Business cycles and the RBC modelmbrzez/Makro_zaawansowana/RBC.pdf · Motivation Business cycles Model Steady state Linearisation Solution, calibration & simulation Application Critique

Motivation Business cycles Model Steady state Linearisation Solution, calibration & simulation Application to fiscal policy Critique Conclusions

Business cycles and the RBC modelAdvanced Macroeconomics

Micha l Brzoza-Brzezina & Jacek Suda

SGH Warsaw School of Economics

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Plan of the Presentation

1 Motivation

2 Business cycles

3 Model

4 Steady state

5 Linearisation

6 Solution, calibration & simulation

7 Application to fiscal policy

8 Critique

9 Conclusions

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Introduction

In the previous lectures we anaylzed growth models

They describe long-run macroeconomic processes...

...but have nothing to say about short & medium termfluctuations

This lecture:

facts about business cyclesthe Real Business Cycle (RBC) modelsome papers around RBC modelssome codinglots of fun :-)

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Motivation

Growth economists often claim that business cycles are not very importantLucas:

”once you have begun to think about economic growth, it is hard

to think about anything else”Indeed fluctuations seem small compared to growth

Figure: US GDP, trend and cycle (HP Filter)

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LOG_GDP Trend Cycle

Hodrick-Prescott Filter (lambda=1600)

Source: Own calculations based on Fred

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Motivation cont’d

But from a”here and now” perspective cyclical fluctuations

matter a lot

Figure: GDP growth (yoy) and unemployment rate in Spain

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Unemployment rate GDP (yoy)

Source: Eurostat

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Motivation cont’d

And if rates are too anonymous, here is the number of people

Figure: Unemployed in Spain

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Source: Eurostat

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Evolution of business cycle models

A long time ago in a galaxy far, far away: IS-LM (Keynes1936, Hicks 1937)

Then comes Lucas (1974) and says:”No, no, no! It wont’t

work unless its microfounded”

And then come Kydland & Prescott (1982) and say:”Hey,

but we already have a nice microfounded model! Let’s make itstochastic”

And then come the New Keynesians (1990s) and say:”Nice

job, but you forgot about important frictions”

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Evolution of thinking about sources of cyclical fluctuations

In the earlier days, no systematic thinking about fluctuations,but rather about particular events:

Keynes: animal spirits responsible for the Great DepressionFriedman & Schwarz: monetary tightening was the culprit

After the microfoundation revolution:

Kydland & Prescott (1982):”Business cycles can be to a large

extent explained by technology shocks”New Keynesians (since 1990s):

”Other shocks (fiscal,

monetary, preference etc.) matter as well”After the financial crisis:

”OMG, financial shocks are so

important!”

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1 Motivation

2 Business cycles

3 Model

4 Steady state

5 Linearisation



8 Critique

9 Conclusions

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How to measure business cycle features?

1 Isolate the cyclical component from the data2 Calculate various moments:

1 standard deviations2 correlations3 autocorrelations

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The cyclical component

Popular (but not unique) definition: data component with aperiod from 6 to 32 quarters

How can this be extracted from the data?

Many options. Most popular:

Band-pass filterHoddrick-Prescott filter

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The Band-pass filter

Able to (approximately) extract only selected frequencies fromthe data

US GDP detrended with Baxter-King (1999) filter (6-32 quarters)

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.04

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Fixed Length Sym m etric (Baxter-King) Filter

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1.2

.00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50

Actual Ideal

Frequency Response Function

cycles/period

Source: Own calculations

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The Hoddrick-Prescott filter

Hodrick-Prescott (HP) filter very popular in applications(though also criticised, see e.g. Hamilton (2017)

Let yt denote the log of a time series variable, τt its trendcomponent and ct its cyclical component, so that yt ≡ τt + ct .

Then there exists a τt that solves:

min

{T∑t=1

(yt − τt)2 + λ

T−1∑t=2

[(τt+1 − τt)− (τt − τt−1)]2}

The first term penalizes the deviations from trend, the secondthe variability of the trend growth rate.

λ determines the smoothness of the trend

For quarterly data λ = 1600 extracts preriods below/above 32quarters

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Exercise: calculate business cycle moments (1)

Calculate basic business cycle moments from US data

Go to https://fred.stlouisfed.org/

Download at quarterly frequency (if data is monthly: Editgraph → Modify frequency) since 1947:

Real Gross Domestic Product: Billions of Chained 2012Dollars, Quarterly, Seasonally Adjusted Annual RateReal Gross Private Domestic Investment: Billions of Chained2012 Dollars, Quarterly, Seasonally Adjusted Annual RateReal Personal Consumption Expenditures: Billions of Chained2012 Dollars, Quarterly, Seasonally Adjusted Annual RateUnemployment Rate: Seasonally AdjustedConsumer Price Index for All Urban Consumers: All Items inU.S. City Average, Seasonally Adjusted

Never forget about seasonal adjustment!!! (Junk in,junk out)

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Exercise: calculate business cycle moments (2)

Save in Excell, import to Matlab

Take logs of all series except unemployment rate (it is alreadyin percent, just divide it by 100)

Apply the Hoddrick-Prescott filter (hpfilter.m) or theBand-pass filter (bpass.m) to all series except unemploymentrate (it has no trend)

For the cyclical components and the unemployment ratecalculate:

standard deviationscorrelations with GDP1st order autocorrelations

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Moments in the US economy

Main business cycle moments in the US economy

Source: Kydland & Prescott (1982)

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Business cycle properties: main takeaways

Consumption in less volatile than GDP

Investments are more volatile than GDP

Both are strongly procyclical

So are hours

Unemployment is countercyclical

Inflation is mildly procyclical

These properties characterize business cycles in mostdeveloped economies

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1 Motivation

2 Business cycles

3 Model

4 Steady state

5 Linearisation



8 Critique

9 Conclusions

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The history and idea of RBC

Lucas (1976) critique:

macro models should be microfoundedand should give an explicit role to expectations

Economists start looking for microfounded business cyclemodels

Kydland and Prescott (1982) come up with stochastic variantof the Ramsey economy

The only force to drive the business cycle are stochastictechnology shocks

Expectations are rational

They claim that these shocks are able to explain a domoniantshare of cyclical fluctuations

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Why study RBC?

RBC is the cornerstone of modern business cycle analysis

Has all the basic ingredients of a dynamic stochastic generalequilibrium model

And is simple

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Is RBC too simple?

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

It gives primacy to technology shocks as the source ofeconomic fluctuations.

It exhibits complete monetary neutrality and Ricardianequivalence (and sets role for monetary and fiscal policy).

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RBC as benchmark

The RBC model should be seen as a benchmark.

If a model with optimizing agents and instantaneous marketclearing can explain the business cycle, no need forimperfections such as sticky prices to explain macroeconomicfluctuations.

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Households - the problem

A representative household maximises lifetime utility

max{lt+i},{ct+i},{kt+i},{it+i}

Et

∞∑i=0

βt+i

[c1−σt+i − 1

1− σ− lt+i

1+ϕ

1 + ϕ

]

subject to the budget constraint

ct + it = wt lt + rtkt−1 + Divt

and the capital accumulation rule

kt = (1− δ)kt−1 + it

The household rents labour and capital to firms and receivesas compensation the real wage wt and the rental rate rt .

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Lagrangean

First, to simplify substitute for investment

ct + kt = wt lt + (1 + rt − δ)kt−1 + Divt

where rt − δ is the real interest rate.

The household chooses ct , lt and kt . Write down theLagrangean:

Lt = Et

∞∑i=0

βt+i

[(c1−σt+i − 1

1− σ−

l1+ϕt+i

1 + ϕ

)−λt+i

(ct+i + kt+i − wt+i lt+i − (1 + rt+i − δ)kt−1+i − Divt+i

)]

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First order conditions

ct : ∂L∂ct

= c−σt − λt = 0

lt : do it yourself :-)

kt : do it yourself :-)

Note that additionally we have a transversality condition (TVC)limt→∞λtkt = 0

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Equilibrium conditions - Euler equation

λt = Et [λt+1(1 + rt+1 − δ)] substitute for λt to get

c−σt = βEt [c−σt+1(1 + rt+1 − δ)]

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

In equilibrium the disutility from one unit less consumed todayequals expected discounted utility of consuming (1 + rt+1 − δ)units tomorrow.

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Equilibrium conditions - consumption vs. leisure

lϕtc−σt

= wt

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

In equilibrium the utility of one unit more of work should beequal to the utility from consuming the compensation (realwage).

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Firms - the problem

A representative firm uses capital and labour hired fromhouseholds to produce a unique good yt .

Its objective is to maximise profits:

Divt = yt − wt lt − rtkt−1

subject to technologyyt = ztk

αt−1l

1−αt

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Firms - equilibrium condition for labour

Substitute for production to get:

Divt = ztkαt−1l

1−αt − wt lt − rtkt−1

First order condition for labour is:

lt : δDivtδlt

= (1− α)ztkαt−1l

−αt − wt = 0

In equilibrium the marginal product of labour equals the realwage

To simplify things substitute from production funtion:

(1− α) ytlt = wt

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Firms - equilibrium condition for capital

kt−1 : do it yourself

In equilibrium the marginal product of capital equals the realrental rate of capital.

To simplify things substitute from production funtion:

α ytkt−1

= rt

In the RBC setting profits are zero (perfect competition). Tosee this substitute rt and wt into the profit function.

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Productivity and market clearing

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

zt = exp(εt)zρt−1

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 itgenerates the business cycle

The goods market clears:

ct + it = yt

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Equilibrium conditions - summary

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

c−σt = βEt [c−σt+1(1 + rt+1 − δ)] (1)

lϕtc−σt

= wt (2)

kt = (1− δ)kt−1 + it (3)

ct + it = yt (4)

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Equilibrium conditions - cont’d

yt = ztkαt−1l

1−αt (5)

(1− α)ytlt

= wt (6)

αyt

kt−1= rt (7)

zt = exp(εt)zρt−1 (8)

This can be solved

But two problems arise:

equations are non-linearthese are expectational difference equations

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1 Motivation

2 Business cycles

3 Model

4 Steady state

5 Linearisation



8 Critique

9 Conclusions

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Interest rate and productivity

In the deterministic steady state uncertainty disappears andall variables 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 + rt+1 − δ)]

(css)−σ = β(css)−σ(1 + r ss − δ)

Simplifyingr ss = β−1 − (1− δ) (10)

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Capital-labour and investment-capital ratios

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

αztk

αt−1l

1−αt

kt−1= rt

In the steady state

r ss = α(kss)α−1(l ss)1−α = α(kssl ss

)α−1

Rearranging we can obtain the formula for the capital-labourratio

kss

l ss=( r ssα

) 1α−1

(11)

where r ss is given by (10).

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Wage and investment-capital ratio

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

wt = (1− α)ztk

αt−1l

1−αt

lt= (1− α)zt

kαt−1

lαt

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)

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Consumption-labour ratio

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

ct + it = ztkαt−1l

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)

where kss/l ss is given by (11).38 / 109


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)

where kss/l ss is given by (11).39 / 109


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).40 / 109



1 Motivation

2 Business cycles

3 Model

4 Steady state

5 Linearisation



8 Critique

9 Conclusions

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Non-linear vs. linear models

We have a system of non-linear difference equations

Two options

1 take it as it is and let Dynare linearise it

2 linearise by hand

Linearising on your own is tedious but has some advantages:

1 some parameters may disappear

2 the system is easier to understand

3 solution in Dynare easier (steady state is known)

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Log-linearisation

Log-linearisation allows to change non-linear equations intolinear equations

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

Two steps:

Express variables as log deviation from steady state using theidentity:

xt = x ssxtx ss

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

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Useful tricks

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

Recall: f (xt) w f (xt | x0) + f′(xt | x0) (xt − x0)

This yields (derive #1 and #3 yourself)

xt = x ssexp(xt) ≈ 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)

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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

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Log-linearise Euler

c−σt = βEt [c−σt+1(1 + rt+1 − δ)]


1 = β(1 + r ss − δ)

So let’s linearise:

(css)−σ (1− σct) =β (css)−σ Et [(1− σct+1)(1− δ + r ss(1 + rt+1))]

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Log-linearise Euler cont’d

Substitute for r ss :

1− σct = βEt

[(1− σct+1)(1− δ + ( 1

β − 1 + δ)(1 + rt+1))]

1− σct = Et [(1− σct+1)(β − βδ + (1− β + βδ)(1 + rt+1))]

1− σct = Et [(1− σct+1)(1 + (1− β(1− δ))rt+1)] Multiply and

drop higher order terms:

1− σct = 1− σEt ct+1 + (1− β(1− δ))Et rt+1

Rearange terms:

σ(Et ct+1 − ct) = (1− β(1− δ))Et rt+1

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Log-linearise market clearing condition

ct + it = yt


css + i ss = y ss

Let’s linearise:

css(1 + ct) + i ss(1 + it) = y ss(1 + yt)

Substract steady state equation:

(1− i ss

y ss )ct + i ss

y ss it = yt

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Log-linearise firm’s equilibrium conditions

For labour

(1− α) ytlt = wt

yt − lt = wt

For capital

α ytkt−1

= rt

yt − kt−1 = rt

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Log-linearise production function

yt = ztkαt−1l

1−αt

Do it yourself :-)

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Log-linearise capital accumulation equation

kt = (1− δ)kt−1 + it

Do it yourself :-)

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Log-linearise shock process

zt = exp(εt)zρt−1

zt = ρzt−1 + εt

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Steady state values

Our equations contain steady state ratio i ss

y ss .

These are determined by our parameters

From the Euler equation:

r ss = β−1 − (1− δ)

and from the equilibrium condition for capital:

r sskss = αy ss

kss

y ss=

α

r ss=

α

β−1 − (1− δ)

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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)

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Log-linearised system

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

Have to solve it

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1 Motivation

2 Business cycles

3 Model

4 Steady state

5 Linearisation



8 Critique

9 Conclusions

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Solving linear DSGE models

Solving a DSGE model means changing the system of forwardlooking diference equations that we have ...

... into a VAR system

Dynare will solve it for you

If you want to learn it, there are several techniques for solvingsuch systems

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

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Blanchard & Kahn 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 : vector k × 1 of shocks (with mean equal to 0 every period)

A1, A0: (n + m)× (n + m) matrices

γ: (n + m)× k matrix

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Blanchard & Kahn condition cont’d

Assume that A1 is invertible. Then:[Xt+1

EtPt+1

]= A

[Xt

Pt

]+ A−1

1 γZt+1

where A = A−11 A0

The Blanchard-Kahn condition says that for the model tohave a unique solution, the number of eigenvalues of A lyingoutside the unit circle (unstable roots) must equal the numberof forward looking variables (jumpers)

Dynare solves the model and checks fulfilment of theBlanchard-Kahn condition

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Calibration

α = α ykky = rk

y

This is the share of capital remuneration in GDP

On which we have statistics: α ≈ 0.33 in the US economy

δ = ik =

iyky

≈ 0.22 = 0.10 in annual terms. Hence, in quarterly

terms δ ≈ 0.025

If we added bonds to the model we could show that β = 1rb

isthe inverse of the real interest rate.

A usual assumption from the RBC literature is 4% annual realinterest rate. In quarterly terms this means β ≈ 0.99.

Calibration of technology to match moments for GDP

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Simulations

Go to Matlab, set path to Dynare, open RBC.mod,

Run stochastic simulation (asymptotic) and analyse moments,

How does the model behave compared to the data moments?

Generate a 200-period long stochastic simulation. Plotoutput. Does it resemble business cycles?

Generate impulse response and check how (and understandwhy) technology affects all model variables.

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RBC - impulse response to technology shock

Figure: Effects of a technology shock in the RBC model

5 10 15 20 25 30 35 400

2

4

6

810-3 y

5 10 15 20 25 30 35 400

0.2

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0.8

1

1.210-3 c

5 10 15 20 25 30 35 40

0

0.005

0.01

0.015

0.02

0.025

0.03i

5 10 15 20 25 30 35 400

0.5

1

1.5

210-3 k

5 10 15 20 25 30 35 40-1

0

1

2

3

410-3 l

5 10 15 20 25 30 35 400

1

2

3

4

5

610-3 z


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RBC - first conclusions

Looks like we have a (very simple) model that reflects mainfeatures of the business cycle

At the same time the model constitutes a usefull benchmark

the model is microfoundedone shock explains all fluctuations

It can be extended in various directions

But even the simple RBC can be used for various purposes

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1 Motivation

2 Business cycles

3 Model

4 Steady state

5 Linearisation



8 Critique

9 Conclusions

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The role of Governments

Governments are big players in modern economies

Source: Eurostat

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Functions of governments

Main functions of governemnts:

provision of public goodsincome redistributionstabilization policies

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Provision of public goods

Public good is something freely available to everyone, e.g.

law and orderstreetlightsdefenceeducationenvironmental protection

These goods could be provided by the private sector, but in anunsatisfactory quality/ quantity/ widespread availability, e.g.:

externalities are not priced-in by the private sectorfree rider problem (e.g. streetlights)

These are so called”market failures”

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Externalities

Informally: Effects of an activity that affect other people

Formally (in modeling sense): variables that agents influence,but do not take it into account in the optimization process(e.g. because of very small impact of individual)

Negative externalities:

pollution, noise

Positive externalities:

synergic effects of education

Pricing in externalities: e.g. selling (and trading) rights topollute requires government intervention

Given formal definition DSGE models are well designed todiscuss externalities

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Income redistribution

Factors of production are paid their marginal product (?)

But this results in very unequal distribution of wealth andincome ...

... which need not be preferred from the social point of view.

Standard measure: Gini coefficient

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Gini coefficient

Standard (but not unique) measure of inequalityEquals (twice) the field under Lorenz curve

Source: http://www.urbansim.org/docs/tutorials/lorenz-curve.pdf

G=0 complete income inequality (one person holdseverything)G=1 complete income equality (everybody has the same)

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Income distribution

Source: United Nations Human Development Report

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Role of redistribution

Source: Forster and Pearson, “Income Distribution and Poverty in the OECD Area: Trends and Driving Forces”OECD Economic Studies. No. 34, 2001.

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Stabilization policies

There are business cycle fluctuations

Which may be perceived as inefficient

The government has policies that can smooth the businesscycle:

fiscal policy (expenditure, taxes)monetary policy

In the context of the RBC model we will speak about fiscalpolicy:

effects of government spending shocksdesireability

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Macroeconomic effects of stabilization policies

Important question in macro (at least since Keynes): what isthe effect of a government expenditure shock on the economy

Key concept - fiscal multiplier m = ∆y∆g

In”hardcore” keynesian models multiplier can be very high.

m = 11−MPC . If MPC = 0.8 then m = 5. (see Chapter 5.1 of

Romer textbook)

But this is”ceteris paribus”. In the real world there are many

frictions and adjustments and the multiplier is probably muchlower:

capacity constraintstax (or debt) adjustmentshousehold adjustment of labor supply and savingsmonetary policy reaction

The RBC model can take (some of) them into account

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Empirical evidence

Before we check what the RBC model has to say, how aboutempirical evidence?

Not so simple - endogeneity and expectations!!!

One needs identifying restrictions in VARs or exogenousinstruments

Blanchard & Perotti (2002) use institutional informationabout tax, transfer and spending programs in the US

Next they estimate a SVAR model on quarterly US data

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Empirical evidence - Blanchard & Perotti (2002)

Tax shocks crowd out consumption and investments

Spending shocks crowd out investment, crowd in consumption

Spending multiplier higher than tax multiplier

Spending multiplier around one

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Empirical evidence cont’d

But identifying exogenous spending/ tax shocks is difficult

As well as controling for the effect of expectations

E.g. Ramey (2011) finds:

most components of consumption fall after a positive shock togovernment spendinggovernment spending multiplier is 0.6-1.1

All in all, empirical evidence is far from conclusive

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Application of RBC models: government policy

Baxter & King (1993) check what the RBC model has to sayabout effects of government spending

We (they) ask the folowing questions:

what is the effect of a temporary change in governmentexpenditure?what is the effect of a permanent change in governmentexpenditure?how does the financing decission affect the results?

The RBC model is a good laboratory to analyse these issues(provided that we add a government to the model)

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Adding government expenditure to the model (lump-sumtaxes)

Assume that there exists a government, which taxeshouseholds (lump-sum) and buys goods

In the simplest variant (used here) they are waisted. But theycan also affect utility or form public capital

Compared with the baseline model, four modifications aremade:

Household budget constraint becomes:ct + it + tt = wt lt + rtkt−1 + DivtResource constraint becomes: ct + it + gt = ytFiscal rule (balanced budget): gt = ttGovernment spending rule (AR(1) process):

gt = gρgt−1 (g y ss)1−ρg exp(εgt )

where tt are lump-sum taxes and gt is governmentexpenditure.

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Modified equilibrium conditions

No changes to household or firm FOCs

This is the nature of lump-sum taxes, they do not affectagent’s behavior

Resource constraint becomes: ct + it + gt = yt

Fiscal rule (balanced budget): gt = tt

Government spending rule (AR(1) process):gt = g

ρgt−1 (g y ss)1−ρg exp(εgt )

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Simulations -temporary rise

Temporary increase in govermnet spending (four quarter rise)financed by lump-sum taxes

Figure: Effects of a temporary governemmnt spending shock in the RBCmodel

0 5 10 15 20 25 30 35 40-9

-8

-7

-6

-5

-4

-3

-2

-110-4

cons

0 5 10 15 20 25 30 35 40-4

-2

0

2

4

6

810-4

y

0 5 10 15 20 25 30 35 400

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

gov

0 5 10 15 20 25 30 35 40-6

-5

-4

-3

-2

-1

0

110-3

inv

0 5 10 15 20 25 30 35 400

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

labor (%)

0 5 10 15 20 25 30 35 40-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

real wage (%)


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Simulations - permanent rise

Permanent increase in govermnet spending financed bylump-sum taxes

Figure: Effects of a permanent governemmnt spending shock in the RBCmodel

0 50 100 150 200-4.6

-4.4

-4.2

-4

-3.8

-3.6

-3.4

-3.2

-3

-2.810-3

cons

0 50 100 150 2004

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

510-3

y

0 50 100 150 2000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

gov

0 50 100 150 2001.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.110-3

inv

0 50 100 150 2000.15

0.155

0.16

0.165

0.17

0.175

0.18

0.185

labor (%)

0 50 100 150 200-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

real wage


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A nice, little exercise :-)

Code rbc g lumpsum.mod is the RBC model with governmentexpenditure financed by lump-sum taxation

Calculate the i periods ahead government spending multipliermt,i ≡ yt+i−yt

gt+i−gton impact (i.e. one period ahead) in the temporary exerciseon impact and in the long-run (say after 200 periods) in thepermanent exercise

And explain the difference

Now, that you understand why the multiplier is positive, howcan you raise it even further?

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Adding government expenditure to the model(distortionary taxes)

Assume now that the government taxes household income andbuys goods

Again, they are waisted.

Compared with the baseline model, three modifications aremade:

Household budget constraint becomes:ct + it = (1− τt) (wt lt + rtkt−1) + DivtResource constraint becomes: ct + it + gt = ytFiscal rule (balanced budget): τt (wt lt + rtkt−1) = gtGovernment spending rule (AR(1) process):

gt = gρgt−1 (g y ss)1−ρg exp(εgt )

where τt is the income tax rate and gt is governmentexpenditure.

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Modified equilibrium conditions

This time household FOCs are affected

This is the nature of distortionary taxes, they affect agent’sbehavior

Euler: c−σt = βEt [c−σt+1(1 + (1− τt+1)rt+1 − δ)]

Labor choice: lϕtc−σt

= (1− τt)wt

Resource constraint becomes: ct + it + gt = yt

Fiscal rule (balanced budget): gt = τt (wt lt + rtkt−1)

Government spending rule (AR(1) process):gt = g

ρgt−1 (g y ss)1−ρg exp(εgt )

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And another nice exercise :-)

Code rbc g distortion.mod is the RBC model with governmentexpenditure financed by income taxation

Unfortunately someone deleted the dynamic equation

Restore the code so that the model assumes distortionarytaxation

Run the simulations (temporary and permanent increase ingovernment spending financed with distortionary taxes)

What is the impact on the economy? How does it differ fromthe previous one? Why?

This exercise is decribed in Baxter & King (1993), you cancheck if you get sth similar.

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Or a homework :-)

Code rbc g lumpsum.mod is the RBC model with governmentexpenditure financed by lump-sum taxation

And derivations in RBC gov derivations.pdf derive the RBCwith distortionary taxation as well (or maybe you want toderive yourself?)

Modify the code so that the model assumes distortionarytaxation now

Run the simulations (temporary and permanent increase ingovernment spending financed with distortionary taxes)

What is the impact on the economy? How does it differ fromthe previous one? Why?

This exercise is decribed in Baxter & King (1993), you cancheck if you get sth similar.

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1 Motivation

2 Business cycles

3 Model

4 Steady state

5 Linearisation



8 Critique

9 Conclusions

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Some problems with RBC models

RBC models have some welcome features

But also some unwelcome

possibly overstated role of TFP shocksno role for stabilization policysmall cost of business cycle fluctuationsno role for monetary policyno role for demand shocksvoluntary unemployment

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Role of TFP shocks

Our calibration of technology shocks was done to match thes.d. and auroreggression of output with GDP

Let us instead construct a TFP series from the data andcheck its properties

The exercise follows (approximately) Solow (1957) andPrescott (1986)

File data Solow residual.mat contains three series (US:1Q1950 - 3Q2019, seasonally adjusted):

GDPCapital (adjusted for utilization as in Solow 1957)Labor hours in nonfarm business activity

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Exercise: role of TFP shocks (1)

Write Matlab code that:

calculates the Solow residual (assume CD production functionwith α = 0.33)calculate its cyclical component (use hpfilter.m)estimate regression: zt = ρzt−1 + εtcalculate standard deviation σ (εt)see next slide for some formulae

Use the estimated ρ and σ (εt) to recalibrate the RBC model

Compare again the moments for output in the model and forGDP (cyclical component) in the data

What fraction of fluctuations can the model explain now?

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Exercise: role of TFP shocks - useful formulae

Some useful formulae:

Solow residual:

Zt = Yt/(Kαt L

1−αt

)OLS estimator:

β =(XTX

)−1XT y , where β is the vector of estimated

parameters, X is the matrix of explanatory and y the vector ofdependent variables

Standard deviation of residuals:

σ(εt) =√

εTt εtn−k−1 where n is sample size and k the number of

explanatory variables

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Exercise: role of TFP shocks (2)

Go back to the Matlab code:

calculate the growth rate of the Solow residual (original, beforedetrending)plot ithow often is it negative?what does this imply in economic terms?

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Role of TFP shocks - some conclusions

TFP shocks calculated as Solow residuals do not explain100% of cyclical fluctuations of output (just about 75%)Solow residual declines almost 25% of sampleProbably even Solow residual overstates variability of TFPOther shocks might be necessary to explain business cycles toa satisfactory degreeSolow residual in the US (qoq)

0 50 100 150 200 250 300-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025


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Role for stabilization policy

The RBC economy fluctuates around steady state (orbalanced growth path)

Are these fluctuations efficient?

In other words: should policymakers react to them and imposea different allocation than the market does?

Introduce”social planer”: an imaginary agent who maximizes

to welfare instead of agents (decentralize equilibrium)

”Social planer” can for instance help overcome the negative

impact of externalities on welfare

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Social planer: problem

The social planer seeks to maximize

max{lt+i},{ct+i},{kt+i},{it+i}

Et

∞∑i=0

βt+i

[c1−σt+i − 1

1− σ− lt+i

1+ϕ

1 + ϕ

]subject to the resource constraint of the economy

yt = ct + it

the capital accumulation rule

kt = (1− δ)kt−1 + it

and technology

yt = ztkαt−1l

1−αt

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Social planer: Lagranean

After substituting for investments and output the Lagrangeanis

Lt = Et

∞∑i=0

[βt+i

(c1−σt+i − 1

1− σ−

l1+ϕt+i

1 + ϕ

)−µt+i

(ct+i + kt+i − (1− δ)kt−1+i − zt+ik

αt−1+i l

1−αt+i

)]

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Social planer: first order conditions

ct : ∂L∂ct

= βtc−σt − µt = 0

lt : ∂L∂lt

= −βt lϕt + µtzt(1− α)kαt−1l−αt = 0

kt : ∂L∂kt

= −µt + (1− δ)Etµt+1 + Etµt+1zt+1αkα−1t l1−αt+1 = 0

Additionally we have a transversality condition (TVC)limt→∞µtkt = 0

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Social planer: equilibrium conditions

Substituting for µt and using production function we get:

lϕt = c−σt (1− α) ytlt

c−σ = Etc−σt+1

(1− δ + α yt+1

kt

)Note that these imply the same allocation as the decentralizedequilibrium.

To see this substitute

(6) into (2)(7) into (1)

Except for prices wt and rt , which are absent in the socialplaner’s problem.

On the top we have the same production function, capitalacumulation law, resource constraint and TFP process.

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Social planer: consequences

The decentralized allocation coincides with the socialoptimum:

The government cannot improve upon the market solutionIn particular countercyclical policy can only reduce welfare

This is what the 1st Fundamental Welfare Theorem tells us: acompetitive equilibrium leads to an efficient allocation.

If we study a competitive economy we do not need to considereach agent’s first order conditions. Instead we can use thefirst order conditions of the social planner.

This is a formal statement of Adam Smith’s”invisible hand”

concept.

Of course assumptions need not always be fulfilled:monopolistic power, externalities etc. allow for improvementsupon decentralized equilibrium.

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Cost of business cycle fluctuations

Another important topic widely discussed in the literature:how important are BC fluctuations?

Microfounded models have a clear metric to evaluate thisquestion

How much do fluctuations lower a representative household’swelfare?

In other words, how much would the HH like to pay to live ina world w/o fluctuations?

Lucas (1987; 2003) offers a calculation

Simplified variant - allows for analytical solution.

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Cost of business cycle fluctuations: intro

Assume consumption (endowment) follows ct = e−σ2

2 εt

where εt ∼ Λ(0, σ2) (i.e. εt follows log-normal distribution)

Preferences are given by CRRA utility funcion:

Et∑∞

t=0 βt c

1−γt

1−γHow much more consumption would agents like to have (in aworld with fluctuations) to be indifferent to a world w/ofluctuations?

Solve for λ:

Et

∞∑t=0

βt[(1 + λ)ct ]

1−γ

1− γ=∞∑t=0

βt[css ]1−γ

1− γ

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Cost of business cycle fluctuations: solution (1)

Note that css = 1

And recall :-) that for a log-normal distributionE (xa) = eaµ+a2σ2/2 . Then we have

∞∑t=0

βt(1 + λ)1−γEtc

1−γt

1− γ=∞∑t=0

βt1

1− γ

∞∑t=0

βt(1 + λ)1−γEt

(e−

σ2

2 εt

)1−γ

1− γ=

1

(1− γ) (1− β)

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Cost of business cycle fluctuations: solution (2)

∞∑t=0

βt(1 + λ)1−γe−

σ2

2(1−γ)e(1−γ)2σ2/2

1− γ=

1

(1− γ) (1− β)

(1 + λ)1−γe−σ2

2(1−γ)e(1−γ)2σ2/2

(1− γ) (1− β)=

1

(1− γ) (1− β)

(1 + λ)1−γeσ2

2 (1−2γ+γ2−(1−γ)) = 1

(1 + λ)1−γeσ2

2γ(γ−1) = 1

ln(1 + λ)1−γ + ln eσ2

2γ(γ−1) = ln 1

λ ' σ2

2γ

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Cost of business cycle fluctuations: result and way forward

So (quite intuitively) compensation λ depends on risk σ2 andrisk aversion γ

US consumption volatility (quarterly, HP detrended) for1959-2020 is σ = 0.0061

λ ' σ2

2 γ = (0.0061)2/2 = 0.000018

Lucas provides somewhat different number based on annualdata & linear trend: λ = 0.0005

Both are negligibly small

Do we miss something?

unemploymentcredit constraints

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Other points of critique

So far we have:

TFP shocks are not able to explain 100% of fluctuationsGoverment policy cannot improve upon the decentralized(market) allocationCost of fluctuations is negligible

Some extra points (see Rebelo 2005 for a nice overview andmore details):

What caused the Great Depression? RBC model suggestsnegative technology shocks. Really???What are the effects of monetary policy on the economy? RBCmodel (if MP is added) suggests that it affects only prices.Really?What causes large cyclical swings of labour hours? RBC modelsuggests voluntary decissions (and high elasticity of laborsuppply). Really?

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Consequences of the critique

This critique should be taken seriously but not panically

The RBC model should be (and usually is) considered as ausufull benchmark of a frictionless, competitive economy

Some questions can be asked in this framework

Many others require a richer specification (which often buildson the RBC core)

Think of economic modeling (research) in the spirit ofOkham’s razor:

if a small (simple) model can explain the problem then do notlook for a a bigger (more complicated) oneif it cannot, look for the simplest (though realistic)modification that solves the problem (e.g. explains empiricalfacts)

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1 Motivation

2 Business cycles

3 Model

4 Steady state

5 Linearisation



8 Critique

9 Conclusions

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Conclusions

The RBC model is the simplest microfounded business cyclemodel

It explains cyclical fluctuations as resulting from technologyshocks

Quite succesful in some areas (e.g. main moments) but hasseveral problematic features

We think of RBC as a useful frictionless benchmark andbuilding block of more complicated models

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Business cycles and the RBC modelmbrzez/Makro_zaawansowana/RBC.pdf · Motivation Business cycles Model Steady state Linearisation Solution, calibration & simulation Application Critique

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