Lectures in Monetary Economics - Harris Dellasharrisdellas.net/teaching/mon_econ/download/2010_szg_slides_2.pdf · Lectures in Monetary Economics Lecture 2 The RBC model Lectures

Lectures in Monetary Economics Lecture 2 The RBC model

Lectures in Monetary EconomicsLecture 2

The RBC model

Harris Dellas

Department of EconomicsUniversity of Bern

December 9, 2009

Lectures in Monetary Economics Lecture 2 The RBC modelThe RBC model

The RBC model

I The RBC constitutes the methodological foundation of theNK model.

I It is a micro-founded DSGE model with rational agents,flexible prices and competitive markets.

I It has good empirical properties (in terms of the matchbetween the model implied and empirical pdf of the data).


A simple, 2 period example

Consumption-savings choice

Utility:u(C1) + βu(C2) (1)

Budget constraint:

P1Y1 = P1C1 + B (2)

P2Y2 + RB = P2C2 (3)

B > 0 means lending in the first period.

P2C2 = P2Y2 + R(P1Y1 − P1C1) (4)


Euler equation (or dynamic IS curve)

uc1 = βRuc2P1

P2= βRuc2

1

π⇒ 1 = βr

uc2

uc1(5)

The Euler equation plays a critical role in the monetarytransmission mechanism:An increase in the real interest rate decreases current spending(consumption).


Work decision

Utilityu(C1, h1) + βu(C2, h2) (6)

The supply of labor

− uh1

uc1=

W1

P1= w1 (7)

The marginal rate of substitution between consumption and leisureequals the real wage.


The demand for labor

dY1

dh1= MPL1 =

W1

P1= w1 (8)

Combine demand and supply of labor

− uh1

uc1= MPL1 (9)

This equation will prove very useful for understanding optimalmonetary policy in the NK model.


A multi-period version of the RBC model with uncertainty and arepresentative agent.

V = E0[∞∑

t=0

βtU(Ct , ht)] (10)

Flow budget constraint:

PtCt + Bt = Rt−1Bt−1 + Wtht + Πt (11)

U(Ct , ht) =1

1− γC 1−γ

t − Nt

1 + σh1+σt (12)

Yt = Ath1−αt (13)

Nt is a preference and At a technology (productivity) shock.


The general equilibrium solution of the model forY ,C , h,w = W /P, r = R/π is obtained by solving

Nthσt

C−γt

= wt

C−γt = βEtrtC−γt+1

wt = (1− α)Ath−αt

Yt = Ath1−αt

Yt = Ct

At+1 = A1−ρa

Aρat εa,t+1

Nt+1 = N1−ρν

Nρνt εg ,t+1

(14)

bt ≡ Bt/Pt = 0


Nonlinear, dynamic, stochastic equations that can only be solvedanalytically in special cases.

In practice we solve an approximate version of the system, typicallya log-linear approximation around the steady state.

The steady state can be ”easily” derived by settingAt = A,Nt = N,Ct = Ct+1 = C . . .


The log-linearized system around the steady state takes the form

wt = σht + γct + νt

0 = γEt ct+1 − γct + Et rt

wt = at − αht

yt = at + (1− α)ht (15)

yt = ct

at+1 = ρaat + εa,t+1

νt+1 = ρν νt + εν,t+1

where for variable x we define x = x−x∗

x∗ ≈ log x − log x∗ as thepercentage deviation of x from its steady state value, x∗.


In state space form

A0Etxt+1 = A1xt + B0et+1 (16)

When A0 is invertible,

Etxt+1 = A−10 A1xt + A−1

0 B0et+1 ⇒Etxt+1 = Axt + Bet+1 (17)

The Blanchard-Khan (1980) method: Partition the state variablesof the system into backward (s) and forward looking (z) variables.[

st+1

Etzt+1

]= A

[stzt

]+ Bet+1 (18)


The properties of the solution The Blanchard-Khan criterion :

n = the number of eigenvalues of A that lie outside the unit circlef = number of the forward looking variables.

I If n=f there exists a unique rational expectations solution tothe system

I If n< f the system has multiple solutions1 (indeterminacy).

I If If n> f then the system has no solution (all dynamic pathsare explosive, violating the transversality condition).

1In this case one needs to use alternative methods to solve the system, forinstance, Sims, 2000.


If A0 is not invertible then the system

A0Etxt+1 = A1xt + B0et+1

can be solved using the QZ decomposition:

∃Q,Λ,Z ,Ω s.t.Q ′ΛZ ′ = A0,Q′ΩZ ′ = A1, Λ,Ω upper triangular

(see Klein, 2002, Sims, 2000).


The software of choice: DynareBasic structure of Dynare// declarationsvar x , y , . . . ;varexo ea, ev , . . . ;parameters alpha, beta, . . . ;// parameter valuesalp = ; bet = ; . . .// model equationsmodel;exp(v) ∗ exp(c ∗ gam) ∗ exp(h ∗ sig)− exp(w) = 0; // consumption(1− alp) ∗ exp(a) ∗ exp(h ∗ (−alp))− exp(w) = 0; // work. . .end;


//steady state solutioninitval;c = log(..); h = log(..); . . .end;steady;check;// stochastic structure shocks;varea = ..; varev =; end ;//simulationsstoch simul(dr algo=0,periods=1000, irf=20, nocorr, nofunctions,order=1) c y h w;


POLICY AND TRANSITION FUNCTIONS:c y h w r

Constant -0.12 -0.12 -0.18 -0.36 0.01a(-1) 0.81 0.81 -0.20 1.02 -0.06v(-1) -0.26 -0.26 -0.40 0.14 0.01ea 0.86 0.86 -0.21 1.07 -0.06ev -0.27 -0.27 -0.43 0.15 0.02


MOMENTS OF SIMULATED VARIABLES:VARIABLE STD. DEV. AUTOCORc 0.028145 0.9568y 0.028145 0.9568h 0.007386 0.9578w 0.035377 0.9571r 0.002111 0.9568


Empirical evaluation of the simple model.

It has some decent properties: Procyclical wages, consumption andemployment.

But without investment it cannot match the most importantstylized facts.


A more general version of the model with capital

ProductionYt = AtK

αt h1−α

t (19)

The capital stock, K , accumulates according to

Kt+1 = (1− δ)Kt + It (20)


Nthσt

C−γt

= wt

C−γt = βEtrtC−γt+1

wt = (1− α)Yt

ht

C−γt = βEtC−γt+1(qt+1 + (1− δ))

qt = αYt

Kt(21)

Yt = AtKαt h1−α

t

Yt = Ct + It + Gt

Kt+1 = (1− δ)Kt + It

At+1 = A1−ρa

Aρat εa,t+1

Nt+1 = N1−ρν

Nρνt εν,t+1

Gt+1 = G1−ρg

Gρgt εg ,t+1


TASK: Compute the steady state of this model. Log-linearize

around the steady state and then solve the model (or simply input

your equations and steady state solution into dynare and let it

solve the model). Report the moments, IRFs and variance

decomposition. Use the same parameter values as in the model

without investment with the additionδ G/Y ρg Σg

0.08 0.2 0.95 0.02

where G/Y is the steady state ratio of government spending to

GDP. What are the main properties of the model? Any comments?


Main implications of the RBC model

I Supply shocks as the main source of macroeconomic volatility.A single supply shock can account for most of macroeconomicfluctuations.

I Money ”neutrality”

Galı’s, 1999, criticism of the RBC model:The RBC model implies that technology shocks lead to procyclicalmovements in employment, productivity and real wages of the typeobserved in the data.But what is the conditional effect of supply shocks on employmentin the data?Galı, 1999: In response to a positive technology shock, laborproductivity rises more than output while employment shows apersistent decline. Hence, supply shocks cannot be the drivingforce of macroeconomic fluctuations.


The difficulty of identifying technology shocks in the data.Galı’s identification scheme: Only technology shocks can have apermanent effects on the level of labor productivity (identificationbased on Blanchard and Quah, 1989).[

∆xt

nt

]=

[C11(L) C12(L)C21(L) C22(L)

] [ept

eTt

](22)

xt = yt − nt , xt is the log of labor productivity.The long term identifying restriction

∑j c12(j) = 0 implies that ep

t

and eTt are shocks with and without a permanent effect on labor

productivity respectively. The former is taken to represent thetechnology shock.


Figure: Technology Shocks and Employment


Figure: IRFs to a Technology Shocks


Response to the findings of Galı:

I Dispute the ability of the particular identification schemesused to truly identify technology shocks (Chari, Kehoe andMcGrattan, 2005). Type of data stationarity, power of longterm restrictions, etc.

I Play defense and argue that the new Keynesian model isequally incapable of matching these stylized facts (Dotsey,1999).

I Suggest plausible, flexible price models that can reproducethese stylized facts. What is needed is models that have eithersluggish aggregate demand or some other demanddiscouraging mechanism (such as low trade elasticities).


Collard and Dellas (C-D), 2007 EJ.

The role of low trade elasticities.

An RBC model of an open economy with low trade elasticities andsluggish capital adjustment can produce the correct patterns.


Figure: C-D, 2007 EJ, Impulse Response to a Technology Shock: Data

(a) Hours in difference

5 10 15 20−2

0

2

4

6x 10

−3

Horizon

Output

5 10 15 201

2

3

4

5

6x 10

−3

Horizon

Productivity

5 10 15 20−6

−4

−2

0

2

4x 10

−3

Horizon

Hours

(b) Linearly detrended hours

5 10 15 200

2

4

6

8x 10

−3

Horizon

Output

5 10 15 201

2

3

4

5

6x 10

−3

Horizon

Productivity

5 10 15 20−4

−2

0

2

4x 10

−3

Horizon

Hours


Figure: C-D, 2007 EJ, Impulse Response Function to a 1 s.d.technological shock: Model vs Data

5 10 15 20−2

0

2

4

6

8x 10

−3

Horizon

Output

5 10 15 20−6

−4

−2

0

2

4

6x 10

−3

Horizon

Hours

5 10 15 200

0.002

0.004

0.006

0.008

0.01

Horizon

Productivity

Low ElasticityHigh Elasticity


Table: Conditional Correlations

Corr(·,∆y/h) Corr(·,∆y)∆h RER NX ∆h RER NX

Flexible, low elasticity

All -0.094 0.110 0.035 0.279 0.081 0.075Techno. -0.415 0.153 0.013 -0.340 0.156 0.022Other 0.029 -0.154 0.150 0.971 -0.152 0.149

Flexible, high elasticity

All 0.048 0.060 -0.013 0.436 0.072 -0.008Techno. 0.042 0.106 -0.093 0.261 0.149 -0.129Other 0.189 -0.177 0.174 0.914 -0.153 0.151


The upshot: The RBC can meet Galı’s challenge

But are there any other reasons to want to abandon the RBCmodel?

The belief that money is not neutral due to

I either imperfect information problems a la Lucas

I or nominal rigidities (price or wage)

The NK model relies on the latter.


I Empirical evidence on real effects of money (Walsh ch 1.3).

I Empirical evidence on nominal stickiness.

1. Fundamental difficulty: A constant price does not mean a rigidprice! A variable price does not mean a flexible price!

2. Direct evidence: Bils and Klenow, 2004, (4-6 months) Dhyneet al., 2005, Nakamura and Steinsson, 2007 (8-11 months).

3. Nominal wage rigidity (Akerlof, 1995)

I Rather limited support for the existence of significant nominalrigidities.

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Lectures in Monetary Economics - Harris Dellasharrisdellas.net/teaching/mon_econ/download/2010_szg_slides_2.pdf · Lectures in Monetary Economics Lecture 2 The RBC model Lectures