles Stationarization We want a stationary equilibrium (technical reasons) Deflate the model for the growth component Γ t On the example: x t = X t /Γ t
25 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 6  Deflated Equilibrium θc t = (1 α) y t 1 h t h t 1 = β [ ( 1 yt+1 c t γ E t + 1 δ k t+1 c t+1 γk t+1 = y t c t + (1 δ)k t y t = A t k α t h1 α t lim s βsγk t+1+s c t+s = )]
26 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 61 4 Solving the Model 4.1 In General Non linear system of stochastic finite difference equations under rational expectations Very complicated In general no analytical solution, need to rely on numerical approximation methods
27 Franck Portier TSE Macro II Chapter 3 Real Business Cycles The Nice Analytical Case U(c, l) = log(c) + θ log(l), γ = 1 (not needed) and δ = 1 Equilibrium θc t = (1 α) y t 1 h t [ h t 1 1 = βe t c t c t+1 k t+1 = y t c t y t = A t k α t h1 α t ( yt+1 k t+1 lim s βsγk t+1+s c t+s = )]
28 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 63 The solution is: c t = (1 αβ)y t h t = h = 1 α 1 α+θ(1 αβ) k t+1 = αβκa t k α t with κ = h 1 α
29 Franck Portier TSE Macro II Chapter 3 Real Business Cycles Numerical solution Unfortunately: No closed form solution in general Have to adopt a numerical approach Log linearize the equilibrium around the steady state (limits?) Solve the linearized model using standard techniques
30 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 65 The solution to the log linearize version of the model takes the form X t+1 = M x X t + M z Z t (1) Z t+1 = ρz t + ε t+1 (2) Y t = P x X t + P z Z t (3) where X t, Z t and Y t collect, respectively, the state variables, the shocks and the variables of interest Resembles a VAR model
31 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 66 In the basic RBC model X t = {k t } (4) Z t = {a t } (5) Y t = {y t, c t, i t, h t } (6) Let s evaluate the quantitative ability of the model to account for and explain the cycle.
32 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 67 c t c t = c c 6 k t = k k k t
33 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 68 5 Quantitative Evaluation 5.1 Calibration Need to assign numerical values to the parameters This is the calibration step What does calibration mean? Make explicit use of the model to set the parameters A lot of discipline, but no systematic recipe Have to set (α, β, θ, δ, γ, ρ, σ, A)
34 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 69 Use data: (k/y, c/y, i/y, h, wh/y, r) Compute (k/y, c/y, i/y, h, wh/y, r) in the model
35 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 7 In the data y =.9% per quarter γ = 1.9. In the data i/k =.76 on annual data. Use capital accumulation to get annual depreciation. i/k(model) = gamma (1 δ) δ =.1 In the data wh/y =.6, in the model wh/y = 1 α α =.4. In the data k/y = 3.32 on annual data. equation Then β =.98 β = γ αy/k + (1 δ) Using the Euler
36 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 71 h =.31, such that θ = (1 α)(1 h) hc/y
37 Franck Portier TSE Macro II Chapter 3 Real Business Cycles Can the Business Cycle be Driven by Capital Dynamics? Want to see whether capital dynamics can account for BC. Perfect foresights dynamics We study a case with fixed labor (h = h) and a variable labor case (U(c, 1 h))
38 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 73 Capital shock  Fixed hours Output Consumption Investment Capital.4 Hours worked Labor productivity (Wages)
39 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 74 Capital shock  Variable hours Output Consumption Investment Capital.4 Hours worked Labor productivity (Wages)
40 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 75 The answer is clearly that capital dynamics cannot be the story for the business cycle. Solow (1957): capital accumulation accounts for 1/8th of output growth. Technical progress, not capital accumulation, is the engine of growth. At the business frequency: transitional dynamics does not conform to the data (c and i for ex). More (shocks?) is needed to understand the BC
41 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 76 That s why the RBC literature proposes technological shocks (Brock & Mirman, 1972)
42 Franck Portier TSE Macro II Chapter 3 Real Business Cycles Technological Shocks How to calibrate the shocks? We have log(a t ) = log(y t ) α log(k t ) (1 α) log(h t ) How to get k t? Use Capital accumulation with k = (k/y)y Estimate log(a t ) = ρ log(a t 1 ) + ε t We get ρ =.95 and σ =.79. We set A = 1 without loss
43 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 78 of generality.
44 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 79 A good fit with estimated shocks.1.5 Output Data Model.4.2 Hours Worked Consumption Investment
45 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 8 A first success? Accounts for the main events in the data The model correctly predicts the data: corr(y, y m )=.75, corr(c, c m )=.73, corr(i, i m )=.7. BUT: corr(h, h m )=.6 Let s compute unconditional moments in the model (simulate HP filtercompute moments)
46 Franck Portier TSE Macro II Chapter 3 Real Business Cycles Results from Model Simulations Variable σ( ) σ( )/σ(y) ρ(, y) ρ(, h) Auto Output Consumption Investment Hours worked Labor productivity (model in yellow)
47 Franck Portier TSE Macro II Chapter 3 Real Business Cycles A success(?) The model correctly predicts the amplitude, serial correlation and relative variability of fluctuations It accounts for a large part of output volatility Correct ranking of the volatility of c, i, y,... Large serial correlation, although it is smaller than in the data. But
48 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 83 C and N are not volatile enough w (and r) are to too procyclical
49 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 84 6 Criticisms 6.1 In General The research on RBC became so successful because Propose a coherent platform to analyse growth and cycles It somewhat fails such that there is room for work Main victory: methodological (part of the toolkit of macroeconomists) Main criticisms
50 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 85 incorrect/implausible calibration of parameters (IES, Labor supply) need for sensibility analysis counterfactual prediction for some prices: real wage is strongly procyclical in the model, CRRA preferences are not compatible with the equity premium price level is too strongly countercyclical 6.2 The Measure of Technological Shocks Key problem: Are technological shocks at the source of BC?
51 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 86 Prescott [1986]: Technology shocks account for 7% of output volatility, but Too volatile Little evidence of large supply shocks (except oil prices) Recessions have to be explained by technological regressions Measurement problems (contamination by demand shocks if increasing returns, imperfect competition, labor hoarding) ; in the growth accounting literature, the SR was
52 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 87 a measure of our ignorance, now it is the engine of the model. 6.3 The Need for Persistent Shocks Recall that log(a t ) = ρ log(a t 1 ) + ε t and ρ is large Why do we need so persistent shocks? Because the model possesses weak propagation mechanisms
53 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 88 Let s see that in details
54 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 89 IRF to A Technological Shock Technology shock
55 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 9 IRF to A Technological Shock Output Consumption Investment Capital 1 Hours worked Labor productivity (Wages)
56 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 91 IRF to A Technological Shock: Temporary vs Persistent Technology shock
57 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 92 IRF to A Technological Shock: Temporary vs Persistent Output Consumption Investment Capital 1.5 Hours worked Labor productivity (Wages)
58 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 93 IRF to A Technological Shock: Persistent vs Permanent Technology shock
59 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 94 IRF to A Technological Shock: Persistent vs Permanent Output Consumption Investment Capital 1 Hours worked Labor productivity (Wages)
60 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 95 How to improve the model: Introducing additional shocks Improving the propagation mechanisms
61 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 96 7 Solving the puzzles? 7.1 Adding Demand Shocks Government Expenditures Change the Household s budget constraint: B t+1 + C t + I t + T t (1 + r t 1 )B t + W t h t + z t K t Government Balanced Budget: T t = G t Government expenditures are modeled as log(g t ) = ρ log(g t 1 ) + (1 ρ g ) log(g) + ε g,t
62 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 97 with ε g,t N (, σ 2 t ). G/Y =.2, ρ g =.97, σ g =.2.
63 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 98 IRF to A Government Spending Shock Government shock
64 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 99 IRF to A Government Spending Shock Output Consumption Investment x 1 3 Capital.2 Hours worked Labor productivity (Wages)
65 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 1 Technological and Government Spending Model Variable σ( ) σ( )/σ(y) ρ(, y) ρ(, h) Output Consumption Investment Hours worked Labor productivity (model in yellow)
66 Franck Portier TSE Macro II Chapter 3 Real Business Cycles Labor indivisibility Hansen [1985]: work a fixed amount of hours or does not Preferences U(C t, 1 h t ) = { log(c e it ) + θ log(1 h ) if she works log(cit u) + θ log(1) if not Randomly drawn with probability π t U t = π it (log(c e it ) + log(1 n )) + (1 π it ) (log(c u it ) + log(1) ) = π it log(c e it ) + (1 π it) log(c u it ) θh t where h it = π it n
67 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 12 There exists full insurance { Cit e + τ ta it + Kit+1 e (z t + 1 δ)k t + w t h Cit u + τ ta it + Kit+1 u (z t + 1 δ)k t + A it if she works if not Risk neutral insurance companies: τ t = π t
68 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 13 Labor indivisibility Utility collapses to the rest remains unchanged U(C t, 1 h t ) = log(c t ) h t
69 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 14 IRF to A Technological Shock  Indivisible Labor Model Output Consumption Investment Capital 2 Hours worked Labor productivity (Wages)
70 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 15 IRF to A Technological Shock  Indivisible Labor Model Output Consumption Investment Capital 2 Hours worked Labor productivity (Wages)
71 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 16 Indivisible Labor Model Variable σ( ) σ( )/σ(y) ρ(, y) ρ(, h) Output Consumption Investment Hours worked Labor productivity (model in yellow)
72 Franck Portier TSE Macro II Chapter 3 Real Business Cycles 17 Contents 1 Introduction 2 2 Measuring the Business Cycle Trend versus Cycle Cycle: Output Gap Growth Cycle Trend Cycle The Hodrick Prescott Filter The HP filter at work
73 Franck Portier TSE Macro II Chapter 3 Real Business Cycles U.S. Business Cycles What are Business Cycles? Main Real Aggregates Moments A Model to Replicate Those Facts The Standard Real Business Cycle (RBC) Model The Household The Firm Equilibrium
74 Franck Portier TSE Macro II Chapter 3 Real Business Cycles An Analytical Example Stationarization Solving the Model In General The Nice Analytical Case Numerical solution Quantitative Evaluation Calibration
75 Franck Portier TSE Macro II Chapter 3 Real Business Cycles Can the Business Cycle be Driven by Capital Dynamics? Technological Shocks Results from Model Simulations A success(?) Criticisms In General The Measure of Technological Shocks The Need for Persistent Shocks
76 Franck Portier TSE Macro II Chapter 3 Real Business Cycles Solving the puzzles? Adding Demand Shocks Labor indivisibility
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