the green solow model - a lecture course presentation

A presentation within the lecture course Growth Theory

in summer term 2011

The Green Solow Model investigating sustainable economic growth

by W. A. Brock and M. S. Taylor

05.05.2011 Florian Hage| Eberhard-Karls-Universität Tübingen | SQ-Seminar in Growth Theory Summerterm 2011

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• Over the last decades the emission of pollutants as a result of economic activities has become an increasingly important field of research

• It has been hypothesised that a relationship holds for many forms of environmental degradation which is called Environmental Kuznets Curve (EKC) 1

• If EKC exists, economic growth might be a mean to environmental improvement, i.e. as countries develop economically, moving from lower to higher levels of income per capita, over all levels of environmental degradation – such as pollution – will eventually fall 2

• Empirical examination of cross-country data has displayed a relationship between one economy’s emission levels and income per capita

Motivation

2

Motivation and empirics Agenda


Overview The Green Solow Model

Comparative steady state analysis Conclusion and critiques

1 Kuznets, S. (1955), pp. 1-28 2 Perman, R. et. al (2003), pp. 36 f.

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Emissions and GDP per capita by year

3 05.05.2011 Florian Hage| Eberhard-Karls-Universität Tübingen | SQ-Seminar in Growth Theory Summerterm 2011

Figure 1 from Florian Hage, data from various sources one can find in the appendix




Take the early to mid 1970s as the start of serious pollution regulation

0

100

200

300

400

500

600

0

50

100

150

200

1950 1975 2000

GD

P P

er C

apit

a in

Lev

els

No

rmal

ise

d b

y 1

95

0

Tota

l Em

issi

on

s in

Lev

els

No

rmal

ise

d b

y 1

95

0

Time

Germany Pollutant Emissions and GDP

Sulfur Emissions / pc (kg)

CO2 Emissions / pc (t)

GDP/pc

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Take the early to mid 1970s as the start of serious pollution regulation






0

100

200

300

0

50

100

150

1950 1975 2000

GD

P P

er C

apit

a in

Lev

els

No

rmal

ise

d b

y 1

95

0

Tota

l Em

issi

on

s in

Lev

els

No

rmal

ise

d b

y 1

95

0

Time

United States Pollutant Emissions and GDP



GDP/pc

Emissions and GDP per capita by year

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• Structure of the Green Solow Model

• Setting up the model 3

• The balanced growth path

• The EKC

• Comparative steady state analysis

• Conclusion and critiques

• Appendix – A contemporary EKC for Germany (1950 ̶ 2006)

– β: speed of adjustment

– Additional comparative steady state analysis

• Literature

Agenda





3 Brock, W. A.; Taylor, M. S. (2010)

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• The Green Solow Model combines the core model of modern growth theory, the SOLOW-SWAN Model 4, and one key finding in environmental economics, the EKC

• How are inputs related to the output and what is itself used for ?

General structure of the Green Solow Model

6

B

L

K

E

Y C

I = S

A


General Structure Setting up the model The balanced growth path The EKC



4 Barrow, R.; Sala-i-Martin, X. (2004), p. 23 f.

Ω

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• The production function is of type COBB-DOUGLAS and given by

• Capital accumulates via investments and depreciates at rate δ

• Additional exogeneous parameters such as population growth, labour augmenting technologies (e.g. computers etc.), emission growth and pollution intensity Ω with its technological progress in abatement (e.g. advanced filter systems, renewables etc.), are represented by

Setting up the model

7

1, BLKBLKFY

KsYKSKIK

BgB

nLL

B

A

E

g

EgE


10




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• Every unit of economic activity generates Ω units of pollution, whereby actual emission E equals pollution created minus pollution abated

A denotes the abatement level as a function of economic activity Y and the economy’s efforts at abatement YA.

where θ = YA/Y denotes the proportion of economic activity at abatement

Setting up the model

8

1

/,11

/,1

,

Y

YYAY

YYAYY

YYAYE

A

A

A

AYE





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• In order to control for the effective population size from now on small letters denote per capita units, i.e.

where y = (1 – θ ) f (k) represents that income per capita which is not used up for abatement

• Since capital per capita as a pure input factor evolves over time by

the amount of capital per capita accumulates over time

The balanced growth path

9

kskk

BLKk

kkfBLKFy

/

111,/1

kngkskBL

KB

1~





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10

Ein Spiel wird als supermodular bezeichnet, wenn die eigene Grenzgewinnfunktion eines Spielers (eigene Gewinnfunktion nach der eigenen Strategievariablen differenziert) durch eine Erhöhung der strategischen Variablen jedes anderen Spielers erhöht wird.

Approaching the balanced growth path gives the steady state


Necessary conditions:

• Steady state

• Inada conditions hold for Y = F(K,BL)

• , otherwise the accumulation process does not start

01~

kngksk B

gross savings function per capita

capital diminishing function per capita

00 k




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• We then find the steady state capital per capita

• The corresponding income per capita

• On the balanced growth path aggregate GDP, consumption and capital all grow at rate

• But the objective of the Green Solow Model is to relate income levels to environmental quality


05.05.2011 11 Florian Hage| Eberhard-Karls-Universität Tübingen | SQ-Seminar in Growth Theory Summerterm 2011

1

1

1

ng

sk

B

11

ng

sky

B

ngggg BKCY




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• Derivation of the growth rate of emissions

where

Taking logs and differentiating with respect to time yields 5

in steady state

• Sustainable growth in steady state is realised if



1YE kBLBLKFY ),(

0

~

EBA gngg

k

k

L

L

B

B

E

E

0~k

AB gng

What does “sustainability” mean here ?

We define sustainable growth as a balanced growth path generating both rising income per capita and an improving environment due to decreasing growth rates in aggregate emissions E.




5 Ferrara, M.; Guerrini, L. (2009), p. 48.

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• How do we derive an EKC ?

We write aggregate emissions at any time t by

Differentiate with respect to time yields

The EKC


tgEetkLBtE )()1()( 000

initial conditions

Percentage rate of change of emissions

with

)()1(

~1 ngksg

k

kg

E

EBEE

kkyy~




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The EKC – sustainable growth

14

EΒ gδng

δngΒ

11 ks

T

B

)(Tk k

)(Tk

BLKk /

BLKk /


Level of

emissions

E

E

gE < 0

k(0) < k(T ) < k*

growth rate of aggregate emissions Ė/E is at first positive but turns negative in finite time

k(T ) < k(0) < k*

growth rate of aggregate emissions Ė/E is negative for all times t; same along the balanced growth path

k




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The EKC – unsustainable growth

15

EΒ gδng

δngΒ

11 ks

T

B

)(Tkk

)(Tk

BLKk /

BLKk /


Level of

emissions

E

E

gE > 0

k(0) < k* < k(T )

Although Ė/E declines over time t, growth rate of aggregate emissions remains positive when k approaches the steady state k* and keeps on growing along the balanced growth path

k




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The EKC


Intermediate critiques

• So far examination of the emission profile

• Just very little evidence about the actual emissions level and its income per capita – especially at k(T )

• Model might cause confusion such that countries with identical parameters (s, n, δ,…) show same emission levels; that is in fact not the case

• Starting conditions such as k(0), Ω(0), B(0) and L(0) determine the model also; consequently we cannot conclude from emissions to income levels

Now examination of the emission’s peak T




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• The amount of capital at T can be found by solving for k which yields

• How long does it take to reach peak emissions T ?

We rewrite the process of capital accumulation as a function of time:

where k(t) is an exponentially weighted average of the economy’s initial

capital per capita k(0) and its steady state level k*, the speed of adjustment

The EKC – peak of aggregate emissions T


0

~

k

kg

E

EE

1

1

/

1)(

EB gng

sTk

1

1

)1()1()0(1)( tt ekektk




0)1( ngB

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• From k(t) we obtain at t = T and then solving for T an implicit function for the time it takes to reach the peak level of aggregate emissions:



)1()1(

)1()1(

)(

)0(ln

1

Tkk

kkT

Comparative stability

Calendar time T needed to reach peak emissions …

0/ T

0)0(/ )1()1(

kkT

0)(/ )1()1(

TkkT

… is declining in the speed of convergence

… is increasing in the gap between k* and k(0)

… is increasing the closer T and B are located to each other




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• Consequently, peak emissions given by E(t) with t = T :

It is apparent from E(T ) that if we compare two economies with same exogeneous parameters (s, gB, n, δ, α, and thus same k*), these economies will neither share the same income per capita nor the same peak level of emissions



TgTT

B

Tg

E

E

eekeng

sLB

eTkLBTE

1)1(000

000

)0(1)1(

)1(

)()1()(

)1( k

)(Ty




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• When exactly is the level of peak emissions E(T ) reached ?

Examining the growth rate of emissions again

with

here it becomes apparent that the changing rage of capital accumulation is generating the dynamics of the Green Solow Model. The only effect of is its impact on growth of output per capita gy, thus gy,t changes over time. In the long run



ytyt

gg

,lim

0)(

)(~

)(

)(~

tk

tkngg

tk

tk

L

L

B

B

E

E

BA

tyg ,

0

0

,

,

ty

ty

g

g

)(/)(~

tktk




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21

Ag

yg

)(

)(~

,tk

tkngg Bty

T t

t


growth

rates

)(tE

Mechanics

t < T : gy,t > gA and gE > 0

t = T :

gy,t = gA and gE = 0

t > T : gy,t < gA and gE < 0

which again represents our sustainability assumption

T

Level of

emissions




Figure following Stefanski, R. (2010), p. 6.

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Characterisation of the EKC profile

t < T :

Rising total emissions because abatement is not enough to outweigh extra pollution caused by faster growth of GDP

t = T :

Peak emissions are reached when the rate at which emissions are created via output growth gy,t are exactly offset by the rate at which they are abated gA

t > T :

Falling growth of emission because improvements in emission intensity Ω outweigh the additional pollution created by production

0, EAty ggg

0, EAty ggg

0, EAty ggg

Aty gg ,




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• How are aggregate emissions E(t) related to output y(t) ?

Substituting into E(t) we obtain E[ϕ(y)]:

which represents a parametric relationship between aggregate emissions and income per capita that we refer to as an EKC

The EKC – relation of aggregate emissions and output


)(1)()1()(000 )0(1

)1()1()]([

ygyy

B

Eeekeng

sLByE

)(tyy

t

y )(yt

y

t

)(yt

0)( y




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• The derivative of E[ϕ(y)] with respect to output per capita y yields:

which too represents the EKC we derived previously.

The EKC – relation of aggregate emissions and output


Ttfor

Ttfor

Ttfor

yyEy

yE

0

0

0

)()]([)]([

Ag

yg

tyg ,

y(T ) y

y

growth

rates

)]([ yE

Level of

emissions

y(T )




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Comparative steady state analysis

25

)(TkBLKk /

Lower initial conditions Ω(0), B(0), L(0) and k(0) :

• Affect E(t) and y(t) directly • But no impact on steady

state magnitudes of k* and y* nor on long run growth rates


Level of

emissions

Initial conditions Savings rate Abatement intensity Technological progress in abatement

k

tgtt

B

tg

E

E

eekeng

sLB

etkLBtE

1)1(000

000

)0(1)1(

)1(

)()1()(

)1( k

)(ty



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26

EΒ gδng

δngΒ

11 ks

)(Tk k

)(Tk

BLKk /

BLKk /

Increase in savings rate s :

• Faster capital accumulation • Magnitudes of k(T ), k* and

y(k) increase • Impact on pollution path • Thus calendar time T

increases, i.e. the process slows down

• In steady state growth rate of emissions and income per capita remain unchanged


Level of

emissions

E

E

k

T

B




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27

EΒ gδng

δngΒ

11 ks

)(Tk k

)(Tk

BLKk /

BLKk /

Increase in abatement intensity θ, e.g. due to tighter environmental policy:

• Slows down capital accumulation via smaller I

• Magnitudes of k(T ), k* and y(k) decrease

• Impact on pollution path • Thus calendar time T

shortens • In steady state growth

rate of emissions and income per capita remain unchanged


Level of

emissions

E

E

k

T

B




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28

EΒ gδng

δngΒ

11 ks

)(Tk k

)(Tk

BLKk /

BLKk /

Increase in abatement intensity θ, e.g. due to tighter environmental policy:

Although θ increases gA remains constant, i.e. tighter environmental policy cannot turn an unsustainable economy in a sustainable one Emission reduction is obtained by a decrease in k and y, NOT because of increasingly effective abatement


Level of

emissions

E

E

k

T

B




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29

EΒ gδng

δngΒ

11 ks

)(Tk k

)(Tk

BLKk /

BLKk /

Increase in technological progress at abatement gA:

• gE = gB + n ‒ gA decreases • Magnitudes of k(T ), and

y(T) decrease, but k* remains unchanged

• Impact on pollution path • Thus calendar time T

decreases • In steady state growth

rate of emissions decreases while growth rate of income per capita remain unchanged


Level of

emissions

E

E

k

T

B




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Conclusion and critiques




• In fact we found an Environmental Kuznets Curve (EKC) plotting data

• We identified three qualitatively different sets of determinants a) Initial conditions affect E(t) and y(t) directly but not in steady state or in the long run b) savings rate s accelerates capital accumulation while abatement intensity θ does not create sustainable economies and emission reduction is not obtained by increasingly effective abatement c) gA decreases gE , while gB + n increases steady state growth rate of emissions

• Putting this together, EKC has shown, as countries develop economically, moving from lower to higher levels of income per capita, levels of environmental degradation – such as pollution – will eventually fall

• Empirical examination of cross-country data has verified this relationship for certain pollutants

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Vielen Dank für ihre Aufmerksamkeit ! Thanχ for your attention!


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A contemporary EKC for Germany (1950 ̶̶ 2006)



0

50

100

150

200

60

90,0

342

07

65

99,9

180

17

71

44,1

143

93

76

97,2

480

42

82

33,4

754

24

9096

,82

667

96

92,6

084

37

10

187

,826

65

10

571

,585

56

11

261

,929

87

12

091

,341

93

12

478

,829

31

12

902

,462

91

13

158

,879

09

1384

4,3

692

14

414

,591

88

14

731

,373

75

14

745

,625

61

15

479

,666

27

16

383

,406

56

17

009

,194

36

17

382

,082

51

18

016

,180

27

18

778

,226

31

18

929

,763

01

18

894

,574

54

1990

3,9

702

20

512

,921

23

21

114

,594

35

21

958

,814

08

22

148

,267

69

22

202

,512

91

22

031

,532

74

22

485

,381

15

23

198

,328

06

2375

7,8

842

24

274

,559

14

24

639

,185

66

25

359

,060

24

25

983

,626

53

24

996

,143

15

26

127

,058

82

26

505

,303

17

2611

9,4

586

26

720

,642

84

27

145

,957

92

27

336

,466

04

27

788

,884

16

2829

1,5

981

28

842

,656

12

29

726

,878

76

30

061

,398

67

30

035

,801

98

29

952

,936

65

3026

0,6

614

3049

6

31

511

,651

36

Emis

sio

ns

Per

Cap

ita

in L

evel

s

GDP Per Capita

Germany Pollutant Emissions by GDP pc



6 000 30 000 20 000 10 000

A contemporary EKC for Germany Speed of adjustment Comparative steady state analysis with respect to g_B or n Literature

Appendix

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• In order to investigate the peak of aggregate emissions we have to find T

• Economies with identical parameters but different initial conditions converge over time t to the same balanced growth path (β-convergence)

• We apply a TAYLOR-series-approximation of first order of the fundamental equation around the steady state to find the speed of adjustment β :

• We find that β is independent from the abatement intensity θ and savings rate s



kngksk B 1~

kkngkkf B )1(~

)(

β > 0

Appendix A contemporary EKC for Germany Speed of adjustment Comparative steady state analysis with respect to g_B or n Literature

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34

EΒ gδng

δngΒ

11 ks

)(Tk k

)(Tk

BLKk /

BLKk /

Increase in technological progress at production gB or population growth n:

• gE = gB + n ‒ gA increases • Magnitudes of k(T ), and

y(T) decrease, but k* remains unchanged

• Impact on pollution path and calendar time T depends on k(0)


Level of

emissions

E

E

k

T

B


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• BARROW, R. J.; SALA-I-MARTIN, X. (2004). Economic Growth. Second Edition. The MIT Press. Cambridge, Massachussets, London, England.

• BROCK, W. A. and TAYLOR, M. S. (2010). The Green Solow Model. Springer Science+Business Media, J Econ Growth (2010) 15: pp. 127–153.

• FERRARA, M. and GUERRINI, L. (2009). More on the Green Solow Model with Logistic Population Change. WSEAS Transactions on Mathematics. Issue 2, Volume 8, February 2009: pp. 41–50.

• KUZNETS, S. (1955). Economic growth and income inequality. The American Economic Review. Volume XLV, No. 1, March 1955: pp. 1–28.

• PERMAN, R. et al. (2003). Natural Resources and Environmental Economics. Pearson / Addison Wesley, 3rd edition. Harlow, UK.

• SOLOW, R. (1957). Technical change and the aggregate production function. Review of Economics and Statistics, 1957, 39(3), 312–320.

• STEFANSKI, R. (2010). On the mechanics of the “Green Solow Model“. University of Oxford..

Literature



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• CO2-data: Complete reference CDIAC (Carbon Dioxide Information Analysis Center). Link to complete reference http://cdiac.ornl.gov/ftp/ndp030/CSV-FILES/

• Sulfur-data: The data is based on the following article: STERN D. I. (2006). Reversal in the trend of global anthropogenic sulfur emissions. Global Environmental Change. Volume 16: pp. 207-220. The article is avilable at: http://www.sterndavidi.com/Publications/GEC2006.pdf The data we used is available at: http://www.sterndavidi.com/datasite.html

• GDP-data: Gross Domestic Product per capita by Purchasing Power Parities (in international dollars, fixed 2005 prices). The inflation and differences in the cost of living between countries has been taken into account. Main sources: Cross-country data for 2005 is mainly based on the 2005 round of the International Comparison Program.

Literature



http://cdiac.ornl.gov/ftp/ndp030/CSV-FILES/





http://www.sterndavidi.com/Publications/GEC2006.pdf



http://www.sterndavidi.com/datasite.html

the green solow model - a lecture course presentation

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