Growth in a Finite

World

Sustainability and the Exponential

Function

Growth in a Finite World

Sustainability and the Exponential

Function

Lecture Series in Sustainability

Science

by

Toni Menninger MSc

http://www.slideshare.net/amenning/ toni.menninger@gmail.com

Growth in a Finite World 1. Growth and Sustainability: A systems theory perspective

The Human sphere as a subsystem of the ecosphere

2. Dimensions of Growth – a historical perspective

3. Quantifying Growth

• Actual (absolute) change

• Fractional (relative) change

• Average rates of growth

4. Growth Models

• Linear growth

• Exponential growth

• Logistic growth

Growth in a Finite World 5. Exponential Growth

• Doubling Time

• Rule of 70

• The power of the powers of 2

• The Logarithmic Plot

6. Summary

7. Further Readings

8. Appendix: The Mathematics of Exponential Growth

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/

Al Bartlett, author of

“The Essential

Exponential”

Dr. Al Bartlett (1923-2013), a physics professor

at the University of Colorado at Boulder since

1950, dedicated much of his career to educating

the public about the implications of exponential

growth. A video of his presentation “Arithmetic,

Population and Energy” is available online.

http://www.albartlett.org/

Growth and

Sustainability: a

Systems Theory

perspective

Growth and Sustainability: a

Systems Theory perspective

• What is growth, and why do

we need to think about it?

• What comes to mind when

you think of “growth”?

• Are there limits to growth?

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, a founder of Ecological Economics)

Growth and Sustainability: a

Systems Theory perspective

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.

• We rely on ecosystem services (e. g. clean

water, waste decomposition).

• Human activity is governed and constrained

by the laws of nature (e. g. conservation of

energy, material cycles).

The Human system (Anthroposphere) is a

“subsystem of a larger ecosystem that is finite,

non-growing, and materially closed”.

• The Anthroposphere has been expanding for

thousands of years. This expansion is driven by

several factors such as population, consumption

or affluence, and technological change (“I=PAT”

equation).

• A subsystem of a materially closed system

cannot materially grow beyond the limits of the

larger system: an equilibrium must be reached.

Growth and Sustainability: a

Systems Theory perspective

The Human Sphere as a Subsystem of

the Ecosphere

G r o w i n g E c o n o m i c S u b s y s t e m

R e c y c l e d M a t t e r

E n e r g y

R e s o u r c e s

E n e r g y

R e s o u r c e s

S o l a r E n e r g y

W a s t e H e a t

Sink Functions

Source Functions

Finite Global Ecosystem

(After Robert Costanza,

Gund Institute of Ecological Economics)

Resource consum-

ption and waste

disposal must be in

balance with the

earth’s ecological

capacity.

S o l a r E n e r g y

Finite Global Ecosystem

Recent history is

characterized by a

dramatic expansion

of the human

“footprint”

Thousands of years ago:

“Empty world” W a s t e H e a t

The Human Sphere as a Subsystem of

the Ecosphere

The Human Sphere as a Subsystem of

Planet Earth

G r o w i n g E c o n o m i c S u b s y s t e m

R e c y c l e d M a t t e r

E n e r g y

R e s o u r c e s

E n e r g y

R e s o u r c e s

S o l a r E n e r g y

W a s t e H e a t

Sink Functions

Source Functions

Finite Global Ecosystem

Recent history is

characterized by a

dramatic expansion

of the human

“footprint”

Hundreds of years ago?

G r o w i n g E c o n o m i c

S u b s y s t e m

Recycled Matter

R e s o u r c e s

S o l a r E n e r g y

W a s t e H e a t

E n e r g y E n e r g y

R e s o u r c e s

Sink Functions

Source Functions

Finite Global Ecosystem

Dramatic expansion of

the human footprint:

Humanity takes up an

ever increasing share of

the global ecosystem,

causes planetary scale

environmental change

“Full world”: Have we reached

the limits to growth?

The Human Sphere as a Subsystem of

the Ecosphere

Famous 1972 report was an early application

of computer aided systems modeling

The Human Sphere as a Subsystem of

the Ecosphere

Ecological Footprint: by current estimates, we

overuse the planet by 50% (footprintnetwork.org)

http://www.footprintnetwork.org/en/index.php/GFN/page/world_footprint/

The Human Sphere as a Subsystem of

the Ecosphere

Dimensions of

Growth

How human impact

has multiplied since the

industrial revolution

Dimensions of Growth: Raw Materials

Raw material use in US: more than ten-fold

increase since 1900

http://pubs.usgs.gov/annrev/ar-23-107/

Dimensions of Growth: Cement Production

World cement production:

50-fold increase since 1926

U.S. Geological Survey Data Series 140

Dimensions of Growth: Copper Production

World Copper production:

50-fold increase since 1900

U.S. Geological Survey Data Series 140

Dimensions of Growth: Fisheries

Fisheries: six-fold increase since 1950

Source: FAO, 2004. http://earthtrends.wri.org/updates/node/140

Dimensions of Growth: Fertilizer

Nitrogen fertilizer: nine-fold increase since 1960

Source: UNEP 2011. https://na.unep.net/geas/getUNEPPageWithArticleIDScript.php?article_id=81

Dimensions of Growth: Energy

US electricity consumption: almost ten-fold

increase since 1950

http://www.energyliteracy.com/?p=142

Dimensions of Growth: Primary Energy

US primary energy consumption: more than

ten-fold increase since 1900

http://www.theenergysite.info/Markets_Demand.html

World primary energy: twenty-fold increase

since 1850, mostly fossil fuels

Dimensions of Growth: Primary Energy

World petroleum consumption: more than ten-

fold increase since 1930

Dimensions of Growth: Petroleum

http://www.americanscientist.org/issues/id.6381/issue.aspx

World passenger car fleet: more than ten-fold

increase since 1950

Dimensions of Growth: Passenger cars

http://www.mindfully.org/Energy/2003/Americans%20Drive%20Further-May03.htm

IMF projects further quadrupling of world wide

car fleet by 2050

Dimensions of Growth: Passenger cars

http://www.planetizen.com/node/41801

Dimensions of Growth: Greenhouse gas

emissions

CO2 emissions: Ten-fold increase since 1900

Dimensions of Growth: Population

World Population:

from 1 billion in 1800 to 7 billion in 2012

0

1,000,000,000

2,000,000,000

3,000,000,000

4,000,000,000

5,000,000,000

6,000,000,000

7,000,000,000

1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

Global population since AD 1000

Dimensions of Growth: A historical

perspective

• Humans have always manipulated their

environment to extract resources and create

favorable conditions.

• The scale of human impact on the ecosphere has

vastly increased, especially since the industrial

revolution: we are causing planetary scale

environmental change, notably alteration of

atmospheric composition, climate change, alteration

of global material cycles (nitrogen, carbon, water),

mass species extinction, large-scale alteration of

vegetation cover, …

Dimensions of Growth: A historical

perspective

• Humans have always manipulated their

environment to extract resources and create

favorable conditions.

• The scale of human impact on the ecosphere has

vastly increased, especially since the industrial

revolution.

• Growth strategies that were successful in an “empty

world” are unsustainable in today’s “full world”.

• Today’s socio-economic institutions are still shaped

by the “growth” paradigm of the past. “Sustainable

growth” has become a buzzword yet it is unclear

what it means.

Quantifying growth

Learn how to calculate and

interpret

• Actual (absolute) change

• Fractional (relative) change

• Average rates of change

Quantifying growth

Two ways of looking at the growth of a

quantity:

• Actual (absolute) change: by how

many units has the quantity

increased?

• Fractional (relative) change: by

what fraction or percentage has the

quantity increased?

Quantifying growth

Absolute change - Example: US

Census

2000: 281.4 million people

2010: 308.7 million people

Increase = 308.7m - 281.4m =

27.3 million people over 10 years

Quantifying growth

Relative change - Example: US

Census

2000: 281.4 million people

2010: 308.7 million people

Ratio: Find the solution together with your neighbor

Fractional (percent) increase: Find the solution together with your neighbor

Quantifying growth

Relative change - Example: US

Census

2000: 281.4 million people

2010: 308.7 million people

Factor of increase =

Ratio of final value to initial value =

308.7/ 281.4 = 1.097

Fractional increase:

(ratio-1)*100% = 9.7%

Quantifying growth

Note on language use

When the price of a product increases from $10 to $30, we

can say the price has increased by the factor 3 (the ratio of

new price to old price), it has tripled, a three-fold increase,

or it has increased by 200%.

When we refer to a “percent increase” or “fractional

increase”, we always mean the difference between new

value and base value (initial value) divided by the base value:

percent increase =

(new value – base value) / base value*100 =

(new value / base value - 1) * 100.

Quantifying growth

Actual versus fractional change

In many contexts, fractional change is the more

useful concept because it allows to quantify

change independently of the base level. Only

so is it meaningful to compare the rate of growth

of different entities (e. g. different countries,

different sectors of the economy). Socio-

economic indicators are often reported as

fractional rates of change: GDP, consumer

spending, the stock market, home prices, tuition

…

Quantifying growth

Average Rates of Growth

To make comparison between different time

periods meaningful, growth rates must be

averaged (usually annualized).

Year Population Increase Fractional

increase

1900 76.1 m

2000 281.4 m 205.3 m 170%

2010 308.7 m 27.3 m 9.7%

Example: US Census

Quantifying growth

Average Rates of Growth: Absolute

Average yearly increase:

𝒊𝒏𝒄𝒓𝒆𝒂𝒔𝒆 𝒐𝒗𝒆𝒓 𝒕𝒊𝒎𝒆 = 𝒇𝒊𝒏𝒂𝒍 𝒗𝒂𝒍𝒖𝒆 – 𝒃𝒂𝒔𝒆 𝒗𝒂𝒍𝒖𝒆

𝒕𝒊𝒎𝒆

Year Population Increase Avg yearly

increase

1900 76.1 m

2000 281.4 m 205.3 m 2.05 m

2010 308.7 m 27.3 m 2.73 m

Example: US Census

Quantifying growth

Average Rates of Growth: Fractional

Average percent growth rate:

𝒈𝒓𝒐𝒘𝒕𝒉 𝒓𝒂𝒕𝒆 = ln 𝒇𝒊𝒏𝒂𝒍 𝒗𝒂𝒍𝒖𝒆/𝒃𝒂𝒔𝒆 𝒗𝒂𝒍𝒖𝒆

𝒕𝒊𝒎𝒆× 𝟏𝟎𝟎%

Take the natural logarithm of the ratio (quotient)

between final value and base value, divide by the

number of time units, and multiply by 100.

The average growth rate is measured in inverse time units,

often in percent per year. The annual growth rate is often

denoted p. a. = per annum.

Quantifying growth

Average Rates of Growth

Average yearly (annualized) percent growth rate:

𝒈𝒓𝒐𝒘𝒕𝒉 𝒓𝒂𝒕𝒆 = ln 𝒇𝒊𝒏𝒂𝒍 𝒗𝒂𝒍𝒖𝒆/𝒃𝒂𝒔𝒆 𝒗𝒂𝒍𝒖𝒆

𝒕𝒊𝒎𝒆× 𝟏𝟎𝟎%

Example: US Census

Year Population in

million

Fractional

increase

Ratio final/

base value

Avg. growth

rate per year

1900 76.1 m

2000 281.4 m 270% 3.70 1.3%

2010 308.7 m 9.7% 1.097 0.9%

Quantifying growth

Average Rates of Growth

To make comparison between different time

periods meaningful, growth rates must be

averaged (usually annualized).

Year Population in

million

Avg yearly

increase

Avg. growth

rate per year

1900 76.1 m

2000 281.4 m 2.05 m 1.3%

2010 308.7 m 2.73 m 0.9%

Example: US Census

Why has absolute

growth increased

but fractional

growth declined?

Quantifying growth

Example: GDP

U.S. GDP (Gross Domestic Product)

quintupled from $3.1 trillion in 1960 to $15.9

trillion in 2013 (*).

Average growth rate

= ln(15.9 / 3.1)/ 53 *100% = 3.1% p. a. (p. a. means per annum = per year)

(*) In 2009 chained dollars (adjusted for inflation);

source: U.S. Bureau of Economic Analysis).

Quantifying growth

Example: Per capita GDP

US GDP growth 1960-2012: 3.1% p. a.

US population growth 1960-2012:

180m to 314m 1.1% p. a.

Per capita GDP growth:

3.1% - 1.1% = 2.0% p. a.

Quantifying growth

Example: Healthcare cost

U.S. National Health Expenditures (NHE)

increased from $307.8 billion in 1970 to

$2,155.9 billion in 2008 (*).

That is a seven-fold increase over 38 years.

Average growth rate

= ln(7) / 38 * 100%= 5.1% p. a.

(*) figures include private and public spending,

adjusted for inflation; source: Health Affairs.

Quantifying growth

Example: Primary Energy

Between 1975 and 2012, World

Primary Energy use increased 116%.

Average growth rate

= ln(2.16) / 37 * 100%= 2.1% p. a.

Exercise: Calculate per capita growth

Source: BP Statistical Review of World Energy.

Growth models

• Linear growth

• Exponential growth

• Logistic growth

Growth models

Linear (arithmetic) growth:

Constant increase per unit of time = straight line

The Lily Pond Parable

If a pond lily doubles its leaf

area every day and it takes

30 days to completely cover a

pond, on what day will the

pond be 1/2 covered?

Discuss with your neighbor

Growth models

Exponential (geometric) growth

• 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 grows proportionally.

• Exponential growth will eventually

overtake any linear or polynomial

growth function.

Growth models

• Linear

• Cubic

• Exponential

Growth models

Exponential (geometric) growth

Growth models

Exponential (geometric) growth will eventually

overtake any power (or polynomial) function.

• Cubic

• 10th power

• Exponential

Examples of exponential growth:

• Biological reproduction: organisms

reproducing at regular generation

periods will, under favorable condi-

tions, multiply exponentially. Expo-

nential growth 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

Learn to understand and apply

• The doubling time

• The Rule of 70

• The power of the powers of 2

• Logarithmic plots

Exponential Growth

Continuous exponential growth of a quantity N

over time t at constant fractional rate p is

described by the exponential function

𝑵 𝒕 = 𝑵𝟎𝒆𝒑𝒕 or 𝑵𝟎exp (𝒑𝒕)

The growth rate can be calculated as

𝒑 = ln 𝑵(𝒕)/𝑵𝟎

𝒕

This is the same as the average growth rate

formula introduced earlier. Refer to full

mathematical treatment in the appendix.

Exponential Growth

Exponential growth characterized by:

• Constant fractional growth rate

• Doubles in a fixed time period,

called the Doubling Time T2.

• “Rule of 70”: The doubling time

can be estimated by dividing 70 by

the percent growth rate.

(Why? Because 100*ln 2=69.3. Refer to appendix.)

Exponential Growth

“Rule of 70” When steady exponential growth

occurs, the doubling time can be

estimated by dividing 70 by the

percent growth rate p:

𝑻𝟐 ≈ 𝟕𝟎 / 𝒑

Conversely, 𝒑 ≈ 𝟕𝟎 / 𝑻𝟐

Exponential Growth

Doubling Time =

70 over percent

growth rate

Growth

rate in %

1 1.4 2 3 3.5 4 7 10

Doubling

time

70 50 35 23 20 17.5 10 7

Exponential Growth

Note on Exponential Decay

Everything said about exponential growth

also applies to exponential decay, where

the “growth” rate is negative. While

sustained exponential growth does not

seem to occur in nature, exponential decay

does: radioactive decay, for example.

Instead of a doubling time, we now refer to

the half-life of a decay process.

Exponential Growth

• US population 1900-2010:

Average growth rate 1.3% p.a.

Doubling time = 70/1.3 = 55 years

• Current US population growth rate 0.8%

Doubling time = 70/0.8 = 78 years

Doubling in 78 years will occur only IF current

growth rate remains constant!

Doubling time: Examples

Exponential Growth

The doubling time can conversely be

used to estimate the growth rate:

• US population quadrupled in 110 years

Two doublings

Doubling time = 55 years

Annual growth rate ≈ 70/55=1.3%

(as calculated before).

Exponential Growth

• US economic growth since 1960

averaged 3.1% per year

Doubling time = 70/3.1 = 23 years

• Health expenditures 1970-2008:

5.1% growth per year

Doubling time = 14 years

Discuss: Can these trends continue?

Doubling time: Examples

Exponential Growth

Health expenditures: 5.1% growth per year

Doubling time = 14 years!

• What happens when one economic sector grows

faster than the overall economy?

Health care system share of GDP increased

from 5% in 1960 to 17.2% in 2012

• What happens when a subsystem grows faster

than the overall system? Can the trend continue?

Doubling time: Examples

Exponential Growth

Exponential growth implies a

fixed Doubling Time T2.

What does

this mean? Example: 7% growth

• 7% yearly growth: T2 = 70/7 = 10 years

• After 10 years: x2 (100% increase)

• After 20 years: x4 (300% increase)

• After 30 years: x8 (700% increase)

• After 40 years: x16 (1,500% increase)

• … How much after 100 years?

Exponential Growth

• 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

Exponential Growth

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.

Exponential Growth

Exercise: how many doublings has the human

population undergone?

Make a guess!

Try to estimate:

• Initial population: minimum 2 (not to be taken literally)

• Current population: 7 billion

• 1 billion is about … doublings

• Fill in the details.

The power of the powers of 2

Exponential Growth

Exercise: how many doublings has the human

population undergone?

Estimate:

• Initial population: minimum 2 (not to be taken literally)

• Current population: 7 billion

• 1 billion is about 30 doublings

• 2x2x2=8

Answer: at most 32 doublings.

• What did you guess?

• How many more doublings can the earth support?

The power of the powers of 2

Exponential Growth

Growth over a life time

A human life span is roughly 70 years. What are

the consequences of 70 years of steady growth at

an annual rate p%?

The doubling time is T2 =70/p, so exactly p

doublings will be observed.

So the multiplier over 70 years is 2p.

The power of the powers of 2

Exponential Growth

Growth over a life time

A human life span is about 70 years. What are the

consequences of 70 years of steady growth at an

annual rate p%? The multiplier over 70 years is 2p.

Example p=3%: Multiply by 23 = 8.

3% per year is often considered a moderate rate of

growth (e. g. in terms of desired economic growth)

yet it amounts to a tremendous 700% increase

within a human life span.

The power of the powers of 2

Exponential Growth

Exercise: The consequences of 3.5% p. a.

steady exponential growth

• What is the doubling time?

• How long will it take to increase four-fold, sixteen-fold.

1000-fold?

• Make a guess first, then work it out using the rule of 70!

The power of the powers of 2

Exponential Growth

The consequences of a 3.5% growth rate:

• Doubling time: T2 = 20 years

• 200 years = 10 x T2 corresponds to a multiplier of 1000.

• 200 years may seem long from an individual perspective

but is a short period in history.

• 200 years is less than the history of industrial society,

and less than the age of the United States.

• The Roman Empire lasted about 700 years.

• Sustainability is sometimes defined as the imperative to

“think seven generations ahead” – about 200 years - in

the decisions we make today.

The power of the powers of 2

Exponential Growth

The consequences of a 3.5% growth rate:

• Doubling time: T2 = 20 years

• 200 years = 10 x T2 corresponds to a multiplier of 1000.

Some economic models assume a long term growth rate

on the order of 3-4% p.a.. Can you imagine the economic

system to grow 1000 fold? What would that mean?

• 1000 times the cars, roads, houses, airports, sewage

treatment plants, factories, power plants?

• 1000 times the resource use and pollution?

• What is it that could/would/should grow 1000 times?

The power of the powers of 2

Exponential Growth

The consequences of 3.5% yearly growth:

• Doubling time: T2 = 20 years

• 200 years = 10 x T2 corresponds to a multiplier of 1000.

Our difficulty in grasping the long term

consequences of seemingly “low” to “moderate”

exponential growth is what Al Bartlett referred to

as humanity’s “greatest shortcoming”.

The power of the powers of 2

Exponential Growth

Caution!

Extrapolating current growth trends into the

future (for example using doubling times) is

usually not permissible because trends change

over time. Doubling times are only indicative of

what would happen if the trend continued. It

would be questionable to base policy decisions

on such trends – although this is often done.

Example: Population growth rates have changed

dramatically over time.

Exponential Growth

How can we recognize exponential

growth in a time series?

• Inspect the data?

• Analyze the data?

• Inspect the graph?

Exponential Growth

How can we recognize exponential

growth in a time series?

• Inspect the data

Example: Census population of Georgia

This data series is roughly consistent with

exponential growth, doubling time 35-40 years.

Year 1960 1970 1980 1990 2000 2010Pop. in million 3.9 4.6 5.4 6.5 8.2 9.7

Year 1960 1970 1980 1990 2000 2010

Population

in million

3.9 4.6 5.4 6.5 8.2 9.7

Exponential Growth

How can we recognize exponential

growth in a time series?

• Analyze the data

Example: Census population of Georgia

Fractional growth rates are fairly consistent, with the

1990s somewhat higher.

Year 1960 1970 1980 1990 2000 2010Pop. in million 3.9 4.6 5.4 6.5 8.2 9.7

Year 1960 1970 1980 1990 2000 2010

Population

in million

3.9 4.6 5.4 6.5 8.2 9.7

% increase 18% 17% 20% 26% 18%

Exponential Growth

How can we recognize exponential

growth in a time series?

• Inspect the graph?

Caution! It is difficult to judge growth rates

from the appearance of a graph.

Example: population growth

Exponential Growth

0

1,000,000,000

2,000,000,000

3,000,000,000

4,000,000,000

5,000,000,000

6,000,000,000

7,000,000,000

1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

Global population since AD 1000

Steady exponential growth?

Population Growth

0

0.5

1

1.5

2

1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

Global population fractional growth rates, AD 1000 to present

Steady exponential growth? No. Growth

rates have changed dramatically over time!

Population Growth

How can we recognize exponential

growth in a time series?

• Inspect the graph?

We need to plot the data to logarithmic

scale. This makes an exponential function

appear as a straight line. The slope

corresponds to the growth rate.

The Logarithmic Scale

The semi-logarithmic plot makes an exponential function

appear as a straight line (red line). The slope corresponds

to the growth rate. A function that grows slower than

exponential gives a concave graph (green line).

It is easy to

change to

logarithmic

scale in a

spreadsheet

software.

The Logarithmic Scale

Example: Global Population

The slope in a

semi-logarithmic

plot corresponds

to the growth

rate.

Can you identify

distinct phases

of population

growth?

The Logarithmic Scale

Population growth

accelerated with the

onset of the industrial

revolution and reached

a peak about 1970.

Population growth was

hyper- (faster than)

exponential in that

period (evident in the

semi-logarithmic graph

being convex ).

Since 1970, the rate of

growth is in decline but

still exceptionally high

by historical standards.

The Logarithmic Scale

In a semi-logarithmic plot, the slope corresponds to the

growth rate.

Example: Primary Energy Use since 1975

Here, a trend line

was fitted to show

that the data are

consistent with

exponential growth

at about 2.% p. a.

Data source: BP Statistical Review of World Energy

The Logarithmic Scale

In a semi-logarithmic plot, the slope corresponds to the

growth rate.

Example: Economic Growth since 1970

North America,

East Asia, World

Where are growth

rates highest?

Are data consistent

with exponential

growth (straight

line)? Are trends

changing?

Plotted logarithmically using the Google Public Data explorer

The Logarithmic Scale

Which of our case studies exhibit

sustained exponential growth?

• Global Population: no, growth rates are falling

and stabilization is expected by mid or late 21st

century

• Global Energy Use: yes, 2% p. a.

• Global Economic Output: yes, 3% p. a.

• Trends are regionally very different – see

following exercise!

The Logarithmic Scale

Exercise: Use the Google Public Data Explorer

Go to www.google.com/publicdata/explore?ds=d5bncppjof8f9_. You are

now on an interactive interface which with you can explore the World

Development Indicators, a wealth of data compiled by the World Bank

for the last 50 years or so. Find data on the thematic menu on the left or

by typing a key word into the search box. Once you have selected a

data series, you can choose for which countries or regions you want it

displayed. You can choose between different

chart types (line chart, bar chart, map chart,

bubble chart) on the task bar:

Spend some time to familiarize yourself with

the data explorer the interface. Look for interesting data sets, try

out the different chart types, especially the

bubble chart, and find out something you always wanted to know.

The Logarithmic Scale

Exercise: Use the Google Public Data Explorer

Look up some data sets such as population, energy use, cereal

production, Gross National Income (GNI, in constant 2000$). For

each, look up the world-wide numbers. Examine how they changed

over time. Try to visualize the magnitude of the numbers. Compare

the data for your own and selected other countries and regions. For

some indicators, you can examine both the per capita and the

aggregate values. For each indicator, read and understand the

definition. Understand the units in which each is measured.

Plot the data to linear and logarithmic scale.

Calculate and compare growth rates. For

example, how do energy use or cereal production

compare with population growth? Identify

exponential growth. Use all the techniques you

have learned to explore relevant real world data!

The Logarithmic Scale

• Exponential growth is characterized by a constant growth

rate and doubling time.

• Rule of 70:

Doubling Time = 70 over percent growth rate

• Logarithmic plots make growth rates visible.

• 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.

Summary

• Al Bartlett (1993): Arithmetic of Growth

• Herman E. Daly (1997): Beyond Growth: The

Economics of Sustainable Development

• Herman E. Daly (2012): Eight Fallacies about Growth

• Charles A. S. Hall and John W. Day, Jr. (2009):

Revisiting the Limits to Growth After Peak Oil

• Richard Heinberg (2011): The End of Growth:

Adapting to Our New Economic Reality

• Tim Jackson (2011): Prosperity Without Growth:

Economics for a Finite Planet

• Toni Menninger (2014): Exponential Growth, Doubling

Time, and the Rule of 70

• Tom Murphy (2011): Galactic-Scale Energy, Do the

Math

Further readings

Growth in a Finite World Sustainability and the Exponential Function

This presentation is part of the Lecture Series in Sustainability

Science. © 4/2014 by Toni Menninger MSc. Use of this material for

educational purposes with attribution permitted. Questions or

comments please email toni.menninger@gmail.com.

Related lectures and problem sets available at

http://www.slideshare.net/amenning/presentations/:

• The Human Population Challenge

• Energy Sustainability

• World Hunger and Food Security

• Economics and Ecology

• Exponential Growth, Doubling Time, and the Rule of 70

• Case Studies and Practice Problems for Sustainability Education:

• Agricultural Productivity, Food Security, and Biofuels

• Growth and Sustainability

… and more to come!

Appendix:

The Mathematics of

Exponential Growth

Mathematics of Exponential Growth

A quantity is said to grow exponentially at a constant (steady) rate if it increases by

a fixed percentage per unit of time. In other words, the increase per unit of time is

proportional to the quantity itself, in contrast with other types of growth (e. g.

arithmetic, logistic). Geometric growth is another word for exponential growth.

Examples

• Compound interest: the interest is a fixed proportion of the account balance,

compounded in fixed time intervals (years, months, days). One can imagine the

interval becoming smaller and smaller until interest is added continuously. This is

known as continuous compounding.

• Biological reproduction: cells dividing at regular time intervals, organisms

reproducing at regular generation periods will, under favorable conditions,

multiply exponentially. Exponential growth in a population occurs when birth and

death rate are constant, and the former exceeds the latter.

• Economics: economic output is often assumed to grow exponentially because

productive capacity roughly depends on the size of the economy. Current

macroeconomic models do not incorporate resource constraints.

• The inverse process is known as exponential decay (e. g. radioactive decay), or

“negative growth”.

Mathematics of Exponential Growth

We introduce

N : a quantity, e. g. a population count, an amount of money, or a rate of

production or resource use

N0 : the initial value of N

p : the rate of growth per unit of time, as a decimal

p% : the rate of growth as a percentage (=100 p)

t : the time period in units of time, e. g. in days or years

N(t) : the value of N after time t has elapsed.

Assume a savings account with an initial deposit of $100 carries 6% interest

compounded annually. Then N0 = $100, p = 0.06, p%= 6, and N(15) would be the

amount accumulated after 15 years (if left untouched and the interest rate remains

constant).

The first year earns $6 interest, so 𝑵 𝟏 = $𝟏𝟎𝟎 + $𝟔 = $𝟏𝟎𝟔 = $𝟏𝟎𝟎 × 𝟏. 𝟎𝟔.

The second year, we have 𝑵 𝟐 = $𝟏𝟎𝟔 × 𝟏. 𝟎𝟔 = $𝟏𝟎𝟎 × 𝟏. 𝟎𝟔𝟐 .

After t years, 𝑵 𝒕 = $𝟏𝟎𝟎 × 𝟏. 𝟎𝟔𝒕.

Mathematics of Exponential Growth

The exponential growth equation

The general formula for discrete compounding is:

𝑵 𝒕 = 𝑵𝟎 (𝟏 + 𝒑)𝒕.

In most real world situations, variables like population don’t make discrete jumps (e.

g. once a year) but grow continuously. Continuous compounding is described by

the exponential function:

𝑵 𝒕 = 𝑵𝟎 𝒆𝒑𝒕 or 𝑵 𝒕 = 𝑵𝟎 𝒆𝒙𝒑(𝒑𝒕)

where e = 2.718… is the base of the natural logarithm. In practice, both formulas

give approximately the same results for small growth rates but the continuous

growth model is preferable to discrete compounding because it is both more realistic

and mathematically more convenient.

The exponential growth formula contains three variables. Whenever two of them are

known, the third can be calculated using simple formulas. Often, one is interested

more in the relative growth 𝑵(𝒕)/𝑵𝟎 than in the absolute value of N(t). In that case,

one can get rid of 𝑵𝟎 by setting 𝑵𝟎 =1=100%.

Mathematics of Exponential Growth

Solving the exponential growth equation

𝑵 𝒕 = 𝑵𝟎 𝒆𝒑𝒕

Case 1: A quantity is growing at a known growth rate for a known period of time, by

what factor does it grow? Answer:

𝑵 𝒕

𝑵𝟎

= 𝒆𝒑𝒕

Case 2: A quantity grows at a known rate p. After what period of time has it grown

by a given factor? The equation is solved by taking the natural logarithm (written ln)

on both sides. Answer:

𝒍𝒏( 𝑵(𝒕)/𝑵𝟎) = 𝒑𝒕 → 𝒕 =𝒍𝒏(𝑵(𝒕)/𝑵𝟎)

𝒑

Case 3: In a known period of time, a quantity increases by a known factor. Find the

(average) growth rate.

Answer:

𝒑 =𝐥𝐧( 𝑵(𝒕)/𝑵𝟎)

𝒕=

𝒍𝒏 𝑵 𝒕 − 𝐥𝐧 𝑵𝟎

𝒕

Mathematics of Exponential Growth

The doubling time and rule of 70

To grasp the power of the exponential growth process, consider that if it doubles

within a certain time period, it will double again after the same period. And again and

again. The doubling time, denoted T2, can be calculated using equation (4) by

substituting

𝑵 𝒕

𝑵𝟎= 𝟐 → 𝑻𝟐 =

𝐥𝐧( 𝟐)

𝒑=

𝟎. 𝟔𝟗𝟑

𝒑=

𝟔𝟗. 𝟑

𝟏𝟎𝟎 𝒑

A convenient approximation is

𝑻𝟐 ≈𝟕𝟎

𝒑%

Thus, the doubling time of an exponential growth process can be estimated by

dividing 70 by the percentage growth rate. This is known as the “rule of 70” and

allows estimating the consequences of exponential growth with little effort.

Mathematics of Exponential Growth

Thousand-fold increase

Knowing the doubling time, it follows that after twice that period, the increase is

fourfold; after three times the doubling time, eightfold. After 𝑡 = 𝑛 × 𝑇2 time units, n

doublings will have been observed, giving a multiplication factor of 2n. It is

convenient to remember 210 = 1024 ≈ 1000 = 103 . After ten doubling times,

exponential growth will have exceeded a factor of 1000:

𝑻𝟏𝟎𝟎𝟎 ≈ 𝟏𝟎 × 𝑻𝟐 ≈𝟕𝟎𝟎

𝒑%

For a 7% growth rate, the doubling time is a decade and the time of thousand-

fold increase is a century.

Growth over a life time

A human life span is roughly 70 years. What are the consequences of 70 years of

steady growth at an annual rate p? From 𝑇2 ≈70

𝑝% follows that 70 years

encompass almost exactly p% doubling times and the total aggregate growth will be

𝑵(𝒕)

𝑵𝟎≈ 𝟐𝒑%

This is another convenient rule to remember.

Mathematics of Exponential Growth

Per capita growth

If two time series Q(t) and N(t) both follow an exponential growth pattern with

growth rates q and p, then the quotient also grows or contracts exponentially. The

growth rate is simply the difference of the growth rates, and Q(t)/N(t) grows if

𝒒 − 𝒑 > 𝟎:

𝑸 𝒕 = 𝒆𝒒𝒕, 𝑵 𝒕 = 𝒆𝒑𝒕 → 𝑸 𝒕

𝑵 𝒕= 𝒆 𝒒−𝒑 𝒕

A typical application is per capita growth, where N is a population and Q might be

energy use or economic output. Q could also indicate a subset of N, for example a

sector of the economy, and Q/N would indicate Q as a share of the total economy.

Cumulative sum of exponential growth

If a rate of resource use R(t), such as the rate of energy use, grows exponentially,

then the cumulative resource consumption also grows at least exponentially. During

each doubling time, the aggregate resource use is twice that of the preceding

doubling time, and at least as much of the resource is used as has been used

during the entire prior history. A startling fact to consider!

Mathematics of Exponential Growth

Summary table

A few simple rules, especially the rule of 70, are often sufficient to get a good

estimate of the effects of growth. The following table summarizes the results for a

range of growth rates.

Semi-logarithmic graphs

For a given time series, it is not usually obvious whether it belongs to an exponential

process. A semi-logarithmic plot helps to visually assess its growth characteristics.

Steady exponential growth will show as a straight line on the graph. Line segments

of different slope indicate a change in growth rates.

For full mathematical treatment, see: Exponential Growth, Doubling Time, and the Rule of 70

Growth rate in % 0.5 1 1.4 2 3 3.5 4 5 7 10

Doubling time T2 140 70 50 35 23 20 17.5 14 10 7

Growth per 70 years 1.4 2 2.6 4 8 11.3 16 32 128 1024

T1000 1400 700 500 350 233 200 175 140 100 70