The Green Solow Model investigating sustainable economic growth by W. W. A. Brock and an d M. S. Taylor Taylor A presentation within the lecture course Growth Theory in summer term 2011
Overview The Green Solow Model Comparative steady state analysis Conclusion and critiques
Motivation and empirics Agenda
Motivation •
•
•
•
Over the last decades the emission of pollutants as a result of economic activities has become an increasingly important important field of research It has been hypothesised that a relationship holds for many forms 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 examination of cross-country data has displayed a relationship between one economy’s economy’s emission levels and income per capita
1 2
Kuznets, S. (1955), pp. 1-28 Perman, R. et. al (2003 ), pp. 36 f.
Overview The Green Solow Model Comparative steady state analysis Conclusion and critiques
Motivation and empirics Agenda
Emissions and GDP per capita by year Germany Pollutant Emissions and GDP
200 0 5 9 1 y b d150 e s i l a m r o N s l e 100 v e L n i s n o i s s i 50 m E l a t o T
600
500
400
300
200 Take the early to mid 1970s as the start of serious pollution regulation
100
0
0 5 9 1 y b d e s i l a m r o N s l e v e L n i a t i p a C r e P P D G
0 1950
1975
2000
Time
Figure 1 from Florian Hage, data from various sources one can find in the appendix
Sulfur Emissions / pc (kg) CO2 Emissions / pc (t) GDP/pc
Overview The Green Solow Model Comparative steady state analysis Conclusion and critiques
Motivation and empirics Agenda
Emissions and GDP per capita by year United States Pollutant Emissions and GDP
150 0 5 9 1 y b d e s i l a 100 m r o N s l e v e L n i s n 50 o i s s i m E l a t o T
300 0 5 9 1 y b d e s i 200 l a m r o N s l e v e L n i a t 100 i p a C r e P P D G
Take the early to mid 1970s as the start of serious pollution regulation
0
0 1950
1975
2000
Time
Figure 2 from Florian Hage, data from various sources one can find in the appendix
Sulfur Emissions / pc (kg) CO2 Emissions / pc (t) GDP/pc
Overview The Green Solow Model Comparative steady state analysis Conclusion and critiques
Motivation and empirics Agenda
Agenda •
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
3
Brock, W. A.; Taylor, M. S. (2010)
Overview The Green Solow Model Comparative steady state analysis Conclusion and critiques
General Structure Setting up the model The balanced growth path The EKC
General structure structure of the Green Solow Model •
•
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 ?
Ω
E A B Y
C
L I = S K
4
Barrow, Barrow, R.; Sala- i-Martin, X. (2004), p. 2 3 f.
Overview The Green Solow Model Comparative steady state analysis Conclusion and critiques
General Structure Setting up the model The balanced growth path The EKC
Setting up the model •
The production function is of type COBB-DOUGLAS and given by
Y F K , BL K •
BL 1
0 1
Capital accumulates via investments and depreciates at rate δ
I K S K sY K K •
Additional exogeneous exogeneous parameters parameters such as population growth, labour augmenting technologies technologies (e.g. computers computers etc.), etc.), emission growth growth and pollution intensity Ω with its technological progress in abatement (e.g. advanced filter systems, renewables etc.), are represented by L nL
g E E E
B g B B
g A
Overview The Green Solow Model Comparative steady state analysis Conclusion and critiques
General Structure Setting up the model The balanced growth path The EKC
Setting up the model •
Every Every unit of economic activity generates Ω units of pollution, whereby actual emission E equals pollution created minus pollution abated E Y A level as a function of economic activity Y and A denotes the abatement level A. the economy’s efforts at abatement Y A
E Y A Y , Y
A Y A AY Y 1, Y / Y
Y 1 A 1, Y A / Y Y 1
A /Y denotes the proportion of economic activity at where θ = Y abatement
Overview The Green Solow Model Comparative steady state analysis Conclusion and critiques
General Structure Setting up the model The balanced growth path The EKC
The balanced growth path •
In order to control for the effective population size from now on small letters letters denote per capita units, i.e. y 1 F K / BL,1 1 f k 1 k k K / BL
where y = (1 – θ represents that income per capita which is not – θ ) f (k ) represents used up for abatement •
Since capital per capita as a pure input factor evolves over time by
sk k k the amount of capital per capita accumulates acc umulates over time
K ~ k s1 k g B n k BL
Overview The Green Solow Model Comparative steady state analysis Conclusion and critiques
General Structure Setting up the model The balanced growth path The EKC
The balanced growth path Approaching the balanced growth path gives the steady state Ein Spiel wird als supermodular bezeichnet, wenn die eigene Grenzgewinnfunktion eines Spielers (eigene Gewinnfunktion nach der Necessary conditions: eigenen Strategievariablen differenziert) durch eine Erhöhung der strategischen Variablen jedes anderen Spielers erhöht wird. •
Steady state
~ k s1 k g B n k 0 gross savings function per capita
•
•
capital diminishing function per capita
Inada conditions hold for Y = F (K BL , BL) k 0 0 , otherwise the accumulation process does not start
Overview The Green Solow Model Comparative steady state analysis Conclusion and critiques
General Structure Setting up the model The balanced growth path The EKC
The balanced growth path •
We then find the steady state capital per capita 1
s1 1 k g B n
•
The corresponding income per capita
y •
•
s1 k g n 1
B
On the balanced growth path aggregate GDP, consumption and capital all grow at rate gY g C g K g B n But the objective of the Green Solow Model is to relate income levels to environmental quality
Overview The Green Solow Model Comparative steady state analysis Conclusion and critiques
General Structure Setting up the model The balanced growth path The EKC
The balanced growth path What does “sustainability” mean here ?
We define sustainable growth as a balanced growth generating both rising income per capita and an path generating improving environment due to decreasing growth rates in aggrega ag gregate te emissions E . •
Derivation of the growth rate of emissions E Y 1 where Y F ( K , BL) BL k
Taking logs and differentiating with respect to time yields 5 ~ B L k ~ 0 in steady state k E B L k g A g B n g E 0
E
•
Sustainable growth in steady state is realised if g B n g A 5
Ferrara, M.; Guerrini, L. (2009), p. 48.
Overview The Green Solow Model Comparative steady state analysis Conclusion and critiques
General Structure Setting up the model The balanced growth path The EKC
The EKC •
How do we derive an EKC ? We write aggregate emissions at any time t by g E t
E (t ) 0 B0 L0 (1 )k (t ) e initial conditions
Differentiate with respect to time yields Percentage rate of change of emissions
E E
g E
~ k
k
g E s(1 )k 1 ( g B n )
~ with y y k k
Overview The Green Solow Model Comparative steady state analysis Conclusion and critiques
General Structure Setting up the model The balanced growth path The EKC
The EKC – sustainable growth g E < 0
E E
k (0) (0) < k (T ) < k *
s1 k 1 T
growth rate of aggregate emissions Ė/E is at first positive but turns negative in finite time
B
k (T ) k (T ) < k (0) (0) < k *
g Β n δ g E
Level of emissions
g Β n δ k K / BL
k
growth rate of aggregate emissions Ė/E is negative for all times t ; same along the balanced growth path
k (T )
k
k K / BL
Overview The Green Solow Model Comparative steady state analysis Conclusion and critiques
General Structure Setting up the model The balanced growth path The EKC
The EKC – unsust unsustainable ainable growth g E > 0
E E
k (0) (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 growing along the balanced growth path
s1 k 1 B
g Β n δ T
Level of emissions
k (T )
k (T )
k
k
g Β n δ g E k K / BL
k K / BL
Overview The Green Solow Model Comparative steady state analysis Conclusion and critiques
General Structure Setting up the model The balanced growth path The EKC
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) (0), Ω(0), B(0) and L(0) determine the model also; consequently we cannot conclude from emissions to income levels examination of Now examination
the emission’s peak T
Overview The Green Solow Model Comparative steady state analysis Conclusion and critiques
General Structure Setting up the model The balanced growth path The EKC
The EKC – peak of aggregat aggregate e emissions T •
E
~ k
g E 0 The amount of capital at T can be found by solving E k for k which yields 1
1 g n g / B E
k (T ) •
s1
How long does it take to reach peak emissions T ? We rewrite rewrite the process of capital accumulation as a function of time: 1
(1 ) t k (0) (1 ) e t 1 k (t ) k 1 e
where k (t ) is an exponentially exponentially weighted average of the economy’s initial capital per capita k (0) (0) and its steady state level k *, the speed of adjustment (1 ) g B n 0
Overview The Green Solow Model Comparative steady state analysis Conclusion and critiques
General Structure Setting up the model The balanced growth path The EKC
The EKC – peak of aggregat aggregate e emissions T •
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 k k (0) (1 ) T ln ( 1 ) (1 ) k k ( T )
Comparative stability Calendar time T needed to reach peak emissions …
T / 0
… is declining in the speed of convergence
(1 ) T / k (0) (1 ) 0 k (1 ) T / k k (T ) (1 ) 0
… is increasing in the gap (0) between k * and k (0) … is increasing the closer T and B are located to each other
Overview The Green Solow Model Comparative steady state analysis Conclusion and critiques
General Structure Setting up the model The balanced growth path The EKC
The EKC – peak of aggregat aggregate e emissions T •
Consequently, peak emissions given by E (t ) with t = T : g E T
E (T ) 0 B0 L0 (1 )k (T ) e
s(1 ) T (1 ) T 1 g E T 1 e e e 0 B0 L0 (1 ) k (0) g B n
(1 )
k
y (T )
It is apparent from E (T ) that if we compare two economies with same exogeneous parameters (s, g B, n, δ, α, and thus same k *), *), these economies will neither share the same income per capita nor the same peak level of emissions
Overview The Green Solow Model Comparative steady state analysis Conclusion and critiques
General Structure Setting up the model The balanced growth path The EKC
The EKC – peak of aggregat aggregate e emissions T •
When exactly exactly is the level of peak emissions E (T ) reached ? Examining the growth rate of emissions again ~ B L k (t ) E B L k (t )
E
~ (t ) k 0 g A g B n k (t )
g y ,t
with g y ,t 0 g y ,t 0
here it becomes apparent that the changing rage of capital accumulation is generating the dynamics of the Green Solow Model. The only effect effect of ~ k (t ) / k (t ) is its impact on growth of output per capita g y, thus g y,t changes over time. In the long run lim g y ,t g y t
Overview The Green Solow Model Comparative steady state analysis Conclusion and critiques
General Structure Setting up the model The balanced growth path The EKC
The EKC – peak of aggregat aggregate e emissions T Mechanics
growth rates
g y,t g B n
~ k (t )
k (t )
t < T : g y,t > g A and g E > 0
g A
t = T : g y,t = g A and g E = 0
g y
t > T : g y,t < g A and g E < 0
Level of emissions
T
which again represents represents our sustainability assumption Figure following Stefanski, Stefanski, R. (2010), p. 6.
t
E (t )
T
t
Overview The Green Solow Model Comparative steady state analysis Conclusion and critiques
General Structure Setting up the model The balanced growth path The EKC
The EKC – peak of aggregat aggregate e emissions T Characterisation of the EKC profile t < T :
g y ,t g A g E 0 Rising total emissions because abatement is not enough to outweigh extra pollution caused by faster growth of GDP
t = T :
g y ,t g A g E 0 g y ,t g A Peak emissions are reached when the rate at which emissions are created via output growth g y,t are exactly offset by the rate at which they are abated g A
t > T :
g y ,t g A g E 0 Falling growth of emission because improvements in emission intensity Ω outweigh the additional pollution created by production
Overview The Green Solow Model Comparative steady state analysis Conclusion and critiques
General Structure Setting up the model The balanced growth path The EKC
The EKC – relation of aggregate emissions and output •
How are aggregate emissions E (t ) related related to output y(t ) ? y
y y (t )
t
t ( y )
t
( y ) 0
y
Substituting t ( y ) into E (t ) we obtain E [ϕ( y)]:
s(1 ) (1 ) ( y ) 1 g E ( y ) ( y ) 1 e E [ ( y )] 0 B0 L0 (1 ) e e k (0) g B n
which represents a parametric relationship between aggregate emissions and income per capita that we refer refer to as an a n EKC
Overview The Green Solow Model Comparative steady state analysis Conclusion and critiques
General Structure Setting up the model The balanced growth path The EKC
The EKC – relation of aggregate emissions and output •
The derivative of E E [ϕ( y)] with respect to output per capita y yields:
growth rates
g y ,t
g A
E [ ( y )] E [ ( y)] ( y ) y 0 for t T 0 for t T 0 for t T
g y Level of emissions
y
y(T )
which too represents the EKC we derived previously. E [ ( y )] y y(T )
Overview The Green Solow Model Comparative Comparativ e steady state analysis Conclusion and critiques
Initial conditions Savings rate Abatement intensity Technological progress in abatement
Comparative steady state analysis g E t
E (t ) 0 B0 L0 (1 )k (t ) e
s(1 ) t (1 ) t 1 g E t 1 e e e 0 B0 L0 (1 ) k (0) g B n
(1 )
k
Lower initial conditions Ω(0), B(0), L(0) and k (0) (0) : • •
y (t ) Level of emissions
Affect E (t ) and y(t ) directly But no impact on steady state state magnitudes of k * and y* nor on long run growth rates
k (T )
k
k K / BL
Overview The Green Solow Model Comparative Comparativ e steady state analysis Conclusion and critiques
Initial conditions Savings rate Abatement intensity Technological progress in abatement
Comparative steady state analysis Increase in savings rate s : • •
• •
•
E E
Faster Faster capital accumulation Magnitudes of k (T ), k * and y(k ) increase Impact on pollution path Thus calendar time T increases, i.e. the Level of emissions process slows down In steady state growth rate rate of emissions and income per capita remain unchanged
s1 k 1 T
g Β n δ g E B
k (T )
k (T )
g Β n δ k K / BL
k
k
k K / BL
Overview The Green Solow Model Comparative Comparativ e steady state analysis Conclusion and critiques
Initial conditions Savings rate Abatement intensity Technological progress in abatement
Comparative steady state analysis Increase in abatement abatement intensity θ , e.g. due to tighter environmental policy: •
•
• •
•
E E
Slows down capital accumulation via smaller I Magnitudes of k (T ), k * and y(k ) decrease Level of Impact on pollution path emissions Thus calendar time T shortens In steady state growth rate rate of emissions and income per capita remain unchanged
s1 k 1 T
g Β n δ g E B
k (T )
k (T )
g Β n δ k K / BL
k
k
k K / BL
Overview The Green Solow Model Comparative Comparativ e steady state analysis Conclusion and critiques
Initial conditions Savings rate Abatement intensity Technological progress in abatement
Comparative steady state analysis Increase in abatement abatement intensity θ , e.g. due to tighter environmental policy:
E E
Although θ increases g A remains constant, i.e. tighter environmental policy cannot turn an Level of unsustainable economy emissions in a sustainable one Emission reduction is obtained by a decrease in k and y, NOT because of increasingly effective abatement
s1 k 1 T
g Β n δ g E B
k (T )
k (T )
g Β n δ k K / BL
k
k
k K / BL
Overview The Green Solow Model Comparative Comparativ e steady state analysis Conclusion and critiques
Initial conditions Savings rate Abatement intensity Technological progress in abatement
Comparative steady state analysis Increase in technological progress at abatement g A:
E E
s1 k 1 g Β n δ g E
T • •
• •
•
g E = g B + n ‒ g A decreases Magnitudes of k (T ), and y(T ) decrease, but k * remains unchanged Impact on pollution path Level of emissions Thus calendar time T decreases In steady state growth rate of emissions decreases while growth rate rate of income per capita remain unchanged
B
k (T )
k (T )
g Β n δ k K / BL
k
k
k K / BL
Overview The Green Solow Model Comparative steady state analysis Conclusion and critiques
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 affect E (t ) and y(t ) directly but not in steady state state or in the long run b) savings rate s accelerates capital accumulation while abatement abatement intensity θ does not create sustainable economies and emission reduction is not obtained by increasingly effective abatement abatement c) g A decreases g E , while g B + n increases steady state growth rate of emissions Putting this together, EKC has shown, as countries develop economically, moving from lower to higher levels levels of income per capita, levels of environmental degradation – such as pollution – will eventually fall Empirical examination examination of cross-country data has verified verified this relationship for certain pollutants
χ for für ThanDank your attention! Vielen fü r ihre Aufmerksamkeit Aufmerksamkeit !
Appendix
A contemporary EKC for Germany Speed of adjustment Comparative steady state state analysis with respect to g_B or n Literature
A contemporary EKC for Germany (1950 ̶ 2006) Germany Pollutant Emissions by GDP pc 200 s l e v e L n i a t i p a C r e P s n o i s s i m E
150
Sulfur Emissions / pc (kg)
100
CO2 Emissions / pc (t)
50
0
7 7 3 2 4 7 7 5 6 7 3 1 1 9 2 8 5 1 7 6 6 1 7 1 1 4 2 3 5 8 9 1 4 5 6 2 4 6 4 3 5 2 7 6 4 2 4 6 1 2 6 7 8 5 4 6 6 0 1 9 4 2 6 3 6 5 8 9 3 9 0 9 8 7 6 2 5 3 5 2 3 0 5 0 2 3 0 6 9 7 1 0 4 1 6 2 5 1 8 1 8 8 9 0 1 8 1 7 6 9 6 1 9 3 2 0 3 0 4 6 4 6 5 9 1 9 2 9 6 1 3 5 6 6 4 2 0 6 3 4 7 1 4 4 7 2 2 1 8 8 9 5 0 6 3 8 3 5 2 7 6 4 9 6 8 8 1 6 6 4 1 4 8 4 8 5 2 8 2 8 2 4 2 6 7 3 9 7 2 6 0 9 8 8 2 6 7 9 2 9 1 6 1 3 8 2 8 5 8 6 2 4 5 0 4 4 5 6 8 5 5 7 9 0 3 6 0 5 , , , , , , 3 3 1 1 4 7 8 6 , 0 8 , 5 , 9 , 3 , 8 , 4 , 8 , 4 5 , 3 , 6 , 6 , 4 , 1 , 0 , 1 , 2 , 7 , 5 , 3 9 , 5 , 8 , 2 , 5 , 5 , 3 , 3 , 7 5 , 1 , 0 , 6 , 1 , 0 , 3 , 9 6 , 9 , 4 , 8 , 1 6 , 8 , 3 , 8 , 9 , 0 , 0 , 9 , 1 , 2 , 4 , 6 6 , 7 1 1 1 8 2 8 4 4 1 5 9 3 9 2 6 8 9 4 0 2 4 8 8 2 1 5 8 5 4 9 9 3 6 7 5 1 0 5 6 8 9 2 6 1 5 2 6 1
6 000 000
10 000
20 000
30 000
0 9 4 7 3 9 2 8 7 6 9 7 0 5 8 1 3 4 7 8 0 8 1 7 2 9 9 1 1 5 4 0 3 8 9 7 7 3 5 8 9 2 0 1 2 4 3 8 2 4 2 6 3 5 2 9 9 4 9 3 0 9 1 5 2 0 4 9 1 3 4 7 7 4 3 0 3 0 7 9 8 9 5 1 9 1 2 0 4 1 3 2 6 3 9 9 1 5 6 7 1 3 7 8 8 7 0 0 9 0 0 5 1 6 2 9 6 0 0 1 2 2 2 3 1 4 4 4 5 6 7 7 8 8 8 8 1 0 1 1 2 2 2 2 3 2 4 4 5 5 4 6 6 2 6 7 7 7 2 8 9 0 0 9 3 6 6 7 7 8 9 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 2
1 5 1 3
GDP Per Capita
Figure 3 from Florian Hage, data from various sources one can find in the appendix
Appendix
A contemporary EKC for Germany Speed of adjustment Comparative steady state state analysis with respect to g_B or n Literature
The EKC – peak of aggregat aggregate e emissions T •
•
•
In order to investigate the peak of aggregate emissions we have to find T Economies with identical parameters but different different initial conditions converge over time t to the same balanced growth path ( β -convergence) β -convergence) We apply a TAYLOR-series-approximation of first order of the fundamental ~ equation k s1 k g B n k around the steady state to find the speed of adjustment ad justment β : ~ f (k ) k (1 ) g B n k k
β > 0
•
We find that β is independent from the abatement intensity θ and savings rate s
Appendix
A contemporary EKC for Germany Speed of adjustment Comparative steady state state analysis with respect to g_B or n Literature
Comparative steady state analysis Increase in technological progress at production g B or population growth n: • •
•
E E
g E = g B + n ‒ g A increases Magnitudes of k (T ), and y(T ) decrease, but k * remains unchanged Level of emissions Impact on pollution path and calendar time T depends on k (0) (0)
s1 k 1 g Β n δ g E
T B
k (T )
k (T )
g Β n δ k K / BL
k
k
k K / BL
Appendix
A contemporary EKC for Germany Speed of adjustment Comparative steady state state analysis with respect to g_B or n Literature
Literature •
•
•
•
•
•
•
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..
Appendix
A contemporary EKC for Germany Speed of adjustment Comparative steady state state analysis with respect to g_B or n Literature
Literature •
•
•
CO2-data: Complete reference 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: S TERN D. I. (2006). Reversal in the trend of Environmental Change. Volume 16: pp. global anthropogenic anthropogenic sulfur emissions . Global Environmental 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 differences in the cost of living between countries has been taken into account. Main sources: Cross-country data for for 2005 is mainly based on the 2005 round of the International Comparison Program.
