Real Business Cycle
Jin Cao1
1Munich Graduate School of Economics
Ludwig-Maximilians-Universität München (WS07/08)
Outline
MotivationFacts of Business CyclesResearch Question and RoadmapDistinguished Features of RBC Models
RBC: Solving a Baseline ModelModel SpecificationsCharacterizing the Dynamic SystemLinear Rational Expectation System
Calibration and SimulationCalibrationSolving for Rational Expectation EquilibriumNumerical Simulation
Motivation: Facts of Business Cycles
I Business cycle: Major macro indicators, consumption, laborsupply, investment... fluctuating along with the boom andbust of the economic growth. Some fairly robust facts:
I Output movements in different sectors of the economy exhibita high degree of coherence;
I Investment is about three times more volatile than GDP;I Consumption is about half as volatile as GDP;I Employment is about eighty percent as volatile as GDP;I The capital stock is much less variable than output;I Consumption-output ratio is strongly counter-cyclical and
investment-output ratio is strongly pro-cyclical.
I An instantaneous question: Are we able to explain?
How Far can We Go with What We Leaned?
I What have we (or people in 1980) learned so far?I Dynamic? Yes, as we always do!I Stochastic? Yes, asset pricing, optimization under uncertainty.I General equilibrium? Yes!
I Partial equilibrium (Solow)I General equilibrium (Ramsey-Cass-Koopmans)I General equilibrium with labor-leisure choice (Ex. 2-4).
I Then: How many facts can be explained by these theories?RBC Models:
I Combine those widely accepted;I Do “the reasonable things to do”, go on with the best practices;I Prescott: Progress, don’t regress!
What Makes an RBC Model?
I Micro-Based (RCK): Infinitely lived representative agent;
I Dynamic optimization. Employment in the economy isdetermined by the individual’s labor-leisure choice;
I Driving force of business cycles: Exogenous shocks totechnology. Transmission machanism: Productivity shock →Real interest rate / wage →Decision in labor supply /Consumption → Investment →Future capital stock...
I The markets are complete and there is no asymmetricinformation. All the agents have rational expectation and themarkets clear continuously: The business cycle is REAL!
Preferences, Technology, and Uncertainty
I The representative agent in this economy considers thefollowing dynamic optimization problem
max{ct ,lt ,kt}+∞
t=0
+∞
∑t=0
βt [θ lnct +(1−θ) ln(1− lt)] ,
s.t. ct +kt+1 = ztkαt l1−α
t +(1−δ )kt ,
lnzt+1 = ρ lnzt + εt+1.
I Neoclassical production function f (kt , lt) = kαt l1−α
t withproductivity shock: εt+1 ∼ N(0,σ2), ρ ∈ (0,1).
The Optimality Conditions
I Define Bellman equation
V (k ,z) = max(c,l ,k ′)
{θ lnc +(1−θ) ln(1− l)+βE
[V (k ′,z ′)
]},
s.t. c +k ′ = zkα l1−α +(1−δ )k .
The first order conditions are
ct =θ
1−θ(1− lt)(1−α)ztkα
t l−αt ,
1ct
= βEt
{1
ct+1
[αzt+1kα−1
t+1 l1−α
t+1 +(1−δ )]}
,
ct = ztkαt l1−α
t +(1−δ )kt −kt+1.
Benchmark: The Steady State
I The steady state for this economy is featured byI The technology shock is constant, zt = 1, ∀t;I Capital, labor, and consumption are constant, i.e. kt = k∗,
ct = c∗, and lt = l∗, ∀t.
I Then βand θ can be determined by the other steady statevariables.
c∗ =θ
1−θ(1− l∗)(1−α)k∗α l∗−α ,
1 = β
[α
y∗
k∗+(1−δ )
],
c∗ = y∗−δk∗.
Log-Linearization and Taylor Expansion
I Difficulty: System is non-linear, hard to say anything about theequilibrium path
I Does the equilibrium path exist at all?I If it exists, how does it look like?
I Remember that we can log-linearize a function f (X ) aroundX ∗
f (X ) = f (e lnX ) = f (X ∗)+ f ′(X ∗)X ∗(lnX − lnX ∗).
I Define X = lnX − lnX ∗ = ln(1+ X−X ∗
X ∗
)≈ X−X ∗
X ∗ — X is thepercentage deviation from X ∗! A normalization.
Log-Linearization and Taylor Expansion
I Linearized system around the steady state — But then?
ct =−(
l∗
1− l∗+α
)lt +α kt + zt ,
−ct =−Et [ct+1]+βαk∗α−1l∗1−α{(α−1)Et
[kt+1
]+Et [zt+1]+(1−α)Et
[lt+1
]},
kt+1 =(k∗α−1l∗1−α
)zt +
[αk∗α−1l∗1−α +(1−δ )
]kt +(1−α)k∗α−1l∗1−α lt−
c∗
k∗ct ,
ρ zt = Et [zt+1] .
Linear Expectation System
I Separate the realized variables from the expected values —Linear expectation system
1 −αl∗
1−l∗ +α −1−1 0 0 0− c∗
k∗ αk∗α−1l∗1−α +(1−δ) (1−α)k∗α−1l∗1−α k∗α−1l∗1−α
0 0 0 ρ
ctktltzt
=
0 0 0 0−1 βαk∗α−1l∗1−α (α−1) βαk∗α−1l∗1−α (1−α) βαk∗α−1l∗1−α
0 1 0 00 0 0 1
Et
ct+1kt+1lt+1zt+1
.
Finding the Plausible Equilibria
I Now we have already built up the relation between xt andEt [xt+1]: Axt = BEt [xt+1]. Jordan decomposition
xt = A−1BEt [xt+1] ,
xt = VDV−1Et [xt+1] ,
V−1xt = DV−1Et [xt+1] .
I D is a diagonal matrix whose diagonal elements are theeigenvalues of A−1B. Now: Which eigenvalue relates to asensible equilibrium path? “Solution concept” — REE!
Calibration: Relate the Model to Real World
I The central question still remains: Could we explain the Factsby the model at hand?
I To do this, relate the model to real world data and explicitlysolve the linear expectation system!
I Some economic facts: α = 13 , c∗
y∗ = 23 , l∗ = 0.3,
δ = 0.025(quarterly), k∗y∗ = 11. Feed them into the steady
state equations and get β and θ , then calculate c∗, k∗, y∗.
I From US (1946-1986) I got c∗ = 0.79, k∗ = 10.90, l∗ = 0.29,y∗ = 1.06, β = 0.99, δ = 0.025, α = 0.36, θ = 0.32, ρ = 0.95.
Calibrated Linear Expectation System
I Now feed these numbers into our linear expectation system
1 −0.36 0.77 −1−1 0 0 0
−0.072 1.010 0.062 0.0970 0 0 0.95
ctktltzt
=
0 0 0 0−1 −0.022 0.022 0.0350 1 0 00 0 0 1
Et
ct+1kt+1lt+1zt+1
,
ctktltzt
=
1.000 0.0220 −0.0220 −0.03500.1468 0.9657 −0.0032 −0.1850−1.2301 0.4229 0.0271 1.3260
0 0 0 1.0526
Et
ct+1kt+1lt+1zt+1
I and then apply Jordan decomposition.
Jordan Decomposition
0 0 0 46.4423
−1.6329 −0.2479 0.0359 −45.8191−1.6611 0.9464 0.0365 0.62711.2629 −0.4547 0.9725 −1.2629
ctktltzt
=
1.0526 0 0 0
0 1.0493 0 00 0 0.9434 00 0 0 0
0 0 0 46.4423−1.6329 −0.2479 0.0359 −45.8191−1.6611 0.9464 0.0365 0.62711.2629 −0.4547 0.9725 −1.2629
Et
ct+1kt+1lt+1zt+1
.
I The eigenvalues of A−1B are λ1 = 1.0526, λ2 = 1.0493,λ3 = 0.9434, λ4 = 0.
I Question: Which one(s) relate(s) to sensible equilibriumpath(s)?
Solution Concept: Rational Expectation Equilibrium
I Rational expectation equilibrium (REE): Determinancy out ofperfect foresight!
d1,td2,td3,td4,t
=
1.0526 0 0 0
0 1.0493 0 00 0 0.9434 00 0 0 0
Et
d1,t+1d2,t+1d3,t+1d4,t+1
,
di ,t = limj→+∞
λji Et
[di ,t+j
].
I The only determinancy comes from λ3 = 0.9434, therefore theonly REE is d3,t = 0.
Rational Expectation System
I Now we can write down the complete stable system underrational expectation equilibrium
I REE: d3,t = 0
ct = 0.570kt +0.022lt +0.378zt ;
I Intratemporal efficiency condition:
0 = ct −0.36kt +0.77lt − zt .
I Intertemporal efficiency condition:
kt+1 =−0.072ct +1.010kt +0.062lt +0.097zt ;
I State (kt ,zt) → control (ct , lt) →next period kt+1 ...
Duplicate the Business Cycle under Continuous Shocks
I Feed lnzt+1 = ρ lnzt + εt+1 (σ = 0.007) into the linearrational expectation system under MATLAB / Mathematica(c-green, y -blue, i-red)Show@yplot, cplot, iplotD
50 100 150 200
-0.2
-0.15
-0.1
-0.05
0.05
0.1
0.15
Graphics
RBC_class.nb 5
Evolution after One-Time Shock
I One-time shock: lnz0 = 0.007, lnzt+1 = lnρzt , ∀t ≥ 0
10 20 30 400
2
4x 10
−3 c
10 20 30 400
0.005
0.01i
10 20 30 400
0.05
0.1k
10 20 30 40−2
0
2x 10
−3 l
10 20 30 400
0.01
0.02y
10 20 30 400
0.005
0.01z
Numerical Result: How Successful We are Now?
I Then I feed 2000 observations in MATLAB and see the precisestandard deviations and correlations — RBC does a great jobin reproducing the reality!
Time Series Data Model
, ,Output 1.71 1.00 1.37 1.00Consumption 0.84 0.76 0.43 0.89Investment 5.38 0.90 4.29 0.99Capital 0.62 0.08 0.38 0.15Employment 1.65 0.88 0.72 0.98
Summary: Solved Problems
I What have we done so far?I A genuine RBC model adapted from Kydland & Prescott
(1982), and Hansen (1985);I Solution methods proposed by Blanchard & Kahn (1980), Sims
(1980), Campbell (1994), Uhlig (1997);I Solution concept: Rational expectation equilibrium a la Muth
(1961), Lucas (1976).
I What may make us satisfied so far?I A remarkable job on accounting for main business cycle facts;I Substantial shift in interpretation of business cycles:
fluctuations as efficient adjustment to productivity, rather thantemporary breakdowns of the market mechanism!
I Therefore, “First Best Solution”!
Summary: Critiques and Unsettled Issues
I Open questions:I No role for government! No justification for government
intervention.I Where do the productivity shocks come from?
I Extensions and Current Research:I Genuine RBC research is already inactive;I Many extensions since then: Labor market imperfections,
government expenditure shocks, “home production”, variablecapital utilization...
I Many active studies based on RBC: Monetary models (!), RBCin open economy, heterogenous agents, DSGE econometrics ...
For Further Reading... I
Stadler, G. W.Real Business Cycles.Journal of Economics Literature, 32, December, 1750–1783,1994.
Cooley, T. F. (Editor)Frontiers of Business Cycle Research.New Jersey: Princeton University Press, 1995.