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.