This presentation, accessible to the general public and specifically designed for students of sustainability, explores the dramatic growth of the human sphere on planet Earth with its limited resources, and presents the mathematical tools for understanding the exponential function.
The lecture is accompanied by the article "Exponential Growth, Doubling Time, and the Rule of 70" (http://www.slideshare.net/amenning/exponential-growthmath) and a collection of practice problems and case studies (http://www.slideshare.net/amenning/exponential-growth-casestudies).
The presentation "The Human Population Challenge" is suitable as a follow-up lecture.
Presented by: Dr Rosie Day, Senior Lecturer of Environment and Society in the School of Geography, Earth and Environmental Sciences at the University of Birmingham
SCIENCE, TECHNOLOGY AND INNOVATION FOR SUSTAINABLE DEVELOPMENT IN ASIA AND TH...Lausanne Montreux Congress
Investment in science, technology and innovation (STI) needs to be the backbone of productivity-led economic
recovery and sustainable development. Despite significant increases in productivity over the past few decades,
economic growth in developing economies of Asia and the Pacific has been primarily driven by factor
accumulation. However, the average rate of productivity growth slowed between the periods 2000-2007 and
2008-2014 by 65 per cent, which has contributed to the current economic slowdown, potentially undermining
efforts to effectively pursue the 2030 Agenda for Sustainable Development.
Carbon Footprint is a measure of organization's Greenhouse Gases emmissions. Many organizations nowadays are conscious with their carbon footprint.
This consciousness led to the development of PAS 2050, a standard developed by the British Standards Institute to assess the lifecycle GHG emissions of goods and services.
Restoring Earth
Despite heightened awareness of global warming, the proliferation of long-term commitments combined with short-term actions are failing to prevent the spiral towards catastrophic climate change and the continued collapse of biodiversity. As reality bites so does the pressure for more effective collective action to dramatically evolve politics, business and consumption.
This Future Agenda insight explores the challenges ahead and argues that, rather than focusing on the treating the symptoms it would be wiser to tackle the root causes of climate change. For example, by re-thinking mainstream economic principles which are not yet set up to account for the environmental damage that is being created. It suggests that a fundamental new form of measurement is required to ensure that we live within our planetary boundaries.
For more details https://www.futureagenda.org/foresights/restoring-earth/
GreenBiz 17 Tutorial Slides: "Putting Circular Economy Principles to Work"GreenBiz Group
The Circular Economy presents an incredible opportunity for transformative growth and change, but taking the concept from theory to implementation requires what can be a challenging exercise in rethinking systems. Join this session to explore how to put these principles into practice. Actual business case studies will set the stage for a group brainstorming session on how to apply circular models to various business scenarios. Key concepts to be covered include:
How adopting a Circular Economy mindset can deliver value to your business
The key drivers for a Circular Economy business
How to integrate lifecycle thinking and circular economy
How to select the right metrics and quantify circularity
What Europe’s Circular Package means for your company
GreenBiz 17 In-Depth Tutorials are intensive half-day sessions held prior to the start of the conference. These are designed to offer participants an opportunity to dive deeper into a topic of interest and develop tangible knowledge and skills. In addition, attendees will have a greater opportunity to network with their peers in these interactive sessions. Concurrent tutorials will be held the morning of Tuesday, February 14, and are available only to those who purchase an All Access Pass.
Exponential Growth, Doubling Time, and the Rule of 70Toni Menninger
Understanding exponential growth is of critical importance in sustainability, resource conservation, and economics. This article provides a rigorous yet accessible introduction to this essential concept. It also provides a selection of practice problems that will help students apply and deepen their understanding of the material.
This article accompanies my lecture presentation "Growth in a Finite World - Sustainability and the Exponential Function" (http://www.slideshare.net/amenning/growth-in-a-finite-world-sustainability-and-the-exponential-function). Also refer to Case Studies for Sustainability Education: Understanding Exponential Growth (http://www.slideshare.net/amenning/exponential-growth-casestudies).
Presented by: Dr Rosie Day, Senior Lecturer of Environment and Society in the School of Geography, Earth and Environmental Sciences at the University of Birmingham
SCIENCE, TECHNOLOGY AND INNOVATION FOR SUSTAINABLE DEVELOPMENT IN ASIA AND TH...Lausanne Montreux Congress
Investment in science, technology and innovation (STI) needs to be the backbone of productivity-led economic
recovery and sustainable development. Despite significant increases in productivity over the past few decades,
economic growth in developing economies of Asia and the Pacific has been primarily driven by factor
accumulation. However, the average rate of productivity growth slowed between the periods 2000-2007 and
2008-2014 by 65 per cent, which has contributed to the current economic slowdown, potentially undermining
efforts to effectively pursue the 2030 Agenda for Sustainable Development.
Carbon Footprint is a measure of organization's Greenhouse Gases emmissions. Many organizations nowadays are conscious with their carbon footprint.
This consciousness led to the development of PAS 2050, a standard developed by the British Standards Institute to assess the lifecycle GHG emissions of goods and services.
Restoring Earth
Despite heightened awareness of global warming, the proliferation of long-term commitments combined with short-term actions are failing to prevent the spiral towards catastrophic climate change and the continued collapse of biodiversity. As reality bites so does the pressure for more effective collective action to dramatically evolve politics, business and consumption.
This Future Agenda insight explores the challenges ahead and argues that, rather than focusing on the treating the symptoms it would be wiser to tackle the root causes of climate change. For example, by re-thinking mainstream economic principles which are not yet set up to account for the environmental damage that is being created. It suggests that a fundamental new form of measurement is required to ensure that we live within our planetary boundaries.
For more details https://www.futureagenda.org/foresights/restoring-earth/
GreenBiz 17 Tutorial Slides: "Putting Circular Economy Principles to Work"GreenBiz Group
The Circular Economy presents an incredible opportunity for transformative growth and change, but taking the concept from theory to implementation requires what can be a challenging exercise in rethinking systems. Join this session to explore how to put these principles into practice. Actual business case studies will set the stage for a group brainstorming session on how to apply circular models to various business scenarios. Key concepts to be covered include:
How adopting a Circular Economy mindset can deliver value to your business
The key drivers for a Circular Economy business
How to integrate lifecycle thinking and circular economy
How to select the right metrics and quantify circularity
What Europe’s Circular Package means for your company
GreenBiz 17 In-Depth Tutorials are intensive half-day sessions held prior to the start of the conference. These are designed to offer participants an opportunity to dive deeper into a topic of interest and develop tangible knowledge and skills. In addition, attendees will have a greater opportunity to network with their peers in these interactive sessions. Concurrent tutorials will be held the morning of Tuesday, February 14, and are available only to those who purchase an All Access Pass.
Exponential Growth, Doubling Time, and the Rule of 70Toni Menninger
Understanding exponential growth is of critical importance in sustainability, resource conservation, and economics. This article provides a rigorous yet accessible introduction to this essential concept. It also provides a selection of practice problems that will help students apply and deepen their understanding of the material.
This article accompanies my lecture presentation "Growth in a Finite World - Sustainability and the Exponential Function" (http://www.slideshare.net/amenning/growth-in-a-finite-world-sustainability-and-the-exponential-function). Also refer to Case Studies for Sustainability Education: Understanding Exponential Growth (http://www.slideshare.net/amenning/exponential-growth-casestudies).
• The Global Food System: Sustainability and Food Security
• The Global Carbon Cycle and CO2 Buildup in the Atmosphere
• The Climate System and Global Warming
For an introduction to stock and flow diagrams, see the book Thinking in Systems by Donella Meadows.
• The Global Food System: Sustainability and Food Security • The Global Carbon Cycle and CO2 Buildup in the Atmosphere • The Climate System and Global Warming For an introduction to stock and flow diagrams, see the book Thinking in Systems by Donella Meadows.
Dr. Marty Matlock - The Science of Sustainability: It is Not a Monometric Con...John Blue
The Science of Sustainability: It is Not a Monometric Concept - Dr. Marty Matlock, Executive Director, Office for Sustainability; Professor, Biological and Agricultural Engineering, University of Arkansas
, from the 2014 Global Roundtable for Sustainable Beef (GRSB), November 2 -5, 2014, São Paulo, Brazil.
More presentations at http://trufflemedia.com/agmedia/conference/2014-global-roundtable-sustainable-beef
Sustainability Science in a Global LandscapeElsevier
Science, technology and innovation have long been recognized as the basis for socioeconomic development. They are also core contributors to sustainable development and to meeting the SDGs. The UN has called for a “seat for science” on the High-Level Political Forum that deals with the UN’s sustainable development agenda, to ensure that “science is not just an observer but an advisor to policymakers.” This report is part of a broader, on-going effort to provide more evidence and analysis on the role of science, technology and innovation in the global agenda of sustainable development.
Read more about it on Elsevier Connect: http://www.elsevier.com/connect/sustainability-science-takes-the-stage-before-un-globalgoals-summit
Development, Environment and Sustainabilty–the triumvirate on Geographical FrameProf Ashis Sarkar
Development, Environment and Sustainability form the triumvirate of present day World. If human is to survive and development is to remain sustainable, the geographical issues and concerns should be the thrust of analysis.
This presentation tells about how climate change is happening due to the population and its impact on the environment in terms of ecological impacts etc.
GEOGRAPHICAL DIMENSIONS OF ‘DEVELOPMENT – ENVIRONMENT INTERRELATION’Prof Ashis Sarkar
The debate of 'environment vs. development' is seriously global and contemporary. It has its own geographical dimension as development is region-specific and time-specific.
DEVELOPMENT VS ENVIRONMENT IN GEOGRAPHICAL FRAMEWORKProf Ashis Sarkar
Development is a big word and is often related to environmental degradation. But how and why? What should be the way out are the issues in which it is based on.
TOO4TO Module 5 / Sustainable Resource Management: Part 1TOO4TO
This presentation is part of the Sustainable Management: Tools for Tomorrow (TOO4TO) learning materials. It covers the following topic: Sustainable Resource Management (Module 5). The material consists of 3 parts. This presentation covers Part 1.
You can find all TOO4TO Modules and their presentations here: https://too4to.eu/e-learning-course/
TOO4TO was a 35-month EU-funded Erasmus+ project, running until August 2023 in co-operation with European strategic partner institutions of the Gdańsk University of Technology (Poland), the Kaunas University of Technology (Lithuania), Turku University of Applied Sciences (Finland) and Global Impact Grid (Germany).
TOO4TO aims to increase the skills, competencies and awareness of future managers and employees with available tools and methods that can provide sustainable management and, as a result, support sustainable development in the EU and beyond.
Read more about the project here: https://too4to.eu/
This project has been funded with support from the European Commission. Its whole content reflects the views only of the author, and the Commission cannot be held responsible for any use which may be made of the information contained therein. PROJECT NUMBER 2020-1-PL01-KA203-082076
This presentation is based on Dr. Jeffrey Sach's online course "The Age of Sustainable Development". Effectively consider this a white paper on "Introduction to Sustainable Development". For the higher-quality version, check out:
http://decklaration.com/susdev
A photograph of the decisive decade we are facing, the perfect storm of environmental, economic and growth crisis we are facing and some possible ways to help the transition from this old unsustainable system to a new world order sustained by a new approach of global prosperity, justice and sustainability.
Descriptive and Inferential Statistical Methods: Analysis of Voting and Elect...Toni Menninger
A presentation highlighting the relevance of statistical methods for the analysis and forecast of elections:
* Voter Turnout by Income, Age, and Gender, with detailed graphs and explanations
* Polls and Election Forecasting, with explanation of the 95% confidence interval
* Representativeness and random sampling
* Aggregated election forecast models
Galton's Juenger - Auftrieb fuer Biologismus und wissenschaftlichen Rassismus...Toni Menninger
A German language review of Richard Herrnstein's and Charles Murray's The Bell Curve (1994), a widely debated book advocating racist and social darwinist explanations of social inequality. The review was published in Psychologische Literaturumschau, 1996. Also reviewed is Russell Jacoby & Naomi Glauberman (eds.): The Bell Curve Debate. History, Documents, Opinions, 1994, and Stefan Kuehl: The Nazi Connection. Eugenics, American
Racism and German National Socialism, 1994.
Frenet Curves and Successor Curves: Generic Parametrizations of the Helix and...Toni Menninger
In classical curve theory, the geometry of a curve in three dimensions is essentially characterized by their invariants, curvature and torsion. When they are given, the problem of finding a corresponding curve is known as 'solving natural equations'. Explicit solutions are known only for a handful of curve classes, including notably the plane curves and general helices.
This paper shows constructively how to solve the natural equations explicitly for an infinite series of curve classes. For every Frenet curve, a family of successor curves can be constructed which have the tangent of the original curve as principal normal. Helices are exactly the successor curves of plane curves and applying the successor transformation to helices leads to slant helices, a class of curves that has received considerable attention in recent years as a natural extension of the concept of general helices.
The present paper gives for the first time a generic characterization of the slant helix in three-dimensional Euclidian space in terms of its curvature and torsion, and derives an explicit arc-length parametrization of its tangent vector. These results expand on and put into perspective earlier work on Salkowski curves and curves of constant precession, both of which are subclasses of the slant helix.
The paper also, for the benefit of novices and teachers, provides a novel and generalized presentation of the theory of Frenet curves, which is not restricted to curves with positive curvature. Bishop frames are examined along with Frenet frames and Darboux frames as a useful tool in the theory of space curves. The closed curve problem receives attention as well.
Exponential Growth: Case Studies for Sustainability EducationToni Menninger
Understanding exponential growth is of critical importance in sustainability, resource conservation, and economics. This work contains a collection of practice problems and realistic case studies developed for the teaching of sustainability science and conservation, with an emphasis on learning and applying the concepts of exponential growth. The exercises are designed to foster quantitative competence (numeracy) as well as critical thinking and systems thinking. Students learn to work with tools such as spreadsheet software and online databases and practice the application of basic but powerful quantitative analyses techniques. The case studies are based on recent, high quality data and explore questions of high relevance for the study and application of sustainability science.
This work is related to the Growth in a finite world presentation (http://www.slideshare.net/amenning/growth-in-a-finite-world-sustainability-and-the-exponential-function).
A lecture in Quantitative Sustainability
It is often claimed that agricultural productivity needs to be increased in order to feed a growing world population. Food security depends on several factors besides the productivity, including waste/efficiency, energy crops, meat consumption, and global justice and equity. This lecture explores the issue of food security in its many dimensions and teaches how to use a high-level systems approach in sustainability science.
Sustainable Agriculture, Food Security, Corn Ethanol: Quantitative Study Prob...Toni Menninger
The problems below are a selection of real world problems developed for the teaching of sustainability/conservation related College classes. The exercises are designed to foster quantitative competence (numeracy) as well as critical thinking and systems thinking. They are basic but realistic and all data used are taken from the published scientific literature and from public online databases maintained by official organizations such as FAO and EIA.
No advanced Mathematics is required, yet these problems are challenging for most students. Many students need help to overcome a certain math anxiety or even phobia. These exercises must be accompanied by intensive discussion, assistance, and feedback. Students who complete these assignments successfully experience the power of even basic quantitative methods. They learn that informed citizens do not have to rely solely upon the advice of experts – with reasonable effort they can gather and interpret information and come up with approximate answers to important, non-trivial real world questions.
¨Uber die Darstellung von Raumkurven durch ihre InvariantenToni Menninger
A treatise in classical curve theory featuring the development of the complete theory of Frenet frames and the Frenet equations, and the derivation of explicit representations of important curve classes (Helices, Curves of Constant Precession, Slant Helices)
Utilizing geospatial analysis of U.S. Census data for studying the dynamics o...Toni Menninger
Geographically referenced US census data provide a large amount of information about the extent of urbanization and land consumption. Population count, the number of housing units and their vacancy rates, and demographic and economic parameters such as racial composition and household income, and their change over time, can be examined at different levels of geographic resolution to observe patterns of urban flight, suburbanization, reurbanization, and sprawl. This paper will review the literature on prior application of census data in a geospatial setting. It will identify strengths and weaknesses and address methodological challenges of census-based approaches to the study of urbanization. To this end, a detailed overview of the geographic structure of U.S. Census data and its evolution is provided. Ecological Fallacies and the Modifiable Areal Unit Problem (MAUP) are discussed and the Population Weighted Density as a more robust alternative to crude population density is introduced. Of special interest will be literature comparing and/or integrating census data with alternative methodologies, e.g. based on Remote Sensing. The general purpose of this paper is to lay the groundwork for the optimal use of high resolution census data in studying urbanization in the United States.
Keywords
Sprawl, Urban sprawl, City, Population Density, Population Weighted Density, Census, US Census, Census Geographies, Urbanization, Suburbanization, Urban flight, Reurbanization, Land Consumption, Land Use, Land Use Efficiency, LULC, Remote Sensing, Geospatial Analysis, GIS, Growth, Urban Growth, Spatial Distribution of Population, City Limits, Urban Extent, Built Environment, Urban Form, Areal Interpolation, Scale, Spatial Scale, Longitudinal Study, Dasymmetric Mapping, Ecological Fallacy, MAUP, Modifiable Areal Unit Problem, Metrics
The "Tragedy of the Commons" is one of the most influential scientific publications ever yet it is widely misunderstood. The short presentation provides a critical appraisal and links to read more.
The Human Population Challenge: From “Population Bomb” to “Demographic Crisis”Toni Menninger
A presentation about the Human Population Challenge developed for students in sustainability, including current data, basic demographic concepts, and a discussion of sustainability related issues.
The presentation "Growth in a Finite World" is closely related and precedes this lecture. The presentation "Energy Sustainability" is also suitable as a follow-up lecture.
This presentation is an introduction to the sustainable energy challenge. It gives an overview over fossil fuels, the laws of energy, energy efficiency and conservation, and renewable energy sources. The focus is on providing students with the scientific tools for understanding the magnitude of the challenge and analyzing potential solutions.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
Delivering Micro-Credentials in Technical and Vocational Education and TrainingAG2 Design
Explore how micro-credentials are transforming Technical and Vocational Education and Training (TVET) with this comprehensive slide deck. Discover what micro-credentials are, their importance in TVET, the advantages they offer, and the insights from industry experts. Additionally, learn about the top software applications available for creating and managing micro-credentials. This presentation also includes valuable resources and a discussion on the future of these specialised certifications.
For more detailed information on delivering micro-credentials in TVET, visit this https://tvettrainer.com/delivering-micro-credentials-in-tvet/
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
Growth in a Finite World - Sustainability and the Exponential Function
1. Growth in a Finite
World
Sustainability and the Exponential
Function
2. 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
3. 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
4. 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
5. 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/
6. 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/
8. 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?
9. 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
10. 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).
11. 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
12. The Human Sphere as a Subsystem of
the Ecosphere
Growing
Economic
Subsystem
Recycled
Matter
Energy
Resources
Energy
Resources
Solar
Energy
Waste Heat
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.
13. Solar
Energy
Finite Global Ecosystem
Recent history is
characterized by a
dramatic expansion
of the human
“footprint”
Thousands of years ago:
“Empty world” Waste Heat
The Human Sphere as a Subsystem of
the Ecosphere
14. The Human Sphere as a Subsystem of
Planet Earth
Growing
Economic
Subsystem
Recycled
Matter
Energy
Resources
Energy
Resources
Solar
Energy
Waste Heat
Sink
Functions
Source
Functions
Finite Global Ecosystem
Recent history is
characterized by a
dramatic expansion
of the human
“footprint”
Hundreds of years ago?
16. Famous 1972 report was an early application
of computer aided systems modeling
The Human Sphere as a Subsystem of
the Ecosphere
17. 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
19. 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/
20. Dimensions of Growth: Cement Production
World cement production:
50-fold increase since 1926
U.S. Geological Survey Data Series 140
21. Dimensions of Growth: Copper Production
World Copper production:
50-fold increase since 1900
U.S. Geological Survey Data Series 140
22. Dimensions of Growth: Fisheries
Fisheries: six-fold increase since 1950
Source: FAO, 2004. http://earthtrends.wri.org/updates/node/140
23. Dimensions of Growth: Fertilizer
Nitrogen fertilizer: nine-fold increase since 1960
Source: UNEP 2011. https://na.unep.net/geas/getUNEPPageWithArticleIDScript.php?article_id=81
24. Dimensions of Growth: Energy
US electricity consumption: almost ten-fold
increase since 1950
http://www.energyliteracy.com/?p=142
25. Dimensions of Growth: Primary Energy
US primary energy consumption: more than
ten-fold increase since 1900
http://www.theenergysite.info/Markets_Demand.html
26. World primary energy: twenty-fold increase
since 1850, mostly fossil fuels
Dimensions of Growth: Primary Energy
27. World petroleum consumption: more than ten-
fold increase since 1930
Dimensions of Growth: Petroleum
http://www.americanscientist.org/issues/id.6381/issue.aspx
28. 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
29. IMF projects further quadrupling of world wide
car fleet by 2050
Dimensions of Growth: Passenger cars
http://www.planetizen.com/node/41801
30. Dimensions of Growth: Greenhouse gas
emissions
CO2 emissions: Ten-fold increase since 1900
31. 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
32. 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, …
33. 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.
34. Quantifying growth
Learn how to calculate and
interpret
• Actual (absolute) change
• Fractional (relative) change
• Average rates of change
35. 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?
36. 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
37. 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
38. 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%
39. 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.
40. 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
…
41. 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
42. 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
43. 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.
44. 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%
45. 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?
46. 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).
47. 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.
48. 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.
49. 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.
52. 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
53. • 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
56. 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
57. 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
58. 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
59. In nature, no sustained material
growth over long time periods has
ever been observed.
Growth models
60. Learn to understand and apply
• The doubling time
• The Rule of 70
• The power of the powers of 2
• Logarithmic plots
Exponential Growth
61. Continuous exponential growth of a quantity N
over time t at constant fractional rate p is
described by the exponential function
𝑵 𝒕 = 𝑵 𝟎 𝒆 𝒑𝒕 or 𝑵 𝟎exp(𝒑𝒕)
The growth rate can be calculated as
𝒑 =
ln 𝑵(𝒕)/𝑵 𝟎
𝒕
This is the same as the average growth rate
formula introduced earlier. Refer to full
mathematical treatment in the appendix.
Exponential Growth
62. 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
63. “Rule of 70”
When steady exponential growth
occurs, the doubling time can be
estimated by dividing 70 by the
percent growth rate p:
𝑻 𝟐 ≈
𝟕𝟎 / 𝒑
Conversely, 𝒑 ≈ 𝟕𝟎 / 𝑻 𝟐
Exponential Growth
64. 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
65. 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
66. • 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
67. 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
68. • 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
69. 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
70. 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
71. • 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
72. 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
73. 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
74. 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
75. 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
76. 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
77. 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
78. 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
79. 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
80. 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
81. 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
82. How can we recognize exponential
growth in a time series?
• Inspect the data?
• Analyze the data?
• Inspect the graph?
Exponential Growth
83. 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 2010
Pop. inmillion 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
84. 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 2010
Pop. inmillion 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
85. 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
87. 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
88. 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
89. 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
90. 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
91. 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
92. 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
93. 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
94. 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
95. 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
96. 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
97. • 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
98. • 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
101. 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”.
102. 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.
103. Mathematics of Exponential Growth
The exponential growth equation
The general formula for discrete compounding is:
𝑵 𝒕 = 𝑵 𝟎 (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 = 𝑵 𝟎 𝒆 𝒑𝒕
or 𝑵 𝒕 = 𝑵 𝟎 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)/𝑵 𝟎 than in the absolute value of N(t). In that case,
one can get rid of 𝑵 𝟎 by setting 𝑵 𝟎 =1=100%.
104. Mathematics of Exponential Growth
Solving the exponential growth equation
𝑵 𝒕 = 𝑵 𝟎 𝒆 𝒑𝒕
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
𝑵 𝟎
= 𝒆 𝒑𝒕
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)/𝑵 𝟎) = 𝒑𝒕 → 𝒕 =
𝒍𝒏(𝑵(𝒕)/𝑵 𝟎)
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)/𝑵 𝟎)
𝒕
=
𝒍𝒏 𝑵 𝒕 − 𝐥𝐧 𝑵 𝟎
𝒕
105. 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
𝑵 𝟎
= 𝟐 → 𝑻 𝟐 =
𝐥𝐧( 𝟐)
𝒑
=
𝟎. 𝟔𝟗𝟑
𝒑
=
𝟔𝟗. 𝟑
𝟏𝟎𝟎 𝒑
A convenient approximation is
𝑻 𝟐 ≈
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.
106. 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:
𝑻 𝟏𝟎𝟎𝟎 ≈ 𝟏𝟎 × 𝑻 𝟐 ≈
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)
𝑵 𝟎
≈ 𝟐 𝒑%
This is another convenient rule to remember.
107. 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 = 𝒆 𝒒𝒕
, 𝑵 𝒕 = 𝒆 𝒑𝒕
→
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!
108. 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