The RBC model - web.sgh.waw.plweb.sgh.waw.pl/~mbrzez/Adv_Macro/3_RBC.pdf · The RBC model Michal Brzoza-Brzezina Warsaw School of Economics Advanced Macro MBB (SGH) RBC Advanced Macro

The RBC model

Micha l Brzoza-Brzezina

Warsaw School of Economics

Advanced Macro

MBB (SGH) RBC Advanced Macro 1 / 56

Plan of the Presentation

1 Trend and cycle

2 Introduction to RBC

3 The model

4 Steady state

5 Log-linearisation

6 Solution

7 Simulations

8 SummaryMBB (SGH) RBC Advanced Macro 2 / 56

What are business cycles?

GDP per capita in developed economies since 19th century:

Stable growth trend

Long-run growth rate approximately constantSubstantial deviations from the growth trend in the short run - businesscycles

Definition of business cycles: high to medium frequency fluctuationsin economic activity (usually proxied by GDP)

After every expansion comes a recessionBut timing and amplitude irregularDifficult to predict

Business cycles are not restricted to GDP, they are typical for virtuallyall macrovariables


Extracting cyclical information from the data

how do we define a trend?

linear trend applied to logs (consistently with expotential growth)

filters operating in the frequency domain: extracting componentswithin a given frequency range

the most popular proxy for trend in the business cycle literature: hpfilter applied to logs

owed to hodrick and prescott (1980)

for any time series {yt}Tt=1, the trend component {yt}Tt=1 minimizes:

T∑t=1

(yt − yt)2 + λ

T−1∑t=2

[(yt+1 − yt)− (yt − yt−1)]2

where: λ ≥ 0 is a smoothing parameterλ penalizes variations in the trend slope:

λ → ∞: linear trendλ → 0: original dataλ = 1600: standard value for quarterly data



1 Trend and cycle


3 The model

4 Steady state

5 Log-linearisation

6 Solution

7 Simulations


Origins of the RBC revolution

1970s: dissatisfaction with the standard Keynesian macroeconomics

The Lucas (1976) critique:

Lack of microfoundations and expectations in the Keynesian modelIn particular: if firms’ and households’ actions depend on theirexpectations about future policy, it is impossible to analyze the effectsof this policy in a model that does not include expectations in anexplicit formMicrofoundations and rational expectations important

Kydland and Prescott (1982): a very stylized macromodel withmicrofoundations and rational expectations performs surprisingly wellin terms of its fit to the data


RBC - some basics

The basic RBC model assumes an economy featuring perfectlyfunctioning competitive markets and rational expectations.

It gives primacy to technology shocks as the source of economicfluctuations.

It exhibits complete monetary neutrality and Ricardian equivalence(and sets role for monetary and fiscal policy)

In a nutshell - the standard Ramsey model with:

Endogenous (but constant in the long run, i.e. n = 0) labour supplyStochastic productivity (i.e. real shocks)



1 Trend and cycle


3 The model

4 Steady state

5 Log-linearisation

6 Solution

7 Simulations


RBC model - basic features

General equilibrium model

No market imperfections or adjustment costs =⇒ allocations arePareto optimal

The only source of uncertainty: stochastic productivity shocks

In a nutshell - the standard Ramsey model with:

Endogenous (but constant in the long run, i.e. n = 0) labour supplyStochastic productivity (i.e. real shocks)


Households - the problem

A representative household maximises lifetime utility

Et

∞∑i=0

βt+i

[c1−σt+i

1− σ− lt+i

1+ϕ

1 + ϕ

]subject to the budget constraint

ct + it = wt lt + Rk,tkt

and the capital accumulation rule

kt+1 = (1− δ)kt + it

The household rents labour and capital to firms and receives ascompensation the real wage wt and the rental rate Rk,t .

Note that additionally we have a transversality condition (TVC)

limt→∞

kt

∞∏s=0

1

1 + rs= 0


Lagrangean

First, to simplify substitute for investment

ct + kt+1 = wt lt + (1 + Rk,t − δ)kt

denote rt ≡ Rk,t − δ as the real interest rate.

The household chooses ct , lt and kt+1. Write down the Lagrangean:

Lt = Et

∞∑i=0

βt+i

[( c1−σt+i

1− σ−

l1+ϕt+i

1 + ϕ

)−λt+i

(ct+i + kt+1+i − wt+i lt+i − (1 + Rk,t+i − δ)kt+i

)]


Expectations

Important:

Because of uncertainty (stochastic shocks), households do not makedeterministic plans for consumption and labour supplyAny decision at time t depends on expectations concerning the relevantvariables in the next periodsThese expectations take into account all information available at themoment of decision makingWe denote these expectations using the expetations operator Et

Stochastic shocks make households revise their expectations everyperiodExpectations are rational, i.e. households know the structure of theeconomy and characteristics (but not realizations) of stochastic shocks


Expectations cont’d

Basic rules for expectations

Et (Xt + Yt) = EtXt + EtYt

Et (kXt) = kEtXt

Et (k + Xt) = k + EtXt

and, unless Xt and Yt are independent:

Et (XtYt) 6= EtXtEtYt


First order conditions

Sequence of events: at time t the households, having kt units ofcapital, observe stochastic disturbance εt (and so productivity zt),form rational expectations about future productivity levels, makeoptimal work, consumption and investment decisions, whichdetermine capital stock at period t + 1 etc.

First order conditions:

ct : Do it yourself :-)

lt : Do it yourself :-)

kt+1 : Do it yourself :-)


Equilibrium conditions - Euler equation

λt = βEt [λt+1(1 + Rk,t+1 − δ)] substitute for λt to get

c−σt = βEt [c−σt+1(1 + Rk,t+1 − δ)]

This is the Euler equation. It determines the household’sintertemporal choice (how much to consume today, how much tosave).

In equilibrium the disutility from one unit less consumed today equalsexpected discounted utility of consuming (1 + Rk,t+1 − δ) unitstomorrow.


Equilibrium conditions - consumption vs. leisure

lϕtc−σt

= wt

This equation determines the household’s intratemporal choice (howmuch to consume, how much to work).

In equilibrium the utility of one unit more of work should be equal tothe utility from consuming the compensation (real wage).


Firms - the problem

A representative firm uses capital and labour hired from households toproduce a unique good yt .

Its objective is to maximise profits:

Divt = yt − wt lt − rtkt

subject to technologyyt = ztk

αt l

1−αt


Firms - equilibrium condition for labour

Substitute for production to get:

Divt = ztkαt l

1−αt − wt lt − Rk,tkt

First order condition for labour is:

lt : δDivtδlt

= (1− α)ztkαt l−αt − wt = 0

In equilibrium the marginal product of labour equals the real wage

To simplify things substitute from production funtion:

(1− α) ytlt = wt


Firms - equilibrium condition for capital

kt : do it yourself

In equilibrium the marginal product of capital equals the real rentalrate of capital.

To simplify things substitute from production funtion:

α ytkt

= Rk,t

In the RBC setting profits are zero (perfect competition). To see thissubstitute Rk,t and wt into the profit function.


Productivity and market clearing

It is assumed that productivity zt follows an AR(1) process:

zt = exp(εt)zρt−1

zt = ρzt−1 + εt

where εt is a productivity shock

This is the only stochastic process in the basic RBC model

Together with the internal persistence of the model it generates thebusiness cycle

The goods market clears:

ct + it = yt


Equilibrium conditions - summary

Now we have a system of 8 equations with 8 endogeneous variables(c, r , l , w , k, i , y , z):

c−σt = βEt [c−σt+1(1 + Rk,t+1 − δ)] (1)

lϕtc−σt

= wt (2)

kt+1 = (1− δ)kt + it (3)

ct + it = yt (4)


Equilibrium conditions - cont’d

yt = ztkαt l

1−αt (5)

(1− α)ytlt

= wt (6)

αytkt

= rt (7)

zt = exp(εt)zρt−1 (8)


Simplified version

Note that the system can be simplified to 3 equations

c−σt = βEt

{c−σt+1

(αzt+1k

α−1t+1 l

1−αt+1 + 1− δ

)}lϕtc−σt

= (1− α)ztkαt l−αt

kt+1 = (1− δ)kt + ztkαt L

1−αt − ct



1 Trend and cycle


3 The model

4 Steady state

5 Log-linearisation

6 Solution

7 Simulations


Interest rate and productivity

In the deterministic steady state uncertainty disappears and allvariables are constant, for example ct = ct+1 = css .

We assume that in the steady state z is constant and

zss = 1 (9)

From (1) we get

c−σt = βEt [c−σt+1(1 + Rk,t+1 − δ)]

(css)−σ = β(css)−σ(1 + Rssk − δ)

SimplifyingRssk = β−1 − (1− δ) (10)


Capital-labour and investment-capital ratios

Substituting form the production function (5) for yt into (7) we get

αztk

αt l

1−αt

kt= rt

In the steady state

Rssk = α(kss)α−1(l ss)1−α = α

(kssl ss

)α−1Rearranging we can obtain formula for capital-labour ratio

kss

l ss=(Rss

k

α

) 1α−1

(11)

where r ss is given by (10).


Wage and investment-capital ratio

Substituting form the production function (5) for yt into (6) we get

wt = (1− α)ztk

αt l

1−αt

lt= (1− α)zt

kαtlαt

which in the steady state becomes

w ss = (1− α)(kssl ss

)α(12)

where kss/l ss is given by (11).

From (3) in the steady state we have

kss = (1− δ)kss + i ss

which simplifies toi ss

kss= δ (13)


Consumption-labour ratio

Substituting from (5) for yt into (4) we get

ct + it = ztkαt l

1−αt

which in the steady state becomes

css + i ss = (kss)α(l ss)1−α =(kssl ss

)αl ss

Dividing by l ss

css

l ss+

i ss

l ss=(kssl ss

)αSubstituting for i ss from (13) and rearranging

css

l ss=(kssl ss

)α− δ k

ss

l ss(14)



Labour

From (2) we havelϕtc−σt

= wt

which in the steady state it becomes

(l ss)ϕ(css)σ = w ss

Substituting for w ss from (12) and for css from (14)

(l ss)ϕ[(kss

l ss

)α− δ k

ss

l ss

]σ(l ss)σ = (1− α)

(kssl ss

)αSolving with respect to l ss we get

l ss =

[(1− α)

(kss

l ss

)α[(kss

l ss

)α− δ kss

l ss

]σ] 1ϕ+σ

(15)



Remaining variables

To obtain css , note

css =css

l ssl ss (16)

where css/l ss is given by (14) and l ss is given by (15).

We can obtain capital kss from

kss =kss

l ssl ss (17)

where kss/l ss is given by (11) and l ss is given by (15).

To obtain output we use the production function

y ss = (kss)α(l ss)1−α

where kss is given by (17) and l ss is given by (15).

and we can obtain investment form (13)

i ss = δkss

where kss is given by (17).



1 Trend and cycle


3 The model

4 Steady state

5 Log-linearisation

6 Solution

7 Simulations


Non-linear vs. linear models

We have a system of non-linear difference equations

This does not have a closed form solution

Standard solution technique:

log-linear approximation of the model equations around the(non-stochastic) steady-stateapplying an algorithm for solving linear rational expectations models(e.g. Blanchard-Kahn algorithm)

Two options

1 take it as it is and let your software (e.g. Dynare) linearise it2 linearise by hand

Linearising on your own is tedious but has some advantages:

1 some parameters may disappear2 the system is easier to understand3 solution easier (steady state is known)


Log-linearisation

Log-linearisation allows to change non-linear equations into linearequations

This is a valid approximation in the viscinity of a given point (usualythe steady state)!!!

Two steps:

Express variables as log deviation from steady state using the identity:

xt = x ssxtx ss

= x ss exp(ln xt − ln x ss) = x ss exp xt


Useful tricks

Apply first-order series expansion w.r.t. xt around steady state, i.e.around xss = 0

This yields

xt = x ssexp(xt) ≈ x ssexp(xss) + x ssexp(xss)(xt − xss) = x ss(1 + xt)xtyt = x ssexp(xt)y

ssexp(yt) ≈ x ssy ss(1 + xt + yt)

xat ≈ (x ss)a (1 + ˆaxt)

xat ybt ≈ (x ss)a (y ss)b (1 + axt + byt)


Log-linearise labour - consumption choice

lϕtc−σt

= wt

In the steady state:

(l ss)ϕ

(csst )−σ= w ss

Let’s log-linearise:

(l ss)ϕ

(css)−σ(1 + ϕlt + σct) = w ss(1 + wt)

Divide by steady state:

ϕlt + σct = wt


Log-linearise Euler

c−σt = βEt [c−σt+1(1 + Rk,t+1 − δ)]

In the steady state:

1 = β(1 + Rssk − δ)

So let’s linearise:

(css)−σ (1− σct) = β (css)−σ Et

[(1− σct+1)(1− δ + Rss

k (1 + Rk,t+1))]


Log-linearise Euler cont’d

Substitute for Rssk :

1− σct = βEt

[(1− σct+1)(1− δ + ( 1

β − 1 + δ)(1 + Rk,t+1))]

1− σct = Et

[(1− σct+1)(β − βδ + (1− β + βδ)(1 + Rk,t+1))

]1− σct = Et

[(1− σct+1)(1 + (1− β(1− δ))Rk,t+1)

]Multiply and drop

higher order terms:

1− σct = 1− σEt ct+1 + (1− β(1− δ))EtRk,t+1

Rearange terms:

σ(Et ct+1 − ct) = (1− β(1− δ))EtRk,t+1


Log-linearise market clearing condition

ct + it = yt

Do it yourself :-)


Log-linearise firm’s equilibrium conditions

For labour

(1− α) ytlt = wt

Do it yourself :-)

For capital

α ytkt

= rt

Do it yourself :-)


Log-linearise production function

yt = ztkαt l

1−αt

Do it yourself :-)


Log-linearise capital accumulation equation

kt+1 = (1− δ)kt + it

Do it yourself :-)


Log-linearise shock process

zt = exp(εt)zρt−1

zt = ρzt−1 + εt


Steady state values

Our equations contain steady state ratio i ss

y ss .

These are determined by our parameters

From the Euler equation:

Rssk = β−1 − (1− δ)

and from the equilibrium condition for capital:

Rssk kss = αy ss

kss

y ss=

α

Rssk

=α

β−1 − (1− δ)


Steady state values cont’d

From the capital accumulation equation:

δkss = i ss

thusi ss

y ss= δ

kss

y ss=

αδ

β−1 − (1− δ)


Log-linearised system

We now have a system of 8 linear (difference) equations and 8variables

Have to solve it



1 Trend and cycle


3 The model

4 Steady state

5 Log-linearisation

6 Solution

7 Simulations


Solving linear DSGE models

Solving a DSGE model means changing the system of forward lookingdiference equations that we have ...

... into a VAR system

There are several techniques for solving such systems

E.g. Blanchard & Kahn (1980), Sims (2002)


Blanchard & Kahn solution & stability condition

One very important thing is the stability condition

Write the system in state space form:

A1

[Xt+1

EtPt+1

]= A0

[Xt

Pt

]+ γZt+1

where:

Xt : vector n × 1 of state variables (backward-looking)

Pt : vector m × 1 of jumpers (forward-looking)

Zt : wektor k × 1 of shocks (with mean equal to 0 every period)

A1, A0: (n + m)× (n + m) matrices

γ: (n + m)× k matrix


Blanchard & Kahn solution method

Assume A1 is invertible and rewrite the system as:

[Xt+1

EtPt+1

]= A

[Xt

Pt

]+ RZt+1

where:

A ≡ A−11 A0

R ≡ A−11 γ

and apply Jordan decomposition A = CΛC−1 where Λ is a diagonalmatrix of eigenvalues and C is matrix of eigenvectors.


Blanchard & Kahn solution method cont’d

Then (under some conditions stated below)

Pt = −C−122 C21Xt

Xt+1 =(A11 − A12C

−122 C21

)Xt + R1Zt+1

Where the matices A, C and R have been divided into blocks whosesize is determined by the number of stable eigenvalues Λ (lying withinthe unit circle)

This is the recursive solution of the DSGE model

Alternative solution method is based on the QZ decomposition of A0

and A1. Does not require invertibility of A1 (Sims 2002).


Blanchard & Kahn stability conditions

Blanchard & Kahn stated two necessary conditions for obtaining aunique solution:

C22 is of full rankthe number of eigenvalues of A lying outside the unit circle (unstableroots) must equal the number of forward looking variables (jumpers)

If too many unstable roots - no solution

If too few - infinite number of solutions



1 Trend and cycle


3 The model

4 Steady state

5 Log-linearisation

6 Solution

7 Simulations


ConvergenceNotes: grey line - exogenous labour (Ramsey model); black line - endogenous labour choice (RBC model); time unit - quarters;all variables expressed as percentage deviations from their steady-state values

GDP Labour input

Consumption Investment

Capital stock

-0.4

-0.3

-0.2

-0.1

0

0 10 20 30 40 50 60 70 80 90 100

-0.1

0

0.1

0.2

0.3

0.4

0 10 20 30 40 50 60 70 80 90 100

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 10 20 30 40 50 60 70 80 90 100

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 10 20 30 40 50 60 70 80 90 100

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0 10 20 30 40 50 60 70 80 90 100


Impulse responses - technology shockNotes: black line - baseline (ρ = 0.976); dark grey line - permanent productivity shock (ρ = 1); light grey line - no producitityshock inertia (ρ = 0); time unit - quarters; all variables expressed as percentage deviations from their steady-state values

GDP Labour input

Consumption Investment

Capital stock Real wage

Real interest rate Productivity

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 10 20 30 40 50 60 70 80 90 100

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 10 20 30 40 50 60 70 80 90 100

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 10 20 30 40 50 60 70 80 90 100

-1

0

1

2

3

4

5

6

7

0 10 20 30 40 50 60 70 80 90 100

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 10 20 30 40 50 60 70 80 90 100

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 10 20 30 40 50 60 70 80 90 100

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 10 20 30 40 50 60 70 80 90 100

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 10 20 30 40 50 60 70 80 90 100


Moments

Std. deviation Autocorrelation Corr. with GDPData Model Data Model Data Model

GDP 1.50 1.51 0.84 0.72 1.00 1.00Consumption 1.18 0.66 0.85 0.79 0.86 0.93Investment 4.14 4.69 0.90 0.71 0.90 0.98Hours worked 1.26 0.73 0.87 0.71 0.86 0.96Capital 0.54 0.48 0.95 0.96 0.07 0.40Wages 0.78 0.81 0.80 0.77 0.10 0.97Interest rate 1.24 0.23 0.74 0.71 0.49 0.94

Notes: Calculations based on HP-detrended variables


RBC model - summary

Very simple dynamic stochastic general equilibrium model (DSGE)Fluctuations driven by productivity shocks only

No market imperfections or adjustment costs =⇒ decentralizedallocations Pareto optimal, attempts to smooth business cycles harmful

Remarkable success:

Surprisingly good fit to the data (in terms of moment matching)Methodological approach adopted by the New Keynesian school -”great neoclassical synthesis”

Criticism:

Estimation rather than moment matchingOther shocksSome counterfactual implications or questionable calibration choices

Response to criticism and extensions:

Unprecedented development program - numerous refinements andextensionsImportant extension: monopolistic competition and sticky prices (theNew Keynesian school)


Documents

The RBC model - web.sgh.waw.plweb.sgh.waw.pl/~mbrzez/Adv_Macro/3_RBC.pdf · The RBC model Michal Brzoza-Brzezina Warsaw School of Economics Advanced Macro MBB (SGH) RBC Advanced Macro