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

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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......

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.

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  • 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/
  • 7. Growth and Sustainability: a Systems Theory perspective
  • 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?
  • 15. Growing Economic Subsystem Recycled Matter Resources Solar Energy Waste Heat Energy Energy Resources 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
  • 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
  • 18. Dimensions of Growth How human impact has multiplied since the industrial revolution
  • 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.
  • 50. Growth models • Linear growth • Exponential growth • Logistic growth
  • 51. Growth models Linear (arithmetic) growth: Constant increase per unit of time = straight line
  • 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
  • 54. • Linear • Cubic • Exponential Growth models Exponential (geometric) growth
  • 55. Growth models Exponential (geometric) growth will eventually overtake any power (or polynomial) function. • Cubic • 10th power • Exponential
  • 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
  • 86. 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
  • 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
  • 99. 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!
  • 100. Appendix: The Mathematics of Exponential Growth
  • 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