Real Business Cycle Model (RBC) Seyed Ali Madanizadeh November 2013
RBC Model Lucas 1980: “One of the functions of theoretical economics is to provide fully articulated, artificial economic systems that can serve as laboratories in which policies that would be prohibitively expensive to experiment with in actual economies can be tested out at much lower cost. [...] Our task as I see it [...] is to write a FORTRAN program that will accept specific economic policy rules as ‘input’ and will generate as ‘output’ statistics describing the operating characteristics of time series we care about, which are predicted to result from these policies.”
RBC Model A Microfounded general equilibrium macroeconomic model (Proposed by Kydland and Prescott (1982)) Explains the short run effects (Business cycles) Consistent with long run facts A Stochastic dynamic general equilibrium model (DSGE) with rational expectations Fully rational household with capital and labor Firms with stochastic productivity
RBC Model 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 The social planner equilibrium and the competitive equilibrium are identical and admit a unique solution
RBC Model Main policy conclusion: ‡ uctuations of all variable (output, consumption, employment, investment...) are the optimal responses to technology shocks exogenous changes in the economic environment. Shocks are not always desirable. But once they occur, this is the best possible outcome: business cycle ‡ uctuations are the optimal response to technology shocks = > no need for government interventions: it can be only deleterious
RBC Model Furious response from the "people from the Oceans" = > Rogoff: "brilliant theories . . . rst look ridiculous then they become obvious". From mid’80s to mid’90s: ten years lost in useless ideological debates between the Oceans and the Lakes From mid’90s: convergence on methodology: "the RBC approach as the new orthodoxy in macroeconomics"
Measuring the Business Cycles Hodrick-Prescott (H-P) Filter � �� 2 � ∞ � ( y t + 1 − y g � t ) 2 + λ ( y t − y g y t − y g ∑ min t ) − t − 1 { y g t } ∞ t = 0 0 H-P filter suppresses the really low frequency fluctuations � 8 years quarterly data λ = 1600
Measuring the Business Cycles
Measuring the Business Cycles
RBC Model Households: ∞ ∑ c t , k t , b t , x t , h t E 0 max u ( c t , 1 − h t ) t = 0 subject to c t + x t + b t + 1 ≤ w t h t + r t k t + R t b t + π t k t + 1 < ( 1 − δ ) k t + x t ≥ k t 0 k 0 Given : We assume that the consumer is making all time-t choices ( x t , c t , k t + 1 , b t + 1 , h t ) conditional on time t information (all variables subscripted t and below, plus the interest rate on bonds R t + 1 ).
RBC Model Firms K t , H t e z t F ( K t , H t ) − w t H t − r t K t max z t follows an AR(1) process: z t = ρ z t − 1 + ε t where ε t is white noise.
RBC Model Equilibrium: An equilibrium in this economy is a joint distribution of prices and allocations = C t + X t Y t = B t 0
Solving the Model FOC � � = 0 β t u c ( c t , 1 − h t ) − λ t [ c t ] E t : � − β t u l ( c t , 1 − h t ) + w t λ t � = 0 [ h t ] : E t [ k t + 1 ] E t [ λ t ( 1 − δ + r t + 1 ) − λ t ] = 0 : [ b t + 1 ] : E t [ λ t + 1 R t + 1 − λ t ] = 0
Solving the Model: Household Consumption Leisure decision (Interpretation!) u l ( c t , 1 − h t ) = u c ( c t , 1 − h t ) w t Euler Equation u c ( c t , 1 − h t ) = β E t [ u c ( c t + 1 , 1 − h t + 1 )( r t + 1 + 1 − δ )] Bond Price R t + 1 = E t [ r t + 1 ] + 1 − δ
Solving the Model: Firms FOC e z t F H ( K t , H t ) = w t e z t F K ( K t , H t ) r t =
Solving the Model Relative labour supply responds to relative wages between two different periods = > households substitute labour intertemporally Also the interest rate matters for labour supply = > ↑ r = > ↑ h s today, because MPK is high = > crucial channel for employment fluctuations What is the effect of ↑ w or ↑ r?
Solving the Model � � temporary ↑ w ⇒ substitution effect prevails ↑ h s ⇒↓ c t w t (given the intratemporal trade-off between consumption and u l ( c t , 1 − h t ) labour: u c ( c t , 1 − h t ) = w t permanent ↑ w = > income and substitution effects cancel � � c t out, no change in h s t and w t Temporary increase in both w and r = > intertemporal substitution both in labour and consumption = > ↑↑ h s t
Solving the Model The standard neoclassical intratemporal trade-off between consumption and labour u l ( c t , 1 − h t ) u c ( c t , 1 − h t ) = w t hence, for a given wage, C and H tend to move in the opposite direction How one can get both C and H highly pro-cyclical? Highly procyclical real wage (= > productivity shocks!!)
Solving the Model Example U = log c t − h 1 + φ t 1 + φ becomes � w t � 1 φ h t = c t So the elasticity of labor supply w.r.t. real wages = 1 φ : Frisch elsticity
Steps to solve the Model FOCs 1 Steady States 2 Calibration and Estimation 3 Solve for the recursive law of motion 4 Calculate the moments: correlations, and standard deviations 5 for the different variables both for the artificial economy and for the actual economy Compare how well the model economy matches the actual 6 economy’s characteristics Calculate the IRFs in response to different shocks 7
Calibration Use microeconomic studies or theory to find values for all of the parameters Utility Function � ( 1 − h t ) a � 1 − χ − 1 c 1 − α t U ( c t , 1 − h t ) = 1 − χ Production function F ( K , H ) = K θ H 1 − θ
Calibration β : At the non-stochastic steady state, we have R = 1 β . The average real interest rate in the U.S. is usually around 4% annually which is about 1% quarterly β = 0 . 99 θ : 1 − θ will be labor’s share of output, a quantity that can be estimated from the national income accounts θ = 0 . 4 χ : Estimates from micro studies of the typical worker’s intertemporal elasticity of substitution are in the range of χ � 1 χ = 1 u ( c , 1 − h ) = ( 1 − α ) ln c + α ln ( 1 − h )
Calibration α : By solving for the steady states we find that: 1 − h = ( 1 − α ) ( 1 − θ ) y α ch From long run data, 31% of available time is spent working ⇒ ¯ h = 0 . 31 The steady state output to consumption ratio is about 1.33 � y � c ⇒ α = 0 . 64 Cooley and Prescott estimate that depreciation is 4.8% annually, so 1.2% quarterly ( δ = 0 . 012). ν : Use quarterly population growth rate ν = 0 . 012
Calibration ρ and σ ε : This model has perfect competition and constant returns to scale. So z t − z t − 1 is the Solow residual. The average value of the Solow residual gives us our estimate for γ . Cooley and Prescott set γ = 0 . 0156, giving about 1 . 6 % annual TFP growth. Once we subtract out this average, we can estimate an AR(1) model ρ = 0 . 95 and σ ε = 0 . 007
Numerical Solution Once we have set up the model, and calibrated parameters, we next need to find a numerical solution to the model Bellman’s equation, and apply numerical dynamic programming methods. Linear-quadratic approximation around the steady states Log-linearize the model around the steady state
Log Linearization For x ∼ 0 : e x ≈ 1 + x � x t � For x t , let ˆ x t = log be the log-deviation of x t from its x ¯ steady state. Thus, 100 ∗ ˆ x t is (approximately) the percent deviation of x t from ¯ x . Then, x t ≈ ¯ xe ˆ x t = ¯ x ( 1 + ˆ x t ) Formally: first order Taylor expansion, xe ˆ x t g t = g ( x t ) = ¯ � � 1 + g � ( ¯ x ) ¯ x g ( ¯ x ) ( 1 + ˆ g t ) ≈ g t = g ( ¯ x ) x t x ) ˆ g ( ¯ g � ( ¯ x t = g � ¯ x ) ¯ x x ≈ g t ˆ x ) ˆ g ˆ x t g ( ¯
Recursive law of motion: An example As an exmaple, consider the log linearization method. ˆ b is log linearized version of b . We guess a decision rule ˆ γ 1 ˆ k t + 1 = k t + γ 2 z t η 1 ˆ c t = k t + η 2 z t ˆ Then verify by substituting into the FOCs.
Simulation, Estimation and Test Simulation: Under those assumptions we can simulate the model on a computer and we get time series for output, employment, productivity, investment, consumption, and capital. Estimation If we have not yet calibrated some of the parameters, we need to estimate them. Matching moments is a very common approach here. Test We look at the moments of real and simulated data: likee correlation between any of these variables and the relative variance of different variables.
Evaluation To an RBC theorist, these numbers represent success. We’ve managed to write down a very simple model that duplicates many of the properties (moments) of the actual data. There are a few failures though. This model seems to understate the variability of both consumption and hours. The RBC approach to this failing is to investigate why the model doesn’t match, and adjust the model so that it does match.
Recommend
More recommend
