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- 1. Exponential Growth 1 T. Menninger Exponential Growth, Doubling Time, and the Rule of 70 Toni Menninger MSc http://www.slideshare.net/amenning/presentations 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. 1. Definition: A quantity is said to grow exponentially (or geometrically) if it increases by a constant percentage or fractional rate 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). 2. 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, or organisms reproducing at regular generation periods will, under favorable conditions, multiply exponentially because typically the number of offspring is proportional to population size. In ecology, population growth is always limited by food supply, diseases and predators. Economics: many models assume economic output to grow exponentially insofar as productive capacity roughly depends on the size of the economy. Current macroeconomic models unrealistically assume unlimited resource supply. The inverse process is called exponential decay (e. g. radioactive decay), or negative growth. 3. Mathematical treatment We introduce the variables N : a quantity, e. g. a population count, a sum of money, a production or consumption rate t : the time N0 : the initial value of N at time 0. N(t) : the value of N at time t. p : the fractional rate of growth per unit of time, as a decimal p% : the fractional rate of growth as a percentage (=100 p)
- 2. Exponential Growth 2 T. Menninger Assume a savings account with an initial deposit of $100 carries 6% interest compounded annually. Then = $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 (1) = $100 $6 = $106. The second year, we have ( ) = $106 0.06 × $106 = $106 × 1.06 = $100 × 1.06 × 1.06 = $100 × 1.06 . After t years, ( ) = $100 × 1.06 (i. e. 1.06 multiplied with itself t times, or taken to the power of t). The general formula for discrete compounding (i. e. compounding in regular time intervals) is: ( ) = ( ) (1) In most real world situations, variables like population don’t jump once a year but grow continuously. We can imagine the compound interval to become smaller: instead of 6% per year, the account earns 3% twice a year, or 0.5% each month, or 0.016% per day. When the interval is made smaller and smaller, at the limit, a continuous curve is approached, described by the exponential function: ( ) = ( ) = ( ) ( ) ( = . 1 is the base of the natural logarithm). Continuous compounding is more realistic and mathematically more convenient than discrete compounding but the results are about the same (P9). 4. Solving the exponential growth equation The exponential growth formula (2) contains three variables. Whenever two of them are known, the third can be calculated using simple formulas. Often, we are interested more in the relative growth (𝑡)/ than in the absolute value of N(t). In that case, we can get rid of N0 by setting N0=1=100%. Case 1: A quantity is growing at a known growth rate for a known period of time, by what factor does it grow? Answer: ( ) = (3) Case 2: A quantity grows at a known rate. 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 ) = 𝐭 → 𝐭 = 𝐥𝐧(𝐍(𝐭)/𝐍 ) (4) Case 3: In a known period of time, a quantity increases by a known factor. Find the (average) growth rate (see exercise P5). Answer: = 𝒍𝒏( ( )/ ) = 𝒍𝒏 ( ) − 𝒍𝒏 (5) Note: the average growth rate is well-defined regardless of whether growth is exponential.
- 3. Exponential Growth 3 T. Menninger 5. The doubling time 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 ( ) = 𝟐 → 𝑻 𝟐 = 𝒍𝒏( 𝟐) = . 𝟔𝟗𝟑 = 𝟔𝟗. 𝟑 A convenient approximation is 𝑻 𝟐 ≈ 𝟕 % (6) 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. Knowing the doubling time, it follows that after twice that period, the increase is fourfold; after three times the doubling time, eightfold. After time 𝑡 = 𝑛 × 𝑇 , n doublings will have been observed, giving a multiplication factor of 2n . It is convenient to remember 1 = 10 4 ≈ 1000 = 103 . After ten doubling times, exponential growth will have exceeded a factor of 1000: 𝑻 ≈ × 𝑻 𝟐 ≈ 𝟕 % ( ) For a 7% growth rate, the doubling time is a decade and the time of thousand-fold increase is a century. 6. 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 𝑇 ≈ 7 𝑝% follows that 70 years encompass almost exactly p% doubling times and the total aggregate growth will be: ( ) = 𝟐 % ( ) This is another convenient rule to remember. 7. Exponential decay Everything explained so far is analogously valid for positive or negative growth rates. In the latter case, we refer to the half-life instead of doubling time. An example is radioactive decay.
- 4. Exponential Growth 4 T. Menninger 8. 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 − > : 𝑸( ) = 𝒒 , ( ) = → 𝑸( ) ( ) = (𝒒− ) (Likewise, the growth rate of a product is the sum of the growth rates.) 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. (Compare exercise P6). 9. Cumulative sum of exponential growth If a quantity Q(t), such as energy use, grows exponentially, then the cumulative resource consumption is also (at least) exponential and each doubling time, (at least) as much of the resource is used as has been used during the entire prior history (see exercise P7). To see this, observe that each doubling time’s amount is (at least) twice that of the preceding. When adding the total for the first n doubling times, with the amount for the first doubling time (t=0) indexed to 1, we get 𝑻𝒐 𝒂𝒍 = 𝟐 𝟒 𝟖 ⋯ 𝟐 𝒏− = ∑ 𝟐 = 𝟐 𝒏 − 𝒏− = So the consumption during doubling time t = n+1 (namely 𝟐 𝒏 ) is about the same (in fact, more) as the cumulative total up to doubling time n. A startling fact to consider! 10. Summary: growth rates and doubling times Formulas (2), (3), (4) and (5) capture exponential growth mathematically but the simple rules (6), (7) and (8) are often sufficient to get a good estimate of the effects of growth. The following table summarizes these results for a range of growth rates. 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
- 5. Exponential Growth 5 T. Menninger 11. Semi-logarithmic graphs A semi-logarithmic plot helps to visually assess the growth characteristics of a time series. Steady exponential growth shows as a straight line on the graph, with the slope corresponding to the growth rate. Line segments of different slope indicate a change in growth rates (compare exercise P4). Example: This graph of global population exhibits the typical shape of exponential growth but the semi-logarithmic plot reveals how growth rates have fluctuated heavily over time. Population growth, as indicated by the slope of the lower curve, accelerated during the industrial revolution, reached a peak in the second half of the 20th century, and is now in decline but still high. Up to about 1970, population growth was hyper- (faster than) exponential, evident in that part of the semi- logarithmic graph being convex (i. e. the graph lies entirely below a line joining two data points). 12. Further reading: Al Bartlett: “Arithmetic of growth” and other materials on the web site, albartlett.org. Toni Menninger: Growth in a Finite World, slideshare.net/amenning/presentations Toni Menninger: Case Studies for Sustainability Education: Exponential Growth © Toni Menninger MSc, 4/2014. Use of this material for educational purposes with attribution is permitted. For questions or comments or for permission to reuse in a published work, please contact the author: toni.menninger@gmail.com.
- 6. Exponential Growth 6 T. Menninger Practice problems: In-depth exploration of exponential growth These exercises are designed to deepen your understanding of exponential growth and to explore the mathematical properties of the exponential function. P1. 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 will or can happen? g) Derive a rule (analogous to the rule of 70) for estimating the tripling time T3. h) What growth rate implies a 10-fold increase every 100 years? P2. 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? P3. 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) Use these results to explain the differences between linear and exponential growth.
- 7. Exponential Growth 7 T. Menninger P4. Semi-logarithmic graphs The following exercises are easy to do with spreadsheet software such as OpenOffice Calc or Microsoft Excel. Familiarize yourself with their basic functions, especially how to copy or import data into a spreadsheet, perform calculations, and create plots (use the ‘X Y’ or scatter plot). 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) Copy a real world data series, for example US population, into a spread sheet and plot it semi- logarithmically. Add an exponential trend line and display the trend line equation. Use the graph and trend line to examine the growth trends. Is the data series consistent with exponential growth? Do you notice changes in the growth trend? What does the trend line equation tell you? P5. The average growth rate The average (fractional) growth rate (p. 2) of a quantity N(t) is calculated as the natural logarithm of the relative change factor over time: = 𝒍𝒏( ( )/ ( ) − = 𝒍𝒏 ( ) − 𝒍𝒏 ( ) − (9) a) Calculate average growth rates for Lineartown and Expoville (P3), for the whole time period and for each segment, and compare the linear and exponential case. How does the average growth rate of Lineartown change as time increases? b) Plot the linear data semi-logarithmically. How are the slopes of the curve segments related to the growth rates you calculated in a)? 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 (hint: use the right hand side of formula (9)). d) (Requires calculus) Let (𝑡) be a differentiable positive function. Calculate the derivative of 𝑛( (𝑡)). Show that this derivative is the “momentary” fractional growth rate at time 𝑡, i. e. the limit of the average growth rate when the time period is made smaller and smaller. Show that if (𝑡) grows linearly or as a power function (compare P8), the fractional growth rate approaches zero for 𝑡 → . e) Intuitively, one might want to calculate the average growth rate simply as the fractional change per time unit, ( )/ ( ) − − . Would this be a good idea? Hint: look at P2.
- 8. Exponential Growth 8 T. Menninger P6. Per capita growth It is often necessary to analyze the quotient of two quantities, for example per capita production or consumption (compare 8, p. 4). 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/ )? a) In a spreadsheet, create exponential 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 can you say about the behavior of 𝐺𝐷𝑃/ ? What happens in each of the cases 𝑝 > 𝑞, 𝑝 = 𝑞, 𝑝 < 𝑞 ? b) Show algebraically that 𝐺𝐷𝑃/ is exponential with the difference 𝑝 − 𝑞 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/ for the same time period is again the difference 𝑝 − 𝑞. Note: the average growth rate (see P5) is well-defined regardless of whether growth is exponential. d) Finally, assume that GDP and population are both growing linearly at constant rates. What can you say about the rate of change of per capita GDP? Is there a numeric trick similar to the exponential case? Use a spreadsheet as in a) or solve algebraically. P7. Cumulative resource use (Requires basic calculus) Assume that a resource is used at a rate that is growing exponentially. Then, as stated earlier (see 9, p. 4), during each doubling time, twice as much of the resource is used as during the preceding doubling time, and this adds up to (at least) the total cumulative resource use during the entire prior history. Let (𝑡) denote the rate of resource use per time unit at time t, p the fractional growth rate, 𝑇 the doubling time, and (𝑡1, 𝑡 ) the cumulative resource use from time t1 to t2. Note that under discrete compounding, t1 and t2 are integers and 𝑇 must be rounded to the next larger integer. Then can be calculated (in the discrete respectively the continuous case) as ( , 𝟐) = ∑ ( ) 𝟐− ( , 𝟐) = ∫ ( ) − . Show that the following relationships hold under both discrete and continuous compounding: ( , 𝟐) = ( ( 𝟐) − ( )), ( , 𝑻 𝟐) = 𝟐 ( − 𝑻 𝟐, ), and ( , 𝑻 𝟐) > ( , ). Explain the meaning of each of these expressions in plain English! What is the behavior of (0, 𝑡)?
- 9. Exponential Growth 9 T. Menninger P8. Exponential vs. power functions The exponential function and 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 𝑥 > 𝑥 𝑁 for all 𝑥 ≥ 𝑥( ). Can you show this mathematically? Find values for x(10), x(50), x(100). P9. Discrete versus continuous compounding When a variable is assumed to grow in discrete steps at a constant rate, it is described by the formula ( ) = ( ) (1) whereas continuous growth is described by the exponential function ( ) = ( ) In this exercise, we’ll verify that when the growth rate is small, say 𝑝 ≤ 1 %, both expressions give approximately the same results. Note that in the discrete case, p and t are taken as dimensionless numbers, whereas in the continuous case, t has time units and p inverse time units. a) Plot discrete versus continuous compounding for different growth rates and observe the difference. Why does continuous compounding grow faster? b) What is the actual yield over 1 year if a bank account has an interest of 1%, 5%, 10%, 14% p. a. compounded continuously? c) A population has grown by exactly 10% in a year. What is the “average” annual growth rate according to continuous compounding? Can you explain why there is a difference? Compare b). d) Calculate the exact doubling time (do not use the rule of 70) for the discrete and continuous case with growth rates 1%, 5%, 10%, 14%. e) Show that (1 𝑥) ≈ for small values of x. Use the fact that n(1 𝑥) ≈ 𝑥 for small 𝑥. Explain how this finding is related to the rest of this exercise. © Toni Menninger MSc, 4/2014. Use of this material for educational purposes permitted. For permission to reuse in a published work, please contact the author. For questions or comments please contact toni.menninger@gmail.com.