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# Exponential Growth: Case Studies for Sustainability Education

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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).

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### Exponential Growth: Case Studies for Sustainability Education

3. 3. Understanding Exponential Growth 3/17 I. Understanding and exploring exponential growth 1. Warmup exercises a) Use the Rule of 70 to estimate the doubling times for exponential growth of 1%, 2%, and 4%. b) With a growth rate of 3.5% p. a., estimate the quadrupling time. c) A variable increases fourfold/eightfold in N years. What is the doubling time and growth rate? d) Estimate or calculate the growth factor over 70 years for growth rates 1%, 2%, 3%, and 4%. e) What power of two is roughly 1 billion? f) Estimate the number of doublings in population size that humanity has undergone to present. How many more doublings do you think can or will happen? g) Derive a rule (similar to the rule of 70) for estimating the tripling time T3. h) What growth rate implies a 10-fold increase every 100 years? 2. Comparing growth rates The population of country A has grown from 10 million to 20 million in 30 years. Country B grew from 3 million to 12 million in 60 years. What can you conclude about both countries’ fractional growth rates? Is A or B growing faster or are both growing at the same rate? Why is it more appropriate to compare fractional (relative) change rather than absolute change? 3. Linear vs. exponential growth a) Lineartown has 10,000 residents and every year, it gains 200 residents. Under this trend, how long would it take for the town to reach 20,000, 40,000, 60,000, 80,000 residents? b) Expoville has 10,000 residents and every year, it grows by 2%. Under this trend, how long would it take for the town to reach 20,000, 40,000, 80,000 residents? Use the rule of 70 to estimate! c) When Lineartown reaches 80,000, what is Expoville’s population going to be? d) Calculate average growth rates for Lineartown and Expoville, for the whole time period and for each segment, and compare the linear and exponential case. What happens in time to the average growth rate of Lineartown? e) Use these results to explain the differences between linear and exponential growth.
4. 4. Understanding Exponential Growth 4/17 4. Pond Lily Parable and Bacteria Colony Parable Exponential growth is often illustrated with the Pond Lily 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?” The answer that intuitively comes to mind is almost always the wrong one! Can you explain why? The late Physics Professor Al Bartlett developed the following parable of a bacteria colony: “A single bacterium is placed in a nutrient-filled bottle at 11:00 am. It grows and divides into two bacteria at 11:01. Each descendant cell divides in two once every minute. At 12:00 pm all nutrients are gone and all the bacteria die.” a) Explain why the bacteria colony grows exponentially with a doubling time of 1 minute. b) Exactly at what time was half of the food used up? c) You are a mathematically-sophisticated bacterium, and at 11:56 you recognize the impending disaster. You immediately jump on your soapbox and warn that unless your fellow bacteria slow their growth dramatically, the end is just four minutes away. Will anyone believe you? d) Just before disaster strikes… a bacterial space program discovers three more bottles in the lab. With a population redistribution program, how much time do the 3 new bottles buy the colony? e) Is there any hope that future discoveries will allow the colony to continue its exponential growth indefinitely? 5. Semi-logarithmic graphs a) Create exponential test data in a spreadsheet and plot them linearly and semi-logarithmically. Show graphs for different growth rates in the same plot. b) Show that the graph of the exponential function (and only of the exponential function) appears as a straight line when plotted semi-logarithmically, and the slope is the growth rate. c) Show that the average growth rate between two points in time is exactly the slope of the line joining the two points on a semi-logarithmic graph. You will use semi-logarithmic plots extensively to explore real world data in sections II and III.
5. 5. Understanding Exponential Growth 5/17 6. Per capita growth It is often necessary to analyze the quotient of two quantities, for example per capita production or consumption. Assume that a country’s GDP (or some other aggregate quantity) grows at an exponential rate with growth rate p, and the population N also grows exponentially at rate q. What can you conclude about economic output per capita (GDP/N)? a) In a spreadsheet, create test data for GDP, population, and per-capita GDP. Choose different sets of p and q values (e. g. 3% and 1%, 2% and 2%). What do you observe about the behavior of GDP/N? What happens in each of the cases ? b) Show algebraically that GDP/N is exponential with the difference as growth rate. c) Now assume that p and q are average growth rates over a period of time, and verify algebraically that the average growth rate of GDP/N for the same time period is again the difference . Note that the average growth rate is well-defined regardless of whether growth is exponential. d) Finally, assume that GDP and population both are growing linearly at constant rates. What can you say about the growth of per capita GDP? Is there a numeric trick similar to the exponential case? Use a spreadsheet as in a). 7. Exponential vs. power functions The exponential function will eventually grow faster than and overtake any finite power function. That is, for any positive integer N, there exists some x(n) so that 𝑒 for all ( ). Can you show this mathematically? Find values for x(10), x(50), x(100).
6. 6. Understanding Exponential Growth 6/17 II. Growth and Sustainability Case Studies: Population Growth, Resource Use, Food Production, and Economic Growth Consult the appendix for a step by step guide to analyzing and plotting data series in a spreadsheet. Use a single workbook with a separate worksheet for each data series. 8. U.S. Population How has population growth in the US developed over the past century? The U.S. Census Bureau collects and publishes demographic data for the United States. Every ten years since 1790, a full population census has been conducted. The decennial census counts (in million) since 1900 are given in the table. The 2013 census estimate is 316.1 million. Year 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 Pop. (million) 76.2 92.2 106.0 123.2 132.2 151.3 179.3 203.2 226.5 248.7 281.4 308.7 Avg. yearly increase 0.9 Avg. growth rate % p. a. 1.9% a) How many doublings do you observe? Use this to estimate the doubling time and average growth rate (use the rule of 70). b) Copy the data into a spread sheet. Calculate the average increase per year and the average percent growth rate for each decade to complete the table. Calculate the fractional increase and the average percent growth rate for the whole period. Compare to your estimate in a). c) Plot population against time in a regular and a semi-logarithmic graph (use scatter plot with smooth lines and markers). Add trend lines and display their equations. Compare the average growth rate to the slope of the exponential trend line. What do the trend line equations tell you? d) Show the average increase and average growth rate per decade in separate plots. When was the absolute increase highest/lowest? When was the fractional increase highest/lowest? e) Describe the growth trends you observe. Is this time series an example of steady exponential growth? Give reasons! f) Further investigation (optional): Choose a different country or geographic unit that interests you and perform the same analysis. Good data sources are the Google Public Data Explorer (World / USA), the UN Population Division and US Census (International data, US decennial census, Selected historical population tables, Population estimates).
7. 7. Understanding Exponential Growth 7/17 9. World Population We’ll now explore global population growth using the same techniques as in exercise 8. The table shows world population estimates from 1950 - 2010 taken from the United Nations Population Division website, where you can download extensive demographic data by country and region. An alternative data source is the US Census International Programs web site. You can look for data going further back if you like. The most current (2013) world population estimate is 7.1 billion. Year 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 Pop. (billion) 2.53 2.76 3.03 3.33 3.69 4.07 4.45 4.86 5.32 5.74 6.13 6.51 6.92 Avg. yearly increase (million) 46 Avg. growth rate % p. a. 1.8% a) Calculate the overall fractional increase and the overall average growth rate. b) Plot population against time in a linear and a semi-logarithmic graph, with trend lines. c) Calculate the average increase and the average fractional growth rate for each 5-year period to complete the table. Plot these data as well. Describe the trend in growth rates. Is there a change, if yes when? Can you identify these trends using the semi-logarithmic graph? d) When was the absolute increase highest/lowest? When was the fractional increase highest/lowest? e) Which growth model (linear, exponential, logistic) does this data series best correspond to? Is this an example of steady exponential growth? Give reasons! f) Further investigation (optional): Explore the projections for future population growth at the UN web site. What growth model are the UN demographers anticipating for the 21st century? 10. Urbanization Calculate the average growth rates for the following populations: a) London, then the world’s largest city, grew from 1 million to 6.7 million over the 19th century. b) The city of Shenzen in China grew from about 330,000 in 1980 to 10 million in 2009. c) China’s population has gone from 21% urban to 50% urban in 28 years (1982-2010). Total population increased from 1.01 to 1.34 billion in the same period.
8. 8. Understanding Exponential Growth 8/17 11. Primary Energy Use Your task in this case study is to analyze a crucial sustainability challenge, energy use. The term primary energy refers to the sum of all raw sources of technical energy used by people. A very useful data source is the World Bank Word Development Indicators Database available on Google. The Google Public Data Explorer is an interactive interface with tools for creating linear and logarithmic data charts by country or region. Familiarize yourself with the tools and look up some data that interest you. a) The Environment category includes data about energy use, greenhouse gas emissions, population and much more. Look up Global Primary Energy Use. Read the definition and understand the physical units in which the data are provided. Explore the data, look up different countries. Also explore Per Capita Energy Use. Which countries are the biggest energy consumers, in absolute and in per capita terms? b) The table shows the primary energy use data, in Mega-tonnes of oil equivalent (Mtoe, see energy unit converter). Year 1971 1975 1980 1985 1990 1995 2000 2005 2010 2011 Energy use (Mtoe) 5,505 6,095 7,096 7,563 8,574 8,993 9,789 11,190 12,516 12,716 Copy the table into a spread sheet and use the techniques you have learned to analyze them. Create linear and semi-logarithmic plots. Estimate, and then calculate average growth rate and doubling time. Is this an example of steady exponential growth? Give reasons! c) Use comparisons to visualize the order of magnitude of our global energy use:  A modern supertanker can transport on the order of 300,000 metric tonnes of oil. How many supertankers would be required to carry 12,716 Mt of oil?  America’s largest power plant burns 11 Mt of coal per year, equivalent to ca. 7 Mt of oil.  The USGS estimates that the Arctic National Wildlife Refuge contains 5.7 to 16 billion barrels (bbl) of recoverable oil, equivalent to 800-2200 Mtoe (1 toe = 7.4 boe). How does this stock of energy compare to global annual energy use? d) Compare the increase in energy use with population growth over the same time period (exercise 9). How has per capita energy use changed (compare exercise 6)? e) If the growth trend observed in the past were to continue, how long would it take for energy use to double, quadruple? How much cumulative energy use would you expect in that time frame?
9. 9. Understanding Exponential Growth 9/17 f) What physical consequences would you expect from continued exponential growth of energy use? Consider all aspects of the energy economy and infrastructure, including availability, exploration and extraction of energy resources; production, transport and distribution of technical energy; and waste products and pollution. How many doublings are likely to be realistically and sustainably be achieved? g) Look up electricity production and perform the same analysis as above. Electric power output increased from 5.2 PWh in 1971 to 22.2 PWh in 2011 (Peta=1015 ). Convert the units to compare the absolute numbers. Compare the growth rates of electricity generation and primary energy use. Also, compare both growth rates to economic growth (see exercise 16). h) Further investigation (optional): Analyze fossil fuel use and greenhouse gas emissions. Useful data sources include the BP Statistical Review of World Energy and the Energy Information Agency.
10. 10. Understanding Exponential Growth 10/17 12. Resource Use: Copper, Aluminum, Cement and Nitrogen Production Industrialization, urbanization and population growth require ever growing amounts of material resources such as the ones we’ll explore in this case study. Copper and aluminum are non-renewable mineral resources and aluminum production is particularly energy intensive. Cement production is also energy intensive and a significant source of greenhouse gas emissions. Nitrogen is plentiful in the atmosphere but its fixation for use as fertilizer is energy intensive. The anthropogenic release of reactive nitrogen has many adverse effects on the environment and human health, including water pollution and greenhouse gas emissions. Anthropogenic nitrogen addition has significantly altered the planet’s natural nitrogen cycle: it now exceeds the rate of natural nitrogen fixation. The US Geological Survey has collected historical data on production and use of various minerals (USGS: Historical Statistics for Mineral and Material Commodities in the United States). Download data for copper, aluminum, cement, and nitrogen production and any others you are interested in. The world production of copper increased from 0.5 Mt in 1900 to 16.1 Mt in 2011; aluminum from 0.007 Mt in 1900 to 44.4 Mt in 2011; world cement production grew from 62.4 Mt in 1926 to 3,590 Mt in 2011; nitrogen fixation from 2.4 Mt in 1946 to 135 Mt in 2011. [1 Mt (megaton) = 1 million metric tons] For each data series, copy the relevant data into a spread sheet work book (use an extra sheet for each) and perform the following analysis (Note: this exercise doesn’t require much work - you can do a lot with copy/paste!): a) Calculate fractional increase, average growth rate, and doubling time for the whole period, and the post-war period since 1950. b) Create linear and semi-logarithmic plots with exponential trend lines. Determine whether the data fit linear or exponential or other growth models. c) Use the semi-logarithmic graph to identify time periods of faster or slower growth and calculate average growth rates and doubling times for these. In particular look at the periods 1950-1980 and 1980-2011. Do you find a pattern? Can you explain what you observe? d) Identify the last few doubling times, in other words, find the year when half, one quarter etc. of the last value was reached. Are these doubling times consistent with your earlier results? e) For each year, calculate the cumulative production and plot to linear and semi-logarithmic scale. Report the total cumulative amount. What percentage of the total was produced only in
11. 11. Understanding Exponential Growth 11/17 the last 20 years? (For perspective, you may consult exercise P7 in “Exponential Growth, Doubling Time, and the Rule of 70”). f) Based on your results, if current trends persist, how much future growth would you roughly expect by 2050 (give or take a few years)? How much cumulative production/resource use would you expect in that time frame? How does that compare to the past cumulative figure? g) Pick at least one of the materials studied above and do some research on the positive and negative effects their production and use has on human and environmental welfare. How would you judge the sustainability of current growth (give reasons and be as specific as possible)? What ways are there to use these resources more efficiently and reduce the environmental damage?
12. 12. Understanding Exponential Growth 12/17 13. Agricultural Productivity In this case study, you will explore one of the most fundamental problems facing humanity: the provision of sufficient food for a growing population. Most people’s diet is based on cereals. Examine how cereal yield and production has changed globally over the past half century. The FAO (Food and Agriculture Organization of the United Nations) database FAOSTAT provides time-series and cross sectional data relating to food and agriculture for some 200 countries. Go to FAOSTAT. In the domain “Production“ – “Crops”, select region “World + (Total)”, and aggregated item “Cereals, Total + (Total)”. Select all years and the elements “Area harvested”, “Yield”, and “Production”. Click on “Preview” to retrieve the data series and copy or download the data into a spread sheet. The following will focus on yield but you should explore the other data series too. In particular, observe that area harvested has remained almost unchanged (in fact it slightly declined since 1989). Most of the world’s arable land area is already under cultivation and any increase in production must result from increased land productivity. Yield is given in units of hectogram/hectare (Hg/Ha). 10,000 Hg=1 T, 1Ha=10,000 m2 . Average cereal yield has almost tripled from 1.3 T/Ha in 1961 to 3.6 T/Ha in 2012. a) Show the yield data in a scatter plot with smooth lines (linear and semi-logarithmic). b) Which growth model (linear, exponential, logistic) does the data series best correspond to? c) Calculate average growth rates for the whole period, 1961-1990, and 1990-2012, respectively. Interpret these numbers and explain how they relate to your answer in b). d) How long will it take for the yield to double (compared to 2012) if you assume steady exponential growth, using the growth rate calculated in c)? e) How long will it take if you assume a constant linear growth trend? f) Explain the relevance of the results in d) and e) for assessing potential future yield growth. Explain whether it is appropriate to extrapolate from past growth trends. g) Compare cereal production growth with world population growth. For the periods 1961-1990 and 1990-present, what can you conclude about per capita cereal production (see exercise 6)? Has cereal production kept up with population? What do your results imply for global food security in the 21st century? Consider that not all cereal is consumed by humans. What other uses are made of cereal? Can you think of ways to improve food sustainability, other than increasing productivity? h) Compare cereal production growth with nitrogen production (exercise 12). Not all but most industrial nitrogen is used as fertilizer in agriculture. i) Further investigation (optional): FAOSTAT data include food supply, resource use, agri- environmental indicators and more. Explore the resource intensity of food production, e. g. amount of fertilizer or greenhouse gas emissions per food calorie.
13. 13. Understanding Exponential Growth 13/17 14. U.S. Economic Growth In this assignment, we’ll start exploring questions of economic growth and sustainability. Gross Domestic Product (GDP), defined as the aggregate value of all goods and services produced within a national economy evaluated in market prices, is the standard measure of economic output and is generally used to quantify economic growth. You should always use inflation- adjusted figures (‘Real GDP’ or ‘GDP in constant dollars’). Real GDP per capita is widely used as a welfare indicator although this is questioned by many experts (refer to class discussion). The web sites of the U.S. Bureau of Economic Analysis (BEA) and the Federal Reserve (FRED) offer interactive data tools with the capability to download a variety of economic data in spreadsheet format and create graphs. For this assignment, download U.S. Real GDP data from FRED (variable GDPCA) or BEA (NIPA table 1.1.6). The value has increased from \$1.1 in 1929 to \$15.8 trillion in 2013 (measured in chained 2009 dollars). a) Download and investigate the data series, using the techniques explained in previous exercises: create linear and semi-logarithmic plots, look at trend lines, and calculate growth rates. Identify periods of exponential growth and characterize them quantitatively in terms of growth rates and doubling times. Look particularly at the period after 1950. Is GDP consistent with exponential growth? Are there signs that the growth trend is changing? b) Add population data (see exercise 8; you can use annual Census population estimates) and calculate Real GDP per capita (see exercise 6). Perform the same analysis as in a). What is the average growth rate? 15. Health Care Economics The health care sector of the economy is an example of a subsystem that has been growing faster than the overall system. The U.S. Department of Health and Human Services tracks National Health Expenditures (NHE) data. NHE includes both public and private spending on health care. The following table gives the NHE share of GDP since 1960: Year 1960 1970 1980 1990 2000 2005 2010 2012 NHE as share of GDP 5.0% 7.0% 8.9% 12.1% 13.4% 15.5% 17.4% 17.2% a) Use the table and GDP data from exercise 14 to calculate real NHE expenses (as dollar amounts). Make the usual plots and calculations. Is this growth trend exponential, and can you identify changes in the trend?
14. 14. Understanding Exponential Growth 14/17 b) Calculate and compare average growth rate and doubling time for GDP and NHE for the periods 1960-1990 and after 1990. c) How much of the economic growth 2000-2010 has been due to the rising cost of health care? What does your result imply for the sustainability and desirability of economic growth? d) As shown, the share of the US economy (as measured by GDP) accounted for by health expenditures has increased from 5.0% in 1960 to 17.2% in 2012. Assuming that both GDP and health care expenses continue growing at steady rates, in what year would Americans spend all their money on health care? Note that this is a thought experiment, not a predictive analysis! Is it realistic to expect the growth trend of the past half century to continue? Give reasons, based on general principles and on the data! 16. Worldwide Economic Growth since 1970 We’ll next look at global economic growth. The World Development indicators database includes the Gross National Income (GNI) data series, a measure similar to GDP. The table shows GNI since 1970 in trillion dollars, adjusted for inflation. The numbers are given in constant 2000 U.S. dollars and are not adjusted for purchasing power parity (PPP) between currencies. Use the data interface to explore and compare economic growth in different countries and regions. Also explore GDP per capita (adjusted for inflation and PPP). Year 1970 1975 1980 1985 1990 1995 2000 2005 2010 2012 GNI (\$T) 15.51 18.46 22.12 25.26 30.24 33.74 40.2 46.34 51.61 54.33 a) Conduct the same analysis of GNI as with US GDP, create plots and calculate growth rates. b) Analyze GNI per capita (use data from exercise 9 and see exercise 6). 17. Energy Intensity and Economic Decoupling In class, we considered three important questions about economic growth: Is it always desirable? Is it necessary? And is it environmentally sustainable? You have learned early in this class that material growth on our finite planet is subject to biophysical constraints and cannot continue beyond those limits. Some economists propose that economic growth can be “decoupled” from resource use by increasing resource productivity (the efficiency of resource use). In this exercise, you will consider this concept of decoupling in the case of energy use.
15. 15. Understanding Exponential Growth 15/17 Calculate energy use (ex. 11) divided by GNI (ex. 16). This is a measure of the energy intensity of the world economy. Resource intensity can be understood as the inverse of resource use efficiency: a higher value means that more of the resource has to be spent for each unit of economic value produced. Decoupling implies decreasing resource/energy intensity. a) Plot energy intensity and calculate the average rate of change. b) Identify the trends in energy intensity. At what time to you see a break in the trend? c) Compare average rates of change of GNI, energy use, and energy intensity overall and for the time periods identified in b) (also see exercise 6). d) Describe and interpret your results: Is there evidence for “decoupling” global economic growth from fossil fuel use? Do current trends point in the direction of “sustainable” or “green” growth, i. e. economic growth without increasing resource (energy) use? At what rate would the energy intensity have to decrease to achieve this goal? 18. Summary and Conclusions To summarize the results of the case studies you have done, create a table showing average growth rates of cereal production, primary energy use, electricity production, cement, copper, nitrogen, and aluminum production, GNI, and world population, for the time periods 1970-1990 and 1990-present (use the most recent data point available). a) Describe how the growth trends differ before and after 1990. Which of these growth trends are consistent with steady exponential growth? b) Calculate the per capita growth rates (review exercise 6) and compare. c) Calculate the rates of change for energy intensity (energy use / GNI, see ex. 17) and resource intensities (aluminum / GNI etc., see ex. 12). Also compare the growth rates for nitrogen production and cereal production (ex. 12 and 13). d) Interpret your results in terms of their implications for sustainability. Are any of the growth rates observed “sustainable”? (You need to outline the meaning of this term!) What would you expect for the next half century if past trends in population growth and resource use (“business as usual”) continue? How would these developments have to change to enable a transition to a sustainable society and economy?