Growth in a Finite World - Sustainability and the Exponential Function

Growth in a Finite World
Sustainability and the Exponential Function

Sustainability and the Exponential Function
Toni Menninger
Graduate Assistant,
Department of Geosciences,
University of Arkansas
Presentation Spring 2011

Al Bartlett, author of “The Essential Exponential”:"The greatest shortcoming of the human race is our inability to understand the exponential function“http://www.albartlett.org/

Growth and Sustainability: a Systems Theory perspective
We depend on a finite planet
We extract material resources (renewable and nonrenewable)
We dump waste into the environment
Human activity is governed by the laws of nature

Growth and Sustainability
Systems view: The Human system (our material culture, society, technology, economy) is a “subsystem of a larger ecosystem that is finite, non-growing, and materially closed. The ecosystem is open with respect to a flow of solar energy, but that flow is itself finite and non-growing.” (Herman Daly)

The Human Sphere as a Subsystem of Planet Earth
Finite Global Ecosystem
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Resource consum-ption and waste disposal must be in balance with the earth’s ecological capacity.
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Recent history is characterized by a dramatic expan-sion of the human “footprint”
Thousands of years ago?
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Hundreds of years ago?
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Recycled
Matter
Have we reached the limits to growth?
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Famous 1972 report was an early application of computer aided systems modeling

http://www.footprintnetwork.org/en/index.php/GFN/page/world_footprint/
Ecological Footprint: by current estimates, we overuse the planet by 50% (footprintnetwork.org)

Dimensions of Growth: Fisheries
Source: FAO, 2004. http://earthtrends.wri.org/updates/node/140
Fisheries: five-fold increase since 1950

Dimensions of Growth: Raw Materials
http://pubs.usgs.gov/annrev/ar-23-107/
Raw material use in US: more than ten-fold increase since 1900

Dimensions of Growth: Energy
http://www.energyliteracy.com/?p=142
US electricity consumption: almost ten-fold increase since 1950

http://www.theenergysite.info/Markets_Demand.html
US primary energy consumption: more than ten-fold increase since 1900

World primary energy: at least twenty-fold increase since 1850, mostly fossil fuels

http://www.americanscientist.org/issues/id.6381/issue.aspx
World petroleum consumption: more than ten-fold increase since 1930

Dimensions of Growth: Passenger cars
http://www.mindfully.org/Energy/2003/Americans%20Drive%20Further-May03.htm
World passenger car fleet: more than ten-fold increase since 1950

Dimensions of Growth: Passenger cars
http://www.planetizen.com/node/41801
IMF projects quadrupling of car fleet by 2050

Dimensions of Growth: Greenhouse gas emissions
CO2 emissions: Ten-fold increase since 1900

Dimensions of Growth: Population
Population:
from 1 billion in 1800 to 7 billion in 2010

Quantifying growth
Two ways of looking at the growth of a quantity:
Linear change: by how many units has the quantity increased?
Fractional change: by what fractionor percentage has the quantity increased?

Quantifying growth
Linear change - Example: US Census
2010: 308.7 million people
Absolute increase:
increase = 308.7-281.4=27.3 million people over 10 years

Quantifying growth
Fractional change - Example: US Census
Ratio: 308.7/ 281.4= 1.097
Fractional (percent) increase: 9.7%
9.7%=(Ratio-1)*100%

Quantifying growth
Average Rates of Growth
Average yearly increase: increase/time= 27.3 million/10 years=2.73 million per year
Average yearly (annualized) (fractional) growth rate: ln(ratio)/time*100% =0.93%

Quantifying growth
Examples
US population in 1900=76.1 million
1900-2010: Increase of 232.6 million, or 2.1 million per year on average
Ratio=4.06, fractional increase =306%, average yearly growth rate =ln(4.06)/110*100=
1.3%

Quantifying growth
Examples
U.S. GDP (Gross Domestic Product) increased from $2.01 trillion in 1950 to $13.2 trillion in 2010 (*).
Average yearly growth rate =ln(6.57)/60*100%=
3.1%
(*) In 2005 chained dollars; source: bea.gov).

Quantifying growth
Examples
U.S. National Health Expenditures (NHE) increased from $307.8 billion in 1970 to $2,155.9 billion in 2008 (*).
That is a six-fold increase over 38 years. Average yearly growth rate =ln(7)/38*100%=
5.1%
(*) adjusted for inflation; source: Health Affairs.

Growth models
Linear (arithmetic) growth:
Constant increase per unit of time = straight line

Growth models
Exponential (geometric) growth: Constant fractional (percentage) increase per unit of time.
Initially, the population increases by a small amount per unit of time. As the population increases, the increase increases proportionally.
Exponential growth will eventually overtake any linear or polynomial growth function.

Growth models
Linear
Cubic
Exponential

Growth models
Examples of exponential growth:
Biological reproduction: organisms reproducing at regular generation periods will, under favorable condi-tions, multiply exponentially. Expo-nentialgrowth in a population occurs when birth and death rate are constant, and the former exceeds the latter.

Growth models
Examples of exponential growth:
Compound interest: the interest paid on a savings account is a fixed proportion of the account balance, compounded in fixed time intervals. If the real interest rate (corrected for inflation) remains constant, the account balance grows exponentially.

Growth models
In nature, no sustained material growth over long time periods has ever been observed.
-> Natural populations:
Logistic (sigmoid) growth After an initial phase of exponential growth, the growth rate slows as a threshold (Carrying Capacity) is approached. Population may stabilize or decline.

Growth models
In nature, no sustained material growth over long time periods has ever been observed.

Growth models
World population growth since 1950 has been highly exceptional.

Growth models
World population growth since 1950 has been exceptional. Rates are declining but still historically high.

Mathematics of exponential growth
Exponential growth=constant fractional growth rate
Doublesin a fixed time period, called the Doubling Time T2. The doubling time can be estimated by dividing 70 by the percent growth rate.

Doubling Time = 70 over percent growth rate

Doubles in a fixed time period, the Doubling Time T2. The doubling time can be estimated by dividing 70 by the percent growth rate.
7% yearly growth: T2 = 70/7 = 10 years
After 20 years: 300% increase
After 30 years: 700% increase
After 40 years: 1,500% increase
…

Exponential growth: doubles in a fixed time period.
Doubles again after the next doubling time.
After N doubling times have elapsed, the multiplier is 2 to the Nth power - 2N!
2->4->8->16->32->64->128 ->256->512->1024

The power of the powers of 2
After N doubling times have elapsed, the multiplier is 2 to the Nth power - 2N!
2->4->8->16->32->64->128->256->512->1024
The 10th power of 2 is approx. 1,000.
The 20th power of 2 is approx. 1,000,000.
The 30th power of 2 is approx. 1,000,000,000.

Examples
Current US population growth rate 0.9%
-> Doubling time = 70/0.9= 78 years
World population growth rate about 1.2% -> Doubling time = 58 years

Examples
Northwest Arkansas: 424,400 people in 2010 versus 311,100 in 2000 – an increase of 36%
Annualized growth rate: 3.1%
Doubling time: 23 years

Examples
Doubling time can also be used to estimate the growth rate.
US population quadrupled in 110 years.
-> Doubling time=55 years
-> Annual growth rate approx. 70/55=1.3%, as calculated before.

Examples
GDP growth since 1950: 3.1% per year
-> Doubling time = 23 years
-> After 69 years, 7-fold increase.
7 times more consumption, resource use, pollution, etc.?
Health expenditures: 5.1% per year
-> Doubling time = 14 years!

Conclusions
Knowing growth rates and doubling times helps better understand environmental, social, and economic challenges.
Exponential growth becomes unsustainable very quickly. In the real world, exponential growth processes are unusual and don’t last long.

Questions?
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Growth in a Finite World - Sustainability and the Exponential Function

by Toni Menninger

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