1. MICHELLE BLESSING Chapter 23: The Economics of Resources

2. Outline of Major Topics Population growth models: predicting US population Exponential vs. logistic growth Renewable and nonrenewable resources Sustaining renewable resources Dynamical systems and chaos Chaos in biological populations

3. Compound Interest and the Savings Formula A = P(1+r) n Compound interest formula: If a principal P is deposited into an account that pays interest at rate r per year, then after n years the account contains the amount Savings Formula: For a uniform deposit of d per year (deposited at the end of the year) and an interest rate r per year, the amount A accumulated after n years is

4. Growth models: Predicting the US population Rate of growth: birth rate minus death rate (plus net migration) Predict the US population for January 1, 2015 based on the following information:  US population increasing at an average growth rate of.98% per year  US population estimate as of January 1, 2010 (US Census Bureau report): 308.4 million A = P(1+r) n P = ? r = ? n = ?

5. Growth models: Predicting the US population P = 308.4 million, r =.0098 (.98%), n = 5 A = P(1+r) n = (308,400,000)(1 +.0098) 5 = 323.81 million

6. Population Pyramids for 2010

7. Natural Limits to Growth Unlike a bank account accumulating compound interest, population growth has natural limitations  availability of space, food, water, shelter, etc. Carrying capacity : The maximum population size that can be supported by the available resources

8. The Logistic Model The logistic model for population growth takes carrying capacity into account by reducing the annual increase rP by a factor of how close the population size P is to the carrying capacity K: As the population increases toward K, the growth rate decreases. This model is simple and describes only one species, but makes excellent predictions for some populations.

9. Resources: Renewable and Nonrenewable Renewable resources are capable of being replaced or replenished at a rate comparable to the rate they are being used up.  e.g. fish, wildlife populations, forests, solar energy, weekly paycheck Nonrenewable resources do not tend to replace or replenish themselves. They are finite.  e.g. Oil, coal, natural gas, inheritance funds or lottery winnings

10. Nonrenewable Resources Important question: How long will the supply of a resource last? How quickly will it be used up? Assume there is a fixed supply S available Three models for how quickly a resource is used: 1. Fixed rate: the rate of use of the resource remains constant, say, U units per year. Thus the supply will last S/U years. 2. Linearly increasing rate: the rate of use of the resource increases by a fixed percentage each year. 3. Nonlinearly increasing rate: the rate of use of the resource increases by a different percentage each year.

11. Model with a linearly increasing rate U 1 = U U 2 = U 1 + r U 1 = (1+r)U 1 = (1+r)U U 3 = U 2 + r U 2 = (1+r) U 2 = (1+r)(1+r)U = (1+r) 2 U U 4 = U 3 + r U 3 = (1+r) U 3 = (1+r)(1+r) 2 U = (1+r) 3 U. U n = U n-1 + r U n-1 = (1+r) U n-1 = (1+r)(1+r) n-1 U = (1+r) n U

12. Model with a linearly increasing rate The total amount of the resource that has been used up at the end of n years : S = U + (1+r)U + (1+r) 2 U + (1+r) 3 U + … + (1+r) n U = This is the same as the formula for the accumulation of regular deposits plus interest from Chapter 21. Exercise: solve for n!

13. Static Reserve and Exponential Reserve The static reserve is how long the supply S will last at a particular constant annual rate of use U, namely S/U years. The exponential reserve is how long the supply S will last at an initial rate of use U that is increasing by a proportion r each year, namely years.

14. Example: U.S. Coal Reserves Coal accounts for 30% of U.S. energy use and 50% of electricity use! Recoverable U.S. coal resource are predicted to last about 250 years at the current rate of use.  Static reserve is 250 years. How long would the supply last if the rate of use increases 2.25% per year?  Calculate the exponential reserve based on the equation:

15. Example: U.S. Coal Reserves years Clearly, even a small increase in the rate of use each year can make a big difference!

16. Predictions for the Peak of Oil Output

17. Sustaining Renewable Resources We are interested in knowing how much we can harvest and still allow for the resource to replenish itself. An equilibrium population size does not change from year to year. Renewable resources behave like nonrenewable resources when the rate of harvesting significantly exceeds the rate of replenishment,  i.e. fish, wildlife populations, forests, groundwater, soil

18. Sustained-Yield Harvesting In the fishing industry, a sustainable yield is one that can be sustained year after year because the fish population would recover to the same level after each harvest.  For a sustainable yield, the same amount is harvested every year and the population remaining after each year’s harvest is the same. A goal for fishing and timber companies is to harvest the maximum sustainable yield.

19. Using up Renewable Resources When harvesting exceeds the ability of the population to replenish itself, extinction of the resource can result. The “tragedy of the commons” By and large, it has been politically impossible to force a harvesting industry to reduce current harvests to ensure stability in the future Variations in the initial population can lead to chaotic behavior

20. Example: Groundwater Groundwater overdraft occurs when the rate at which water is removed from an aquifer exceeds the rate at which it is replaced.

21. Ogallala Aquifer Largest known aquifer in the world, over 174,000 square miles! Main cause of depletion is irrigation for farming. Recent studies have estimated an average recharge rate for the entire High Plains region of approximately 0.5 of an inch per year. Water table is dropping an average of 2 meters per year. At this rate, the groundwater will be depleted in less than 50 years; over half the total volume will be gone by 2020. Pollution and contamination make the rate of loss even more devastating.

22. Example: Soil Erosion In the USA, soil has recently been eroded at about 17 times the rate at which it forms: about 90% of US cropland is currently losing soil above the sustainable rate.

23. Dynamical Systems and Chaos A dynamical system is a system whose state depends only on its state at previous times. Thus, its behavior is not random, but deterministic. Chaos is complex but deterministic behavior that is sensitive to initial conditions and unpredictable over time.  Determinism: Future behavior of the system is completely determined by its present state, its past history, and known laws; chance is not involved.  Sensitive dependence on initial conditions: a small variation at the beginning can make a big difference later on (“the butterfly effect”)

24. Example of Iterating a Function Define a function f by the following procedure. 1. Start with any number. 2. Double the value of this number. 3. Take the last two digits of this number. 4. Repeat. Ex: 37 and 38 37, 74, 48, 96, etc. 38, 76, 52, 04, etc. Already by the fourth iteration, the sequences are far apart!

25. Chaos in Biological Populations If we measure this year’s population as a fraction x of the carrying capacity, and do the same for next year’s population as a fraction f (x), the logistic model takes the form: f(x) = λ x(1 - x) where λ =1 + r is the amount by which the population is multiplied each year. For different values of the parameter λ and different starting values for the population fraction, many different behaviors occur.  λ = 2.8 and starting population fraction x = 0.36 produces situation a.  λ = 3.1 and starting population fraction x = 0.235 produces situation b.  λ = 3.0 and starting population fraction x = 0.4 produces situation c.  λ = 4.0 and and starting population fraction produced situation d.

26. Discussion and Homework To what extent is the use or misuse of natural resources a technological problem? To what extent is it a social, political, or ethical problem? What kinds of technological or political (policy) solutions seem most promising? What are pother applications of dynamical systems and chaos theory? Other discussion questions? Homework (7 th Edition): #9a, 11